Modulated Apertures and Resolution in Microscopy (SpringerBriefs in Applied Sciences and Technology) 3031475518, 9783031475511

Table of contents :
Preface
Contents
Part I The Impulse Response for Modified Apertures and Resolution in Microscopy
1 Resolution in Microscopy and Defect of Focus with Modulated Apertures
1.1 Introduction
1.2 The Lateral and Axial PSF
1.2.1 Martinez-Corral Filter [2]
1.2.2 Author’s Filter [3]
1.2.3 Lateral Resolution Limit Using a Uniform Circular Aperture
1.2.4 Lateral Resolution Limit Using Modulated Apertures
1.2.5 Axial Resolution
1.2.6 Depth of Focus in Microscopy Provided with a Uniform Circular Aperture [6]
1.2.7 Depth of Focus in Microscopy Provided with Modulated Apertures
References
2 Linear Aperture
2.1 Introduction
2.2 Calculation of the PSF for a Pupil Having Linear Amplitude Variation
2.3 Obstructed Circular Pupil with B/W Equal Areas
References
3 Linear-Quadratic Aperture
3.1 Introduction
3.2 Analysis
3.2.1 The Effect of the Pinhole Diameter on the PSF of a Standard CSLM
3.3 Results and Discussions
3.4 Conclusion
References
4 A Study on Star Aperture and Their Application in Confocal Scanning Laser Microscope
4.1 Introduction
4.2 Theoretical Analysis
4.3 Results and Discussions
4.4 Conclusion
References
5 A Study on Some Transparent Rose Apertures: An Application on the Confocal Laser Scanning Microscope and the Speckle Images
5.1 Introduction
5.1.1 The Aperture Construction and Computation of the PSF
5.1.2 Application on the Confocal Scanning Laser Microscope (CSLM)
5.1.3 Application to Speckle Imaging
5.2 Results and Discussion
5.3 Conclusion
References
6 A New Model of Modulated Aperture [28]
6.1 Background
6.2 Methods
6.3 Theoretical Analysis
6.4 Results
6.5 Discussions
6.6 Conclusions
References
Part II Application of Confocal Laser Microscope and Speckle Images
7 Cauchy Aperture and Its Application in Confocal Microscopy
7.1 Introduction
7.2 Analysis
7.3 Results and Discussion
7.4 Conclusion
References
8 Speckle Images Formed by Diffusers Using Conical and Linear Apertures
8.1 Introduction
8.2 Theoretical Analysis
8.3 Results and Discussion
8.4 Conclusion
References

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SpringerBriefs in Applied Sciences and Technology Abdallah Mohamed Hamed

Modulated Apertures and Resolution in Microscopy

SpringerBriefs in Applied Sciences and Technology

SpringerBriefs present concise summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50 to 125 pages, the series covers a range of content from professional to academic. Typical publications can be: • A timely report of state-of-the art methods • An introduction to or a manual for the application of mathematical or computer techniques • A bridge between new research results, as published in journal articles • A snapshot of a hot or emerging topic • An in-depth case study • A presentation of core concepts that students must understand in order to make independent contributions SpringerBriefs are characterized by fast, global electronic dissemination, standard publishing contracts, standardized manuscript preparation and formatting guidelines, and expedited production schedules. On the one hand, SpringerBriefs in Applied Sciences and Technology are devoted to the publication of fundamentals and applications within the different classical engineering disciplines as well as in interdisciplinary fields that recently emerged between these areas. On the other hand, as the boundary separating fundamental research and applied technology is more and more dissolving, this series is particularly open to trans-disciplinary topics between fundamental science and engineering. Indexed by EI-Compendex, SCOPUS and Springerlink.

Abdallah Mohamed Hamed

Modulated Apertures and Resolution in Microscopy

Abdallah Mohamed Hamed Physics Department Faculty of Science Ain Shams University Cairo, Egypt

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISBN 978-3-031-47551-1 ISBN 978-3-031-47552-8 (eBook) https://doi.org/10.1007/978-3-031-47552-8 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

Since the invention of the laser in 1960, there has been a renaissance in optics and optical electronics. With optics and optical electronics, new applications have been found in all branches of science and engineering. The famous book Introduction to Fourier Optics and Holography by Goodman JW (1968) and Confocal Microscopic Imaging by Sheppard CJR (1978), followed by my recent publications about aperture modulation, led to the presentation of the book Modulated Apertures and Resolution in Microscopy. The book’s content extracted from my recent publications adds information on resolution improvement in microscopy due to the modification in the aperture distribution. We intended this book for senior undergraduate and graduate students in optical sciences. The object of drafting a book on modulated apertures with different distributions is outlined below. When we replaced the circular aperture of uniform transmittance with the modulated gaps, the point spread function (PSF) changed, and the cutoff spatial frequency decreased, leading to a resolution improvement in the conventional optical microscope. In addition, the PSF related to resolution in an optical microscope further improves the resolution in a confocal laser scanning microscope. We study the modified distribution inside the circular aperture using different shapes. These are linear, quadratic, graded index, and black-and-white gaps. That described cracks we used in the computation of the point spread function from the Fourier transform operation. The cutoff spatial frequency obtained from the PSF corresponding to each aperture provides information about the resolution, remembering that the resolution limit cutoff is dependent on the numerical aperture NA and the wavelength of the illumination according to the Rayleigh criterion formula. The book is composed of two parts. Part I is around the fabrication of modulated apertures and computation of the impulse response or the PSF. We presented an introductory chapter (1) on axial and lateral resolution in microscopy. In the following chapters (2–6), the resolution was computed for the modulated apertures and compared with uniform circular and annular gaps.

v

vi

Preface

Part II is concerned with applying the modulated Cauchy aperture in the confocal imaging chapter (7) and speckle imaging chapter (8). Abdallah Mohamed Hamed Ain Shams University Cairo, Egypt

Contents

Part I

The Impulse Response for Modified Apertures and Resolution in Microscopy

1 Resolution in Microscopy and Defect of Focus with Modulated Apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Lateral and Axial PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Martinez-Corral Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Author’s Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Lateral Resolution Limit Using a Uniform Circular Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Lateral Resolution Limit Using Modulated Apertures . . . . . 1.2.5 Axial Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Depth of Focus in Microscopy Provided with a Uniform Circular Aperture . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Depth of Focus in Microscopy Provided with Modulated Apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 4 5 7 7 8 9 11 14

2 Linear Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Calculation of the PSF for a Pupil Having Linear Amplitude Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Obstructed Circular Pupil with B/W Equal Areas . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15

3 Linear-Quadratic Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Effect of the Pinhole Diameter on the PSF of a Standard CSLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22

15 17 19

25

vii

viii

Contents

3.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26 30 30

4 A Study on Star Aperture and Their Application in Confocal Scanning Laser Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 34 36 46 50

5 A Study on Some Transparent Rose Apertures: An Application on the Confocal Laser Scanning Microscope and the Speckle Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Aperture Construction and Computation of the PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Application on the Confocal Scanning Laser Microscope (CSLM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Application to Speckle Imaging . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 A New Model of Modulated Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

51 51 52 52 54 54 63 63 65 65 66 66 67 69 74 75

Application of Confocal Laser Microscope and Speckle Images

7 Cauchy Aperture and Its Application in Confocal Microscopy . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 82 89 90

Contents

ix

8 Speckle Images Formed by Diffusers Using Conical and Linear Apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.2 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Part I

The Impulse Response for Modified Apertures and Resolution in Microscopy

Chapter 1

Resolution in Microscopy and Defect of Focus with Modulated Apertures

1.1 Introduction This chapter presents different linear, quadratic, and graded indexes and black-andwhite pupils. We calculate the point spread function from the Fourier transform operation upon the described apertures. It depends on the numerical aperture NA and the illumination wavelength according to the Rayleigh criterion formula [1].

1.2 The Lateral and Axial PSF Consider the amplitude point spread function (APSF) of a coherent imaging system apodized by a purely absorbing pupil filter. Considering the spherical symmetry of revolution for the circular aperture, we write the PSF in radial coordinates as follows [1]: {1 h(u, w20 ) = 2

P(ρ)exp(− j2π w20 ρ 2 ) J0 (2π uρ)ρdρ

(1.1)

0

P(ρ) is the pupil function, and ρ is the pupil plane’s normalized radial coordinate. U = (ρ 0 /λf ) r corresponds to the transverse radial coordinate expressed in optical units, ρ 0 is the maximum extent of the pupil, f is the focal length of the imaging lens, and J 0 denotes the Bessel function of the 1st kind and zero order. w20 = 2πr20f 2 specifies the amount of defocus measured in wavelength units, and r 20 is the actual radial distance of the defocused cross-section. Next, perform the following nonlinear mapping to separate the axial and transverse distributions (1.3).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. M. Hamed, Modulated Apertures and Resolution in Microscopy, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-3-031-47552-8_1

3

4

1 Resolution in Microscopy and Defect of Focus with Modulated Apertures

ξ = ρ 2 − 0.5, and q(ξ ) = P(ρ) ; Martinez filter[2]

(1.2)

Substituting from Eq. (1.2) into Eq. (1.1), we write: {0.5 h(u, w20 ) = 2

( ) √ q(ξ )exp(− j2π w20 ξ )J0 2π u ξ + 0.5 dξ

(1.3)

−0.5

The axial behavior of the system obtained from Eq. (1.3) at u = 0 is as follows: {0.5 h(0, w20 ) = 2

q(ξ )exp(− j2π w20 ξ )dξ = F T {q(ξ )}

(1.4)

−0.5

where FT is the exact Fourier transform realized by the transformation mentioned in Eq. (1.2). The transverse behavior obtained from Eq. (1.3) at w20 = 0 is as follows: {0.5 h(u, w20 ) = 2

( ) √ q(ξ )J0 2π u ξ + 0.5 dξ.

(1.5)

−0.5

The Martinez-Corral filter uses Eqs. (1.1)–(1.5) by Martinez et al. in the following applications.

1.2.1 Martinez-Corral Filter [2] The Martinez-Corral filter consists of a transparent annulus and a central clear circular aperture of an area less than the area of the annulus. Using the PSF represented by Eqs. (1.4) and (1.5), we obtain the following irradiances: ] [ ]2 sin(π w20 ) 2 2 sin(π μw20 ) +μ Iaxial (w20 ) = π w20 π μw20 sin(π w20 ) sin(π μw20 ) − 2μ cos[2π w20 (∈ −0.5)μ] π w20 π μw20 ( ) √ √ 1 { Itrans (μ) = 2 2 J1 (2π μ) + (0.5− ∈)J1 2π μ (0.5− ∈) π μ ( )}2 √ √ − 0.5 + μ(1− ∈)J1 2π μ 0.5 + (1− ∈)μ ] [ ξ − (∈ −0.5)μ q(ξ ) = rect(ξ ) − rect μ [

(1.6)

(1.7)

1.2 The Lateral and Axial PSF

5

where 0.5 < ε < 1 and 0 < εμ < 0.5, μ is the obscuration parameter, and ε is the asymmetry parameter.

1.2.2 Author’s Filter [3] The aperture under study has a definite number of black-and-white (B/W) annuli of a certain number of circles N = 20, where the center is a clear circular disk, as shown in the figure in ref. [12]. The mathematical representation corresponding to the effective pupil of this aperture is as follows: ( ) ∑ N ρ + P(ρ) = circ ΔPi (ρ), ρ0 i=1

(1.8)

where ΔPi (ρ) = P2i+1 (ρ) − P2i (ρ) is the difference between any two successive circular apertures with an annular shape, N is the total number of circles, and ρ is the radial coordinate in the aperture plane (u, v).

1.2.2.1

Computation of the Transverse APSF and Its Irradiance

Calculation of the amplitude impulse response of the considered aperture or the amplitude point spread function (APSF) by running the Fourier transform upon the pupil represented by Eq. (1.8) to obtain the following equation: } N { 2J1 (α1r ) ∑ 2J1 (α2i+1r ) 2J1 (α2i r ) + − h(w) = (α1r ) (α2i+1 r ) (α2i r ) i=1

(1.9)

( ) ( ) w2i+1 J1 is the Bessel function of the 1st order, where α2i = λ2 w2i , α2i+1 = 2π λ and r is the radial coordinate in the Fourier plane (x, y). For nonequally spaced annuli, Eq. (1.9) is used in the computations suppressing annuli where the obscuration parameter has μ = 0.7895, giving the best transverse resolution. It corresponds to the following parameters: a central disk of radius ρ = 1/19 followed by the dark annulus of μ = 15/19 = 0.7898, followed by the three remaining white–dark–white annuli. The other extreme value of μ = 0.0526 will give a poorer resolution corresponding to the following parameters: a central disk of radius ρ = 1/19 followed by the dark annulus of μ = 1/19 = 0.0526, followed by 12 B/W annuli. The transverse irradiance is the modulus square of the amplitude point spread function represented as follows: Itrans (w) = |h(w)|2

(1.10)

6

1 Resolution in Microscopy and Defect of Focus with Modulated Apertures

In the case of confocal imaging systems, the irradiance or the image of a point becomes: Itrans (w) = |h(w)|4

(1.11)

For an aperture of a skinny annular shape combined with the transparent central disk, Eq. (1.9) becomes: 2J1 (α1r ) ∑ {J0 (α2i+1r )} + (α1r ) i=1 N

h(w) =

1.2.2.2

(1.12)

Computation of the Axial APSF and Its Irradiance

The binary filter is composed of ten transparent annuli and ten black annuli with a central clear disk in a B/W cascaded concentric annular arrangement, as shown in the figure in Ref. [4], represented as follows: q(ξ ) =

N ∑

rect(ξ − nμξ0 ),

(1.13)

n=1

where ξ 0 is the interval width between two successive annuli and μ is the obscuration parameter. The 1D Fourier transform runs upon Eq. (1.13), making use of convolution operations to obtain this result for the axial APSF: N

2 ∑ sin(π w20 ) exp( j2π nμξ0 w20 ). h(w20 ) = (π w20 ) N

(1.14)

n=− 2

This complex function of the axial APSF is decomposed into real and imaginary parts as follows: Real[h(w20 )] = sinc(w20 )

N /2 ∑

cos(2π nμξ0 w20 )

(1.15)

sin(2π nμξ0 w20 )

(1.16)

n=−N /2

Im[h(w20 )] = sinc(w20 )

N /2 ∑ n=−N /2

Taking the modulus square of the axial APSF, we compute the corresponding axial irradiance of the ordinary optical systems to obtain this result:

1.2 The Lateral and Axial PSF

7

⎡ Iaxial (w20 ) = sin c (w20 ).⎣ 2

⎤2

N

2 ∑

n=−

cos(2π nμξ0 w20 )⎦

(1.17)

N 2

since the second summation over the sine function vanishes. In the case of confocal imaging, the axial irradiance is ⎡ Iaxial (w20 ) = sin c4 (w20 ).⎣

⎤4

N

2 ∑

n=−

cos(2π nμξ0 w20 )⎦

(1.18)

N 2

1.2.3 Lateral Resolution Limit Using a Uniform Circular Aperture The PSF corresponding to a uniform circular aperture of numerical aperture NA = a/f, where aperture radius = a, and f is the focal lens of the Fourier transform lens given in Eq. (1.19): P S Fcir (w) =

2J1 (w) , w

(1.19)

where the cutoff reduced coordinate W c = 3.83, deduced from Eq. (1.19). Since Wc = 2π N Arc = 3.83, then: λ Resolution cutoff limit = rc = 0.61λ/NA.

(1.20)

1.2.4 Lateral Resolution Limit Using Modulated Apertures The resolution is affected by the modification made in the aperture for monochromatic illumination at wavelength λ. The following shows in Chaps. (2–8) that the lateral resolution depends upon the overall PSF distribution, giving different cutoffs related to the modulated aperture. A multiplication factor α or β < 1 introduced in the resolution cut gives this new criterion based on the PSF results. rc = 0.61α

λ for modulated linear aperture NA

(1.21)

8

1 Resolution in Microscopy and Defect of Focus with Modulated Apertures

rc = 0.61β

λ for modulated quadratic aperture. NA

(1.22)

For a uniform circular aperture, α = 1. In the next chapters, we show the following: β (quadratic) < α(linear), which was determined exactly from the PSF cutoff plots. Consequently, in Chaps. 2–8, this specific study on the modulated apertures is shown. We use digital techniques to fabricate the modulated apertures and compute the PSF analytically or digitally using the FFT. The computations of PSF plots and images used MATLAB code.

1.2.5 Axial Resolution In an ideal microscope, only light can reach the detector from the focal plane (typically an observer or a CCD), creating a clear image of the plane of the microscope’s sample. Unfortunately, a microscope is not this specific, and light from sources outside the focal plane also reaches the detector; in a thick piece, there may be a significant amount of material, and so spurious signal, between the focal plane and the objective lens. Considering no modification to the microscope, i.e., with a simple wide-field light microscope, the quality of optical sectioning is governed by the same physics as the depth of field effect in photography. For a high numerical aperture lens, equivalent to a wide aperture, the depth of field is small (shallow focus) and gives good optical sectioning. High-magnification objective lenses typically have higher numerical apertures (so better optical sectioning) than low-magnification objectives. The oil immersion objective typically has even larger numerical apertures to improve optical sectioning. The resolution in the depth direction (the “z resolution”) of a standard wide-field microscope depends on the numerical aperture and the wavelength of the light. The computed z resolution is written as follows [1]: ΔZ =

n (N A)2

(1.23)

λ is the wavelength, (n) is the refractive index of the objective lens immersion media, and NA is the numerical aperture. In comparison, lateral resolution is approximated by the Rayleigh criterion as [1, 5]: rc = 0.61α

λ , α = 1 for uniform circular aperture. NA

The diffraction-limited depth of field in Eq. (1.4) shrinks inversely with the square of the numerical aperture. At the same time, the lateral limit of the resolution varied in a manner that is inversely proportional to the first power of the numerical aperture

1.2 The Lateral and Axial PSF

9

Fig. 1.1 a Normalized transverse PSF for the Martinez filter. b Normalized transverse irradiance for the Martinez filter. c Normalized transverse PSF for a circular aperture. d Normalized transverse irradiance for a circular aperture

Eq. (1.2). Thus, the axial resolution and thickness of optical sections are affected by the system’s numerical aperture much more than the lateral resolution of the microscope (Figs. 1.1 and 1.2).

1.2.6 Depth of Focus in Microscopy Provided with a Uniform Circular Aperture [6] The depth of focus (Δz) of a microscope system described by Δz should be a function of the wavelength of light, λ, the numerical aperture of the objective lens NA, the index of refraction, n, and the total magnification of the optical system M. The numerical aperture is defined as follows: NA = n• sinα, where n is the medium’s refraction index between the lens and the specimen studied, and α is the angle illustrated in Fig. 1.3. For a given (λ, NA, and n), we predict Δz. It is easily computed by referring to Fig. 1.3. Consider two light-emitting points along the axis of the optical system: one in-focus at position R and one out-of-focus at position R + Δz. Each point produces

10

1 Resolution in Microscopy and Defect of Focus with Modulated Apertures

Fig. 1.2 N a Normalized axial irradiance for B/W concentric annuli. b Normalized axial irradiance for the Martinez filter. c Normalized axial irradiance for a clear circular aperture. d Normalized axial irradiance for the B/W annulus, Martinez filter, and circular aperture

Fig. 1.3 Two points are in-focus, and one is out-of-focus; each produces a spherical wavefront. The wavefront approaches the entrance pupil of an optical system that subtends an angle α, as shown. The maximum distance between the wavefronts, W, occurs at angle α

1.2 The Lateral and Axial PSF

11

(according to Huygens’ model) a spherical wavefront of light. Given a pupil that subtends an angle α with the in-focus object, the maximal W difference between these two wavefronts will occur at angle α. Using the law of cosines, we have the following: (R + Δz)2 = (R + W )2 + (Δz)2 + 2(Δz)(R + W ) cos(α).

(1.24)

Solving for W, for Δz < < R, reduces to W∼ = Δz (1 − cosα) = 2Δzsin2 (α/2)

(1.25)

In Eq. (1.25), W is set equal to λ/4, which directly gives the equation. ΔZ =

( 4n 1 −

/

λ 1−

( N A )2

)

(1.26)

n

A widely used criterion for characterizing the distortion due to defocusing is the Rayleigh limit. In this context, the maximal path difference may never exceed a quarter wavelength if we wish to keep a diffraction-limited system [3], W ≤ λ/4.

1.2.7 Depth of Focus in Microscopy Provided with Modulated Apertures In our linear, quadratic, and B/W concentric annulus models, we obtain the depth of field based on [6].

1.2.7.1

Case of Linear and Quadratic Apertures

N A = sin(α) = af the numerical aperture for a uniform circular aperture. The linearly distributed pupils varied according to this formula: N A L = ρf . Using Eq. (1.26), we obtain the following defect of focus: ΔZ L =

1 1 λ λ ( )= ( ). √ / ( )2 4 1 − 1 − (N A )2 4 ρ L 1− 1− f

(1.27)

In the case of a low numerical aperture, assuming NA = 0.1 or ρf , an approximate expression for the defect of focus is obtained as follows:

12

1 Resolution in Microscopy and Defect of Focus with Modulated Apertures

( ) f2 1 . ΔZ L = 2 ρ2

(1.28)

Consequently, according to formula (1.13), the focal defect varied with the aperture radius. In the same manner, an expression for the focal defect corresponding to the 2 quadratic aperture of N A Q = ρf is written as follows: ΔZ L =

1 1 λ λ ). ( )= ( / / ) ( ( )2 4 4 2 ρ2 1 − 1 − N AQ 1− 1− f

(1.29)

For low NA, we obtain the approximate expression for the defect of focus as follows: ΔZ Q =

λf 2 (1/ρ)4 . 2

(1.30)

Hence, the focus defect varied in the 4th power of the aperture radius for the quadratic aperture. Consequently, the focus defect depends on the aperture distribution compared with the constant value obtained in a uniform circular aperture. We refer to the above results that ΔZ Q < ΔZ L < ΔZ cir . In the case of low NA, Eqs. (1.13 and 1.15), the differential change in depth of focus is obtained as: ( )2 δ(ΔZ cir − ΔZ L ) a =1− for linear aperture deviation from the constant aperture. ΔZ cir ρ δ(ΔZ cir − ΔZ Q ) = 1 − a 2 /ρ 4 for quadratic aperture deviation from the constant aperture. ΔZ cir

where the approximate ΔZ cir for low NA = 0.1 is computed from Eq. (1.26) as: ΔZ cir =

1.2.7.2

( ) λf 2 1 . 2 a2

Case of B/W Concentric Annular Aperture

Different authors investigated the defect of focus for the ordinary optical system, and we obtained a new formula from [3] extended in our model of B/W concentric annuli [4]. For the arrangement shown in Fig. 1.4 of B/W concentric annuli, we have different wavefront aberrations corresponding to the concentric annuli. Referring to the plot,

1.2 The Lateral and Axial PSF

13

Fig. 1.4 Focus defect in the case of B/W concentric annuli. We consider eight zones for the simplicity of the graph. The total numerical aperture is NA = n sinα. The successive NAs for the concentric annuli are sin α1 , sin α2 , and sin α8 , as shown in Fig. R, the wavefront radius, and w, the aberration wavefront shift

the wavefront aberration (W) increases with concentric annuli: ΔN .A.1 = N .A.2 − N .A.1 ΔN .A.1 < ΔN .A.2 < N .A.3 < . . . Corresponding to wavefront aberration W 1 , W 2, W 3 , respectively. Hence, W 1 < W 2 < W 3 < W 4 < . . . Since the defect of focus computed for a circular aperture as follows: ΔZ =

1 λ ( ) ; n = 1. √ 4 1 − 1 − (N A)2

(1.31)

Following this expression and considering separated independent annuli, we can write: ΔZ 1 =

1 λ ) ( √ 4 1 − 1 − (ΔN A )2 1

(1.32)

where ΔZ 1 is the defect of focus corresponding to the difference between the first two annuli of NA1 and NA2 and ΔN .A.1 = N . A.2 − N . A.1

14

1 Resolution in Microscopy and Defect of Focus with Modulated Apertures

We obtained the defect of focus for the annulus. Finally, we obtained the whole B/W concentric arrangement formula: 1 λ∑ ). ( ΔZ T = √ 4 i=1 1 − 1 − (ΔN A )2 2i N

(1.33)

ΔN .A.2i = N .A.2i − N .A.2i−1 represents the transparent annular width, as shown in Fig. 1.2, and ΔZ T = ΔZ 1 + ΔZ 2 + . . . We considered independent concentric transparent annuli since they were completely separated. For a considerable number of annuli, we replace the summation with the following integration: λ ΔZ = 4

{ρ0 ρ=0

1−

√

1 1 − (N A)2

d(N A)

(1.34)

The pupil radius extends from ρ = 0 at the center along the optical axis to ρ = ρ0 , which is the pupil radius.

References 1. J.W. Goodman, Introduction to Fourier optics and holography (McGraw-Hill Co., New York, 1968) 2. M. Martinez-Corral et al., Tailoring the axial shape of the spread function using Toraldo concept. Opt. Express 10(1), 98–103 (2002) 3. A.M. Hamed, Computation of the lateral and axial point spread functions in confocal imaging systems using binary amplitude mask. Pram J Phys. 66(6), 1037–1048 (2006) 4. A.M. Hamed et al., The point spread function (PSF) using longitudinal black and white strips inside a circular aperture. Int. J. Photonics Opt. Tech. 3(1), 1–9 (2017) 5. R.N. Bracewell, Fourier transform and its applications (McGraw-Hill Co, New York, 1966) 6. I.T. Young, Depth of focus in microscopy, in SCIA 93 Proceedind of the 8 Scandinavian Conference on image analysis (1993)

Chapter 2

Linear Aperture

2.1 Introduction In scanning optical confocal microscopes, the quality of both lenses, the objective lens and collector lens, is equally responsible for the point spread function. Sheppard et al. [1] studied the effect of various obstructions with annular pupils. If the object moves along the optical axis, the energy decay versus the defect of focus follows a sinc4 function, as given for the first time by Normarski [2]. This chapter presents another system of obstructed pupils using lenses with a conic pupil function. We have shown an improvement in the shape of the resultant point spread function [3]. In this chapter, we describe super-resolution pupils by constructing objective lenses with a linear amplitude as a function of the pupil radius. These pupils will give the best resolution, as discussed in the following sections. Additionally, obstructed circular pupils presented an alternative finite number of black and transparent annular zones, resulting in improved point spread function compared with circular uniform pupils. Finally, we showed the resolution in the two-point image.

2.2 Calculation of the PSF for a Pupil Having Linear Amplitude Variation This chapter presents the problem in two dimensions using polar coordinates as follows: The linear amplitude distribution for the pupil under consideration can be expressed as follows:     ρ ρ P(ρ) =  ,if  ≤ 1 ρ0 ρ0 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. M. Hamed, Modulated Apertures and Resolution in Microscopy, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-3-031-47552-8_2

(2.1)

15

16

2 Linear Aperture

We consider this pupil as a hyper-resolving pupil because of its transmission distribution. It has the advantage of attenuating low frequency, so the diffracting object structure is imaged with enhanced contrast because object areas of slowly varying transmission are attenuated [4]. In this way, the low spatial frequency of the image is attenuated, which also gives better contrast to higher frequency components of the image spectrum. Returning now to formula (2.1) and applying the two-dimensional Bessel–Fourier transform leads us to write the point spread function as follows: 2π 1 h 1 (r ) = 2 0

0

   1 kρr j2πρ r cos θ ρdρdθ = 4π ρ 2 J0 dρ, ρ exp − λf f 

(2.2)

0

where k = 2π /λ is the propagation constant. We have obtained the impulse response for the objective lens as follows:  h 1 (w) = 4π

f kr

3

W

∫ w 2 J0 (w)dw

(2.3)

0

with W = k r/f , and w = kρr/f . The solution of the integral (2.3) given by Hamed [3] during the treatment of conic amplitude distribution as a pupil function. The result leads to 

f h 1 (w) = 4π kr

 3   2 Ji (w) w J1 (w) + w J0 (w) − 2

(2.4)

i

(With i = 1, 3, 5, …). Hence, we get the impulse response of one pupil as [5] 

  J1 (w) 2 3 + J0 (w)/w − 2 h 1 (w) = 4π Ji (w)/w . w i

(2.5)

Consequently, the resulting impulse response of the optical system in a confocal microscope is h T (W ) = h 1 (W ) h 2 (W )

 

J1 (w) 2 J0 (w) 2 J0 (w)J1 (w) 2 h T (w) = 16π + +2 2 w w w3 ⎫  2 

 Ji (w) J0 (w) Ji (w) ⎬ J1 (w) + −4 + 2 3 2 w w w w3 ⎭ i i

(2.6)

2.3 Obstructed Circular Pupil with B/W Equal Areas

17

Hence, the intensity spread function is I (w) = |h T (w)|2 = |h 1 (w)h 2 (w)|2 . Formula (2.6) assumes two symmetric pupils; hence,  I (w) = 256π

4

 Ji (w) J0 (w) J1 (w) + − 2 w w2 w3 i

4 .

(2.7)

In the case of two points separated by w 1 , we calculate the image easily to give.   I (w) = |[δ1 (w) + δ2 (w ± w1 )] ∗ h T (w)|2 = h T (w) + h T (w ± w1 )2 ,

(2.8)

where* denotes the convolution operation. In the case of the circular pupil for the objective lens and linear amplitude distribution for the collector lens, or vice versa, we have the total impulse response of the optical system as follows: 

J1 (w) h T (w) = 8π w



  Ji (w) J1 (w) J0 (w) + −2 . w w2 w3 i

(2.9)

In the case of the annular pupil with the former pupil, which shows a radially increasing linear amplitude distribution, we have 

  Ji (w) J1 (w) J0 (w) + h T (w) = const.J0 (w) −2 . w w2 w3 i

(2.10)

Consequently, the image of a point object, in this case, gives:  I (w) = const.[J0 (w)]

2

 Ji (w) J1 (w) J0 (w) + −2 2 w w w3 i

2 .

(2.11)

We constructed computer programs to compute formulae (2.7), (2.9), and (2.11) numerically.

2.3 Obstructed Circular Pupil with B/W Equal Areas Referring to Fig. 2.1, we have a circular pupil with a numerical aperture (N .A. = n  sin θ ≈ ρ/ f ), where θ is the half-angular aperture of the lens under consideration and n’ is the refractive index of the object medium. In this case, we calculate the

18

2 Linear Aperture

Fig. 2.1 Circular pupil obstructed with a finite number of successive black and transparent areas

point spread function easily for a pupil of this form if we take into consideration that each zone is computed from the difference between two consecutive circular pupils as follows: h l (w) = F T {P(B/ W )}.

(2.12)

B/W indicates the succession of black-and-white (transparent) areas. FT. is the Fourier transform operation. We rewrite Eq. (2.12) as follows: h 1 (w) = F T

N 

Pr n = F T {Pr 1 + Pr 2 + · · · + Pr n }

(2.13)

n=1

Since, e.g., Pr 1 = Pr1 − Pr1 (Fig. 2.1), N is the total number of transparent zones constituting the whole pupil. Hence, running the Fourier transformation over Eq. (2.13) gives the point spread function:    

 N  2 2J1 (wn ) 2 2J1 wn h 1 (w) = rn − rn (2.14) wn wn n=1

References

19

W = (2π r/λf )ρ, where λ is the illumination wavelength, f is the lens’s focal length, and r is the radial coordinate in the object plane. This expression (2.14) is valid for the number of annuli constituting the pupil. Hence, by considering two symmetric pupils of the above type satisfying formula (2.13), we can find that the resulting point spread function for the optical coherent confocal microscope gives this general formula: h T (w) = h 1 (w)h 2 (w).

(2.15)

Therefore, the image for a point object, in this case, gives: I (w) = |h T (w)|2

(2.16)

and the image for two points is separated by a distance w1 given previously by formula (2.8), considering that h1 h2 is provided with Eq. (2.15).

References 1. C.J.R. Sheppard, A. Choudhury, Image formation in the scanning microscope. Opt. Acta 24, 1051 (1977) 2. G. Nomarski, J. Opt. Soc. Am. 21, 1166 (1975) 3. J.J. Clair, A.M. Hamed, Theoretical studies on a coherent optical microscope. Optik 64, 133 (1983) 4. K.G. Birch, A spatial frequency filter to remove zero frequency. Opt. Acta 15, 113–117 (1968) 5. A.M. Hamed, J.J. C1air, Image and supe-resolution in coherent optical microscopes. Optik 64, 277 (1983)

Chapter 3

Linear-Quadratic Aperture

3.1 Introduction The enormous work on resolution in microscopy on circular and annular apertures placed before the objective lenses of the confocal microscope [1–10]. We used Gaussian modulation in laser scanning confocal microscopy in Ref. [11–14]. In [13], we presented super-resolution spinning disk confocal microscopy using optical photon reassignment, where the point spread function is considered Gaussian. The effect of the numerical aperture on the interference fringe spacing was studied [14]. We investigated the amplitude modulation of linear, quadratic, and other modulated apertures using speckle photography [15–20], and we applied these apertures to a confocal laser scanning microscope [21–24]. A strategy was presented to enable optical-sectioning microscopy with improved contrast and imaging depth using low-power (0.5–1 mW) diode laser illumination [25]. Recently, in image scanning microscopy (ISM), we replaced the point detector of the confocal microscope with a quadrant detector of the same size, thus using a finite number of detector elements. This implementation offers a resolution close to the CSLM theoretical value. It improves the signal-to-noise ratio by a factor of 1.5 concerning the CSLM without losing the optical sectioning [26]. This chapter presents two models of hyper-resolving apertures proposed starting with computation of the point spread function for these apertures and making other computations of the coherent transfer function, the results, and discussions, followed by a conclusion [27].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. M. Hamed, Modulated Apertures and Resolution in Microscopy, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-3-031-47552-8_3

21

22

3 Linear-Quadratic Aperture

3.2 Analysis The aperture is composed of a quadratic distribution followed by a linear distribution and ends with a transparent uniform annulus, as shown in Fig. 3.1. We describe this aperture as the sum of the following segments: P1 (ρ) = ρ 2 ; 0 ≤ |ρ| < |ρ0 /2| for the quadratic segment

(3.1)

P2 (ρ) = ρ; |ρ0 /2| ≤ |ρ| < |3ρ0 /4| for the linear segment

(3.2)

P3 (ρ) = 1; |3ρ0 /4| ≤ |ρ| < |ρ0 | for the annulus

(3.3)

PT = P1 (ρ) + P2 (ρ) + P3 (ρ)

(3.4)

This aperture is hyper-resolving since it attenuates the low spatial frequency like the annular aperture. Nevertheless, it will gain better contrast than annular, linear, and modulated pupils by replacing the point detector with a quadrant detector [26]. We fabricated this aperture using MATLAB code. Operating the Fourier transform upon the pupil function described in Eqs. (3.1– 3.4), we obtain the PSF as follows: 2π ρ0 h(r ) =

PT (ρ) exp 0

0

   j2π ρr cos(θ ) ρdρdθ − λf

(3.5)

Fig. 3.1 Image (RHS) and the corresponding line plot (LHS) of the described aperture of three different segments. From the center, we construct a quadratic distribution followed by a linear distribution and end with a transparent uniform annulus. We take the plot at the horizontal central section at y = 256 pixels

3.2 Analysis

23

   2π ρ0/2 j2π ρr cos(θ ) ρdρdθ h(r ) = 2 ρ 2 exp − λf 0

0

   2π 3ρ  0 /4 j2π +2 ρr cos(θ ) ρdρdθ ρ exp − λf 0 ρ0 /2

2π ρ0 +2

exp 0 3ρ0 /4

   j2π ρr cos(θ ) ρdρdθ − λf

(3.6)

It is convenient to run the two-dimensional Fourier–Bessel transformation in polar coordinates to obtain: 3ρ    ρ0 /2  0 /4  2πρr 2πρr 2 ρdρ + 4π ρdρ ρ J0 h(r ) = 4π ρ J0 πf λf ρ0 /2

0



ρ0 + 4π

(1) J0 3ρ0 /4

 2πρr ρdρ πf

(3.7)

where J 0 is the zero-order Bessel function and r = (x 2 + y2 )1/2 is the radial coordinate in the Fourier plane. We obtain the PSF as follows:    27π J1 (w2 ) 8J1 (w1 ) J1 (3w0 /4) J1 (w0 ) + − − h 1st model (r ) = 2π w0 3w0 /4 16 w2 27w1        J0 (w2 ) 8J0 (w1 ) 8 + − −2 Ji (w2 )/w23 − Ji (w1 )/w13 27 w22 27w12 i

  J1 w20 2J2 w20 + 2π −  2 (3.8) w0 w0 

2

2

The 2nd model of the hyper-resolving aperture differs from the 1st model in that we consider the quadratic segment replaced by ρ 6 distributions of width 0.4 × radius and the central disk of radius = 0.1 × radius added, as shown in Fig. 3.4. The total radius = 128 pixels. ρ   0 P1 (ρ) = 0.0001; 0 ≤ |ρ| <   for the central dark disk 10 P2 (ρ) = ρ 6 ; |ρ0 /10| ≤ |ρ| < |ρ0 /2| for the segment of ρ 6

(3.9) (3.10)

24

3 Linear-Quadratic Aperture

P3 (ρ) = ρ; |ρ0 /2| ≤ |ρ| < |3ρ0 /4| for the linear segment

(3.11)

P4 (ρ) = 1; |3ρ0 /4| ≤ |ρ| < |ρ0 | for the annulus

(3.12)

PT = P1 (ρ) + P2 (ρ) + P3 (ρ) + P4 (ρ)

(3.13)

The PSF obtained, in an analogous manner to that made in the 1st model, using Eqs. (3.9–3.13) and substituted in Eq. (3.5), we obtain: 2π ρ0/2 h(r ) = 2

ρ 6 exp 0 ρ0/10

   j2π ρr cos(θ ) ρdρdθ − λf

   2π 3ρ  0 /4 j2π +2 r cos(θ ) ρdρdθ ρ exp − λf 0 ρ0 /2



2π ρ0 +2

exp 0 3ρ0 /4

  j2π − ρr cos(θ ) ρdρdθ λf

(3.14)

We solve Eq. (3.14) to obtain the final result for the PSF:    27π J1 (w2 ) 8J1 (w1 ) J1 (3w0 /4) J1 (w0 ) + − − w0 3w0 /4 16 w2 27w1        J0 (w2 ) 8J0 (w1 ) 8 3 3 −2 Ji (w2 )/w2 − + Ji (w1 )/w1 − 27 w22 27w12 i  

   J1 w20 J2 w100 J1 w100 J2 w20 + 2π − w0 − 6  2 −  2 w0 w0 w0 

h 2nd model (r ) = 2π

+ 24

2

 w0

10

 w0

J3 2 J3 10  w0 3 −  w0 3 2

10

− 48

2

 w0

 w0 10

J4 10 J4 2  w0 4 −  w0 4 2

(3.15)

10

The image of a point-like object given for the conventional microscope is as follows: I1st model (r ) = |h 1st model (r )|2 ; for the 1st model

(3.16)

I2nd model (r ) = |h 2nd model (r )|2 ; for the 2nd model

(3.17)

The image of a point in the confocal scanning laser microscope gives the following:

3.2 Analysis

25

I (r )confocal = |h 1st model (r )|4 ; for the 1st model

(3.18)

I (r )confocal = |h 2nd model (r )|4 ; for the 2nd model

(3.19)

The effective P.S.F. in a confocal microscope is as follows: h effective (r ) = h 1 (r )h 2 (r )

(3.20)

h1 represents the 1st lens, and h2 represents the 2nd lens.

3.2.1 The Effect of the Pinhole Diameter on the PSF of a Standard CSLM Considering that the detector pinhole has dimensions, the PSF has the form: h effective (r ) = h 1 (r ).[h 2 (r ) ∗ d(r )]

(3.21)

For a theoretical point detector, d(r) is replaced by a Dirac-delta distribution, d(r) = δ(r). In this case, Eq. (3.21) reduces to Eq. (3.20). Circular and annular pinhole apertures are considered in Eq. (3.21), while the PSF corresponds to the two microscope lenses given in Eqs. (3.8, 3.15). We computed the coherent transfer function (CTF) corresponding to the investigated models of hyper-resolving apertures from the autocorrelation of the two symmetric pupils P1 = P2 . We used a confocal scanning laser microscope. Consequently, C T F(ρ) = P1 (ρ) ∗ P2 (ρ)

(3.22)

where (*) is a symbol for the convolution operation and ρ is the radial coordinate in the aperture plane; ρ = (x 2 + y2 )1/2 . We write the integral form of the above symbolic correlation as follows:  ∞ C T F(x, y) = −∞

  P1 x − x  , y − y  P2 x  , y  d x  dy 

(3.23)

26

3 Linear-Quadratic Aperture

3.3 Results and Discussions The design of the hyper-resolving aperture is composed of three different layers, where its central half has a quadratic distribution, the next quarter has a linear distribution, and the last quarter has a constant annulus, as shown in the right-hand side of Fig. 3.1. We plotted the 2nd model of the hyper-resolving aperture in Fig. 3.2. It consists of four concentric segments. From the center is a dark disk followed by a higher-order distribution ρ 6 , linear and constant segment. We investigated the plot of the absolute value of the PSF for the two different apertures at a specific horizontal line at 256 pixels, considering that the apertures have equal radii of 8 pixels for the two models of the pupil. We showed, referring to Fig. 3.3, that the varied choice of aperture segments must improve the resolution through the attenuation of the central part. “In this study, we propose quadraticlinear-constant components as in the 1st model or using primary black disk and a higher-order radial distribution followed by linear and continuous segments as in the 2nd model. Comparing the two plots, the peak value of the absolute PSF in the 1st model is I max = 153.5, while in the 2nd model, I max = 140.5.” This difference in peak values is due to the central obstruction followed by the higher-order distribution using four concentric segments compared with the 1st model composed of three different parts in a ratio of 2:1:1 from the center. In addition, the full width at half maximum (FWHM) computed from the plots shown in Fig. 3.3 is as follows: For the 1st model, FWHM = (277–237) = 40 pixels, equaling the FWHM for the PSF corresponding to the 2nd model. At the same time, the cutoff frequency of the PSF = 291 pixels/λ f from the center at 256 pixels was reached for the 1st model compared with the cutoff spatial frequency = 289 pixels/λ f for the 2nd model. Hence, the PSF corresponding to the 2nd model gives better resolution than the PSF for the 1st model, assuming monochromatic illumination λ. We obtain the 1st-order

Fig. 3.2 On the left is the described aperture showing total radius of 128 pixels, while on the right is a line plot of the aperture at a horizontal line of 256 pixels. The pupil has a central dark disk, ρ 6 linear, constant segments arranged from the center with a ratio of 0.1: 0.4: 0.25: 0.25 of the radii

3.3 Results and Discussions

27

Fig. 3.3 Image of the PSF is shown in (a), while its intensity plot is shown in (b). The aperture diameter is 16 pixels for the 1st hyper-resolving aperture

Fig. 3.4 Plot of the absolute value of the PSF corresponding to the two models of hyper-resolving apertures at a horizontal line at 256 pixels. We plot the curve for two equal apertures with the same diameter of 18 pixels in the image of dimensions 512 × 512 pixels. In (a), the PSF corresponds to the aperture of quadratic-linear-constant segments arranged from the center with a ratio of 2:1:1, while in (b), the PSF corresponds to the 2nd aperture of the dark disk-ρ6 distribution-linear-constant segments arranged from the center with a ratio of 0.1:0.4:0.25:0.25 of the radii

diffraction pattern with a value of 41 for the 1st model, as shown in Fig. 3.4. At the same time, it has 47 for the 2nd model, as shown in Fig. 3.4b. In addition, we showed that the leg’s strength is apparent in Fig. 3.4b corresponding to the 2nd model compared with the PSF plot of Fig. 3.4a for the 1st model. Consequently, the legs of the PSF or the diffraction pattern corresponding to the 2nd model are more muscular, which may help image the extended objects like for annular apertures. In imaging, we obtain the PSF optically using an aperture placed at the (x, y) plane and a converging lens of focal length ( f ) placed behind the gap in the same plane. The illumination is a coherent laser beam λ; hence, the Fourier plane (u, v) is the focal plane. We realized the FT by the following mathematical equation in spherical coordinates:

28

3 Linear-Quadratic Aperture

ρ = (x 2 + y 2 )1/2 and r = (u 2 + v 2 )1/2 .. . 2π ρ0 h(r ) = 0

0

   j2π ρr cos(θ ) ρdρρθ PT (ρ) exp − λf

From the above FT equation, the radial spatial coordinate (ρ) in the aperture plane is related to the frequency coordinate (r/λf ). Hence, the PSF distribution as a function of (r) gives a cutoff spatial frequency at r = r c . If the numerical aperture NA = ρ 0 /f = 0.5 for moderate NA. Hence, the cutoff spatial coordinate r c becomes ≥ 500 nm, where λ is the wavelength. We compute the normalized PSF for the quadratic aperture compared with that corresponding to the circular aperture plotted in Fig. 3.5a. We showed that the cutoff spatial frequency is narrower for the quadratic aperture than that corresponding to the circular aperture. In this graph, 2π NA (r c /λ)quadratic = 3.202 < 2π NA(r c /λ)circular = 3.99. For λ = 500 nm and numerical aperture NA = 0.5, the corresponding cutoff radial values are (r c )quadratic = 510 nm and (r c )circular = 635 nm. In addition, the linear aperture gives better resolution than the circular aperture. Consequently, the two investigated models assumed taking into consideration the Improvement reached with linear and quadratic apertures. The PSF curves for the two models are plotted in Fig. 3.5b, c. The normalized absolute value of the PSF for the 1st model aperture compared with that corresponding to the circular aperture shown in Fig. 3.5b. We showed that the cutoff spatial frequency is narrower for this aperture than for the circular aperture. In this graph, 2π NA (r c /λ)1st model = 3.30 < 2π NA (r c /λ)circular = 3.99. For λ = 500 nm and NA = 0.5, the corresponding cutoff radial values are (r c )1st model = 525 nm and (r c )circular = 635 nm. Since (r c )1st model = 525 nm < (r c )circular = 635 nm, the narrower value of 525 nm gives better resolution compared with the circular aperture manipulation. The normalized absolute value of the PSF for the 2nd model aperture (blue) compared with that corresponding to the circular aperture (green) is plotted in Fig. 3.5c. We showed that the cutoff spatial frequency is narrower for this aperture than for the circular aperture. In this graph, 2π NA(r c /λ)1st model = 3.20 2 < 2π NA(r c /λ)circular = 3.99. For λ = 500 nm and NA = 0.5, the corresponding cutoff radial values are (r c )2nd model = 510 nm and (r c )circular = 635 nm. Since (r c )2nd model = 510 nm < (r c ) circular = 635 nm, the narrower value of 510 nm gives better resolution compared with the circular aperture manipulation. In Fig. 3.5, 2π NA (r c /λ) 1st model = 3.20 2 < 2π NA (r c /λ)circular = 3.99 shown in (b), 2π NA (r c /λ) 2nd model = 3.20 2 < 2π NA(r c /λ)circular = 3.99 shown in (c), and 2π NA (r c /λ)quadratic = 3.202 shown in (a). We showed that the resolution reached for the 2nd model is more significant than that for the 1st model. We obtained a difference in the cutoff divided by the average

3.3 Results and Discussions

29

a

b

c Fig. 3.5 Normalized absolute value of the PSF for the quadratic model in (a), 1st model in (b), and 2nd model in (c) compared with a uniform circular aperture

30

3 Linear-Quadratic Aperture

cutoff of approximately 2.9%. The resolution improvement in the 2nd model was 21.83% compared with that obtained in the circular uniform aperture. The legs obtained in the 1st and 2nd models attenuated compared with the diffracted legs shown for the quadratic aperture, as shown in Fig. 3.5a–c, while the spatial frequency cutoff, which is an indicator of resolution, in the case of the quadratic gap is similar to that for the 2nd model.

3.4 Conclusion The choice of the described models of hyper-resolving apertures allows for improving the PSF, giving better resolution than the circular and annular apertures. The choice of these apertures attenuates the low spatial frequency. We attribute this to the use of the central obstruction and higher-order ρ 6 distributions. The linear and annular outer segments improve the high spatial frequency in the aperture. Consequently, using these hyper-resolving apertures, the CSLM improves different resolutions compared to the conventional microscope. In addition, the CTF computed from the autocorrelation of the two symmetric hyper-resolving pupils for the two models is essential in investigating the CSLM. We discussed the effect of the pinhole diameter on the PSF of a standard CSLM using the two models of hyper-resolving apertures. Finally, a compromise of resolution and contrast was reached for these suggested apertures compared with circular, annular, and modulated apertures.

References 1. C.J.R. Sheppard, The use of lenses with an annular aperture in scanning optical microscopy. Optik 48, 329–334 (1977) 2. C.J.R. Sheppard, A. Choudhury, Image formation in the scanning microscope. J. Mod. Opt. 24, 1051–1073 (1977) 3. C.J.R. Sheppard, T. Wilson, Depth of field in the scanning microscope. Opt. Lett. 3, 115–117 (1978) 4. C.J.R. Sheppard, T. Wilson, Imaging properties of annular lenses. Appl. Opt. 18, 3764–3769 (1979) 5. C.J.R. Sheppard, T. Wilson, Fourier imaging of phase information in conventional and scanning microscopes. Phil. Trans. Roy. Soc. A295, 513–536 (1980) 6. I.J. Cox, C.J.R. Sheppard, T. Wilson, Improvement in resolution by confocal microscopy. App. opt. 21, 778–781 (1982) 7. C.J.R. Sheppard, X.Q. Mao, Confocal microscopes with slit apertures. J. Mod. Opt. 35, 1169– 1185 (1988) 8. C.J.R. Sheppard, Super-resolution in confocal imaging. Optik 80, 53–54 (1988) 9. C.J.R. Sheppard, M. Gu, Improvement of axial resolution in confocal microscopy using annular pupil. Opt. Commun. 84, 7–13 (1991) 10. M. Gu et al., Optimization of axial resolution in confocal imaging using annular pupils. Optik 93, 87–90 (1993) 11. G. Cox, C.J.R. Sheppard, Practical limits of resolution in confocal and nonlinear microscopy. Microsc. Res. Tech. 63, 18–22 (2004)

References

31

12. C.J.R. Sheppard, T. Wilson, Gaussian beam theory of lenses with annular aperture, I.E.E. J. Microwaves, Optics Acoust. 2, 105–112 (1978) 13. T. Azuma, T. Kei, Super- resolution spinning disk confocal microscopy using optical photon reassignment. Opt. Express 23, 15003–15011 (2015) 14. C.J.R. Sheppard, K.G. Larkin, The effect of numerical aperture on interference fringe spacing. Appl. Opt. 34, 4731–4734 (1995) 15. A.M. Hamed, Numerical speckle images formed by diffusers using modulated conical and linear apertures. J. Modern Opt. 56, 1174–1181 (2009) 16. A.M. Hamed, Formation of speckle image formed for diffusers illuminated by the modulated aperture (circular obstruction). J. Modern Opt. 56, 1633–1642 (2009) 17. A.M. Hamed, Discrimination between speckle images using diffusers modulated by deformed apertures: simulations. J. Opt Eng. 50, 1–7 (2011) 18. A.M. Hamed, Computer generated quadratic and higher-order apertures and its application on numerical speckle images. Opt. Photonics J. 1, 43–51 (2011) 19. A.M. Hamed, Study of graded-index and truncated apertures using speckle images. Precis. Instrum. Mech. 3, 144–152 (2014) 20. A.M. Hamed, T. Al-Saeed, Image analysis of modified Hamming aperture: application on confocal microscopy and holography. J. of Modern Opt. 62, 801–810 (2015) 21. J.J. Clair, A.M. Hamed, Theoretical studies on coherent optical microscopes. Optik 64(1983), 133–141 (1983) 22. A.M. Hamed, J.J. Clair, Image, and super- resolution in coherent optical microscopes. Optik 64, 277–284 (1983) 23. A.M. Hamed, J.J. Clair, Studies on optical properties of confocal scanning optical microscope using pupils with radially transmission ρn distribution. Optik 65, 209–218 (1983) 24. A.M. Hamed, A study on amplitude modulation and an application on confocal imaging. Optik 107, 161–164 (1998) 25. S.Y. Leigh, Modulated-alignment dual-axis (MAD) confocal microscopy for deep optical sectioning in tissues. Biomed. Opt. Express 5, 1709–1720 (2014) 26. M. Castello et al., Image scanning microscopy with a quadrant detector. Opt. Lett. 40, 5355– 5358 (2015) 27. A.M. Hamed, Improvement of point spread function (P.S.F.) using linear-quadratic aperture. Optik 131, 838–849 (2017)

Chapter 4

A Study on Star Aperture and Their Application in Confocal Scanning Laser Microscope

4.1 Introduction The microscope considered in the processing is called CSLM. It is mainly composed of two objective lenses arranged in tandem and has a common short focus where the scanned object is placed. Coherent illumination of the microscope is provided by a laser beam, and a coherent point detector is placed in the imaging plane. This confocal microscope has been studied by Sheppard et al. [1–4]. An explanation for the imaging of confocal microscopy attaining super-resolution in confocal imaging was presented [5, 6]. It was shown early by Sheppard et al. [7] that resolution has been improved by using an annular aperture compared with the open circular aperture. The microscope resolution is dependent on the wavelength of illumination and the numerical aperture NA or the aperture size for a certain focal length; hence, the theoretical limit of resolution is computed as follows: resolution = λ/NA. The distribution in the aperture has little effect on the resolution and contrast Hamed et al. [8–12]. Most confocal microscopes do not produce images in real time with nonlaser light sources. The tandem scanning confocal microscope does produce such images, but because the pinhole apertures of the Nipkow disk must be placed far apart to reduce crosstalk between neighboring pinholes, only 1% or less of the light available for imaging is used. Wilson et al. [13] showed that by using aperture correlation techniques and relaxing the requirement to obtain a pure confocal image directly, one can obtain real-time confocal images with a dramatically increased (25% or even 50%) light budget. In this study, we present a new black-and-white (B/W) concentric star aperture. We have an analytical formula for the point spread function (PSF) computed by operating the Fourier transform upon the aperture. Then, we obtained the PSF by applying the fast Fourier transform (FFT) operation. Second, we computed the CTF for the CSLM provided with the B/W concentric star aperture and compared it with that in the case of transparent star aperture. Finally, we reconstructed the coronary artery image © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. M. Hamed, Modulated Apertures and Resolution in Microscopy, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-3-031-47552-8_4

33

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4 A Study on Star Aperture and Their Application in Confocal Scanning …

using the CSLM provided with the modulated star aperture. A comparison of the reconstructed images is given in the case of circular apertures at different diameters.

4.2 Theoretical Analysis Referring to Fig. 4.1, we propose the following: √ AB = AC = BC = D E = D F = E F = a, and b = a

3 . 2

The aperture is composed of triangle ABC of equal sides and three smaller triangles centrally located at the following points:   √    √   a  a  a 3 a 3 x− a1 , y − a1 , y − b1 , x + b1 , 3 3 3 3  x y where , and a1 b1

 

Fig. 4.1 Geometry of the star aperture is composed of a large rectangle ABC and three small rectangles located at D, E, and F

4.2 Theoretical Analysis

35

√ a a 3 a1 = , and b1 = 3 6 Hence, the aperture is represented as follows: x y + 1 ((x − a1 )/a1 , (y − b1 )/b1 ) (x, y) =  , a b  x y + 2 ((x + a1 )/a1 , (y − b1 )/b1 )) + 3 , a1 b1

(4.1)

where the symbol  represents the triangular function shown in Eq. 4.1. Each of the four triangles in Eq. 4.1 is represented by the autoconvolution of the rectangular functions of widths equal to half the width of the triangular functions. Hence, Eq. 4.1 is rewritten as follows:  x x y y ⊗ rect , , a/2 b/2 a/2 b/2   x − a1 y − b1 x − a1 y − b1 , ⊗ rect1 , + rect1 a1 /2 b1 /2 a1 /2 b1 /2   x + a1 y − b1 x + a1 y − b1 , ⊗ rect2 , + rect2 a1 /2 b1 /2 a1 /2 b1 /2   x x y y , ⊗ rect3 , + rect3 a1 /2 b1 /2 a1 /2 b1 /2 

P(x, y) = rect

(4.2)

Consequently, we compute the PSF corresponding to the star aperture by operating the Fourier transform upon Eq. 4.2. We make use of the properties of Fourier transform and convolution operations to obtain the following formula:   a 2 b2 au bv 2 2 sinc sinc h(u, v) = F T {P(x, y)} = 16 2f 2f   a12 b12 a1 u b1 v 2 2 sinc sinc + 16 2f 2f

 j4 

 exp f (a1 u + b1 v) − j3 − j5 exp + (a1 u + b1 v) + exp (a1 u + b1 v) 4 2 2 16 u v 2f 2f 4 f4



2 −j −j − exp (3a1 u + 5b1 v) − exp (5a1 u + 3b1 v) 2f 2f

 j4 



exp − f (a1 u + b1 v) j5 j3 + exp exp u + b v) u + b v) (a (a 1 1 1 1 164 u 2 v 2 2f 2f

− exp

4 f4



2 j j (3a1 u + 5b1 v) − exp (5a1 u + 3b1 v) 2f 2f

(4.3)

36

4 A Study on Star Aperture and Their Application in Confocal Scanning …

4.3 Results and Discussions A coronary artery image used in the processing is shown in Fig. 4.2. The image dimensions are 512 × 512 pixels. We plotted the concentric black-and-white (B/W) star aperture as shown in Fig. 4.3. Six zones are considered, and the image dimensions are 512 × 512 pixels. The normalized PSF corresponding to the B/W star image shown in Fig. 4.3 is computed using the FFT algorithm and plotted as in Fig. 4.4. The computed total bandwidth is equal to 8 pixels. The PSF has a sharp central peak, and the legs have an irregular distribution of three peaks corresponding to the three transparent zones and then decay randomly. The Fourier spectrum image corresponding to the B/W star aperture shown in Fig. 4.2 is computed and plotted as in Fig. 4.5a. It shows three sharp diffracted images corresponding to the conjugate spikes of the star image. One of the diffracted images is aligned vertically, and the other two images are aligned 600 with the vertical image. The central zone of the Fourier spectrum corresponding to the B/W star aperture is shown in Fig. 4.5b. The noisy pattern that appeared in the image is due to the interference of the different wings of the modulated star pattern. Hence, the central zone looks like a diffuser. A transparent star aperture is plotted in Fig. 4.6 for comparison with the B/W star aperture. The image dimensions are 512 × 512 pixels. The computed normalized PSF has a sharp central peak and is surrounded by two other smaller decaying peaks of 12% and 6% compared to the central peak and then decaying weak peaks, as shown in Fig. 4.7. The computed total bandwidth is equal to 4 pixels. Fig. 4.2 A coronary artery image used in the processing. The image dimensions are 512 × 512 pixels

4.3 Results and Discussions

37

Fig. 4.3 A star aperture of six concentric black-and-white zones. The image dimensions are 512 × 512 pixels

Fig. 4.4 Plot of the normalized PSF corresponding to the B/W star image in Fig. 4.3 is shown

38

4 A Study on Star Aperture and Their Application in Confocal Scanning …

a

b Fig. 4.5 a Fourier spectrum corresponding to the B/W star aperture shown in Fig. 4.3. b Central zone of the Fourier spectrum corresponding to the B/W star aperture shown in Fig. 4.3. The noisy pattern is due to interference of the different wings of the modulated star pattern

4.3 Results and Discussions

39

Fig. 4.6 A transparent star aperture. The image dimensions are 512 × 512 pixels

Fig. 4.7 Plot of the normalized PSF corresponding to the transparent star aperture in Fig. 4.6

The Fourier spectrum image corresponding to the transparent star aperture shown in Fig. 4.6 is computed and plotted as in Fig. 4.8a. The diffraction has a repetitive shape depending on the geometry of the star aperture. The central zone of the Fourier spectrum corresponding to the transparent star aperture is shown in Fig. 4.8b. The image is less noisy compared to the diffracted image shown in Fig. 4.5b in the case of the B/W concentric star aperture.

40

4 A Study on Star Aperture and Their Application in Confocal Scanning …

a

b Fig. 4.8 a Fourier spectrum corresponding to the transparent star aperture shown in Fig. 4.6. b Central zone of the Fourier spectrum corresponding to the transparent star aperture shown in Fig. 4.6

4.3 Results and Discussions

41

Another comparison is made using a circular aperture, as shown in Fig. 4.9, followed by the normalized PSF, as shown in Fig. 4.10. The central peak is surrounded by decaying peaks. The Fourier spectrum is shown in Fig. 4.11a. The central zone of the Fourier spectrum corresponding to the circular aperture has concentric rings originating from the revolution symmetry of the circular shape and is represented in Fig. 4.11b.

Fig. 4.9 A transparent circular aperture. The image dimensions are 512 × 512 pixels

Fig. 4.10 Plot of the normalized PSF corresponding to the circular aperture is shown. The computed total bandwidth is equal to 8 pixels

42

4 A Study on Star Aperture and Their Application in Confocal Scanning …

a

b Fig. 4.11 a Fourier spectrum corresponding to the circular aperture shown in Fig. 4.9. b Central zone of the Fourier spectrum corresponding to the circular aperture

4.3 Results and Discussions

43

We computed the normalized CTF using direct convolution for the B/W concentric star aperture, and we plotted it as in Fig. 4.12a. Six harmonic degraded peaks depending on the aperture geometry are shown in the plot considering the central peak. The normalized CTF using FFT is computed and plotted in Fig. 4.12b. Similar degraded peaks are shown in the plot as expected. The computed image of the CTF for the two concentric B/W star apertures has a repetitive diffracted hexagonal shape and is plotted in Fig. 4.12c.

a

b Fig. 4.12 a Normalized CTF using direct convolution corresponding to the B/W concentric star aperture. Six harmonic degraded peaks are shown in the plot. b Normalized CTF using FFT corresponding to the B/W concentric star aperture. Six degraded peaks are shown in the plot. c Image of the CTF corresponds to the two concentric B/W star apertures

44

4 A Study on Star Aperture and Their Application in Confocal Scanning …

Fig. 4.12 (continued)

For comparison with the transparent star aperture, the computed CTF plot corresponds to the two transparent star apertures using direct convolution, as shown in Fig. 4.13a. A similar plot using FFT is shown in Fig. 4.13b. The computed image is shown in Fig. 4.13c. We showed that the central peak is surrounded by a homogenous shape compared to the repetitive hexagonal shape in Fig. 4.12c. Hence, we discriminate between the two CTF images, which are dependent on the aperture geometry. In another comparison with the circular aperture, the computed CTF has the known shape as in Fig. 4.14a. The total width equals 128 pixels, i.e., two times the aperture diameter. The CTF image is shown in Fig. 4.14b. The normalized CTF uses direct convolution corresponding to the coronary artery image shown in Fig. 4.2. The total width is two times the image width along a certain direction. The asymmetry shown in the plot is due to the geometry of the image, as shown in Fig. 4.15. The reconstructed images corresponding to the original coronary artery image using two B/W concentric star apertures in the CSLM are shown in Fig. 4.16. Using two symmetric transparent star apertures in the CSLM, the reconstructed images corresponding to the original coronary artery image are shown in Fig. 4.17. For comparison, using the CSLM, the reconstructed images corresponding to the original coronary artery image using circular apertures of diameters 64, 128, and 256 pixels are shown in Fig. 4.18. We showed that the resolution is improved by increasing the aperture diameter for a certain focal length.

4.3 Results and Discussions

45

a

b Fig. 4.13 a CTF plot corresponds to the two transparent star apertures using direct convolution. b CTF plot corresponds to the two transparent star apertures using FFT. c Image of the CTF corresponds to the two transparent star apertures

46

4 A Study on Star Aperture and Their Application in Confocal Scanning …

Fig. 4.13 (continued)

4.4 Conclusion The PSF has stronger legs compared to the legs obtained in the PSF corresponding to the transparent star aperture. Hence, the design of a new B/W concentric star aperture is useful for imaging extended objects in confocal microscopy. In addition, we observed a noisy pattern in the central zone of the Fourier spectrum corresponding to the B/W star aperture. This noisy pattern has a random shape similar to a diffuser due to the interference of the different wings of the modulated star pattern. Second, discrimination between the B/W star and the transparent star apertures is shown. The CTF in the case of the B/W star aperture has six decaying peaks compared to the smooth decaying shape in the case of the transparent star aperture. Another comparison between the CTF in the case of the transparent star aperture and that obtained in the case of the circular aperture showed a reasonable difference. Finally, the application of the coronary artery image in the CSLM using different apertures showed that the contrast of the reconstructed images is dependent on the intensity; hence, for greater aperture diameters, we obtain increased contrast. In addition, the resolution of the image is dependent on the PSF, which is affected by the aperture shape and the wavelength of the illumination.

4.4 Conclusion

47

a

b Fig. 4.14 a Plot of the normalized CTF corresponds to the two circular apertures, each with a diameter of 64 pixels. The total width equals 128 pixels, i.e., two times the aperture diameter. b Image of the CTF corresponds to the two circular apertures, each having a diameter of 64 pixels

48

4 A Study on Star Aperture and Their Application in Confocal Scanning …

Fig. 4.15 Normalized CTF using direct convolution corresponding to the coronary artery image shown in Fig. 4.2. The total width is two times the image width along a certain direction. The asymmetry shown in the plot is due to the geometry of the image

Fig. 4.16 Reconstructed images correspond to the original coronary artery image using two B/W concentric star apertures shown in Fig. 4.3

4.4 Conclusion

49

Fig. 4.17 Reconstructed images correspond to the original coronary artery image using two symmetric transparent star apertures shown in Fig. 4.5

Fig. 4.18 Reconstructed images correspond to the original coronary artery image using circular apertures of diameters 64, 128, and 256 pixels

50

4 A Study on Star Aperture and Their Application in Confocal Scanning …

Fig. 4.18 (continued)

References 1. C.J.R. Sheppard, A. Choudhury, Image formation in the scanning microscope. Opt. ActaActa 24, 1051–1073 (1977). https://doi.org/10.1080/713819421 2. C.J.R. Sheppard, T. Wilson, Depth of the field in scanning microscope. Opt. Lett.Lett. 3, 115–117 (1978). https://doi.org/10.1364/OL.3.000115 3. C.J.R. Sheppard, X.Q. Mao, Confocal microscopes with slit apertures. J. Mod. Opt. 35, 1169– 1185 (1988). https://doi.org/10.1080/09500348814551251 4. C.J.R. Sheppard, Super- resolution in confocal imaging. Optik 80, 53–54 (1983) 5. I.J. Cox, C.J.R. Sheppard, T. Wilson, Super- resolution by confocal fluorescence microscopy. Optik 60, 391–396 (1982) 6. I.J. Cox, C.J.R. Sheppard, T. Wilson, Improvement in resolution by nearly confocal microscopy. Appl. Opt. 21, 778–781 (1982). https://doi.org/10.1364/AO.21.000778 7. C.J.R. Sheppard, T. Wilson, Imaging properties of annular lenses. Appl. Opt. 18, 3764–3769 (1979). https://doi.org/10.1364/AO.18.003764 8. A.M. Hamed, J.J. Clair, Image, and super-resolution in optical coherent microscopes. Optik 64, 277–284 (1983) 9. A.M. Hamed, Resolution and contrast in confocal optical scanning microscope. Opt. Laser Technol. 16, 93–96 (1984). https://doi.org/10.1016/0030-3992(84)90061-6 10. A.M. Hamed, J.J. Clair, Studies on optical properties of confocal scanning optical microscope using pupils with radially transmission distribution. Optik 65, 209–218 (1983) 11. A.M. Hamed, Study of graded index and truncated apertures using speckle images. Precis Instrument Mech PIM. 3, 144–152 (2014) 12. A.M. Hamed, Improvement of point spread function (PSF) using linear quadratic aperture. Optik 131, 838–849 (2017). https://doi.org/10.1016/j.ijleo.2016.11.201 13. T. Wilson, R. Skaitis et al., Confocal microscopy by aperture correlation. Opt. Lett.Lett. 21, 1879–1881 (1996)

Chapter 5

A Study on Some Transparent Rose Apertures: An Application on the Confocal Laser Scanning Microscope and the Speckle Images

5.1 Introduction Early authors studied confocal microscopic imaging and suggested aperture modification using annular and Gaussian apertures [1–7], followed by authors [8, 9] using linear, quadratic, and black-and-white concentric apertures. Recently, the author computed the point spread function (PSF) corresponding to the new polynomial, rectangular, and triangular apertures [10–13]. The computation of the average speckle size from the point spread function for triangular apertures is investigated in [10]. The point spread function of hexagonally segmented telescopes by a new symmetrical formulation was investigated [14, 15]. The studies seek to further improve the resolution of the confocal scanning microscope since the two objective lenses are responsible for the resolution. Other methods of modulation based on structured illumination microscopy (SIM) are one of the most significant wide-field super-resolution optical imaging techniques. Conventional SIM utilizes a sinusoidal structured pattern to excite the fluorescent sample, which eventually downregulates higher spatial frequency sample information within the diffraction-limited passband of the microscopy system, as outlined [16, 17]. Stochastic optical reconstruction microscopy (STORM), a widely used super-resolution technique based on the principle of single-molecule localization, is outlined in [18]. STORM achieves a spatial resolution of 20–30 nm, a tenfold improvement compared to conventional optical microscopy.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. M. Hamed, Modulated Apertures and Resolution in Microscopy, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-3-031-47552-8_5

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5 A Study on Some Transparent Rose Apertures: An Application …

5.1.1 The Aperture Construction and Computation of the PSF The rose curve is a graph that is produced from a polar equation in the form of r = a sin (n θ ) or r = a cos (n θ ), where a /= 0 and n is an integer > 1. They are called rose curves because the loops that are formed resemble petals. The number of petals that are present will depend on the value of n. The value of a will determine the length of the petals. Then, the rose aperture of n petals is represented as follows: P(a, θ ) = a cos(n); where n is the number of petals.

(5.1)

For an even value of n: n = 2, we have four petals, and for n = 4, we obtain eight petals. For an odd value of n, for n = 3, we obtain three petals, and for n = 5, we obtain five petals, as shown in Fig. 5.1. The point spread functions (PSFs) corresponding to the rose apertures are obtained by operating the fast Fourier transform (FFT) on the aperture.

5.1.2 Application on the Confocal Scanning Laser Microscope (CSLM) The algorithm used for image detection using the CSLM is as follows Hamed (2023): The confocal scanning laser microscope (CSLM) mainly consists of two confocal objectives arranged in tandem where the object (the coronary arteries) is in the common short focus in the plane (x, y), as shown in Fig. 5.2 The illumination and detection are coherent. The apertures limiting the objective lenses have B/W concentric hexagonal shapes. For confocal imaging, the complex amplitude in the detection plane considering coherent illumination and coherent detection is as follows: A(x, y) = h 1 (x, y)h 2 (x, y) ⊗ g(x, y)

(5.2)

˜ A(u, v) = F T {A(x, y)} = [P1 (u, v) ⊗ P2 (u, v)].G(u, v)

(5.3)

C T F(u, v) = P1 (u, v) ⊗ P2 (u, v)and G(u, v) = F T {A(x, y)}

(5.4)

Inverse FT of Eq. (5.3) gives Eq. (5.2). h 1 represents the PSF corresponding to the 1st objective lens, and h 2 stands for the PSF for the 2nd objective lens.

5.1 Introduction

53

Fig. 5.1 Four transparent rose apertures are surrounded by a dark square background. The aperture in polar coordinates is represented as follows: r = a cos(n); n = 2, 3, 4, and5. The azimuthal range θ = [0 2π ] and a = constant

Fig. 5.2 Setup for the confocal laser scanning microscope (CLSM). SF spatial filter composed of an objective lens O, pinhole P, and converging lens L. L 1 and L 2 : objective lenses of the microscope limited by the apertures P1 and P2 have rose curves. g is the object located in the confocal plane (x, y)

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5 A Study on Some Transparent Rose Apertures: An Application …

Taking the modulus square on the complex amplitude A(x, y), we obtain the image intensity represented as follows: I (x, y) = |h 1 (x, y)h 2 (x, y) ⊕ g(x, y)|2

(5.5)

5.1.3 Application to Speckle Imaging The speckle images are formed by illuminating both the diffuser and the aperture in the plane (x, y) with coherent laser radiation and detecting the images in the focal plane of converging lens L in the plane (u, v). We obtain a digital speckle image by operating the FFT on both the diffuser and the modulated aperture. We summarize the speckle formation as follows: S(u, v) = F T {P(x, y).d(x, y)}.

(5.6)

The Fourier transform of the multiplication in Eq. (5.6) is transformed into the convolution product of the Fourier transform of each term. Then, we obtain: S(u, v) = h(u, v) ⊗ D(u, v)

(5.7)

h(u, v) = F T {P(x, y)} and D(u, v) = F T {d(x, y)}.

(5.8)

Hence, the PSF for a certain modulated aperture will affect the obtained speckle image, as shown in Eq. (5.7).

5.2 Results and Discussion The four transparent rose apertures surrounded by a dark square background are shown in Fig. 5.1, where the aperture in polar coordinates is represented as follows: r = a cos(n); n = 2, 3, 4, and 5. The azimuthal range θ = [0 2π ] and a = constant. The setup for the confocal laser scanning microscope (CLSM) is shown in Fig. 5.2 SF spatial filter composed of an objective lens O, pinhole P, and converging lens L. L 1 and L 2 : objective lenses of the microscope limited by the apertures P1 and P2 have rose curves. g is the object located in the confocal plane (x, y). This microscope is considered in the reconstruction of the COVID-19 image considering the rose apertures in Fig. 5.1 (Table 5.1). Referring to Fig. 5.3a, we show the plot of the normalized PSF corresponding to the rose aperture of n = 2. The total bandwidth = 2 pixels. We observed the 1st secondary peak of high value compared to the central peak of 34% located at 24 pixels followed by harmonic peaks of low values of 2%.

5.2 Results and Discussion

55

Table 5.1 Rose apertures of different n and the corresponding total bandwidth Rose aperture of cos (n θ ) distribution

Total bandwidth in pixels

N =2

2

N =3

4

N =4

6

N =5

2

a

b Fig. 5.3 a Normalized PSF corresponding to the rose aperture of n = 2. The total bandwidth = 2 pixels. b Normalized PSF corresponding to the rose aperture of n = 3. The total bandwidth = 4 pixels. c Normalized PSF corresponding to the rose aperture of n = 4. The total bandwidth = 6 pixels. d Normalized PSF corresponding to the rose aperture of n = 5. The total bandwidth = 2 pixels

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5 A Study on Some Transparent Rose Apertures: An Application …

a

b Fig. 5.3 (continued)

5.2 Results and Discussion

57

Referring to Fig. 5.3b, we show the plot of the normalized PSF corresponding to the rose aperture of n = 3. The total bandwidth = 4 pixels. It is two times the width of the PSF for the aperture of n = 2. We observed a sharp peak of 50% compared to the unity value of 100% corresponding to the central peak and located at 25 pixels. Another secondary peak of weak values of 12% is located at 29 pixels, followed by degraded values of nearly 4 and 2%. Referring to Fig. 5.3c, we show the plot of the normalized PSF corresponding to the rose aperture of n = 4. The total bandwidth = 6 pixels. It is the greatest bandwidth compared to that corresponding to the apertures where n = 2, 3, and 5. We observed a sharp peak of 23% compared to the unity value of 100% corresponding to the central peak and located at 26 pixels. Other weak peaks are randomly distributed oscillating between higher and lower values at nearly 2%. Referring to Fig. 5.3d, we show the plot of the normalized PSF corresponding to the rose aperture of n = 5. The total bandwidth = 2 pixels. We observed two secondary peaks of high values compared to the central peak of 50% and 32% located at 24 pixels and 28 pixels, respectively. A third lower secondary peak of 8% is shown on the plot at 35 pixels. We observed that the central peak of unity is located at 22 pixels for all the plots shown in Fig. 5.3a–d. In general, we showed different PSF plots, as shown in Fig. 5.3a–d, depending on the aperture geometry. Five sharp peaks of different intensities are shown with an aperture of n = 5. The peaks have values of 77, 29, 21, and 20% compared to the central peak of unity located at 1193 pixels shifted from the center by 1193–1024 = 169 pixels. Three peaks to the left of the central peak are shown, and one to the right of the central peak is shown. We computed the coherent transfer function (CTF) corresponding to the apertures in Fig. 5.1, and we obtained the plots shown in Fig. 5.4a–d. The total bandwidth for all the apertures shown in Fig. 5.4 is equal to 2048 pixels and hence two times the aperture width of 1024 pixels, as expected. The CTF shape is dependent on the aperture, which is different for all plots. The reconstructed images using the CSLM are plotted in Fig. 5.5. The reconstructed image of COVID-19 using the CSLM provided with the two objectives of the microscope with rose apertures, which depends on the number of petals (n), as shown in Fig. 5.5a–d. The reconstructed image using the CSLM provided with objectives with a uniform circular aperture is given for comparison in Fig. 5.5e. Different speckle images are obtained by operating the FFT on the cascaded diffuser, rose, and circular apertures. The rose apertures are shown in Fig. 5.1, where n = 2, 3, 4, and 5. The speckle images are shown in Fig. 5.6. It is shown that for certain diffusers limited by a circular pupil, the speckle image is dependent on the geometry of the rose aperture (Figs. 5.6 and 5.7).

58

5 A Study on Some Transparent Rose Apertures: An Application …

a

b Fig. 5.4 a Plot of the normalized CTF for the rose aperture of n = 2 using direct convolution at y = 512 pixels. b Plot of the normalized CTF for the rose aperture of n = 3 using direct convolution at y = 512 pixels. c Plot of the normalized CTF for the rose aperture of n = 4 using direct convolution at y = 512 pixels. Two peaks of values 68 and 31% are to the right of the central peak, and another two peaks of 72% and 48% are to the right of the central peak. d Plot of the normalized CTF for the rose aperture of n = 5 using direct convolution at y = 512 pixels

5.2 Results and Discussion

59

a

b Fig. 5.4 (continued)

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5 A Study on Some Transparent Rose Apertures: An Application …

Fig. 5.5 a Reconstructed image of COVID-19 using the CSLM provided with the two objectives of the microscope with a rose aperture of n = 2. b Reconstructed image of COVID-19 using the CSLM provided with the two objectives of the microscope with a rose aperture of n = 3. c Reconstructed image of COVID-19 using the CSLM provided with the two objectives of the microscope with a rose aperture of n = 4. d Reconstructed image of COVID-19 using the CSLM provided with the two objectives of the microscope with a rose aperture of n = 5. e Reconstructed image of COVID-19 using the CSLM provided with the two objectives of the microscope with a circular aperture

5.2 Results and Discussion

Fig. 5.5 (continued)

61

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5 A Study on Some Transparent Rose Apertures: An Application …

Fig. 5.6 Different speckle images are obtained from the cascaded diffuser, rose, and circular apertures. The rose apertures are shown in Fig. 5.1, where n = 2, 3, 4, and 5

Fig. 5.7 Speckle image obtained from a diffuser and circular aperture of radius = 64 pixels

References

63

5.3 Conclusion First, we computed the PSF corresponding to the rose apertures of different petals of order n = 2, 3, 4, and 5. The bandwidth corresponding to the apertures is increased with the number of petals up to n = 4. Equal bandwidth is obtained in the case of n = 2 and 5. In addition, the PSF shape has a different distribution corresponding to the different apertures. Second, we computed the CTF for all the rose apertures giving shape depending on the number of petals in the rose aperture. The total width of the CTF is equal to two times the aperture diameter. Finally, the reconstructed images of COVID-19 using the CSLM provided with rose apertures of different petals showed different contrasts. The microscope resolution is better for lower values of the number of petals.

References 1. C.J.R. Sheppard, A. Choudhury, Image formation in the scanning microscope. Opt. Acta 24, 1051–1073 (1977). https://doi.org/10.1080/713819421 2. C.J.R. Sheppard, T. Wilson, Depth of the field in scanning microscope. Opt. Lett. 3, 115–117 (1978). https://doi.org/10.1364/OL.3.000115 3. L. Thibon et al., Resolution enhancement in confocal microscopy using Bessel-Gauss beams. Opt. Express 25(3), 2162–2177 (2017). https://doi.org/10.1364/OE.25.002162 4. C.J.R. Sheppard, Superresolution in confocal imaging. Optik 80, 53–54 (1983) 5. I.J. Cox, C.J.R. Sheppard, T. Wilson, Superresolution by confocal fluorescence microscopy. Optik 60, 391–396 (1982) 6. I.J. Cox, C.J.R. Sheppard, T. Wilson, Improvement in resolution by nearly confocal microscopy. Appl. Opt. 21, 778–781 (1982). https://doi.org/10.1364/AO.21.000778 7. C.J.R. Sheppard, T. Wilson, Imaging properties of annular lenses. Appl. Opt. 18, 3764–3769 (1979). https://doi.org/10.1364/AO.18.003764 8. A.M. Hamed, Numerical speckle images formed by diffusers using modulated conical and linear apertures. J. Mod. Opt. 56(10), 1174–1181 (2009). https://doi.org/10.1080/095003409 02985379 9. A.M. Hamed, Improvement of point spread function (P.S.F.) using linear-quadratic aperture. Optik 131, 838–849 (2017). https://doi.org/10.1006/j.ijleo.2016.11.211 10. A.M. Hamed, The computation of the average speckle size from the point spread function for triangular apertures. Eur. Phys. J. Plus 138, 340 (2023). https://doi.org/10.1140/epjp/s13360023-03909-2 11. A.M. Hamed, Contrast of laser speckle images using some modulated apertures. J Phys Pram 95, 122 (2021). https://doi.org/10.1007/s12043-021-02151-8 12. A.M. Hamed, Speckle imaging of annular Hermite Gaussian laser beam. J Phys Pram 95, 202 (2021). https://doi.org/10.1007/s12043-021-02231-9 13. A.M. Hamed, Investigation of a new modulated aperture using speckle techniques. Beni-Suef Univ J Basic Appl Sci 11, 39 (2022). https://doi.org/10.1186/s43088-022-00222-2 14. S. Itoh, T. Matsuo et al., Point spread function of hexagonally segmented telescopes by new symmetrical formulation. MNRAS 483, 119–131 (2019) 15. A.M. Hamed, Application of a hexagonal aperture on the confocal scanning laser microscope. Opt. Quant. Electron. 55(8), 749 (2023)

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16. R. HeintZman, T. Huser, Superresolution structured illumination microscopy. Chem. Rev. 117(23), 13890–13908 (2017). https://doi.org/10.1021/acs.chemrev.7b00218 17. S. Krishnendu, J. Joby, An overview of structured illumination microscopy: recent advances and perspectives. J. Opt. 23 (2021). https://doi.org/10.1088/2040-8986/ac3675 18. J. Quan, H. Xu, A. Ma, et al., Stochastic optical reconstruction microscopy. Curr. Protocol Cytom. 81 (2017). https://doi.org/10.1002/cpcy.23

Chapter 6

A New Model of Modulated Aperture [28]

6.1 Background The speckle effect [1–5] results from the interference of waves of the same frequency, having separate phases and amplitudes, which add together to give a resultant wave whose amplitude and intensity vary randomly. The size of the speckles is a function of the wavelength of the light. In addition, the size of the laser beam illuminates the first surface and the distance between this surface and the plane where the speckle pattern formed. Dainty [3] derives an expression for the mean speckle size as σ = λ z/D, where D is the width of the illuminated area and z is the distance between the object and the location of the speckle pattern. In [6], we proposed multiple-image encryption based on different modulated apertures in an optical setup under a holographic arrangement. The system is a security architecture that uses different pupil aperture masks in the encoding lens to encrypt other images. The technique uses multiplexing to perform high-density holographic storage [7]. Speckle modeled considering the analysis of the statistics of the image of a phase object such as a rough surface. We calculated the image using the imaging system’s coherent transfer function and the angular spectrum representation. This approach gives a three-dimensional image and includes the effects of a high numerical aperture and the finite depth of the structure. We assumed different correlation coefficients. We have fractal distributions, such as exponential and Gaussian correlations [8]. The laser speckle contrast images (LSCI) were investigated in many biomedical publications [9–19] while using modulated apertures as in [20]. We used linear and quadratic apertures to form speckle images [21, 22]. At the same time, we investigate [23] the annular Hermite–Gaussian aperture. Recently, we investigated a slab waveguide that includes a linear substrate, an exponentially graded-index guiding layer, and a power-dependent refractive index covering the medium [24]. At the same time, a three-layer slab waveguide with a graded-index film and a nonlinear substrate is shown [25]. In addition, we give other © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. M. Hamed, Modulated Apertures and Resolution in Microscopy, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-3-031-47552-8_6

65

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6 A New Model of Modulated Aperture [28]

work on a planar waveguide with an exponential grade-index guiding layer and a nonlinear cladding in [26]. We present a numerical analysis in [27] for the proposal of a novel decahedron photonic crystal fiber where the central elliptic core is filled with a highly nonlinear 2D material, molybdenum disulfide (MoS2 ). This chapter presents a new aperture composed of eight equally spaced conic circles that are shifted outside the center and used to form a speckle image. We obtained a reconstruction of the pupil. In addition, we computed the point spread function (PSF) and the autocorrelation corresponding to the cited model. The aim of the work using the cited apertures is to gain better transverse resolution and improved contrast compared with annular pupils that give poorer contrast.

6.2 Methods We fabricated a new aperture in the form of eight equally spaced conic circles and placed them at a constant distance from the center [28]. Four circles were set along the Cartesian coordinates, while the other four were placed along the rotated coordinates by angle = 45. The design of this aperture and other similar arrangements using linear and quadratic distributions was realized using MATLAB code. The following theoretical analysis corresponds to the creation of the conic structure of eight conic circles, followed by the computation of the PSF using the FFT algorithm.

6.3 Theoretical Analysis The eight equally spaced conic apertures shown in Fig. 6.1 are described as follows: PT (x, y) = P1 + P2 + · · · + P8

(6.1)

with the apertures P1,2 found along the x-axis at distances x0 and the apertures P3,4 located along the y-axis at distances y0 represented as follows: P1,2 (x, y) = 1 − P3,4 (x, y) = 1 −

/ /

(x ± x0 )2 + y 2

(6.2)

x 2 + (y ± y0 )2

(6.3)

For coordinates rotated, another four apertures are shown in the exact Fig. 6.1 and are represented as follows: /( P5,6 (x, y) = 1 −

x0 x±√ 2

)2

(

y0 + y±√ 2

)2 (6.4)

6.4 Results

67

Fig. 6.1 Eight conic apertures placed tangent to the annular aperture. The annular width = 16 pixels, and the external radius = 350 pixels. The radius of each conic aperture = 64 pixels and the plot of the conic aperture of radius = 256 pixels placed at 512, 512 pixels

/( P7 (x, y) = 1 − /( P8 (x, y) = 1 −

x0 x−√ 2 x0 x+√ 2

)2

)2

) ( y0 2 + y+√ 2 (

y0 + y−√ 2

(6.5)

)2 (6.5)

The Fourier transform corresponding to this model of eight conic apertures is not easy to solve, so we use the FFT technique to obtain the PSF. Similarly, we computed the PSF for the other eight equally spaced apertures of the linear and quadratic distributions using the FFT technique.

6.4 Results The design of eight conic apertures placed tangent to the annular aperture is shown in Fig. 6.1. The annular width = 16 pixels, and the external radius = 350 pixels. The radius of each conic aperture = 64 pixels. We showed a plot of the conic aperture of radius = 256 pixels placed at 512 × 512 pixels in the RHS of the Fig. 6.1. Eight equally spaced conic apertures were the centers placed at a fixed distance = 268 pixels from the center (512,512), as shown in Fig. 6.2. The radius of each conic aperture = 64 pixels. We computed the PSF using FFT for all apertures. The normalized PSF as a function of radial distance r (pixels) in the Fourier plane for a lens limited by the pupil in Fig. 6.2 is shown in Fig. 6.3. We obtained the same arrangement for the apertures in the Figs. 6.4a, 6.5a for the linear and quadratic distributions. We showed the corresponding plots of normalized PSF in the Figs. 6.4b, 6.5b). Finally, we established

68

6 A New Model of Modulated Aperture [28]

Fig. 6.2 Eight equally spaced conic apertures were the centers placed at a fixed distance = 268 pixels from the center set at (512,512). The radius of each conic aperture = 64 pixels

eight equally spaced uniform circular apertures in Fig. 6.6a for comparison. We obtained the corresponding normalized PSF plot in Fig. 6.6b, and we give the same arrangement surrounded by an annulus and its corresponding normalized PSF in the plot Fig. 6.7a, b. The normalized PSF showed similar shapes for the eight conic, linear, and quadratic apertures. We computed the autocorrelation images and their line plots at the center using FFT techniques for the conic, linear, quadratic, and uniform circular, as shown in Fig. 6.8a–d. The line plot corresponds to the conic’s autocorrelation and the uniform circular models we plotted, as shown in Fig. 6.9a, b. Fig. 6.3 Normalized PSF as a function of radial distance r (pixels) in the Fourier plane for a lens limited by the aperture shown in Fig. 6.2

6.5 Discussions

69

Fig. 6.4 a Eight equally spaced linear apertures were the centers placed at fixed distance = 268 pixels from the center (512,512). The radius of each conic aperture = 64 pixels. b Normalized PSF as a function of radial distance r (pixels) in the Fourier plane for a lens limited by the aperture shown in Fig. 6.4a

6.5 Discussions It shows, referring to the plotted PSF corresponding to the apertures in the Figs. 6.1, 6.2, 6.3, 6.4 and 6.5, a remarkable similarity in the distribution in the central peak. In addition, a noticeable difference was shown in the patterns’ legs for all plots. A

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6 A New Model of Modulated Aperture [28]

Fig. 6.5 a We placed eight equally spaced quadratic apertures at a fixed distance of 268 pixels from the center (512,512). The radius of each conic aperture = 64 pixels. b Normalized PSF as a function of radial distance r (pixels) in the Fourier plane for a lens limited by the aperture shown in Fig. 6.5a

6.5 Discussions

71

Fig. 6.6 a Eight equally spaced uniform circular apertures were the centers placed at fixed distance = 268 pixels from the center set at (512,512). The radius of each conic aperture = 64 pixels. b Normalized PSF as a function of radial distance r (pixels) in the Fourier plane for a lens limited by the aperture shown in Fig. 6.6a

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6 A New Model of Modulated Aperture [28]

Fig. 6.7 a An aperture composed of eight equally spaced uniform circles surrounded by an annulus. The external radius of the full aperture = 352 pixels, the annular width = 16 pixels, and the radius of each circle = 64 pixels. The matrix of dimensions is 1024 × 1024 pixels. b Normalized PSF as a function of radial distance r (pixels) in the Fourier plane for a lens limited by the aperture shown in Fig. 6.5a

noteworthy difference is shown in Fig. 6.5b since the model is surrounded by an annulus, as expected. Consequently, the PSF distribution depends upon the aperture distribution assuming monochromatic light for the illumination. It shows different irregular shapes for the legs of the diffraction pattern. Referring to Fig. 6.9a, b, the autocorrelation plots showed a smooth pattern for the circular arrangement compared to the deformation and shrinking that appeared in the

6.5 Discussions

73

Fig. 6.8 a Autocorrelation image and the plot for the conic aperture. b Autocorrelation image and plot of the linear aperture. c Autocorrelation image and plot for the quadratic aperture. d Autocorrelation image and plot for the uniform circular aperture

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6 A New Model of Modulated Aperture [28]

Fig. 6.9 Autocorrelation of the conic model shown in (a) and that corresponding to the circular uniform model shown in (b)

central peak in the case of the conic model. In addition, the total width corresponding to the autocorrelation is two times the aperture diameter, as expected.

6.6 Conclusions We manufactured apertures in the form of equally spaced conic, linear, and quadratic circles. The normalized PSF is computed for all pupils and has the same shape. In addition, the autocorrelation corresponds to the apertures we computed and plotted. Finally, we calculated and plotted speckle imaging using a diffuser limited by the manufactured pupils. We obtained the reconstructed pupils from the speckle images, which we improved by filtering techniques.

References

75

References 1. J.W. Goodman, Some fundamental properties of speckles. J. Opt. Soc. Am. 66(11), 1145–1150 (1976) 2. M. Francon (ed.), Laser Speckle and Applications in Optics, 1st edn. (Elsevier, 1979). ISBN 978-0-323-16072-8 3. C. Dainty (ed.), Laser Speckle and Related Phenomena, 2nd edn. (Springer, 1984). ISBN 978-0-387-13169-6 4. J.W. Goodman, Statistical properties of laser speckle patterns, in Laser Speckle and Related Phenomena, Vol. 9 in series Topics in Applied Physics, ed. by J.C. Dainty (Springer, Berlin, Heidelberg, 1975), pp. 9–75 5. F. Fercher, J.D. Briers, Flow visualization by single-exposure speckle photography. Opt. Commun.Commun. 37(5), 326–330 (1981) 6. J.F. Barrera, R. Henao et al., Multiple image encryption using an aperture-modulated optical system. Opt. Comm. 261(1), 29–33 (2006) 7. A. Salazar et al., Experimental study of volume speckle in four-wave mixing arrangement. Opt. Comm. 221, 249–256 (2003) 8. C.J.R. Sheppard, Nonparaxial, three-dimensional and fractal speckle, in Proceedings of SPIE 8413, Speckle 2012: V International Conference on Speckle Metrology (2012), p. 841302. https://doi.org/10.1117/12.977790. 9. S. Yuan, A. Devor et al., Determination of optimal exposure time for imaging of blood flow changes with laser speckle contrast imaging. Appl. Opt. 44, 1823 (2005) 10. L.M. Richards, S.M.S. Kazmi et al., Low-cost laser speckle contrast imaging of blood flow using a webcam. Biomed. Opt. Express 4, 2269 (2013) 11. M.A. Davis, S.M.S. Kazmi, A.K. Dunn, Imaging depth and multiple scattering in laser speckle contrast imaging. J. Biomed. Opt. 18, 086001 (2014) 12. S.M.S. Kazmi, L.M. Richards et al., Expanding applications, accuracy, and interpretation of laser speckle contrast imaging of cerebral blood flow. J. Cereb. Blood Flow Metab.Cereb. Blood Flow Metab. 35, 1076–1084 (2015) 13. M.A. Davis, L. Gagnon et al., Sensitivity of laser speckle contrast imaging to flow perturbations in the cortex. Biomed. Opt. Express 7, 759 (2016) 14. A.B. Parthasarathy, E.L. Weber et al., Laser speckle contrast imaging of cerebral blood flow in humans during neurosurgery: a pilot clinical study. J. Biomed. Opt. 15, 066030 (2010) 15. A. Ponticorvo, A.K. Dunn, How to build a laser speckle contrast imaging (LSCI) system to monitor blood flow. J. Vis. Exp. 45, 2004 (2010). https://doi.org/10.3791/2004 16. D.A. Boas, A.K. Dunn, Laser speckle contrast imaging in biomedical optics. J. Biomed. Opt. 15, 011109 (2010) 17. D. Briers, D. Duncan, Laser speckle contrast imaging: theoretical and practical limitations. J. Biomed. Opt. 18(6), 66018 (2013). https://doi.org/10.1117/1.JBO.18.6.066018 18. J.D. Briers, S. Webster, Laser speckle contrast analysis (LASCA): a non-scanning, full-field technique for monitoring capillary blood flow. J. Biomed. Opt. 1(2), 174–179 (1996) 19. O. Thompson, M. Andrews, E. Hirst, Correction for spatial averaging in laser speckle contrast analysis. Biomed. Opt. Express 2(4), 1021–1029 (2011) 20. A.M. Hamed, Contrast of laser speckle images using some modulated apertures. J Phys Pram 95, 122 (2021). https://doi.org/10.1007/s12043-021-02151-8 21. A.M. Hamed, Numerical speckle images formed by diffusers using modulated conical and linear apertures. J. Mod. Opt.Mod. Opt. 56(10), 1174–1181 (2009). https://doi.org/10.1080/ 09500340902985379 22. A.M. Hamed, Improvement of point spread function (P.S.F.) using linear-quadratic aperture. Optik 131, 838–849 (2017). https://doi.org/10.1006/j.ijleo.2016.11.211 23. A.M. Hamed, Speckle imaging of annular Hermite Gaussian laser beam. J. Phys. Pram. 95, 202 (2021). https://doi.org/10.1007/s12043-021-02231-9

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24. A. Taya Sofyan, J. Hussein Aya et al., Dispersion curves of a slab waveguide with a nonlinear covering medium and an exponential graded-index thin film. J. Opt. Soc. Am. 38(11), 3237– 3243 (2021) 25. J. Hussein Aya, M. Nassar Zaher, A. Taya Sofyan, Dispersion properties of slab waveguides with a linear graded-index film and a nonlinear substrate. Microcyst. Technol. 27(7), 2589–2594 (2021) 26. J. Hussein Aya, A. Taya Sofyan, Universal dispersion curves of a planar waveguide with an exponentially graded index guiding layer and a nonlinear cladding. Results Phys. 20, 6 (Article 103734) 27. A. Upadhyay, S. Singh et al., An ultrahigh birefringent and nonlinear decahedron photonic crystal fiber employs molybdenum disulfide: a numerical analysis. Mater. Sci. Eng. B 270, 115236 (2021) 28. A.M. Hamed, Investigation of a new modulated aperture using speckle techniques. Beni-Suef Univ. J. Basic Appl. Sci. 11, 39 (2022). https://doi.org/10.1186/s43088-022-00222-2

Part II

Application of Confocal Laser Microscope and Speckle Images

Chapter 7

Cauchy Aperture and Its Application in Confocal Microscopy

7.1 Introduction The aperture function was theoretically derived as a 2D Cauchy function for perfectly reconstructing all-in-focus images by filtering the focal stack images. The filters were simply designed using the Cauchy function. In the simulation, the all-in-focus image was successfully reconstructed with high quality by the presented filters without region segmentation [1]. Most approaches to estimating a scene’s 3D depth from a single image often model the point spread function (PSF) as a 2D Gaussian function. However, those methods suffer from some noise, and it is difficult to obtain high-quality depth recovery. An efficient blur estimation method based on the Cauchy PSF was proposed by [2] and shows that it is robust to noise, inaccurate edge location, and interference from neighboring edges. The author [3] presented a simple yet effective approach to estimate exactly the amount of spatially varying defocus blur at edges based on a Cauchy distribution model for the PSF. In recent work, several methods have been proposed to recover a depth map from a single image, which does not suffer from the correspondence problem of multiple image matching [4–7]. A hyper-resolving aperture composed of a polynomial distribution was suggested by [8]. The PSF was computed and compared with that corresponding to linear, quadratic, and circular apertures. In addition, the influence of the number of zones on the PSF was discussed. We obtain compromised resolution and contrast for the polynomial apertures compared with uniform circular apertures. An application of a confocal scanning laser microscope using the Siemens star pattern as an object considering the polynomial apertures is given. A new concentric Hamming linear aperture was suggested by [9]. The point spread function (PSF) was computed for concentric Hamming linear apertures and compared with the corresponding circular, conventional Hamming, and obstructed Hamming apertures. In addition, the autocorrelation corresponding to the aperture © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. M. Hamed, Modulated Apertures and Resolution in Microscopy, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-3-031-47552-8_7

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under consideration is computed. Application of the Hamming linear aperture in the formation of modulated speckle images of Ramses II statues using a definite random diffuser is given. Recently, we presented [10] a novel method of PSF computation corresponding to the hexagonal aperture, making use of Fourier optics and convolution operations. The application of a confocal scanning laser microscope provided with hexagonal and circular apertures is given. The PSF of hexagonally segmented telescopes is derived by a new symmetrical formulation derived from Fraunhofer diffraction [11]. In addition, we presented recent work on the contrast of speckle images using triangular apertures, and we computed the average speckle size from the point spread function in [12]. I have obtained a novel formula for the contrast of speckle images considering the Voigt distribution using Fraunhofer diffraction and convolution operations [13].

7.2 Analysis The Cauchy aperture is mathematically represented as follows: Pcauchy (ρ) =

1+β

  ρ 1 ;  2   ≤ 1 and β < 1 ρ0 2 ρ

(7.1)

ρ0

β is a parameter, and ρ is the radial coordinate in the aperture plane with a maximum radius of ρ 0 . We represent the obstructed Cauchy aperture as the difference between the ordinary Cauchy aperture and a circular central zone of maximum radius ρ40 as follows:   4ρ . (7.2) Pobst. (ρ) = Pcauchy (ρ) − circ ρ0 We computed the PSF for the obstructed Cauchy aperture by operating the Fourier transform in polar coordinates as follows: 1 h(r ) = 2

2 ∞ 0 −∞

 2π r cos(ρ) ρdρdθ. P(ρ) exp − j λf

(7.3)

Symbolically, we write the PSF for the obstructed aperture as follows:    4ρ . h(r ) = FT Pcauchy (ρ) − circ ρ0

(7.4)

7.2 Analysis

81

Substituting Eq. 7.1 in Eq. 7.4, we write: ⎤

⎡ ⎢ h(r ) = FT⎣

1+ρ

   4ρ 1 ⎥ = I1 (r ) − I2 (r )  2 ⎦ − FT circ ρ0 2 ρ

(7.5)

ρ0

We solve the two transformations in Eq. (7.5) to obtain the following: We solved the first integral in Eq. (7.5) by applying the Fourier transform, and we obtained the double-sided function represented as in Eq. (7.6).  I1 (r ) =

π λfβ



  2πρ0 |r | . exp − λ fβ

(7.6)

The double-sided function represented by Eq. 7.6 represents the PSF corresponding to the ordinary Cauchy aperture, where r = r; for r > 0, and r = −r; for r < 0. The circular obstruction region has a PSF represented by an Airy disk as follows: I2 (r ) =

2π  ρ0  2J1 (W1 ) r. ; W1 = W1 πf 4

(7.7)

Hence, we obtain from Eqs. (7.5–7.7) the PSF corresponding to the obstructed Cauchy aperture as follows:  h(r ) =

π λfβ



  2πρ0 2J1 (W1 ) |r | − exp − . λ fβ W1

(7.8)

We computed the CTF for the CSLM from the autocorrelation of the pupil function [2]. We write the CTF corresponding to the Cauchy aperture in integral form as follows: ∞  CTF(ρ) = −∞

1 1 + β 2 ρ 2



 1 dρ  . 1 + β 2 (ρ − ρ  )2

(7.9)

We write the CTF symbolically as follows: CTF(ρ) =

1 1 ⊗ . 1 + (β.ρ)2 1 + (β.ρ)2

(7.10)

The algorithm used to compute the image in the CSLM is summarized as follows [10]: confocal imaging A(x, y) = h 1 h 2 ⊗ g(x, y)

(7.11)

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7 Cauchy Aperture and Its Application in Confocal Microscopy

A(u, v) = FT{A(x, y)} = (P1 ⊗ P2 ).G(u, v)

(7.12)

CTF = P1 ⊗ P2 and G(u, v) = FT{g(x, y)}

(7.13)

Inverse FT of (7.12) gives (7.11). We take modulus square of Eq. (7.11) and getting the intensity of the image. The image contrast is computed from the following formula: K =

σ I 

(7.14)

σ root mean square deviation and I  is the mean value of the image intensity.

7.3 Results and Discussion We showed in Fig. 7.1, in the LHS, the image of the Cauchy aperture with dimensions of 256 × 256 pixels. The aperture radius = 64 pixels and β = 0.5. On the RHS, we showed the plot at y = 128 pixels normalized to unity. We showed, as in Fig. 7.2a, the plot of the normalized PSF or the Cauchy Fourier spectrum corresponding to the Cauchy aperture shown in Fig. 7.1. We show the plot of the normalized PSF corresponding to the obstructed Cauchy aperture in Fig. 7.2b. The aperture radius = 64 pixels, and the obstruction central zone radius = 4 pixels. We give another plot of the PSF for the obstruction central zone of radius = 8 pixels, as in Fig. 7.2c. We computed the Cauchy aperture and the

Fig. 7.1 In the LHS, we show an image of the Cauchy aperture with dimensions of 256 × 256 pixels. The radius = 64 pixels and β = 0.5. On the RHS, we showed the plot at y = 128 pixels normalized to unity

7.3 Results and Discussion

83

a

b Fig. 7.2 a Plot of the normalized PSF or the Cauchy Fourier spectrum corresponding to the aperture shown in Fig. 7.1. The aperture radius = 64 pixels. b Plot of the normalized PSF corresponding to the obstructed Cauchy aperture. The aperture radius = 64 pixels, and the obstruction central zone radius = 4 pixels. c Plot of the normalized PSF corresponding to the obstructed Cauchy aperture. The aperture radius = 64 pixels, and the obstruction central zone radius = 8 pixels

corresponding PSF plotted in Figs. 7.1 and 7.2a–c using the fast Fourier transform (FFT) and MATLAB code. In the LHS of Fig. 7.3, we show an image of the obstructed Cauchy aperture with dimensions of 256 × 256 pixels. The obstruction central zone of radius = 4 pixels. The radius = 64 pixels and β = 0.5. On the RHS, we showed the plot of the PSF at y = 128 pixels normalized to unity. We plotted the obstructed Cauchy aperture using Eq. (7.8), and we compared it with the ordinary Cauchy aperture using Eq. (7.6). We showed that the total bandwidth is equal to 53 pixels for the obstructed Cauchy aperture, while it is equal to 108 pixels in the case of the ordinary Cauchy aperture. In addition, we showed the appearance

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7 Cauchy Aperture and Its Application in Confocal Microscopy

c Fig. 7.2 (continued)

Fig. 7.3 In the LHS, we show an image of the obstructed Cauchy aperture with dimensions of 256 × 256 pixels. The obstruction central zone of radius = 4 pixels. The radius = 64 pixels and β = 0.5. On the RHS, we showed the plot of the PSF at y = 128 pixels normalized to unity

of legs as in Fig. 7.4a due to the obstruction of the central zone with a circular part. No legs appear for the PSF corresponding to the ordinary Cauchy aperture, as shown in Fig. 7.4b. We showed in Table 7.1 and Fig. 7.5 the cutoff spatial frequency versus the obstruction central zone. The cutoff spatial frequency is improved by increasing the obstruction zone. Hence, the microscope resolution may be increased by increasing the obstruction of the central zone.

7.3 Results and Discussion

85

Fig. 7.4 a Normalized PSF corresponding to the obstructed Cauchy aperture computed from analytic Eq. (7.8). The total width = 277–224 = 53 pixels. b Normalized PSF corresponding to the Cauchy aperture computed from analytic Eq. (7.6). The total width = 305 − 197 = 108 pixels

86 Table 7.1 Cutoff spatial frequency in pixels versus the obstruction central zone

7 Cauchy Aperture and Its Application in Confocal Microscopy

Obstructed zone in pixels

Cutoff radial distance in pixels

0

29

1

29

2

11

3

9

4

7

5

6

6

5

7

5

8

4

Fig. 7.5 Cutoff spatial frequency in pixels versus the obstruction central zone in pixels

We plot the normalized CTF or the autocorrelation corresponding to the Cauchy aperture as shown in Fig. 7.6a. We plotted the CTF corresponding to the obstructed Cauchy aperture as in Fig. 7.6b. The aperture radius = 64 pixels, and the obstruction central zone radius = 4 pixels. Referring to Fig. 7.6a, b, the total bandwidth for the Cauchy and obstructed Cauchy apertures is invariant and equals two times the aperture diameter. Hence, the bandwidth for the CTF is equal to 256 pixels for the aperture diameter = 128 pixels, as expected. We showed that three peaks are found in the case of an obstructed Cauchy aperture compared with only one peak in the case of a uniform Cauchy aperture. Referring to the CSLM algorithm represented by Eqs. (7.11–7.13), we obtain the reconstructed image for the coronavirus image, as shown in Fig. 7.7a. The two objectives provided obstructed Cauchy apertures. The obstruction central zone of radius = 4 pixels and β = 0.5 is shown in Fig. 7.3. In addition, we obtained the reconstructed images using two ordinary Cauchy apertures, as shown in Fig. 7.7b, and obtained a contrast equal to 0.4737.

7.3 Results and Discussion

87

a

b Fig. 7.6 a Plot of the normalized CTF or the autocorrelation corresponding to the Cauchy aperture shown in Fig. 7.1. b Plot of the normalized CTF corresponding to the obstructed Cauchy aperture. The aperture radius = 64 pixels, and the obstruction central zone radius = 4 pixels

In the case of two objectives providing obstructed Cauchy apertures, we showed that the computed contrast for the reconstructed image is equal to 0.4879 compared with the image contrast of 0.5118, as shown in Fig. 7.7c, considering the two objectives followed by uniform circular apertures. We show the plot of the reconstructed image contrast vs. the obstruction zone in the obstructed Cauchy aperture in pixels in Fig. 7.8. The maximum contrast is equal to 0.4973 at the obstruction zone equal to 14 pixels. The plot is given for two symmetric obstructed Cauchy apertures in the CSLM. The image degradation in the case of obstructed Cauchy apertures in CSLM compared with circular apertures computed from the contrast is only equal to (0.5118−0.4973)×100 % = 2.8%. 0.5118

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7 Cauchy Aperture and Its Application in Confocal Microscopy

Fig. 7.7 a Reconstructed image for the coronavirus image in the CSLM. The two objectives provided obstructed Cauchy apertures. The obstruction central zone of radius = 4 pixels and β = 0.5 is shown in Fig. 7.3. b Reconstructed image for the coronavirus image in the CSLM. The two objectives provided ordinary Cauchy apertures. c Reconstructed image for the coronavirus image in the CSLM. The two objectives are followed by uniform circular apertures. The aperture radius = 64 pixels in all Fig. 7.7a, b

7.4 Conclusion

89

Fig. 7.8 Plot of the reconstructed image contrast vs. the obstruction zone from the Cauchy aperture in pixels. The maximum contrast is equal to 0.4973 at the obstruction zone equal to 14 pixels. The plot is given for two symmetric obstructed Cauchy apertures in the CSLM. The contrast for two circular apertures in the CSLM is equal to 0.5118

The degradation in the case of obstructed Cauchy apertures in the CSLM compared with ordinary Cauchy apertures computed from the contrast is only equal to (0.4973−0.4737)×100 % = 4.75%. Hence, the image contrast in the case of an obstructed 0.4973 Cauchy aperture is better than that corresponding to the case of an ordinary Cauchy aperture by 4.75%.

7.4 Conclusion We showed that the microscope resolution is improved with the manipulation of obstructed Cauchy apertures. Hence, we obtain a compromise of resolution and contrast in the case of obstructed Cauchy apertures compared with the uniform circular apertures in the CSLM. A little degradation in image contrast is shown in the case of obstructed Cauchy apertures. In addition, in the case of obstructed Cauchy apertures, we showed that the image contrast increased with increasing obstructed zone until reaching a maximum at 14 pixels and then decreased. Additionally, we showed legs in the PSF corresponding to the obstructed Cauchy aperture compared with the ordinary Cauchy aperture, which may be useful for imaging extended objects.

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7 Cauchy Aperture and Its Application in Confocal Microscopy

Abbreviations PSF FFT CTF CSLM

Point spread function. Fast Fourier transformation. Coherent transfer function. Confocal scanning laser microscope

References 1. A. Kubota, K. Kodama, Cauchy aperture and perfect reconstruction filters for extending depthof-field from focal stack. IEICE Trans. Inf. Syst. E102–D(11) (2019) 2. Y. Ming, J. Jiang, Moving object detection of infrared video based on Cauchy distribution. J. Infrared Millimeter Terahertz Waves 27, 65–72 (2008) 3. Y. Ming, Depth estimation from a single image based on cauchy distribution model. J. Comput. Commun. 9, 133–142 (2021) 4. H. Yuan, S. Wu et al., Object guided depth map recovery from a single defocused image. Acta Electron. Sin. Electron. Sin. 42, 2009–2015 (2014) 5. D. Hoiem, A. Efros, M. Hebert, Recovering occlusion boundaries from an image. Int. J. Comput. Vis. 91, 328–346 (2011). https://doi.org/10.1007/s11263-010-0400-4 6. S. Bae, F. Durand, Comput. Graph. Forum 26, 571–579 (2007). https://doi.org/10.1111/j.14678659.2007.01080 7. J. Elder, S. Zucker, Local scale control for edge detection and blur estimation. IEEE Trans. Pattern Anal. Mach. Intell.Intell. 20, 699–716 (1998). https://doi.org/10.1109/34.689301 8. A.M. Hamed, A hyper-resolving polynomial aperture and its application in microscopy. BeniSuef Univ. J. Basic Appl. Sci. 11(1), 25 (2022) 9. A.M. Hamed, Image processing of Ramses II statue using speckle photography modulated by a new Hamming linear aperture. Pramana J. Phys. J. Phys. 94(1), 126 (2020) 10. A.M. Hamed, Application of a hexagonal aperture on the confocal scanning laser microscope. Opt. Quant. Electron. 55, 749 (2023a) 11. S. Itoh, T. Matsuo et al., Point spread function of hexagonally segmented telescopes by new symmetrical formulation. MNRAS 483, 119–131 (2019) 12. A.M. Hamed, The computation of the average speckle size from the point spread function for triangular apertures. Eur. Phys. J. Plus 138, 340 (2023b) 13. A.M. Hamed et al., The contrast of speckle images using viscous fluid flows. Eur. Phys. J. Plus 138, 447 (2023)

Chapter 8

Speckle Images Formed by Diffusers Using Conical and Linear Apertures

8.1 Introduction We know that electronic/digital speckle pattern interferometry (ESPI/DSPI) is a promising field with various applications in the measurement of displacement/ deformations, vibration analysis, contouring, nondestructive testing, etc. [1–3]. One of its distinct features is the capability of ESPI/DSPI to display correlation fringes on a TV monitor. Digital speckle interferometry (DSI) does not need photographic film. DSI, we used to study the density field in an acoustic wave [4, 5] for quantitative diagnosis of the speckle intensity. Digital data treatment was based on the direct computer-aided correlation analysis of the temporal evolution of dynamic speckle patterns. In many metrological applications, speckles are used to determine surface roughness properties. Several complex effects and parameters are a concern. Some authors have presented a numerical simulation tool for the synthetic generation of a laser speckle pattern for a nonimaging observer in the far field of an illuminated surface [6, 7]. This chapter presents a numerical speckle image from randomly distributed objects considering aperture modulation. Three different apertures were investigated in [8]. The first aperture has an amplitude variation of conical shape, the second has a linear variation, and the third has a square root of the linearly varied function. The digital speckle images were computed from the superposition of the aperture spread function of the coherent optical system and the classical speckle pattern obtained in the case of a circular aperture. It is interesting to show the resolution of the acquired digital speckle pattern affected by the distribution inside the modulated pupil that gives different PSF. Finally, the autocorrelation intensity of the diffuser is affected by the aperture modulation. We obtain the autocorrelation and plot the corresponding profiles. The obtained results are compared with the circular aperture for illumination.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. M. Hamed, Modulated Apertures and Resolution in Microscopy, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-3-031-47552-8_8

91

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8 Speckle Images Formed by Diffusers Using Conical and Linear Apertures

8.2 Theoretical Analysis A conical aperture is represented as follows: | | | | |ρ| |ρ| | | Pconical (ρ) = 1 − | | with || || ≤ 1 ρ0 ρ0 = zero otherwise

(8.1)

√ where ρ = x 2 + y 2 . The radial coordinate is in the conical aperture. This conical aperture was constructed using the MATLAB program, as shown in Fig. 8.1. The point spread function or the impulse response is calculated by running the two-dimensional Fourier transformation in polar coordinates to obtain the following: {ρ0 {2π ( h(r ) = 2 0

0

| |) ( ) |ρ| ρr | | 1 − | | exp − j2π cos θ ρ dρ dθ ρ0 f

| |) ( ) {ρ0 ( |ρ| 2πρr | | = 4π 1 − | | J0 ρ dρ ρ0 f

(8.2)

0

√ where J0 is the zero-order Bessel function r = u 2 + v 2 and is the radial coordinate in the plane where the speckle image was obtained. (

{ρ0 h(r ) = 4π

ρ J0 0

( ) ( ) {ρ0 ) 2πρr 2πρr 2π dρ − dρ ρ 2 J0 f ρ0 f

(8.3)

0

With the help of the recurrence formulae [1, 2], we finally obtain the following [2]: h(w) =

4πρ02

{[ ∑ ]} 2 i Ji (w) − w J0 (w) w3

(8.4)

. where the reduced coordinate is w = 2πρr f The randomly distributed object d(x, y) multiplied by the conical aperture P(x, y) is written as follows: A(x, y) = d(x, y). Pconical (x, y).

(8.5)

8.2 Theoretical Analysis

93

Fig. 8.1 Computer-generated apertures of dimensions 1024 × 1024

The complex amplitude found in the focal plane of the lens L was obtained by running the Fourier transform upon the complex amplitude A(x, y), Eq. (8.5), to obtain: B(u, v) = FT {A(x, y)} = FT {d(x, y). Pconical (x, y)}

(8.6)

where FT refers to the Fourier transform operation. Making use of the properties of convolution operations [9, 10], Eq. (8.6) becomes: B(u, v) = FT{d(x, y)} ∗ FT{Pconical (x, y)}

(8.7)

B(u, v) = s(u, v) ∗ h(u, v).

(8.8)

The symbol (*) stands for the convolution operation. S(u, v) is the complex amplitude of the speckle pattern formed in the focal plane of the lens L. It is given by s(u, v) = FT{d(x, y)}, while h(u, v) is the amplitude impulse response of the imaging system, which we calculated by operating the

94

8 Speckle Images Formed by Diffusers Using Conical and Linear Apertures

Fourier transformation upon the modulated conical aperture and is given by h(u, v) = FT{P conical(x, y)}. Equation (8.2) is obtained as in Eq. (8.4). The intensity of the speckle image recorded in the Fourier plane for the modulated aperture is given by: I (u, v) = |s(u, v) ∗ h modul (u, v)|2 .

(8.9)

Either we can reconstruct the diffuser multiplied by the modulated aperture or the autocorrelation function of the diffuser affected by modulation. We run the Fourier transform on Eq. (8.5) to reconstruct the diffuser. ) ( A x, y ' = FT{B(u, v)}.

(8.10)

where (x ' , y' ) is the imaging or reconstruction plane. Substituting Eq. (8.8) into Eq. (8.10), we obtain the diffuser function multiplied by the modulated aperture since the Fourier transform of the convolution product transforms into multiplication [9, 10] in the Fourier plane (x ' , y' ). Second, we reconstruct the autocorrelation function of the diffuser. The modulated conical aperture affects the autocorrelation function of the diffuser. We run the Fourier transform on the intensity distribution of the speckle pattern Eq. (8.9) to obtain: ( ) Aauto x ' , y ' = FT{I (u, v)} { } = FT |s(u, v) ∗ h modul (u, v)|2

(8.11)

( '' '' ) { ( ) )} { ( ) ( )} ( Aauto x , y = d x ' , y ' . Pconical . x ' , y ' ∗ d x ' , y ' . Pconical x ' , y ' . (8.12) Special Case If the impulse response of the imaging system is approximated by the Dirac-Delta distribution, i.e., h modul (u, v) = h(u, v) = δ(u, v), which is valid for uniform illumination originating from a laser beam by spatial filtering, then the speckle image becomes precisely the Fourier transform of the diffuser function. Hence, Eq. (8.9) becomes: I (u, v) = |s(u, v) ∗ δ(u, v)|2 = |s(u, v)|2 .

(8.13)

In this case, the reconstructed autocorrelation function of the diffuser is exactly the FT of Eq. (8.13) to obtain: ( ) ( ) ( ) Aauto x ' , y ' = d x ' , y ' ∗ d ∗ x ' , y ' .

(8.14)

The autocorrelation intensity was computed as the modulus square of Eq. (8.14)

8.3 Results and Discussion

95

) | ( )|2 ( I x ' , y ' = | Aauto x ' , y ' | .

(8.15)

Computation of the PSF for radial ρ n distribution of the aperture. ρ n aperture is represented as follows [11, 12]: | | | | |ρ| |ρ| | | Plinear (ρ) = | | with || || ≤ 1. ρ0 ρ0

(8.16)

The point spread function (PSF) or the amplitude impulse response is obtained by computing the Bessel–Fourier transform of Eq. (8.17) as follows: ) ( ρr ρ exp − j2π cosθ ρ dρ dθ ; ρ0 = 1 f 0 0 ) ( 1 kρr dρ = 4π ∫ ρ 2 J0 f 0 {1 {2π

h(r ) = 2

(8.17)

where k = 2π /λ is the propagation constant. W = k r/f and w = ρ k r/f is a reduced coordinate. Finally, the PSF is obtained as: ∑ ] [ J0 (w) 2 i Ji (w) J1 (w) . (8.18) + h(w) = 4π − w w2 w3 This last equation was used to draw the PSF corresponding to a linearly varied aperture.

8.3 Results and Discussion A MATLAB program was constructed to design three different apertures. The first aperture has a conical distribution, the second aperture has a linear variation, and the third has a ρ 1/2 distribution compared with the uniform circular aperture. These apertures are shown in Fig. 8.1a–d. Another MATLAB program we constructed to fabricate a diffuser as a randomly distributed object was used in this study with dimensions 1024 × 1024, as shown in Fig. 8.2. The parts of MATLAB programs used to obtain Figs. 8.3, 8.4, 8.5 and 8.6 we used to obtain the numerical speckle images for the described apertures making use of Eq. (8.9) and represented as shown in Fig. 8.3a–d. The four speckle images are different since they affected different distributions of impulse response PSF, as shown in Fig. 8.7. The profile shapes of the speckle pattern at slice x = [1 256 128 128] and slice y = [1 256 128 128] are plotted as shown in Fig. 8.4 a–d.

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8 Speckle Images Formed by Diffusers Using Conical and Linear Apertures

100 200 300 400 500 600 700 800 900 1000 100

200

300

400

500

600

700

800

900

1000

Fig. 8.2 Randomly distributed object constructed numerically with dimensions 1024 × 1024

Fig. 8.3 Numerical speckle image of the diffuser obtained in the case of conical, linear, r 1/2 , and constant circular apertures for illumination

8.3 Results and Discussion

97

Fig. 8.4 Profile of the speckle image of the diffuser in conical, linear, r 1/2 , and constant circular apertures. The slice x = [1 256 128 128] and slice y = [1 256 128 128]

We obtain the profile of the speckle image for the linear aperture, and we represent it in Fig. 8.4b. It has a resolution much better than the conical aperture Fig. 8.4a. The resolution improvement for the modulated speckle images is attributed to the improvement in the imaging system’s point spread function using modulated apertures. The resolution improvement we obtained for the modulated speckle images is attributed to the improvement in the imaging system’s point spread function using modulated pupils. We represented the autocorrelation intensity of the diffuser in the modulated aperture and modulus square of Eq. (8.12) as shown in Fig. 8.5a–d. It is clear from these images that the diameter of the central part of the autocorrelation intensity is twice the diameter of the completely circular aperture shown in Fig. 8.1a–d, as expected. We showed that the contrast of the autocorrelation intensity peak in the case of the conical aperture is a weak Fig. 8.5a compared with that shown in Fig. 8.5b, c, d obtained for linear, ρ 1/2 , and circular apertures. Finally, the profiles of the autocorrelation intensity of the diffuser are multiplied by the modulated apertures we draw as in Fig. 8.6a–d. The profiles are taken at the same section for slice x = [1 256 128 128] and slice y = [1 256 128 128].

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8 Speckle Images Formed by Diffusers Using Conical and Linear Apertures

Fig. 8.5 a–d Numerical image of the autocorrelation intensity of the diffuser modulated by the conical (a), linear (b), r 1/2 (c), and circular (d) apertures

A graph for PSF is plotted in Fig. 8.7a for the three curves. We obtain the amplitude impulse response, or the point spread function (PSF). We took three different apertures at a particular NA and λ. The red curve was plotted for the conical aperture, while the green curve we devised for the linear variation inside the pupil, and the blue curve was for the uniform circular aperture. The green curve corresponding to the linearly varied gap shows better resolution than the red and blue curves corresponding to the conical and uniform circular apertures. Another graph was plotted using Eq. (8.17) and is represented in Fig. 8.7b, where the PSF curve for the ρ 1/2 distribution shows another improvement in resolution compared with circular and other modified apertures.

8.4 Conclusion

99

Fig. 8.6 a–d Profile of the autocorrelation intensity obtained from Fig. 8.5a–d. The slice x = [1 256 128 128] and slice y = [1 256 128 128]

Fig. 8.7 a, b Amplitude impulse response or the point spread function (PSF) for three different apertures are shown in Fig. 8.7a, while in Fig. 8.7b, the ρ 1/2 distributed aperture is compared with the uniform circular aperture

8.4 Conclusion We computed the autocorrelation intensity of the randomly distributed object using three different apertures for the illumination of the optical system. The first aperture assumes a conical distribution, the second has linear variation, and the third has a

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8 Speckle Images Formed by Diffusers Using Conical and Linear Apertures

ρ 1/2 distribution. It was concluded that the diameter of the autocorrelation peak is two times the diameter of the full aperture, as expected. In addition, it was concluded that the diffuser’s autocorrelation peak in the conical gap has a resolution less than that of the linear pupil. The point spread function improved in the case of the linear aperture compared with the uniform circular and conical aperture at a certain NA and wavelength. The profiles of speckle images showed an improvement in resolution in the case of linear gaps and ρ 1/2 apertures (Fig. 8.4b, c) compared with the profiles obtained in the case of conical apertures and uniformly circular apertures (Fig. 8.4a, d) at slice x = [1 256 128 128] and slice y = [1 256 128 128].

References 1. J. Ohtsubo, H. Fujii, T. Asakura, Surface roughness measurement using speckle pattern. Jpn. J. Appl. Phys.. J. Appl. Phys. 14, 29 (1975) 2. A.M. Hamed et al., Analysis of speckle images to assess surface roughness. Opt. Laser Technol. 36(2004), 249–253 (2004) 3. F.M. Furgiuele et al., Monitoring of biaxial tests of ceramic materials by digital speckle interferometry. Exp. Tech. 19, 15–19 (1995) 4. R.S. Sirohi, A.R. Ganesan, Some applications of digital speckle pattern interferometry (DSPI), in International Conference on IEEE, 11–13 Sept (1989) 5. M. Raffel et al., Particle Image Velocimetry: A Practical Guide (Springer, Berlin, 1998), pp. 189–229. ISBN-3-450-63683-8 6. C. Greated et al., Numerical simulation of acoustic wave tracing. J. Math. Model. Russian Acad. Sci. 15(7), 75–80 (2003) 7. A.M. Hamed, M. Saudy, Computation of surface roughness using optical correlation. Pramana J. Phys. J. Phys. 68, 831–842 (2007) 8. A.M. Hamed, Numerical speckle images formed by diffusers using modulated conical and linear apertures. J. Mod. Opt.Mod. Opt. 56, 1174–1181 (2009). https://doi.org/10.1080/095 00340902985379 9. J.D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, N.Y., 1978) 10. J.W. Goodman, Introduction to Fourier Optics and Holography (McGraw Hill Co., N.Y., 1968) 11. J.J. Clair, A.M. Hamed, Theoretical studies on a coherent optical microscope. Optik 64(1983), 133 (1983) 12. A.M. Hamed, J.J. Clair, Image and super-resolution in coherent optical microscopes. Optik 64, 277–284 (1983)