Azimuthal Walsh Filters, 1st ed. 2020 A Tool to Produce 2D and 3D Light Structures Progress in Optical Science and Photonics Series, Vol. 10

1. Introduction

2. Walsh Functions, Walsh Filters And Self-Similarity

2.1 Introduction

2.2 One Dimensional Walsh Functions

2.3 Two Dimensional Walsh Functions

2.3.1 Rectangular Walsh Functions

2.3.2 Polar Walsh Functions

2.4 Radial Walsh Functions

2.5 Azimuthal Walsh Functions

2.7 Walsh Filters

2.7.1 Radial Walsh Filters

2.7.2 Azimuthal Walsh Filters

2.8 Self-Similarity In Azimuthal Walsh Functions

References

3. Transverse Intensity Distribution On The Far-field Plane Of Azimuthal Walsh Filters

3.1 Introduction

3.2 Analytical Formulation of Far-field Amplitude Distribution along an azimuth for a Single Sector on the Exit Pupil

3.3 Azimuthal Walsh Filters on the Exit Pupil

3.4 Asymmetrical amplitude point spread function on the Far-field Plane due to azimuthal Walsh filter at the Exit Pupil Plane

3.4.1 Case 1 : Zero Order Azimuthal Walsh Filter

3.4.2 Case 2 : First Order Azimuthal Walsh Filter

3.4.3 Case 3 : Second Order Azimuthal Walsh Filter

3.4.4 Case 4 : Third Order Azimuthal Walsh Filter

3.5 Intensity Distribution on the Far-Field Plane

References

4. Self-Similarity in Transverse Intensity Distributions on the Far-Field Plane of Self-Similar Azimuthal Walsh Filter

4.1 Introduction

4.2 Transverse Intensity Distributions for Zero Order Azimuthal Walsh Filter on the Far-Field Plane

4.3 Self-Similarity in Far-Field intensity distributions for Group I Self-Similar members of Azimuthal Walsh Filters

4.4 Self-Similarity in Far-Field intensity distributions for Group IIA Self-Similar members of Azimuthal Walsh Filters

4.5 Self-Similarity in Far-Field intensity distributions for Group IIB Self-Similar members of Azimuthal Walsh Filters

4.6 Self-Similarity in Far-Field intensity distributions for Group IIIA Self-Similar members of Azimuthal Walsh Filters

4.7 Rotational Self-Similarity observed in 2D Transverse intensity distributions at Far-Field Plane for adjacent orders of Azimuthal Walsh Filters

References

5. Intensity Distribution In The Far-field Region of Azimuthal Walsh Filters

5.1 Introduction

5.2 Analytical Formulation of Intensity distribution on axially shifted image planes

5.2.1 Synthesis of Azimuthal Walsh Filters using Azimuthal Walsh Block functions

5.2.2 Evaluation of integral using the concept of concentric equal area zones of Azimuthal Walsh Filters

5.3 Illustrative Results with Discussion

5.3.1 Intensity distribution in the Far-Field region for Zero Order azimuthal Walsh Filters

5.3.2 Intensity distribution in the Far-Field region for First Order azimuthal Walsh Filters

5.3.3 Intensity distribution in the Far-Field region for Second Order azimuthal Walsh Filters

5.3.5 Intensity distribution in the Far-Field region for Fourth Order azimuthal Walsh Filters

5.3.7 Intensity distribution in the Far-Field region for Sixth Order azimuthal Walsh Filters

5.4 Intensity Distributions on Transverse Planes Very Near to the Focus With Azimuthal Walsh Filters

for First Order Azimuthal Walsh Filters

5.4.2 Intensity Distribution in the Immediate Vicinity of the Far-Field Plane  for Second Order Azimuthal Walsh Filters

5.4.3 Intensity Distribution in the Immediate Vicinity of the Far-Field Plane  for Third Order Azimuthal Walsh Filters

5.4.4 Intensity Distribution in the Immediate Vicinity of the Far-Field Plane  for Fourth Order Azimuthal Walsh Filters

5.4.5 Intensity Distribution in the Immediate Vicinity of the Far-Field Plane  for Fifth Order Azimuthal Walsh Filters

5.4.6 Intensity Distribution in the Immediate Vicinity of the Far-Field Plane  for Sixth Order Azimuthal Walsh Filters

5.4.7 Intensity Distribution in the Immediate Vicinity of the Far-Field Plane for Seventh Order Azimuthal Walsh Filters

References

6.1 Introduction

6.2.1 Study of Self-Similarity on Transverse Image Planes shifted towards right or (+) ve side of Far-Field Plane

6.3 Similarity In Transverse Intensity Distributions In The Far-Field Region For Group IIA Self-Similar Members of Azimuthal Walsh Filters

6.3.2 Study Of Self-Similarity on Transverse Image Planes Shifted Towards left or (-) ve side of Far-Field Plane 

References

7. Perspectivas Futuras Del Trabajo De Investigación 

References

Dr. Indrani Bhattacharya is a Post-doctoral researcher associated with Prof. Ayan Banerjee of Light Matter Interaction Lab, Indian Institute of Science Education and Research, IISER, Kolkata and Prof. Vasudevan Lakshminarayanan of School of Optometry, University of Waterloo, Canada. She obtained her Ph.D. from the Department of Applied Optics and Photonics, University of Calcutta. Dr. Bhattacharya is having 20 years industry and 9 years of academic experience. Her research areas include Diffractive Optics, Biomimetics, Optical Tweezers, Point Spread Function Engineering, In-Vivo and In-Vitro Biomedical Applications, Optical Fibre Sensors. She is a member of Optical Society of India, International Society of Optics and Photonics, International Society of Optomechatronics.

Highlights the use of azimuthal Walsh filters as pupil filters under certain conditions

Discusses the generation and synthesis of azimuthal Walsh filters

Describes how the orthogonality and self-similarity of the filters can be used to create 2D and 3D light distributions near the focus