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

2.6 Self-Similarity In 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

5.3.4 Intensity distribution in the Far-Field region for Third Order azimuthal Walsh Filters

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

5.3.6 Intensity distribution in the Far-Field region for Fifth Order azimuthal Walsh Filters

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

5.3.8 Intensity distribution in the Far-Field region for Seventh Order azimuthal Walsh Filters

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

5.4.1 Intensity Distribution in the Immediate Vicinity of the Far-Field Plane

for First Order Azimuthal Walsh Filters

5.4.2 Intensity Distribution in the Immediate Vicinity of the Far-Field Plane for Second