A Light-Ray Approach to Fractional Fourier Optics
Abstract
:1. Introduction
2. Space and Angular Variables and Their Transfers
2.1. Angular Frequency and Light-Ray Representation
2.2. Ray Transfer
- i
- We first assume that (Figure 5). Then (second-order approximation in and ), that is , and we conclude by , and .
- ii
- In the limits of a second-order approximation, we haveWe conclude by , andThe same reasoning in the y–z plane leads to and
- iii
- iv
- We haveWe also have , so thatWe conclude by andThe same reasoning in the y–z leads to andAll the ’s have been determined.In conclusion, Equation (6) is explicitly writtenWith , a more concise form of Equation (23) is
3. Rotations in a Reduced Phase Space
3.1. Defining Reduced Variables and an Angle of Rotation
3.2. Interpreting Rotations in the Reduced Phase Space
4. Link with Diffraction and Fractional Fourier Optics
4.1. General Transfer by Diffraction (Fresnel Phenomenon)
4.2. Fraunhofer Diffraction
5. Link with the Spherical Angular Spectrum
5.1. The Notion of Spherical Angular Spectrum
5.2. Propagation of the Spherical Angular Spectrum
6. Accordance with the Huygens–Fresnel Principle
6.1. Expression with Non-Homogeneous Variables
6.2. Expression with Homogeneous Variables
7. Coherent Imaging
7.1. Imaging by a Refracting Spherical Cap
7.1.1. Matrix form of Snell’s Law (Refraction) [12]
7.1.2. Ray Transfer by a Refracting Spherical Cap
7.1.3. Conjugation Formula and Lateral Magnification
7.1.4. Determination of
7.1.5. Conjugation of Curvature Centers (Double-Conjugation Law [2,10,11])
- The vertex of is the paraxial image of the vertex of ;
- The curvature center of is the paraxial image of the curvature center of .
7.1.6. Determination of
7.1.7. Radius Magnification
7.2. Generalization to Centered Systems
- Since is the paraxial image of through , we obtain that is the paraxial image of through ;
- Since is the paraxial image of through , we obtain that is the paraxial image of through .
7.3. Homogeneous Matrix Representation
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Proof of Equations (37)–(40)
Appendix B. Proof of Equation (46)
- (i)
- We begin with
- (ii)
- Then we consider
- (iii)
- We have
- (iv)
- Since both and are two-dimensional vectors, we have
Appendix C. Proof of Equation (65)
- (i)
- We begin with the exponential depending on . We have
- (ii)
- Factor in . We have
- (iii)
- Factor in . We have
- (iv)
- Differential term. Since both Φ and are two-dimensional variables, we have
- (v)
- The previous changes of variables lead us to write Equation (59) in the form
Appendix D. An Alternative Proof of the Conjugation of Curvature Centers
Appendix E. Checking a21 = 0
Appendix F. An Alternative Proof of the Radius Magnification Law
Appendix G. Homogeneous Imaging Matrix
- (i)
- Composition of transformations. From Equation (A43) we deduce
- (ii)
- Imaging. The spherical cap is the coherent image of is .
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Fogret, É.; Pellat-Finet, P. A Light-Ray Approach to Fractional Fourier Optics. Fractal Fract. 2023, 7, 505. https://doi.org/10.3390/fractalfract7070505
Fogret É, Pellat-Finet P. A Light-Ray Approach to Fractional Fourier Optics. Fractal and Fractional. 2023; 7(7):505. https://doi.org/10.3390/fractalfract7070505
Chicago/Turabian StyleFogret, Éric, and Pierre Pellat-Finet. 2023. "A Light-Ray Approach to Fractional Fourier Optics" Fractal and Fractional 7, no. 7: 505. https://doi.org/10.3390/fractalfract7070505