## Focal shift, optical transfer function, and phase-space representations

JOSA A, Vol. 17, Issue 4, pp. 772-779 (2000)

http://dx.doi.org/10.1364/JOSAA.17.000772

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### Abstract

The focal shift for a lens of finite value of Fresnel number can be defined in terms of the second moment of the intensity distribution in transverse planes. The connection with the optical transfer function is described. The specification of the focused amplitude in terms of the fractional Fourier transform is discussed, and the connections among the fractional Fourier transform, the Wigner distribution, and the ambiguity function are described, leading to a model for effects of Fresnel number in terms of a rotation in phase space. The uncertainty principle is discussed, including the significance of the beam propagation factor M^{2} and the width of optical fiber beam modes. Calculation of the moments in terms of the modulus and the phase of the illuminating wave is presented, and the use of the Kaiser–Teager energy operator is also described.

© 2000 Optical Society of America

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.2590) Fourier optics and signal processing : ABCD transforms

(140.3300) Lasers and laser optics : Laser beam shaping

(260.1960) Physical optics : Diffraction theory

**Citation**

Colin J. R. Sheppard and Kieran G. Larkin, "Focal shift, optical transfer function, and phase-space representations," J. Opt. Soc. Am. A **17**, 772-779 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-4-772

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### References

- S. De Nicola, D. Anderson, and M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
- Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
- A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
- Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
- C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
- R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
- N. Bareket, “Second moment of the diffraction point spread function as an image quality criterion,” J. Opt. Soc. Am. 69, 1311–1312 (1979).
- M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
- R. Martinez-Herrero and P. M. Meijas, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
- E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
- T. Alieva, V. Lopez, F. Aguillo-Lopez, and L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
- H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transform as a tool for analysing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
- H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
- P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
- C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2104 (1998).
- D. Dragoman, “The relation between light diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2117–2124 (1998).
- W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
- A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
- A. W. Lohmann and B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
- L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
- D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. Ser. B 38, 209–219 (1996).
- M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
- K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
- J. Bertrand and P. Bertrand, “Tomographic procedures for constructing phase space representations,” in The Physics of Phase Space, Y. S. Kim and W. W. Zachary, eds. (Springer, New York, 1987).
- W. D. MacMillan, Dynamics of Rigid Bodies (Dover, New York, 1960).
- H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1969).
- M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
- D. Mustard, “Uncertainty principles invariant under the fractional Fourier transform,” J. Aust. Math. Soc. Ser. B, 33, 180–191 (1991).
- A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
- H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
- S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
- P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
- S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
- K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
- S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).
- K. Petermann, “Constraints for fundamental-mode spot size for broadband dispersion-compensated single-mode fibres,” Electron. Lett. 19, 712–714 (1983).
- C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20, 144–145 (1984).
- A. Liang and C.-C. Fan, “Mode-field radius of noncircular field single-mode fibre: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
- P. Grivet, Electron Optics (Pergamon, Oxford, UK, 1965), p. 419.
- S. C. Fleming, “Measurement of mode field radius distribution and evaluation of mode field radius in single mode optical waveguides,” in Tests, Measurements, and Characterization of Electro-Optic Devices and Systems, S. G. Wadekar, ed., Proc. SPIE 1180, 95–106 (1989).
- A. Papoulis, “Apodization for optimum imaging of smooth objects,” J. Opt. Soc. Am. 62, 1423–1429 (1972).
- P. Maragos, J. F. Kaiser, and T. F. Quaterieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
- P. Maragos and A. C. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A 12, 1867–1876 (1995).
- R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, and M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
- C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).

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