Abstract

The Fresnel diffraction integral form of optical wave propagation and the more general Linear Canonical Transforms (LCT) are cast into a matrix transformation form. Taking advantage of recent efficient matrix multiply algorithms, this approach promises an efficient computational and analytical tool that is competitive with FFT based methods but offers better behavior in terms of aliasing, transparent boundary condition, and flexibility in number of sampling points and computational window sizes of the input and output planes being independent. This flexibility makes the method significantly faster than FFT based propagators when only a single point, as in Strehl metrics, or a limited number of points, as in power-in-the-bucket metrics, are needed in the output observation plane.

© 2015 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 3rd Edition (Roberts and Company, 2004).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1969).
  3. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (John Wiley& Sons, 2007).
  4. D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26(1), 109–112 (1990).
    [Crossref]
  5. A. E. Siegman, Lasers (University Science Books, 1986).
  6. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum Press, 1979).
  7. M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12(8), 1772–1783 (1971).
    [Crossref]
  8. B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22(5), 928–937 (2005).
    [Crossref] [PubMed]
  9. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12(4), 743–751 (1995).
    [Crossref]
  10. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientic Computing, 3rd ed. (Cambridge University Press, 2007).
  11. E. O. Brigham, The Fast Fourier Transform, (Prentice Hall, 1974).
  12. Wikipedia contributors, “Matrix multiplication,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Matrix_multiplication&oldid=600028683
  13. D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE Press, 2011).
  14. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (SPIE Press Monograph, 2010).
  15. P. W. Milonni and A. H. Paxton, “Model for the unstable-resonator carbon monoxide electric-discharge laser,” J. Appl. Phys. 49(3), 1012 (1978).
    [Crossref]
  16. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16(9), 624–626 (1991).
    [Crossref] [PubMed]

2005 (1)

1995 (1)

1991 (1)

1990 (1)

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26(1), 109–112 (1990).
[Crossref]

1978 (1)

P. W. Milonni and A. H. Paxton, “Model for the unstable-resonator carbon monoxide electric-discharge laser,” J. Appl. Phys. 49(3), 1012 (1978).
[Crossref]

1971 (1)

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12(8), 1772–1783 (1971).
[Crossref]

Hadley, G. R.

Hennelly, B. M.

Hermansson, B.

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26(1), 109–112 (1990).
[Crossref]

Mendlovic, D.

Milonni, P. W.

P. W. Milonni and A. H. Paxton, “Model for the unstable-resonator carbon monoxide electric-discharge laser,” J. Appl. Phys. 49(3), 1012 (1978).
[Crossref]

Moshinsky, M.

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12(8), 1772–1783 (1971).
[Crossref]

Ozaktas, H. M.

Paxton, A. H.

P. W. Milonni and A. H. Paxton, “Model for the unstable-resonator carbon monoxide electric-discharge laser,” J. Appl. Phys. 49(3), 1012 (1978).
[Crossref]

Quesne, C.

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12(8), 1772–1783 (1971).
[Crossref]

Sheridan, J. T.

Yevick, D.

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26(1), 109–112 (1990).
[Crossref]

IEEE J. Quantum Electron. (1)

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26(1), 109–112 (1990).
[Crossref]

J. Appl. Phys. (1)

P. W. Milonni and A. H. Paxton, “Model for the unstable-resonator carbon monoxide electric-discharge laser,” J. Appl. Phys. 49(3), 1012 (1978).
[Crossref]

J. Math. Phys. (1)

M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12(8), 1772–1783 (1971).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Other (10)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientic Computing, 3rd ed. (Cambridge University Press, 2007).

E. O. Brigham, The Fast Fourier Transform, (Prentice Hall, 1974).

Wikipedia contributors, “Matrix multiplication,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Matrix_multiplication&oldid=600028683

D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE Press, 2011).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (SPIE Press Monograph, 2010).

A. E. Siegman, Lasers (University Science Books, 1986).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum Press, 1979).

J. W. Goodman, Introduction to Fourier Optics, 3rd Edition (Roberts and Company, 2004).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1969).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (John Wiley& Sons, 2007).

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Figures (4)

Fig. 1
Fig. 1 A graphical representation of a system being influenced by a Linear Canonical Transform. A source field, u(x1,y1) is transformed into an observation field U(x2,y2), in the observation plane a distance z from the source plane.
Fig. 2
Fig. 2 (a) The irradiance distribution of a flat-top square laser beam with a half-width of 5 cm propagated a distance of 100 m. (b) A Gaussian beam of 2.5 cm radius propagated a distance of 10 km. In both cases the laser wavelength is one micron.
Fig. 3
Fig. 3 Propagation of a Gaussian beam through the atmosphere. The 1-micron wavelength beam waist radius is 1.5 cm, is first propagated a distance of 5 km (a), then an atmospheric equivalent phase screen (r0 = 5cm) is applied and the modified beam is propagated another 5 km (b). The two-dimensional beam at 10 km is shown as EMA calculated (c), and FFT calculated (d).
Fig. 4
Fig. 4 Speed ratio for EMA over FFT wave-propagator. Solid circles represent actual simulation using MATLAB. Red line is the formula of Eq. (12) for β = 2.8. Blue broken line is for β = 2.4, and green line for β = 3.

Tables (1)

Tables Icon

Table 1 The abcd matrix configuration and corresponding Linear Canonic Transforms

Equations (12)

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U( x 2 , y 2 )= 1 ib u( x 1 , y 1 ) e i π b [ ( a x 1 2 2 x 1 x 2 +d x 2 2 )+( a y 1 2 2 y 1 y 2 +d y 2 2 ) ] d x 1 d y 1
U= H T u H
u= H U H T
H j1,j2 = W j1 1 ib e i π b ( a x j1 2 2 x j1 x j2 +d x j2 2 )
u( x 1 , y 1 ) e i π λz [ ( x n x 1 ) 2 + ( y m y 1 ) 2 ] d x 1 d y 1 j=1 N W j e i π λz ( y m y j ) 2 k=1 N W k e i π λz ( x n x k ) 2 u j,k = j=1 N k=1 N H m,j u j,k H k,n = [ H T uH ] m,k
Δ U x = 1/iλz e iπz/λ u o ( x 1 , y 1 ) x 1 0.5Δ x 1 +0.5Δ e i π λz ( x x 2 ) 2 dx
Δ U y = 1/iλz e iπz/λ u o ( x 1 , y 1 ) y 1 0.5Δ y 1 +0.5Δ e i π λz ( y y 2 ) 2 dy
H( x 1 , x 2 )= e ikz iλz x 1 0.5Δ x 1 +0.5Δ e i π λz ( x 2 x ) 2 dx
H( x 1 , x 2 )= e iπz/λ Δ iλz e iπ λz ( x 2 x 1 ) 2 [ sin(π( x 2 x 1 )Δ/λz π( x 2 x 1 )Δ/λz ]
r effective = a 2 + ( λz/ r o ) 2
Error( x 2 , y 2 ) L 2 Δ 2 12λz max | 2 f( x 1 , y 1 ; x 2 , y 2 ) 2 x 1 + 2 f( x 1 , y 1 ; x 2 , y 2 ) 2 y 1 |
Speedc 1+3 log 2 ( N 2 ) N β2

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