Abstract

The introduction of the fast Fourier transform (FFT) constituted a crucial step towards a faster and more efficient physio-optics modeling and design, since it is a faster version of the Discrete Fourier transform. However, the numerical effort of the operation explodes in the case of field components presenting strong wavefront phases—very typical occurrences in optics— due to the requirement of the FFT that the wrapped phase be well sampled. In this paper, we propose an approximated algorithm to compute the Fourier transform in such a situation. We show that the Fourier transform of fields with strong wavefront phases exhibits a behavior that can be described as a bijective mapping of the amplitude distribution, which is why we name this operation “homeomorphic Fourier transform." We use precisely this characteristic behavior in the mathematical approximation that simplifies the Fourier integral. We present the full theoretical derivation and several numerical applications to demonstrate its advantages in the computing process.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. N. Bracewell, The Fourier transform and its applications (McGraw-Hill, 1986).
  2. J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).
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    [Crossref]
  4. S. Zhang, D. Asoubar, C. Hellmann, and F. Wyrowski, “Propagation of electromagnetic fields between non-parallel planes: a fully vectorial formulation and an efficient implementation,” Appl. Opt. 55(3), 529–538 (2016).
    [Crossref]
  5. A. V. Pfeil and F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47(13), 2335–2350 (2000).
    [Crossref]
  6. M. Kuhn, F. Wyrowski, and C. Hellmann, Non-sequential optical field tracing, Advanced Finite Element Methods and Applications (Springer, 2013257–273
  7. J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comp. 19(90), 297 (1965).
    [Crossref]
  8. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21(14), 2470 (1982).
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  9. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30(25), 3627–3632 (1991).
    [Crossref]
  10. L. Rabiner, R. W. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
    [Crossref]
  11. A. Andrianov, A. Szabo, A. Sergeev, A. Kim, V. Chvykov, and M. Kalashnikov, “Computationally efficient method for fourier transform of highly chirped pulses for laser and parametric amplifier modeling,” Opt. Express 24(23), 25974–25982 (2016).
    [Crossref]
  12. R. Tong and R. W. Cox, “Rotation of nmr images using the 2d chirp-z transform,” Magn. Reson. Med. 41(2), 253–256 (1999).
    [Crossref]
  13. Z. Wang, S. Zhang, O. Baladron-Zorita, C. Hellmann, and F. Wyrowski, “Application of the semi-analytical fourier transform to electromagnetic modeling,” Opt. Express 27(11), 15335–15350 (2019).
    [Crossref]
  14. P. Debye, “Das verhalten von lichtwellen in der nähe eines brennpunktes oder einer brennlinie,” Ann. Phys. 335(14), 755–776 (1909).
    [Crossref]
  15. G. C. Sherman and W. C. Chew, “Aperture and far-field distributions expressed by the debye integral representation of focused fields,” J. Opt. Soc. Am. 72(8), 1076–1083 (1982).
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  16. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge university press, 1995).
  17. J. Brüning and V. W. Guillemin, Mathematics Past and Present: Fourier Integral Operators (Springer, 1994).
  18. J. J. Stamnes, “Waves, rays, and the method of stationary phase,” Opt. Express 10(16), 740–751 (2002).
    [Crossref]
  19. K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. ii. stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1871–1881 (2001).
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  20. R. Rolleston and N. George, “Stationary phase approximations in fresnel-zone magnitude-only reconstructions,” J. Opt. Soc. Am. A 4(1), 148–153 (1987).
    [Crossref]
  21. Physical optics simulation and design software “Wyrowski VirtualLab Fusion,” developed by Wyrowski Photonics UG, distributed by LightTrans GmbH, Jena, Germany.
  22. F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5-6), 449–466 (2011).
    [Crossref]
  23. K. G. Binmore and K. G. Binmore, Mathematical Analysis: a straightforward approach (Cambridge University Press, 1982).

2019 (1)

2016 (2)

2014 (1)

2011 (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5-6), 449–466 (2011).
[Crossref]

2002 (1)

2001 (1)

2000 (1)

A. V. Pfeil and F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47(13), 2335–2350 (2000).
[Crossref]

1999 (1)

R. Tong and R. W. Cox, “Rotation of nmr images using the 2d chirp-z transform,” Magn. Reson. Med. 41(2), 253–256 (1999).
[Crossref]

1991 (1)

1987 (1)

1982 (2)

1969 (1)

L. Rabiner, R. W. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

1965 (1)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comp. 19(90), 297 (1965).
[Crossref]

1909 (1)

P. Debye, “Das verhalten von lichtwellen in der nähe eines brennpunktes oder einer brennlinie,” Ann. Phys. 335(14), 755–776 (1909).
[Crossref]

Andrianov, A.

Asoubar, D.

Baladron-Zorita, O.

Binmore, K. G.

K. G. Binmore and K. G. Binmore, Mathematical Analysis: a straightforward approach (Cambridge University Press, 1982).

K. G. Binmore and K. G. Binmore, Mathematical Analysis: a straightforward approach (Cambridge University Press, 1982).

Bone, D. J.

Bracewell, R. N.

R. N. Bracewell, The Fourier transform and its applications (McGraw-Hill, 1986).

Brüning, J.

J. Brüning and V. W. Guillemin, Mathematics Past and Present: Fourier Integral Operators (Springer, 1994).

Chew, W. C.

Chvykov, V.

Cooley, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comp. 19(90), 297 (1965).
[Crossref]

Cox, R. W.

R. Tong and R. W. Cox, “Rotation of nmr images using the 2d chirp-z transform,” Magn. Reson. Med. 41(2), 253–256 (1999).
[Crossref]

Debye, P.

P. Debye, “Das verhalten von lichtwellen in der nähe eines brennpunktes oder einer brennlinie,” Ann. Phys. 335(14), 755–776 (1909).
[Crossref]

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

Guillemin, V. W.

J. Brüning and V. W. Guillemin, Mathematics Past and Present: Fourier Integral Operators (Springer, 1994).

Hellmann, C.

Itoh, K.

Kalashnikov, M.

Kim, A.

Kuhn, M.

D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Efficient semi-analytical propagation techniques for electromagnetic fields,” J. Opt. Soc. Am. A 31(3), 591–602 (2014).
[Crossref]

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5-6), 449–466 (2011).
[Crossref]

M. Kuhn, F. Wyrowski, and C. Hellmann, Non-sequential optical field tracing, Advanced Finite Element Methods and Applications (Springer, 2013257–273

Larkin, K. G.

Mandel, L.

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge university press, 1995).

Pfeil, A. V.

A. V. Pfeil and F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47(13), 2335–2350 (2000).
[Crossref]

Rabiner, L.

L. Rabiner, R. W. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

Rader, C.

L. Rabiner, R. W. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

Rolleston, R.

Schafer, R. W.

L. Rabiner, R. W. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

Sergeev, A.

Sherman, G. C.

Stamnes, J. J.

Szabo, A.

Tong, R.

R. Tong and R. W. Cox, “Rotation of nmr images using the 2d chirp-z transform,” Magn. Reson. Med. 41(2), 253–256 (1999).
[Crossref]

Tukey, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comp. 19(90), 297 (1965).
[Crossref]

Wang, Z.

Wolf, E.

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge university press, 1995).

Wyrowski, F.

Zhang, S.

Ann. Phys. (1)

P. Debye, “Das verhalten von lichtwellen in der nähe eines brennpunktes oder einer brennlinie,” Ann. Phys. 335(14), 755–776 (1909).
[Crossref]

Appl. Opt. (3)

IEEE Trans. Audio Electroacoust. (1)

L. Rabiner, R. W. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

J. Mod. Opt. (2)

A. V. Pfeil and F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47(13), 2335–2350 (2000).
[Crossref]

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5-6), 449–466 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Magn. Reson. Med. (1)

R. Tong and R. W. Cox, “Rotation of nmr images using the 2d chirp-z transform,” Magn. Reson. Med. 41(2), 253–256 (1999).
[Crossref]

Math. Comp. (1)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex fourier series,” Math. Comp. 19(90), 297 (1965).
[Crossref]

Opt. Express (3)

Other (7)

K. G. Binmore and K. G. Binmore, Mathematical Analysis: a straightforward approach (Cambridge University Press, 1982).

M. Kuhn, F. Wyrowski, and C. Hellmann, Non-sequential optical field tracing, Advanced Finite Element Methods and Applications (Springer, 2013257–273

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge university press, 1995).

J. Brüning and V. W. Guillemin, Mathematics Past and Present: Fourier Integral Operators (Springer, 1994).

R. N. Bracewell, The Fourier transform and its applications (McGraw-Hill, 1986).

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

Physical optics simulation and design software “Wyrowski VirtualLab Fusion,” developed by Wyrowski Photonics UG, distributed by LightTrans GmbH, Jena, Germany.

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Figures (10)

Fig. 1.
Fig. 1. Fast Fourier transform of a field with house-shaped amplitude and different values of the spherical-phase radius. Panel (a) shows the amplitude distribution of the $E_x$ component in the spatial domain, common starting point to all results. A spherical phase with radius of curvature $R$ is then superimposed on this amplitude, and its Fourier transform computed. Panels (b) to (g) present the $\tilde {E}_x$ component (i.e. the result of said Fourier-transform operation) for different values of $R$ , as indicated below each of the individual panels. The required numbers of sampling points of fast Fourier transform for each case are given in Tab. 1.
Fig. 2.
Fig. 2. Homeomorphic Fourier transform of the same house-shaped amplitude with a superimposed spherical phase in comparison to Fig. 1. Panel (a) shows the amplitude distribution of the $E_x$ component in the spatial domain. Panels (b) to (g) present the result of Homeomorphic Fourier-transform operation for different values of $R$ . Please note the window size of panel (b) and the scale of panels (b) to (d) are different from Fig. 1. The required numbers of sampling points of homeomorphic Fourier transform for each case are given in Tab. 1.
Fig. 3.
Fig. 3. Comparison of the numerical effort of the homeomorphic and the Fast Fourier transforms (HFT and FFT respectively, for short) for the case of a house-shaped field with a spherical phase, for different values of the numerical aperture (NA) of said spherical phase. The abscissa records increasing values of the NA, which correspond to decreasing values of the radius of curvature $R$ of the spherical phase. The ordinate corresponds to a logarithmic scale of the number of sampling points. The dashed line for the number of sampling points used to sample the function in the case of the HFT operation, and, the solid line for the number of sampling points necessitated by the computation of the FFT. Finally, the CPU time at two specific positions, where $R = 10$ mm and $R = 5$ mm, are pointed out.
Fig. 4.
Fig. 4. Homeomorphic Fourier transform (FFT results are not shown, since deviation between HFT and FFT just 0.2 %) of a field with amplitude shaped in the form of a letter sequence spelling “Light”, in the cases of different aberrant phases. Panel (a) shows the amplitude distribution of the $E_x$ component in the space domain, common starting point to all results. Different aberrations are then superimposed on this amplitude, and its Fourier transform computed. The remaining panels present the $\tilde {E}_x$ component (i.e. the result of said Fourier-transform operation) for (b) positive defocus, (c) negative defocus, (d) positive $x$ astigmatism, (e) negative $x$ astigmatism, (f) positive $y$ astigmatism and (g) negative $y$ astigmatism.
Fig. 5.
Fig. 5. Homeomorphic Fourier transform (FFT results are not shown, since deviation between HFT and FFT just 0.2 %) of a field with a general bijective wavefront phase. Panel (a) should the amplitude distribution of the $E_x$ component in the space domain, common starting point to all results. A spherical wavefront with different aberrations is then superimposed on this amplitude, and the Fourier transform computed. The remaining panels present the $\tilde {E}_x$ component (i.e. the result of said Fourier-transform operation) for (b) a purely spherical phase, (c) spherical phase and trefoil $x$ , (d) spherical phase and coma $x$ , (e) spherical phase, trefoil $x$ and coma $x$ , (f) spherical phase and tetrafoil $y$ , (g) spherical phase and tertiary astigmatism $y$ , (h) spherical phase, tetrafoil $y$ and tertiary astigmatism $y$ , (i) spherical phase and pentafoil $x$ , (j) spherical phase and tertiary spherical, and (k) spherical phase, pentafoil $x$ and tertiary spherical. The corresponding Zernike terms are indicated under each panel.
Fig. 6.
Fig. 6. Fourier transform of a field with a non-bijective wavefront phase. Panel (a) shows the amplitude of the $E_x$ component in the space domain, panel (b) shows the result of the Fourier transform as obtained via the rigorous Fast Fourier Transform, and panel (c) shows the result of the Fourier transform as per the approximation provided by the homeomorphic Fourier transform with bijective regularization of the mapping relation.
Fig. 7.
Fig. 7. Analysis of the wavefront phase for the simulation task: Homeomorphic Fourier transform of a field with a non-bijective wavefront phase. Panel (a) shows the initial smooth phase, panel (b) the residual phase after performing the fitting, panel (c) the phase from which the bijective mapping will be obtained, panel (e) serves to examine the homeomorphism condition for the fitted phase (the violation of the condition is evident from the change in sign of the second derivatives) and panel (f), analogously to (e), serves to examine how the same condition is fulfilled in the case of the final regularized phase employed to compute the bijective mapping.
Fig. 8.
Fig. 8. Spherical phase function $\psi \! \left ( {\boldsymbol{\rho}}\right )$ and its partial derivative functions.
Fig. 9.
Fig. 9. (a) The blue curve corresponds to the intersection of the first derivative of the spherical phase function $\psi _x\!\left ({\boldsymbol{\rho}}\right )$ with the constant plane given by $\psi _x\!\left ({\boldsymbol{\rho}}\right ) = k_{xi}$ . (b) Along the path ${\boldsymbol{\rho}}\!\left (t\right ) = \left [v\!\left (t\right ), y\!\left (t\right )\right ]$ , we can find another parametric curve on the surface $\psi _y\!\left ({\boldsymbol{\rho}}\right )$ .
Fig. 10.
Fig. 10. Example of the third kind of smooth function phase. At $P_1$ , $P_2$ and $P_3$ the second derivative of $\psi \!\left ({\boldsymbol{\rho}}\right )$ is equal to zero. Based on these three points, the definition domain is divided into four segments. Comparing the variation of the derivatives between two neighboring segments, we can analyze the continuity of the zero position.

Tables (4)

Tables Icon

Table 1. Comparison of the homeomorphic Fourier transform and the fast Fourier transform: number of sampling points and deviation. The numerical aperture (NA) is defined as n sin θ = n a a 2 + R 2 . Here, a is the radius of the aperture and n is the refractive index of the surrounding medium. Corresponding field distribution results are respectively shown in Fig. 1 and Fig. 2.

Tables Icon

Table 2. Simulation parameters of the aberrant phase for the example presented in Section 3.2.

Tables Icon

Table 3. Simulation parameters of the aberrant phase for the example in Section 3.3.

Tables Icon

Table 4. Simulation parameters of the aberrant phase for the example in Section 3.4.

Equations (22)

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V ( ρ ) = | V ( ρ ) | exp { i arg [ V ( ρ ) ] } = | V ( ρ ) | exp { i γ ( ρ ) }
V ( ρ ) = | V ( ρ ) | exp { i [ ϕ ( ρ ) + ψ ( ρ ) ] } = U ( ρ ) exp [ i ψ ( ρ ) ] ,
V ~ ( κ ) = 1 2 π V ( ρ ) exp ( i ρ κ ) d 2 ρ = = 1 2 π U ( ρ ) exp [ i ψ ( ρ ) i ρ κ ] d 2 ρ ,
V ~ ( κ ) a [ ρ ( κ ) ] U [ ρ ( κ ) ] exp { i ψ [ ρ ( κ ) ] i κ ρ ( κ ) } ,
ψ ( ρ ) = κ ,
a ( ρ ) = { i ψ x x ( ρ ) i ψ x x ( ρ ) ψ x y 2 ( ρ ) ψ x x ( ρ ) ψ y y ( ρ ) , ψ x x ( ρ ) 0 1 | ψ x y ( ρ ) | , ψ x x ( ρ ) = 0 ,
V ~ ( κ ) = | V ~ ( κ ) | exp { i arg [ V ~ ( κ ) ] } = A ~ ( κ ) exp [ i ψ ~ ( κ ) ]
{ A ~ ( κ ) = a [ ρ ( κ ) ] U [ ρ ( κ ) ] ψ ~ ( κ ) = ψ [ ρ ( κ ) ] κ ρ ( κ ) .
V ( ρ ) = 1 2 π V ~ ( κ ) exp ( i ρ κ ) d 2 κ = = 1 2 π A ~ ( κ ) exp [ i ψ ~ ( κ ) + i ρ κ ] d 2 κ .
~ ψ ~ ( κ ) = ρ .
V ( ρ ) = a ~ [ κ ( ρ ) ] A ~ [ κ ( ρ ) ] exp { i ψ ~ [ κ ( ρ ) ] + i ρ κ ( ρ ) } ,
a ~ ( κ ) = { i ψ ~ k x k x ( κ ) i ψ ~ k x k x ( κ ) ψ ~ k x k y 2 ( κ ) ψ ~ k x k x ( κ ) ψ ~ k y k y ( κ ) , ψ ~ k x k x ( κ ) 0 1 | ψ ~ k x k x ( κ ) | , ψ ~ k x k x ( κ ) = 0 .
{ U ( ρ ) = a ~ [ κ ( ρ ) ] A ~ [ κ ( ρ ) ] ψ ( ρ ) = ψ ~ [ κ ( ρ ) ] + ρ κ ( ρ ) ,
V ( ρ ) = U ( ρ ) exp [ i ψ ( ρ ) ] = = U ( ρ ) exp [ i ψ res ( ρ ) ] exp [ i ψ map ( ρ ) ] .
κ map = ψ map ( ρ ) .
σ := x , y | V ~ FFT ( κ ) V ~ HFT ( κ ) | 2 x , y | V ~ FFT ( κ ) | 2
ψ ( ρ ) = ψ Zer ( ρ ) = k m = 0 M n = 0 N c n m Z n m ( r , θ ) ,
κ = f ( ρ ) = ψ ( ρ ) .
κ = ( k x , k y ) := { k x = x ψ ( ρ ) = ψ x ( ρ ) , k y = y ψ ( ρ ) = ψ y ( ρ ) .
d ψ x [ x ( t ) , y ( t ) ] d t = ψ x x d x d t + ψ x y d y d t = 0 .
d ψ y [ x ( t ) , y ( t ) ] d t = ψ x y d x d t + ψ y y d y d t ,
d ψ y [ x ( t ) , y ( t ) ] d t = { ψ x y 2 ψ x x ψ y y ψ x x d y d t , ψ x x 0 ψ x y d x d t , ψ x x = 0

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