Abstract

As more complicated microscope systems are engineered, the amount of effects taken into account rises steadily. In this context we experienced the need for a simulation approach, that will deliver the intensity distribution in space and time for scanning laser microscopes. To achieve this goal, the frequency space representation of microscope objectives was used and adapted to determine their solution of the electromagnetic wave equation. We describe the steps necessary to efficiently implement an approach to simulate multidimensional solutions of the wave equation. This includes the connection between the back focal plane and the Fourier space representation as well as a proper interpolation method for the latter. The error-potential of our least erroneous interpolation, the power of hann (POH) interpolation, is compared to other common interpolation methods. Finally we demonstrate the current potential of the approach by simulating an “expanding” optical vortex focus.

© 2015 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7th ed.
    [Crossref]
  2. R.-A. Lorbeer and A. Heisterkamp, “Three dimensional numerical simulation of complex optical systems using the coherenttransfer function,” Proc. SPIE - Adv. Micro. Tech. 7367, 73671K1 (2009).
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    [Crossref]
  4. J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer tomography,” IEEE Trans. Med. Imag. 4, 200–207 (1985).
    [Crossref]
  5. S. Matej and I. G. Kazantsev, “Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters,” IEEE Trans. Med. Imag. 25, 845–854 (2006).
    [Crossref]
  6. H. Schomberg and J. Timmer, “The gridding method for image reconstruction by Fourier transformation,” IEEE Trans. Med. Imag. 14, 596–607 (1995).
    [Crossref]
  7. F. Kreuder, M. Grass, V. Rasche, H. Braunisch, and O. Dissel, “Fast calculation of the X-ray transform from the radial derivative of the 3D Radon transform using gridding and fast backprojection techniques,” Proc. EMBEC 37, 1042–1043 (1999).
  8. K. Fourmont, “Non-equispaced fast Fourier transforms with applications to tomography,” J. Fourier Anal. Appl. 9, 431–450 (2003).
    [Crossref]
  9. D. Ruijters, B. M. ter Haar Romeny, and P. Suetens, “Efficient gpu-based texture interpolation using uniform b-splines,” J. Graph. GPU Game Tools 13, 61–69 (2008).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  13. A. Nuttall, “Some windows with very good sidelobe behavior,” IEEE Trans. Image Process. 29, 84–91 (1981).
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    [Crossref] [PubMed]
  15. K. Chan and S. Tang, “Selection of convolution kernel in non-uniform fast Fourier transform for Fourier domain optical coherence tomography,” Opt. Express 19, 26891–26904 (2011).
    [Crossref]
  16. E. H. W. Meijering, K. J. Zuiderveld, and M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
    [Crossref]
  17. G. A. Swartzlander, “Achromatic optical vortex lens,” Opt. Lett. 31, 2042–20444 (2006).
    [Crossref] [PubMed]

2011 (1)

2009 (1)

R.-A. Lorbeer and A. Heisterkamp, “Three dimensional numerical simulation of complex optical systems using the coherenttransfer function,” Proc. SPIE - Adv. Micro. Tech. 7367, 73671K1 (2009).

2008 (1)

D. Ruijters, B. M. ter Haar Romeny, and P. Suetens, “Efficient gpu-based texture interpolation using uniform b-splines,” J. Graph. GPU Game Tools 13, 61–69 (2008).
[Crossref]

2006 (2)

S. Matej and I. G. Kazantsev, “Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters,” IEEE Trans. Med. Imag. 25, 845–854 (2006).
[Crossref]

G. A. Swartzlander, “Achromatic optical vortex lens,” Opt. Lett. 31, 2042–20444 (2006).
[Crossref] [PubMed]

2003 (1)

K. Fourmont, “Non-equispaced fast Fourier transforms with applications to tomography,” J. Fourier Anal. Appl. 9, 431–450 (2003).
[Crossref]

1999 (4)

F. Kreuder, M. Grass, V. Rasche, H. Braunisch, and O. Dissel, “Fast calculation of the X-ray transform from the radial derivative of the 3D Radon transform using gridding and fast backprojection techniques,” Proc. EMBEC 37, 1042–1043 (1999).

E. H. W. Meijering, K. J. Zuiderveld, and M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[Crossref]

N. a. Thacker, A. Jackson, D. Moriarty, and E. Vokurka, “Improved quality of re-sliced MR images using re-normalized sinc interpolation,” J. Magn. Reson. Im. 10, 582–588 (1999).
[Crossref]

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imag. 18, 1049–1075 (1999).
[Crossref]

1996 (1)

B. Reddy and B. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[Crossref] [PubMed]

1995 (1)

H. Schomberg and J. Timmer, “The gridding method for image reconstruction by Fourier transformation,” IEEE Trans. Med. Imag. 14, 596–607 (1995).
[Crossref]

1992 (1)

J. P. Boyd, “A fast algorithm for Chebyshev, Fourier, and sine interpolation onto an irregular grid,” J. Comput. Phys. 103, 243–257 (1992).
[Crossref]

1985 (1)

J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer tomography,” IEEE Trans. Med. Imag. 4, 200–207 (1985).
[Crossref]

1981 (1)

A. Nuttall, “Some windows with very good sidelobe behavior,” IEEE Trans. Image Process. 29, 84–91 (1981).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7th ed.
[Crossref]

Boyd, J. P.

J. P. Boyd, “A fast algorithm for Chebyshev, Fourier, and sine interpolation onto an irregular grid,” J. Comput. Phys. 103, 243–257 (1992).
[Crossref]

Braunisch, H.

F. Kreuder, M. Grass, V. Rasche, H. Braunisch, and O. Dissel, “Fast calculation of the X-ray transform from the radial derivative of the 3D Radon transform using gridding and fast backprojection techniques,” Proc. EMBEC 37, 1042–1043 (1999).

Chan, K.

Chatterji, B.

B. Reddy and B. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[Crossref] [PubMed]

Dissel, O.

F. Kreuder, M. Grass, V. Rasche, H. Braunisch, and O. Dissel, “Fast calculation of the X-ray transform from the radial derivative of the 3D Radon transform using gridding and fast backprojection techniques,” Proc. EMBEC 37, 1042–1043 (1999).

Fourmont, K.

K. Fourmont, “Non-equispaced fast Fourier transforms with applications to tomography,” J. Fourier Anal. Appl. 9, 431–450 (2003).
[Crossref]

Gönner, C.

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imag. 18, 1049–1075 (1999).
[Crossref]

Grass, M.

F. Kreuder, M. Grass, V. Rasche, H. Braunisch, and O. Dissel, “Fast calculation of the X-ray transform from the radial derivative of the 3D Radon transform using gridding and fast backprojection techniques,” Proc. EMBEC 37, 1042–1043 (1999).

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).
[Crossref]

Heisterkamp, A.

R.-A. Lorbeer and A. Heisterkamp, “Three dimensional numerical simulation of complex optical systems using the coherenttransfer function,” Proc. SPIE - Adv. Micro. Tech. 7367, 73671K1 (2009).

Jackson, A.

N. a. Thacker, A. Jackson, D. Moriarty, and E. Vokurka, “Improved quality of re-sliced MR images using re-normalized sinc interpolation,” J. Magn. Reson. Im. 10, 582–588 (1999).
[Crossref]

Kazantsev, I. G.

S. Matej and I. G. Kazantsev, “Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters,” IEEE Trans. Med. Imag. 25, 845–854 (2006).
[Crossref]

Kreuder, F.

F. Kreuder, M. Grass, V. Rasche, H. Braunisch, and O. Dissel, “Fast calculation of the X-ray transform from the radial derivative of the 3D Radon transform using gridding and fast backprojection techniques,” Proc. EMBEC 37, 1042–1043 (1999).

Lehmann, T. M.

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imag. 18, 1049–1075 (1999).
[Crossref]

Lorbeer, R.-A.

R.-A. Lorbeer and A. Heisterkamp, “Three dimensional numerical simulation of complex optical systems using the coherenttransfer function,” Proc. SPIE - Adv. Micro. Tech. 7367, 73671K1 (2009).

Matej, S.

S. Matej and I. G. Kazantsev, “Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters,” IEEE Trans. Med. Imag. 25, 845–854 (2006).
[Crossref]

Meijering, E. H. W.

E. H. W. Meijering, K. J. Zuiderveld, and M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[Crossref]

Moriarty, D.

N. a. Thacker, A. Jackson, D. Moriarty, and E. Vokurka, “Improved quality of re-sliced MR images using re-normalized sinc interpolation,” J. Magn. Reson. Im. 10, 582–588 (1999).
[Crossref]

Nuttall, A.

A. Nuttall, “Some windows with very good sidelobe behavior,” IEEE Trans. Image Process. 29, 84–91 (1981).

O’Sullivan, J. D.

J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer tomography,” IEEE Trans. Med. Imag. 4, 200–207 (1985).
[Crossref]

Rasche, V.

F. Kreuder, M. Grass, V. Rasche, H. Braunisch, and O. Dissel, “Fast calculation of the X-ray transform from the radial derivative of the 3D Radon transform using gridding and fast backprojection techniques,” Proc. EMBEC 37, 1042–1043 (1999).

Reddy, B.

B. Reddy and B. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[Crossref] [PubMed]

Ruijters, D.

D. Ruijters, B. M. ter Haar Romeny, and P. Suetens, “Efficient gpu-based texture interpolation using uniform b-splines,” J. Graph. GPU Game Tools 13, 61–69 (2008).
[Crossref]

Schomberg, H.

H. Schomberg and J. Timmer, “The gridding method for image reconstruction by Fourier transformation,” IEEE Trans. Med. Imag. 14, 596–607 (1995).
[Crossref]

Spitzer, K.

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imag. 18, 1049–1075 (1999).
[Crossref]

Suetens, P.

D. Ruijters, B. M. ter Haar Romeny, and P. Suetens, “Efficient gpu-based texture interpolation using uniform b-splines,” J. Graph. GPU Game Tools 13, 61–69 (2008).
[Crossref]

Swartzlander, G. A.

Tang, S.

ter Haar Romeny, B. M.

D. Ruijters, B. M. ter Haar Romeny, and P. Suetens, “Efficient gpu-based texture interpolation using uniform b-splines,” J. Graph. GPU Game Tools 13, 61–69 (2008).
[Crossref]

Thacker, N. a.

N. a. Thacker, A. Jackson, D. Moriarty, and E. Vokurka, “Improved quality of re-sliced MR images using re-normalized sinc interpolation,” J. Magn. Reson. Im. 10, 582–588 (1999).
[Crossref]

Timmer, J.

H. Schomberg and J. Timmer, “The gridding method for image reconstruction by Fourier transformation,” IEEE Trans. Med. Imag. 14, 596–607 (1995).
[Crossref]

Viergever, M. A.

E. H. W. Meijering, K. J. Zuiderveld, and M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[Crossref]

Vokurka, E.

N. a. Thacker, A. Jackson, D. Moriarty, and E. Vokurka, “Improved quality of re-sliced MR images using re-normalized sinc interpolation,” J. Magn. Reson. Im. 10, 582–588 (1999).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7th ed.
[Crossref]

Zuiderveld, K. J.

E. H. W. Meijering, K. J. Zuiderveld, and M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[Crossref]

IEEE Trans. Image Process. (3)

A. Nuttall, “Some windows with very good sidelobe behavior,” IEEE Trans. Image Process. 29, 84–91 (1981).

B. Reddy and B. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[Crossref] [PubMed]

E. H. W. Meijering, K. J. Zuiderveld, and M. A. Viergever, “Image reconstruction by convolution with symmetrical piecewise nth-order polynomial kernels,” IEEE Trans. Image Process. 8, 192–201 (1999).
[Crossref]

IEEE Trans. Med. Imag. (4)

T. M. Lehmann, C. Gönner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imag. 18, 1049–1075 (1999).
[Crossref]

J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer tomography,” IEEE Trans. Med. Imag. 4, 200–207 (1985).
[Crossref]

S. Matej and I. G. Kazantsev, “Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters,” IEEE Trans. Med. Imag. 25, 845–854 (2006).
[Crossref]

H. Schomberg and J. Timmer, “The gridding method for image reconstruction by Fourier transformation,” IEEE Trans. Med. Imag. 14, 596–607 (1995).
[Crossref]

J. Comput. Phys. (1)

J. P. Boyd, “A fast algorithm for Chebyshev, Fourier, and sine interpolation onto an irregular grid,” J. Comput. Phys. 103, 243–257 (1992).
[Crossref]

J. Fourier Anal. Appl. (1)

K. Fourmont, “Non-equispaced fast Fourier transforms with applications to tomography,” J. Fourier Anal. Appl. 9, 431–450 (2003).
[Crossref]

J. Graph. GPU Game Tools (1)

D. Ruijters, B. M. ter Haar Romeny, and P. Suetens, “Efficient gpu-based texture interpolation using uniform b-splines,” J. Graph. GPU Game Tools 13, 61–69 (2008).
[Crossref]

J. Magn. Reson. Im. (1)

N. a. Thacker, A. Jackson, D. Moriarty, and E. Vokurka, “Improved quality of re-sliced MR images using re-normalized sinc interpolation,” J. Magn. Reson. Im. 10, 582–588 (1999).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. EMBEC (1)

F. Kreuder, M. Grass, V. Rasche, H. Braunisch, and O. Dissel, “Fast calculation of the X-ray transform from the radial derivative of the 3D Radon transform using gridding and fast backprojection techniques,” Proc. EMBEC 37, 1042–1043 (1999).

Proc. SPIE - Adv. Micro. Tech. (1)

R.-A. Lorbeer and A. Heisterkamp, “Three dimensional numerical simulation of complex optical systems using the coherenttransfer function,” Proc. SPIE - Adv. Micro. Tech. 7367, 73671K1 (2009).

Other (2)

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).
[Crossref]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7th ed.
[Crossref]

Supplementary Material (1)

» Media 1: MOV (3269 KB)     

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Figures (5)

Figure 1
Figure 1 Impulse response for multiple widths of the POH interpolation. a) Several POH kernels at the central grid position. b) Impulse response after Fourier transformation with identical color code. The resulting normed amplitude show the behavior of a POH H H 2 2 kernel for H = 4, 6 and 12. The first drop in frequency domain stays consistent on the Nyquist frequency 0.5 1 grid points. For bigger H the kernel broadens and drops at steeper rates for extreme frequencies above Nyquist.
Fig. 2
Fig. 2 Impulse response for multiple positions pi of the POH interpolation. a) POH 6 2 kernels at three different positions. The dots indicate the values used for the interpolation. b) Impulse response after Fourier transformation with identical color code. The resulting amplitude and phase distributions differ from position to position. For the amplitude distribution a constant (0 dB) would represent the ideal frequency response. The phase response should result in a linear phase with a slope proportional to the displacement Δ spanning from −90 to 90 deg for Δ = ±0.5. Differences from the ideal expected behavior have to be compensated by normalization. This approach is limited by the variations e.g. in the extreme frequencies of the normed amplitude.
Fig. 3
Fig. 3 Amplitude and phase error at all frequencies after a one dimensional FFT with normalization to constant errors. a) Maximum deviation from the ideal amplitude of 1 in any of the 500 tested interpolation positions after normalization to average amplitude after the FFT. b) Maximum deviation from the ideal phase −i(kr) in any of the 500 tested positions after subtraction of the average phase error after the FFT.
Fig. 4
Fig. 4 Top: POH 6 2 interpolation method compared to oversampled version. Bottom: POH 6 2 interpolation method compared to equidistant sampling of an analytical | E ˜ ( k , ω ) |. a) & d): Results of the POH 6 2 interpolation method. b) & e) The absolute error calculated as the difference between the standard simulation and the reference simulation, both with their maximum intensity normed to one. c) & f) Relative error calculated by dividing the absolute errors (b & e) by their reference simulations.
Fig. 5
Fig. 5 Simulation of a spectral vortex. Side by side view of the x,z-plane and the x,y-plane (indicated by the red dashed lines) with the time evolving from top left to bottom right. The diameter of the vortex reduces over time. After a complete collapse it starts expanding again. Scale bar represents 10 μm. (see Media 1

Equations (15)

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Δ E ( r , t ) = n 2 c 2 t 2 E ( r , t )
E ( r , t ) = d k d ω E ˜ ( k , ω ) e i ( k r ω t )
k ( ρ ) = ( cos ϕ sin Θ sin ϕ sin Θ cos Θ ) | k | = ( ρ x f ρ y f 1 ( ρ f ) 2 ) ω n ( ω ) c ,
E ˜ ( k ( ρ ) , ω ) = S ( ρ ) E BFP ( ρ , ω )
V i = F I ( s = x des x i Δ x )
F I ( s ) = { NN ( s ) = { 1 1 2 s < 1 2 0 otherwise CS ( s ) = { 0 | s | 2 1 6 ( 2 | s | ) 3 1 | s | < 2 2 3 1 2 | s | 2 ( 2 | s | ) | s | < 1 SWS H , ξ ( s ) = { sin ( π s ξ ) π s i = 0 n a i cos ( 2 π i s H ) | s | < H 2 0 otherwise POH H L ( s ) = { cos 2 L ( π s H ) | s | < H 2 0 otherwise
δ S BFP = | ρ x ρ × ρ y ρ | = 1 f 2
δ S KS = | ρ x k × ρ y k | = | k | f 2 cos ( θ )
P δ S BFP | E BFP | 2 = δ S KS | E KS | 2
| E KS | = | E BFP | S ( ρ ) = | E BFP | δ S BFP δ S KS = | E BFP | cos ( θ ) | k |
P δ S BFP ( | E BFP | num δ S BFP δ S ) 2 = δ S KS ( | E KS | num δ S KS δ S ) 2
| E KS | num = | E BFP | num S num ( ρ ) = | E BFP | num δ S K S δ S BFP = | E BFP | num | k | cos ( θ )
E ( k , r ) = d k x δ ( k x k ) e i ( k x r ) = e i ( k r )
| E ˜ ( k , ω ) | = cos θ | k | sin 2 ( π k k min k max k min ) cos 2 ( π 2 sin θ NA ) ,
NN < CS < POH 6 2 < SSW 11 < SSW 31 < POH 12 5

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