Abstract

We discuss a general theoretical framework for representing and propagating fully coherent, fully incoherent, and the intermediate regime of partially coherent submillimeter-wave fields by means of general sampled basis functions, which may have any degree of completeness. Partially coherent fields arise when finite-throughput systems induce coherence on incoherent fields. This powerful extension to traditional modal analysis methods by using undercomplete Gaussian–Hermite modes can be employed to analyze and optimize such Gaussian quasi-optical techniques. We focus on one particular basis set, the Gabor basis, which consists of overlapping translated and modulated Gaussian beams. We present high-accuracy numerical results from field reconstructions and propagations. In particular, we perform one-dimensional analyses illustrating the Van Cittert–Zernike theorem and then extend our simulations to two dimensions, including simple models of horn and bolometer arrays. Our methods and results are of practical importance as a method for analyzing terahertz fields, which are often partially coherent and diffraction limited so that ray tracing is inaccurate and physical optics computationally prohibitive.

© 2004 Optical Society of America

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References

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  1. S. Withington, G. Yassin, “Power coupled between partially coherent vector fields in different states of coherence,” J. Opt. Soc. Am. A 18, 3061–3071 (2001).
    [CrossRef]
  2. S. Withington, J. A. Murphy, “Modal analysis of partially-coherent submillimeter-wave quasioptical systems,” in Eighth International Symposium on Space Terahertz Technology, R. Blundell, E. Tong, eds. (Harvard–Smithsonian Center for Astrophysics, Cambridge, Mass., 1997), pp. 446–456.
  3. J. A. Murphy, S. Withington, “Gaussian beam mode analysis of multi-beam quasi-optical systems,” in Conference on Advanced Technology MMW, Radio, and Terahertz Telescopes, T. G. Phillips, ed., Proc. SPIE3357, 97–104 (1998).
    [CrossRef]
  4. S. Withington, M. P. Hobson, R. H. Berry, “Representing the behavior of partially coherent optical systems using over-complete basis sets,” manuscript available from the author (R. Berry, rhb21@mrao.cam.ac.uk).
  5. G. Golub, C. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. PressLondon, 1996).
  6. S. Qiu, H. G. Feichtinger, “Structure of the Gabor matrix and efficient numerical algorithms for discrete Gabor expansions,” in Visual Communications and Image Processing ’94, A. K. Katsaggelos, ed., Proc. SPIE2308, 1146–1157 (1994).
    [CrossRef]
  7. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]
  8. H. G. Feichtinger, N. Kaiblinger, P. Prinz, “POCS approach to Gabor analysis,” in Sixth International Workshop on Digital Image Processing and Computer Graphics: Applications in Humanities and Natural Sciences, E. Wenger, L. I. Dimitrov, eds., Proc. SPIE3346, 18–29 (1998).
    [CrossRef]
  9. S. Qiu, H. G. Feichtinger, “The under-sampled discrete Gabor transform,” IEEE Trans. Signal Process. 46, 1221–1228 (1998).
    [CrossRef]
  10. S. Qiu, “Discrete Gabor structure and optimal representation,” IEEE Trans. Signal Process. 34, 2259–2268 (1995).
  11. H. Feichtinger, T. Stohmer, Gabor Analysis and Algorithms: Theory and Applications (Birkhauser, Boston, 1998).
  12. P. F. Goldsmith, Quasioptical Systems (IEEE Press, New York, 1998).
  13. M. R. Rayner, C. Rieckmann, C. G. Parini, “Diffracted Gaussian beam analysis of quasi-optical multireflector systems,” in Proceedings of the Millennium Conference on Antennas and Propagation AP2000 (ESA-ESTEC, Noordwijle, The Netherlands, 2000).
  14. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).
  15. M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Proc. IEEE 68, 538–539 (1980).
    [CrossRef]

2001 (1)

1998 (1)

S. Qiu, H. G. Feichtinger, “The under-sampled discrete Gabor transform,” IEEE Trans. Signal Process. 46, 1221–1228 (1998).
[CrossRef]

1995 (1)

S. Qiu, “Discrete Gabor structure and optimal representation,” IEEE Trans. Signal Process. 34, 2259–2268 (1995).

1990 (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

1980 (1)

M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).

Bastiaans, M. J.

M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

Berry, R.

S. Withington, M. P. Hobson, R. H. Berry, “Representing the behavior of partially coherent optical systems using over-complete basis sets,” manuscript available from the author (R. Berry, rhb21@mrao.cam.ac.uk).

Berry, R. H.

S. Withington, M. P. Hobson, R. H. Berry, “Representing the behavior of partially coherent optical systems using over-complete basis sets,” manuscript available from the author (R. Berry, rhb21@mrao.cam.ac.uk).

Daubechies, I.

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

Feichtinger, H.

H. Feichtinger, T. Stohmer, Gabor Analysis and Algorithms: Theory and Applications (Birkhauser, Boston, 1998).

Feichtinger, H. G.

S. Qiu, H. G. Feichtinger, “The under-sampled discrete Gabor transform,” IEEE Trans. Signal Process. 46, 1221–1228 (1998).
[CrossRef]

S. Qiu, H. G. Feichtinger, “Structure of the Gabor matrix and efficient numerical algorithms for discrete Gabor expansions,” in Visual Communications and Image Processing ’94, A. K. Katsaggelos, ed., Proc. SPIE2308, 1146–1157 (1994).
[CrossRef]

H. G. Feichtinger, N. Kaiblinger, P. Prinz, “POCS approach to Gabor analysis,” in Sixth International Workshop on Digital Image Processing and Computer Graphics: Applications in Humanities and Natural Sciences, E. Wenger, L. I. Dimitrov, eds., Proc. SPIE3346, 18–29 (1998).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).

Goldsmith, P. F.

P. F. Goldsmith, Quasioptical Systems (IEEE Press, New York, 1998).

Golub, G.

G. Golub, C. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. PressLondon, 1996).

Hobson, M. P.

S. Withington, M. P. Hobson, R. H. Berry, “Representing the behavior of partially coherent optical systems using over-complete basis sets,” manuscript available from the author (R. Berry, rhb21@mrao.cam.ac.uk).

Kaiblinger, N.

H. G. Feichtinger, N. Kaiblinger, P. Prinz, “POCS approach to Gabor analysis,” in Sixth International Workshop on Digital Image Processing and Computer Graphics: Applications in Humanities and Natural Sciences, E. Wenger, L. I. Dimitrov, eds., Proc. SPIE3346, 18–29 (1998).
[CrossRef]

Murphy, J. A.

S. Withington, J. A. Murphy, “Modal analysis of partially-coherent submillimeter-wave quasioptical systems,” in Eighth International Symposium on Space Terahertz Technology, R. Blundell, E. Tong, eds. (Harvard–Smithsonian Center for Astrophysics, Cambridge, Mass., 1997), pp. 446–456.

J. A. Murphy, S. Withington, “Gaussian beam mode analysis of multi-beam quasi-optical systems,” in Conference on Advanced Technology MMW, Radio, and Terahertz Telescopes, T. G. Phillips, ed., Proc. SPIE3357, 97–104 (1998).
[CrossRef]

Parini, C. G.

M. R. Rayner, C. Rieckmann, C. G. Parini, “Diffracted Gaussian beam analysis of quasi-optical multireflector systems,” in Proceedings of the Millennium Conference on Antennas and Propagation AP2000 (ESA-ESTEC, Noordwijle, The Netherlands, 2000).

Prinz, P.

H. G. Feichtinger, N. Kaiblinger, P. Prinz, “POCS approach to Gabor analysis,” in Sixth International Workshop on Digital Image Processing and Computer Graphics: Applications in Humanities and Natural Sciences, E. Wenger, L. I. Dimitrov, eds., Proc. SPIE3346, 18–29 (1998).
[CrossRef]

Qiu, S.

S. Qiu, H. G. Feichtinger, “The under-sampled discrete Gabor transform,” IEEE Trans. Signal Process. 46, 1221–1228 (1998).
[CrossRef]

S. Qiu, “Discrete Gabor structure and optimal representation,” IEEE Trans. Signal Process. 34, 2259–2268 (1995).

S. Qiu, H. G. Feichtinger, “Structure of the Gabor matrix and efficient numerical algorithms for discrete Gabor expansions,” in Visual Communications and Image Processing ’94, A. K. Katsaggelos, ed., Proc. SPIE2308, 1146–1157 (1994).
[CrossRef]

Rayner, M. R.

M. R. Rayner, C. Rieckmann, C. G. Parini, “Diffracted Gaussian beam analysis of quasi-optical multireflector systems,” in Proceedings of the Millennium Conference on Antennas and Propagation AP2000 (ESA-ESTEC, Noordwijle, The Netherlands, 2000).

Rieckmann, C.

M. R. Rayner, C. Rieckmann, C. G. Parini, “Diffracted Gaussian beam analysis of quasi-optical multireflector systems,” in Proceedings of the Millennium Conference on Antennas and Propagation AP2000 (ESA-ESTEC, Noordwijle, The Netherlands, 2000).

Stohmer, T.

H. Feichtinger, T. Stohmer, Gabor Analysis and Algorithms: Theory and Applications (Birkhauser, Boston, 1998).

van Loan, C.

G. Golub, C. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. PressLondon, 1996).

Withington, S.

S. Withington, G. Yassin, “Power coupled between partially coherent vector fields in different states of coherence,” J. Opt. Soc. Am. A 18, 3061–3071 (2001).
[CrossRef]

S. Withington, J. A. Murphy, “Modal analysis of partially-coherent submillimeter-wave quasioptical systems,” in Eighth International Symposium on Space Terahertz Technology, R. Blundell, E. Tong, eds. (Harvard–Smithsonian Center for Astrophysics, Cambridge, Mass., 1997), pp. 446–456.

J. A. Murphy, S. Withington, “Gaussian beam mode analysis of multi-beam quasi-optical systems,” in Conference on Advanced Technology MMW, Radio, and Terahertz Telescopes, T. G. Phillips, ed., Proc. SPIE3357, 97–104 (1998).
[CrossRef]

S. Withington, M. P. Hobson, R. H. Berry, “Representing the behavior of partially coherent optical systems using over-complete basis sets,” manuscript available from the author (R. Berry, rhb21@mrao.cam.ac.uk).

Yassin, G.

IEEE Trans. Inf. Theory (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

IEEE Trans. Signal Process. (2)

S. Qiu, H. G. Feichtinger, “The under-sampled discrete Gabor transform,” IEEE Trans. Signal Process. 46, 1221–1228 (1998).
[CrossRef]

S. Qiu, “Discrete Gabor structure and optimal representation,” IEEE Trans. Signal Process. 34, 2259–2268 (1995).

J. Inst. Electr. Eng. (London) (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

Other (9)

H. G. Feichtinger, N. Kaiblinger, P. Prinz, “POCS approach to Gabor analysis,” in Sixth International Workshop on Digital Image Processing and Computer Graphics: Applications in Humanities and Natural Sciences, E. Wenger, L. I. Dimitrov, eds., Proc. SPIE3346, 18–29 (1998).
[CrossRef]

S. Withington, J. A. Murphy, “Modal analysis of partially-coherent submillimeter-wave quasioptical systems,” in Eighth International Symposium on Space Terahertz Technology, R. Blundell, E. Tong, eds. (Harvard–Smithsonian Center for Astrophysics, Cambridge, Mass., 1997), pp. 446–456.

J. A. Murphy, S. Withington, “Gaussian beam mode analysis of multi-beam quasi-optical systems,” in Conference on Advanced Technology MMW, Radio, and Terahertz Telescopes, T. G. Phillips, ed., Proc. SPIE3357, 97–104 (1998).
[CrossRef]

S. Withington, M. P. Hobson, R. H. Berry, “Representing the behavior of partially coherent optical systems using over-complete basis sets,” manuscript available from the author (R. Berry, rhb21@mrao.cam.ac.uk).

G. Golub, C. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. PressLondon, 1996).

S. Qiu, H. G. Feichtinger, “Structure of the Gabor matrix and efficient numerical algorithms for discrete Gabor expansions,” in Visual Communications and Image Processing ’94, A. K. Katsaggelos, ed., Proc. SPIE2308, 1146–1157 (1994).
[CrossRef]

H. Feichtinger, T. Stohmer, Gabor Analysis and Algorithms: Theory and Applications (Birkhauser, Boston, 1998).

P. F. Goldsmith, Quasioptical Systems (IEEE Press, New York, 1998).

M. R. Rayner, C. Rieckmann, C. G. Parini, “Diffracted Gaussian beam analysis of quasi-optical multireflector systems,” in Proceedings of the Millennium Conference on Antennas and Propagation AP2000 (ESA-ESTEC, Noordwijle, The Netherlands, 2000).

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Figures (10)

Fig. 1
Fig. 1

Degree of completeness of Gabor frames with the parameters N = 144 , a { 2 ,   3 ,   4 ,   6 ,   8 ,   9 ,   12 ,   16 ,   18 ,   24 ,   36 ,   48 ,   72 ,   144 } , and b { 6 ,   8 ,   9 ,   12 ,   16 ,   18 ,   24 ,   36 ,   48 ,   72 ,   144 } ; the mother atom associated with each frame was taken to satisfy Eq. (33). An important feature is the region where the lines tend to a value of unity on the vertical axis, i.e., completeness. It is this region that is shown in the inset.

Fig. 2
Fig. 2

Largest and smallest eigenvalues of the matrices S and R for the Gabor frames with parameters N = 144 , a = 12 , b { 2 ,   3 ,   4 ,   6 ,   8 ,   9 ,   12 ,   16 ,   18 ,   24 ,   36 ,   72 } , and a mother atom of the form (33). The critically sampled frame is not complete. As expected, the largest eigenvalue is the same for both the S and R matrices.

Fig. 3
Fig. 3

Largest and smallest eigenvalues of the matrices S and R for the Gabor frames with parameters N = 144 , a = 9 , b { 2 ,   3 ,   4 ,   6 ,   8 ,   9 ,   12 ,   16 ,   18 ,   24 ,   36 ,   72 } , and a mother atom of the form (33). The critically sampled frame is complete. As expected, the largest eigenvalue is the same for both the S and R matrices.

Fig. 4
Fig. 4

Real parts of dual functions associated with undersampled frames. The frame parameters are included on the plots in the form g ( N ,   a ,   b ) .

Fig. 5
Fig. 5

Real parts of dual functions associated with critically sampled frames. The frame parameters are included on the plots in the form g ( N ,   a ,   b ) .

Fig. 6
Fig. 6

Real parts of dual functions associated with oversampled frames. The frame parameters are included on the plots in the form g ( N ,   a ,   b ) .

Fig. 7
Fig. 7

(top left) Two coherent Gaussian sources at the input plane, (top right) the coherent Gaussian sources at the image (Fourier) plane, (bottom left) two mutually incoherent and self-incoherent Gaussian sources at the input plane, (bottom right) the incoherent Gaussian sources at the image plane.

Fig. 8
Fig. 8

An incoherent top hat was propagated to the Fourier plane, where decreasingly small apertures were simulated. Propagation of the limited field was then performed to the next Fourier plane. Solid curves, intensity; dashed curves, coherence function. In each plot the fraction (t) of samples passing through the simulated, on-axis symmetric aperture is indicated. Qualitatively the throughput of the system decreases from top to bottom and left to right. Plots of relative intensity are labeled with g ( N ,   a ,   b ) .

Fig. 9
Fig. 9

Bolometer array: left, basis-function reconstruction of the intensity at the input plane; center, intensity at the Fourier plane; right, result of placing a limiting 7 × 7-pixel aperture on axis at the Fourier plane and transforming back to the image plane.

Fig. 10
Fig. 10

Same as Fig. 9, but for a horn array.

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

x = k = 1 M a k e k .
E = [ e 1 e k e M ] ,
Ea = x
E = U Σ V ,
S = k = 1 M e k e k = EE ,
R = k = 1 M e k e k = E E ,
S = U Σ Σ U ,
R = V Σ Σ V .
SU = U Σ Σ ,
RV = V Σ Σ ,
= S ˜ - 1 S - I ,
Sx = k ( e k x ) e k ,
S ˜ - 1 S I ,
S ˜ - 1 Sx = k ( e k x ) S ˜ - 1 e k x .
e ˜ k = S ˜ - 1 e k .
x k b k e ˜ k .
S S ˜ - 1 = I .
S S ˜ - 1 x = k e k e k S ˜ - 1 x x .
e ˜ k = e k S ˜ - 1 .
x k a k e k ,
W = xx .
W = xx = k = 1 M h = 1 M b k a h * e ˜ k e h = k = 1 M h = 1 M C kh e ˜ k e h ,
C kh = b k a h * = e k W e ˜ h .
C kh = e k e ˜ h .
C kh = δ kh .
W = xx = k = 1 M h = 1 M a k a h * e k e h = k = 1 M h = 1 M C kh e k e h ,
C kh = e ˜ k W e ˜ h .
W = k = 1 M h = 1 M δ kh e k e h = k = 1 M e k e k ,
T va g ( j ) = g ( ( j - va ) mod   N ) , v Z ,
M vb g ( j ) = exp - 2 π ij ( ( v + v 0 ) mod   N ) b N g ( j ) ,
v Z ,
S jk = n = 0 a ˜ - 1 m = 0 b ˜ - 1 M mb T na g ( j ) [ M mb T na g ( k ) ] .
g ˜ mn = S - 1 g mn = S - 1 T na M mb g = T na M mb S - 1 g = T na M mb g ˜ .
g ( x ) = A 0 exp - π   x 2 2 a 2 ,
= S g ˜ - g .

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