Abstract

Active millimeter wave imaging systems have become a promising candidate for indoor security applications and industrial inspection. However, there is a lack of simulation tools at the system level. We introduce and evaluate two modeling approaches that are applied to active millimeter wave imaging systems. The first approach originates in Fourier optics and concerns the calculation in the spatial frequency domain. The second approach is based on wave propagation and corresponds to calculation in the spatial domain. We compare the two approaches in the case of both rough and smooth objects and point out that the spatial frequency domain calculation may suffer from a large error in amplitude of 50% in the case of rough objects. The comparison demonstrates that the concepts of point-spread function and f-number should be applied with careful consideration in coherent millimeter wave imaging systems. In the case of indoor applications, the near-field effect should be considered, and this is included in the spatial domain calculation.

© 2009 Optical Society of America

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References

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  1. E. N. Grossman, A. Luukanen, and A. J. Miller, “Terahertz active direct detection imagers,” Proc. SPIE 5411, 68-77 (2004).
    [CrossRef]
  2. B. Grafulla-Gonzalez, K. Lebart, and A. R. Harvey, “Physical optics modeling of millimeter-wave personnel scanners,” Pattern Recogn. Lett. 27, 1852-1862 (2006).
    [CrossRef]
  3. F. Qi, I. Ocket, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Millimeter wave imaging: system modeling and phenomena discussion,” in Proceedings of 19th International Conference on Applied Electromagnetics and Communications, September 24-26, 2007, Dubrovnik (2007), pp. 283-286.
  4. G. N. Sinclair, R. N. Anderton, and R. Appleby, “Outdoor passive millimeter wave security screening,” in Proceedings of IEEE 35th International Carnahan Conference on Security Technology (IEEE, 2001), pp. 172-179.
  5. L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39-50 (2003).
    [CrossRef]
  6. Z. Luo, J. Xiong, and J. Yang, “A model-based analyzing and calculating of the focal plane array for passive millimeter wave imaging system,” in Proceedings of International Conference on Circuits and Systems Proceedings, June 25-28, 2006, Guilin, China, pp. 629-632.
    [CrossRef]
  7. L. Zhang, J. Stiens, A. Elhawil, and R. Vounckx, “Multispectral illumination and image processing techniques for active millimeter-wave concealed object detection,” Appl. Opt. 47, 6357-6365 (2008).
    [CrossRef] [PubMed]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  9. P. R. Goldsmith, C. T. Hsieh, G. R. Huguenin, J. Kapitzky, and E. L. Moore, “Focal plane imaging systems for millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 41, 1664-1675 (1993).
    [CrossRef]
  10. R. Appleby and H. B. Wallace, “Standoff detection of weapons and contraband in the 100 GHzto1 THz region,” IEEE Trans. Antennas Propag. 55, 2944-2956 (2007).
    [CrossRef]
  11. J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).
  12. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).
  13. S. Mezouari and A. R. Harvey, “Validity of Fresnel and Fraunhofer approximations in scalar diffraction,” J. Opt. A, Pure Appl. Opt. 5, 86-91 (2003).
    [CrossRef]
  14. T. Hasegawa, M. Hoshino, and T. Iwasaki, “Simulation of diffraction fields for microwave imaging,” in Proceedings of International Symposium on Electromagnetic Compatibility, May 17-21, 1999, Tokyo (1999) pp. 428-431.
  15. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979).
    [CrossRef]
  16. J. E. Harvey, D. Bogunovic, and A. Krywonos, “Aberrations of diffracted wave fields: distortion,” Appl. Opt. 42, 1167-1174 (2003).
    [CrossRef] [PubMed]
  17. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7-14 (1981).
    [CrossRef]
  18. F. D. Feiock, “Wave propagation in optical systems with large apertures,” J. Opt. Soc. Am. 68, 485-489 (1978).
    [CrossRef]
  19. F. Qi, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Inaccuracies in Fourier optics,” in Proceedings of URSI-Benelux Forum, June 8, 2009, Delft (2009).
  20. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  21. F. Qi, D. Schreurs, V. Tavakol, and B. Nauwelaers, “Comparison of optical and millimeter wave imaging on speckle,” IEEE International Symposium on Antennas and Propagation (IEEE, 2008), pp. 1-4.
  22. S. Zhang, Near Field Optical Microscope and Its Applications (Science Press, 2000).
  23. V. Tavakol, F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “System modeling for active mm-wave imaging systems using an enhanced calculation method,” in Proceedings of European Radar Conference, October 30-31, 2008, Amsterdam (2008) pp. 56-59.

2008

2007

R. Appleby and H. B. Wallace, “Standoff detection of weapons and contraband in the 100 GHzto1 THz region,” IEEE Trans. Antennas Propag. 55, 2944-2956 (2007).
[CrossRef]

2006

B. Grafulla-Gonzalez, K. Lebart, and A. R. Harvey, “Physical optics modeling of millimeter-wave personnel scanners,” Pattern Recogn. Lett. 27, 1852-1862 (2006).
[CrossRef]

2004

E. N. Grossman, A. Luukanen, and A. J. Miller, “Terahertz active direct detection imagers,” Proc. SPIE 5411, 68-77 (2004).
[CrossRef]

2003

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39-50 (2003).
[CrossRef]

S. Mezouari and A. R. Harvey, “Validity of Fresnel and Fraunhofer approximations in scalar diffraction,” J. Opt. A, Pure Appl. Opt. 5, 86-91 (2003).
[CrossRef]

J. E. Harvey, D. Bogunovic, and A. Krywonos, “Aberrations of diffracted wave fields: distortion,” Appl. Opt. 42, 1167-1174 (2003).
[CrossRef] [PubMed]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1993

P. R. Goldsmith, C. T. Hsieh, G. R. Huguenin, J. Kapitzky, and E. L. Moore, “Focal plane imaging systems for millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 41, 1664-1675 (1993).
[CrossRef]

1981

1979

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

1978

Anderton, R. N.

G. N. Sinclair, R. N. Anderton, and R. Appleby, “Outdoor passive millimeter wave security screening,” in Proceedings of IEEE 35th International Carnahan Conference on Security Technology (IEEE, 2001), pp. 172-179.

Appleby, R.

R. Appleby and H. B. Wallace, “Standoff detection of weapons and contraband in the 100 GHzto1 THz region,” IEEE Trans. Antennas Propag. 55, 2944-2956 (2007).
[CrossRef]

G. N. Sinclair, R. N. Anderton, and R. Appleby, “Outdoor passive millimeter wave security screening,” in Proceedings of IEEE 35th International Carnahan Conference on Security Technology (IEEE, 2001), pp. 172-179.

Bogunovic, D.

Elhawil, A.

Feiock, F. D.

Goldsmith, P. R.

P. R. Goldsmith, C. T. Hsieh, G. R. Huguenin, J. Kapitzky, and E. L. Moore, “Focal plane imaging systems for millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 41, 1664-1675 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

Grafulla-Gonzalez, B.

B. Grafulla-Gonzalez, K. Lebart, and A. R. Harvey, “Physical optics modeling of millimeter-wave personnel scanners,” Pattern Recogn. Lett. 27, 1852-1862 (2006).
[CrossRef]

Grossman, E. N.

E. N. Grossman, A. Luukanen, and A. J. Miller, “Terahertz active direct detection imagers,” Proc. SPIE 5411, 68-77 (2004).
[CrossRef]

Harvey, A. R.

B. Grafulla-Gonzalez, K. Lebart, and A. R. Harvey, “Physical optics modeling of millimeter-wave personnel scanners,” Pattern Recogn. Lett. 27, 1852-1862 (2006).
[CrossRef]

S. Mezouari and A. R. Harvey, “Validity of Fresnel and Fraunhofer approximations in scalar diffraction,” J. Opt. A, Pure Appl. Opt. 5, 86-91 (2003).
[CrossRef]

Harvey, J. E.

J. E. Harvey, D. Bogunovic, and A. Krywonos, “Aberrations of diffracted wave fields: distortion,” Appl. Opt. 42, 1167-1174 (2003).
[CrossRef] [PubMed]

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

Hasegawa, T.

T. Hasegawa, M. Hoshino, and T. Iwasaki, “Simulation of diffraction fields for microwave imaging,” in Proceedings of International Symposium on Electromagnetic Compatibility, May 17-21, 1999, Tokyo (1999) pp. 428-431.

Hoshino, M.

T. Hasegawa, M. Hoshino, and T. Iwasaki, “Simulation of diffraction fields for microwave imaging,” in Proceedings of International Symposium on Electromagnetic Compatibility, May 17-21, 1999, Tokyo (1999) pp. 428-431.

Hsieh, C. T.

P. R. Goldsmith, C. T. Hsieh, G. R. Huguenin, J. Kapitzky, and E. L. Moore, “Focal plane imaging systems for millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 41, 1664-1675 (1993).
[CrossRef]

Huguenin, G. R.

P. R. Goldsmith, C. T. Hsieh, G. R. Huguenin, J. Kapitzky, and E. L. Moore, “Focal plane imaging systems for millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 41, 1664-1675 (1993).
[CrossRef]

Iwasaki, T.

T. Hasegawa, M. Hoshino, and T. Iwasaki, “Simulation of diffraction fields for microwave imaging,” in Proceedings of International Symposium on Electromagnetic Compatibility, May 17-21, 1999, Tokyo (1999) pp. 428-431.

Kapitzky, J.

P. R. Goldsmith, C. T. Hsieh, G. R. Huguenin, J. Kapitzky, and E. L. Moore, “Focal plane imaging systems for millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 41, 1664-1675 (1993).
[CrossRef]

Krywonos, A.

Lebart, K.

B. Grafulla-Gonzalez, K. Lebart, and A. R. Harvey, “Physical optics modeling of millimeter-wave personnel scanners,” Pattern Recogn. Lett. 27, 1852-1862 (2006).
[CrossRef]

Luo, Z.

Z. Luo, J. Xiong, and J. Yang, “A model-based analyzing and calculating of the focal plane array for passive millimeter wave imaging system,” in Proceedings of International Conference on Circuits and Systems Proceedings, June 25-28, 2006, Guilin, China, pp. 629-632.
[CrossRef]

Luukanen, A.

E. N. Grossman, A. Luukanen, and A. J. Miller, “Terahertz active direct detection imagers,” Proc. SPIE 5411, 68-77 (2004).
[CrossRef]

Mezouari, S.

S. Mezouari and A. R. Harvey, “Validity of Fresnel and Fraunhofer approximations in scalar diffraction,” J. Opt. A, Pure Appl. Opt. 5, 86-91 (2003).
[CrossRef]

Miller, A. J.

E. N. Grossman, A. Luukanen, and A. J. Miller, “Terahertz active direct detection imagers,” Proc. SPIE 5411, 68-77 (2004).
[CrossRef]

Moffa, P.

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39-50 (2003).
[CrossRef]

Moore, E. L.

P. R. Goldsmith, C. T. Hsieh, G. R. Huguenin, J. Kapitzky, and E. L. Moore, “Focal plane imaging systems for millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 41, 1664-1675 (1993).
[CrossRef]

Nauwelaers, B.

F. Qi, I. Ocket, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Millimeter wave imaging: system modeling and phenomena discussion,” in Proceedings of 19th International Conference on Applied Electromagnetics and Communications, September 24-26, 2007, Dubrovnik (2007), pp. 283-286.

F. Qi, D. Schreurs, V. Tavakol, and B. Nauwelaers, “Comparison of optical and millimeter wave imaging on speckle,” IEEE International Symposium on Antennas and Propagation (IEEE, 2008), pp. 1-4.

F. Qi, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Inaccuracies in Fourier optics,” in Proceedings of URSI-Benelux Forum, June 8, 2009, Delft (2009).

V. Tavakol, F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “System modeling for active mm-wave imaging systems using an enhanced calculation method,” in Proceedings of European Radar Conference, October 30-31, 2008, Amsterdam (2008) pp. 56-59.

Ocket, I.

V. Tavakol, F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “System modeling for active mm-wave imaging systems using an enhanced calculation method,” in Proceedings of European Radar Conference, October 30-31, 2008, Amsterdam (2008) pp. 56-59.

F. Qi, I. Ocket, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Millimeter wave imaging: system modeling and phenomena discussion,” in Proceedings of 19th International Conference on Applied Electromagnetics and Communications, September 24-26, 2007, Dubrovnik (2007), pp. 283-286.

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Qi, F.

F. Qi, D. Schreurs, V. Tavakol, and B. Nauwelaers, “Comparison of optical and millimeter wave imaging on speckle,” IEEE International Symposium on Antennas and Propagation (IEEE, 2008), pp. 1-4.

V. Tavakol, F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “System modeling for active mm-wave imaging systems using an enhanced calculation method,” in Proceedings of European Radar Conference, October 30-31, 2008, Amsterdam (2008) pp. 56-59.

F. Qi, I. Ocket, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Millimeter wave imaging: system modeling and phenomena discussion,” in Proceedings of 19th International Conference on Applied Electromagnetics and Communications, September 24-26, 2007, Dubrovnik (2007), pp. 283-286.

F. Qi, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Inaccuracies in Fourier optics,” in Proceedings of URSI-Benelux Forum, June 8, 2009, Delft (2009).

Schreurs, D.

F. Qi, I. Ocket, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Millimeter wave imaging: system modeling and phenomena discussion,” in Proceedings of 19th International Conference on Applied Electromagnetics and Communications, September 24-26, 2007, Dubrovnik (2007), pp. 283-286.

V. Tavakol, F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “System modeling for active mm-wave imaging systems using an enhanced calculation method,” in Proceedings of European Radar Conference, October 30-31, 2008, Amsterdam (2008) pp. 56-59.

F. Qi, D. Schreurs, V. Tavakol, and B. Nauwelaers, “Comparison of optical and millimeter wave imaging on speckle,” IEEE International Symposium on Antennas and Propagation (IEEE, 2008), pp. 1-4.

F. Qi, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Inaccuracies in Fourier optics,” in Proceedings of URSI-Benelux Forum, June 8, 2009, Delft (2009).

Shoucri, M.

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39-50 (2003).
[CrossRef]

Sinclair, G. N.

G. N. Sinclair, R. N. Anderton, and R. Appleby, “Outdoor passive millimeter wave security screening,” in Proceedings of IEEE 35th International Carnahan Conference on Security Technology (IEEE, 2001), pp. 172-179.

Southwell, W. H.

Stiens, J.

Stutzman, W. L.

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).

Tavakol, V.

F. Qi, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Inaccuracies in Fourier optics,” in Proceedings of URSI-Benelux Forum, June 8, 2009, Delft (2009).

F. Qi, I. Ocket, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Millimeter wave imaging: system modeling and phenomena discussion,” in Proceedings of 19th International Conference on Applied Electromagnetics and Communications, September 24-26, 2007, Dubrovnik (2007), pp. 283-286.

V. Tavakol, F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “System modeling for active mm-wave imaging systems using an enhanced calculation method,” in Proceedings of European Radar Conference, October 30-31, 2008, Amsterdam (2008) pp. 56-59.

F. Qi, D. Schreurs, V. Tavakol, and B. Nauwelaers, “Comparison of optical and millimeter wave imaging on speckle,” IEEE International Symposium on Antennas and Propagation (IEEE, 2008), pp. 1-4.

Thiele, G. A.

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).

Vounckx, R.

Wallace, H. B.

R. Appleby and H. B. Wallace, “Standoff detection of weapons and contraband in the 100 GHzto1 THz region,” IEEE Trans. Antennas Propag. 55, 2944-2956 (2007).
[CrossRef]

Xiong, J.

Z. Luo, J. Xiong, and J. Yang, “A model-based analyzing and calculating of the focal plane array for passive millimeter wave imaging system,” in Proceedings of International Conference on Circuits and Systems Proceedings, June 25-28, 2006, Guilin, China, pp. 629-632.
[CrossRef]

Yang, J.

Z. Luo, J. Xiong, and J. Yang, “A model-based analyzing and calculating of the focal plane array for passive millimeter wave imaging system,” in Proceedings of International Conference on Circuits and Systems Proceedings, June 25-28, 2006, Guilin, China, pp. 629-632.
[CrossRef]

Yujiri, L.

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39-50 (2003).
[CrossRef]

Zhang, L.

Zhang, S.

S. Zhang, Near Field Optical Microscope and Its Applications (Science Press, 2000).

Am. J. Phys.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

Appl. Opt.

IEEE Microw. Mag.

L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Microw. Mag. 4(3), 39-50 (2003).
[CrossRef]

IEEE Trans. Antennas Propag.

R. Appleby and H. B. Wallace, “Standoff detection of weapons and contraband in the 100 GHzto1 THz region,” IEEE Trans. Antennas Propag. 55, 2944-2956 (2007).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

P. R. Goldsmith, C. T. Hsieh, G. R. Huguenin, J. Kapitzky, and E. L. Moore, “Focal plane imaging systems for millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 41, 1664-1675 (1993).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

S. Mezouari and A. R. Harvey, “Validity of Fresnel and Fraunhofer approximations in scalar diffraction,” J. Opt. A, Pure Appl. Opt. 5, 86-91 (2003).
[CrossRef]

J. Opt. Soc. Am.

Pattern Recogn. Lett.

B. Grafulla-Gonzalez, K. Lebart, and A. R. Harvey, “Physical optics modeling of millimeter-wave personnel scanners,” Pattern Recogn. Lett. 27, 1852-1862 (2006).
[CrossRef]

Phys. Rev. Lett.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Proc. SPIE

E. N. Grossman, A. Luukanen, and A. J. Miller, “Terahertz active direct detection imagers,” Proc. SPIE 5411, 68-77 (2004).
[CrossRef]

Other

T. Hasegawa, M. Hoshino, and T. Iwasaki, “Simulation of diffraction fields for microwave imaging,” in Proceedings of International Symposium on Electromagnetic Compatibility, May 17-21, 1999, Tokyo (1999) pp. 428-431.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (Wiley, 1998).

F. Qi, D. Schreurs, V. Tavakol, and B. Nauwelaers, “Comparison of optical and millimeter wave imaging on speckle,” IEEE International Symposium on Antennas and Propagation (IEEE, 2008), pp. 1-4.

S. Zhang, Near Field Optical Microscope and Its Applications (Science Press, 2000).

V. Tavakol, F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “System modeling for active mm-wave imaging systems using an enhanced calculation method,” in Proceedings of European Radar Conference, October 30-31, 2008, Amsterdam (2008) pp. 56-59.

F. Qi, I. Ocket, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Millimeter wave imaging: system modeling and phenomena discussion,” in Proceedings of 19th International Conference on Applied Electromagnetics and Communications, September 24-26, 2007, Dubrovnik (2007), pp. 283-286.

G. N. Sinclair, R. N. Anderton, and R. Appleby, “Outdoor passive millimeter wave security screening,” in Proceedings of IEEE 35th International Carnahan Conference on Security Technology (IEEE, 2001), pp. 172-179.

Z. Luo, J. Xiong, and J. Yang, “A model-based analyzing and calculating of the focal plane array for passive millimeter wave imaging system,” in Proceedings of International Conference on Circuits and Systems Proceedings, June 25-28, 2006, Guilin, China, pp. 629-632.
[CrossRef]

F. Qi, V. Tavakol, D. Schreurs, and B. Nauwelaers, “Inaccuracies in Fourier optics,” in Proceedings of URSI-Benelux Forum, June 8, 2009, Delft (2009).

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Figures (7)

Fig. 1
Fig. 1

Imaging system setup, including an object, a lens, and an image plane.

Fig. 2
Fig. 2

Original image as a benchmark.

Fig. 3
Fig. 3

Simulation results by using the two methods for a smooth object: (a) amplitude distribution by Fourier optics; (b) phase distribution by Fourier optics; (c) amplitude distribution by wave calculations; (d) phase distribution by wave calculations.

Fig. 4
Fig. 4

Comparison of the imaging results for both smooth and rough objects accompanied by incident fields on lens: (a) incident fields on lens in case of a smooth object; (b) differential image between imaging results by the two modeling approaches in the case of a smooth object; (c) incident fields on lens in the case of a rough object; (d) differential image between imaging results by the two modeling approaches in the case of a rough object.

Fig. 5
Fig. 5

Simulation results by using the two methods for the case of a rough surface: (a) amplitude distribution by Fourier optics; (b) phase distribution by Fourier optics; (c) amplitude distribution by wave calculations; (d) phase distribution by wave calculations.

Fig. 6
Fig. 6

Degradation of the PSF: (a) PSF for central point; (b) PSF for off-axis point.

Fig. 7
Fig. 7

Imaging the object with the same f-number, but in a more compact system with half the dimension of the lens and half the system scale: (a) amplitude distribution; (b) phase distribution.

Equations (17)

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

M = z 2 z 1 ,
ξ ̃ = M ξ , η ̃ = M η ,
x ̃ = x λ z 2 , y ̃ = y λ z 2 .
U i ( u , v ) = U g ( u , v ) h ( u , v ) ,
with U g ( u , v ) = 1 | M | U o ( u M , v M ) ,
h ( u , v ) = | M | P ( λ z 2 x ̃ , λ z 2 y ̃ ) exp [ j 2 π ( u x ̃ + v y ̃ ) ] d x ̃ d y ̃ ,
or h ( u , v ) = 1 λ 2 z 1 z 2 P ( x , y ) exp [ j 2 π λ z 2 ( u x + v y ) ] d x d y ,
G i ( f X , f Y ) = G g ( f X , f Y ) H ( f X , f Y ) ,
with H ( f X , f Y ) = | M | ( λ z 2 ) P ( λ z 2 f X , λ z 2 f Y ) P ( λ z 2 f X , λ z 2 f Y ) .
H ( f X , f Y ) = circ ( f X 2 + f Y 2 f o ) ,
with f o = w λ z 2 .
E ( P ) = 1 j λ A E ( P 1 ) e j k r r cos θ d s ,
U i ( u , v ) = exp ( j k z 1 ) j λ z 1 exp ( j k z 2 ) j λ z 2 exp [ j k 2 z 2 ( u 2 + v 2 ) ] × P ( x , y ) exp [ j k 2 ( x 2 + y 2 ) ( 1 z 1 + 1 z 2 1 f ) ] × { { U o ( ξ , η ) exp [ j k 2 z 1 ( ξ 2 + η 2 ) ] } exp [ j 2 π λ z 1 ( x ξ + y η ) ] d ξ d η } exp [ j 2 π λ z 2 ( u x + v y ) ] d x d y .
r = z 1 + ( x ξ z ) 2 + ( y η z ) 2 = z + ( x ξ ) 2 + ( y η ) 2 2 z [ ( x ξ ) 2 + ( y η ) 2 ] 2 8 z 3 + ,
U ( x , y ) = e j k z j λ z exp [ j k 2 z ( x 2 + y 2 ) ] { U ( ξ , η ) exp [ j k 2 z ( ξ 2 + η 2 ) ] } exp [ j 2 π λ z ( x ξ + y η ) ] d ξ d η .
θ = tan 1 ( l 2 Z ) .
R = π 2 π 2 | O ( α ) | 2 d α θ 2 θ 2 | O ( α ) | 2 d α 1 .

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