Abstract

In this paper, we present the design of a novel spectral synthesis system that is based on an array of in-plane moving micro gratings, instead of using a conventional out-of-plane micromirror array. Utilizing the unique nondispersive characteristic of the optical phase modulation mechanism based on in-plane movable micro gratings, we demonstrate that the synthetic spectra problem can be greatly simplified and effectively reduced to that of conventional phase retrieval.

©2008 Optical Society of America

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References

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  1. M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, “Synthetic infrared spectra,” Opt. Lett. 22, 1036–1038 (1997).
    [Crossref] [PubMed]
  2. M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, “Synthetic spectra: a tool for correlation spectroscopy,” Appl. Opt. 36, 3342–3348 (1997).
    [Crossref] [PubMed]
  3. T. A. Leskova and A. A. Maradudin, “Synthetic spectra from rough-surface scattering,” Opt. Lett. 30, 2784–2786 (2005).
    [Crossref] [PubMed]
  4. M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express 14, 12590–12612 (2006).
    [Crossref] [PubMed]
  5. H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
    [Crossref]
  6. M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
    [Crossref]
  7. R. Belikov and O. Solgaard, “Optical wavelength filtering by diffraction from a surface relief,” Opt. Lett. 28, 447–449 (2003).
    [Crossref] [PubMed]
  8. G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express 11, 1392–1399 (2003).
    [Crossref] [PubMed]
  9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  10. G. Zhou and F. S. Chau, “Nondispersive optical phase shifter array using microelectromechanical systems based gratings,” Opt. Express 15, 10958–10963 (2007).
    [Crossref] [PubMed]
  11. J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63–90.
  12. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. 30, 304–308 (1979).
    [Crossref]
  13. M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
    [Crossref] [PubMed]
  14. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  15. F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography” J. Opt. Soc. Am. A5, 1058–1065 (1988).
    [Crossref]
  16. J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
  17. G. Zhou, Y. Chen, Z. Wang, and H. Song, “Genetic local search algorithm for optimization design of diffractive optical elements,” Appl. Opt. 38, 4281–4290 (1999).
    [Crossref]
  18. T. J. Suleski and D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers,” Appl. Opt. 34, 7507–7517 (1995).
    [Crossref] [PubMed]
  19. G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
  20. M. A. Wolfe, Numerical Methods for Unconstrained Optimization: an Introduction, (Van Nostrand Reinhold Company Ltd., New York, 1978), Chap. 6, 161–167.
  21. B. Kress and P. Meyrueis, Digital diffractive optics: an introduction to planar diffractive optics and related technology, (John Wiley & Sons. Ltd, New York, 2000), Chap. 1, 35–42.

2006 (1)

2005 (2)

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
[Crossref]

T. A. Leskova and A. A. Maradudin, “Synthetic spectra from rough-surface scattering,” Opt. Lett. 30, 2784–2786 (2005).
[Crossref] [PubMed]

2004 (1)

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
[Crossref]

2003 (2)

1999 (1)

1997 (3)

1995 (1)

1989 (2)

J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

1988 (1)

F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography” J. Opt. Soc. Am. A5, 1058–1065 (1988).
[Crossref]

1982 (1)

1979 (1)

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. 30, 304–308 (1979).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Belikov, R.

Bruck, Y. M.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. 30, 304–308 (1979).
[Crossref]

Bryngdahl, O.

F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography” J. Opt. Soc. Am. A5, 1058–1065 (1988).
[Crossref]

Butler, M. A.

Carhart, G. W.

Chau, F. S.

G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express 11, 1392–1399 (2003).
[Crossref] [PubMed]

G. Zhou and F. S. Chau, “Nondispersive optical phase shifter array using microelectromechanical systems based gratings,” Opt. Express 15, 10958–10963 (2007).
[Crossref] [PubMed]

Chen, Y.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63–90.

Johansen, I. R.

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
[Crossref]

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
[Crossref]

Kravitz, S. H.

Kress, B.

B. Kress and P. Meyrueis, Digital diffractive optics: an introduction to planar diffractive optics and related technology, (John Wiley & Sons. Ltd, New York, 2000), Chap. 1, 35–42.

Lacolle, M.

M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express 14, 12590–12612 (2006).
[Crossref] [PubMed]

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
[Crossref]

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
[Crossref]

Leskova, T. A.

Løvhaugen, O.

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
[Crossref]

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
[Crossref]

Maradudin, A. A.

Meyrueis, P.

B. Kress and P. Meyrueis, Digital diffractive optics: an introduction to planar diffractive optics and related technology, (John Wiley & Sons. Ltd, New York, 2000), Chap. 1, 35–42.

O’Shea, D. C.

Ricco, A. J.

Ricklin, J. C.

Sagberg, H.

M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express 14, 12590–12612 (2006).
[Crossref] [PubMed]

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
[Crossref]

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Senturia, S. D.

Sinclair, M. B.

Sodin, L. G.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. 30, 304–308 (1979).
[Crossref]

Solgaard, O.

M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express 14, 12590–12612 (2006).
[Crossref] [PubMed]

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
[Crossref]

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
[Crossref]

R. Belikov and O. Solgaard, “Optical wavelength filtering by diffraction from a surface relief,” Opt. Lett. 28, 447–449 (2003).
[Crossref] [PubMed]

Song, H.

Sudbø, A. S.

M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express 14, 12590–12612 (2006).
[Crossref] [PubMed]

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
[Crossref]

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
[Crossref]

Suleski, T. J.

Swanson, G. J.

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

Tay, F. E. H.

Turunen, J.

J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).

Vasara, A.

J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).

Veldkamp, W. B.

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

Vorontsov, M. A.

Wang, Z.

Westerholm, J.

J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).

Wolfe, M. A.

M. A. Wolfe, Numerical Methods for Unconstrained Optimization: an Introduction, (Van Nostrand Reinhold Company Ltd., New York, 1978), Chap. 6, 161–167.

Wyrowski, F.

F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography” J. Opt. Soc. Am. A5, 1058–1065 (1988).
[Crossref]

Zhou, G.

Zubrzycki, W. J.

Appl. Opt. (4)

IEEE J. Sel. Top. Quantum. Electron. (1)

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005).
[Crossref]

J. Opt. Soc. Am. (1)

F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography” J. Opt. Soc. Am. A5, 1058–1065 (1988).
[Crossref]

Opt. Comm. (1)

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. 30, 304–308 (1979).
[Crossref]

Opt. Eng. (2)

J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

Opt. Express (2)

Opt. Lett. (4)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other (4)

M. A. Wolfe, Numerical Methods for Unconstrained Optimization: an Introduction, (Van Nostrand Reinhold Company Ltd., New York, 1978), Chap. 6, 161–167.

B. Kress and P. Meyrueis, Digital diffractive optics: an introduction to planar diffractive optics and related technology, (John Wiley & Sons. Ltd, New York, 2000), Chap. 1, 35–42.

G. Zhou and F. S. Chau, “Nondispersive optical phase shifter array using microelectromechanical systems based gratings,” Opt. Express 15, 10958–10963 (2007).
[Crossref] [PubMed]

J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63–90.

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Figures (9)

Fig. 1.
Fig. 1. Spectral synthesis using a programmable micromirror array.

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Fig. 2.
Fig. 2. (a) 3D view and (b) top view of a spectral synthesis system using a micro grating array.

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Fig. 3.
Fig. 3. Spectral resolution and bandwidth vs. number of grating elements and element width.

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Fig. 4.
Fig. 4. (a) Schematic showing grating elements in the array’s local coordinate system (b) shift of the sinc function values using an auxiliary grating profile.

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Fig. 5.
Fig. 5. Block diagram showing the proposed gradient search method for spectral synthesis.

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Fig. 6.
Fig. 6. Alternative spectral synthesis system using in-plane moving gratings.

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Fig. 7.
Fig. 7. (a) Targeted intensity spectrum of the first design example (b) calculated intensity spectrum of the outgoing light wave from the optimized spectral synthesis system.

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Fig. 8.
Fig. 8. (a) Targeted intensity spectrum of the second design example (b) calculated intensity spectrum of the outgoing light wave from the optimized spectral synthesis system.

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Fig. 9.
Fig. 9. Calculated intensity spectrum of the outgoing light wave from the optimized spectral synthesis system with 1024 grating elements for the desired spectrum shown in Fig. 8(a).

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Equations (29)

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U ( u ) = C ( u ) ∑ m = 1 M exp ⁡ ( − i 4 π d m u ) exp ⁡ [ − i 2 π m w sin ⁡ θ u ] ,
U in ( u ) = U 0 ( u ) exp ⁡ [ i 2 π u ( sin ⁡ θ i x − cos ⁡ θ i y ) ] ,
exp ⁡ [ i 2 π u h ( x ) ] = ∑ k C k ( u ) exp ⁡ [ i 2 π k x p ] ,
U 1 ( u ) = U 0 ( u ) η ( u ) exp ⁡ { i 2 π [ ( u sin ⁡ θ i − 1 p ) x + u cos ⁡ θ d y ] } ,
U 2 , m ( u ) = U 0 ( u ) η ( u ) exp ⁡ { i 2 π [ − ( u sin ⁡ θ i − 1 p ) x ′ + u cos ⁡ θ d l ] } exp ⁡ [ i 2 π u h ( x ′ − d m ) ] ,
exp ⁡ [ i 2 π u h ( x ′ − d m ) ] = ∑ k C k ( u ) exp ⁡ [ i 2 π k ( x ′ − d m ) p ] ,
U 2 , m ( u ) = U 0 ( u ) η 2 ( u ) exp ⁡ ( − i 2 π u sin ⁡ θ i x ′ ) exp ⁡ ( i 2 π u cos ⁡ θ d l ) exp ⁡ ( i 2 π d m p ) .
U 2 ( u ) = U 0 ( u ) η 2 ( u ) exp ⁡ ( − i 2 π u sin ⁡ θ i x ′ ) exp ⁡ ( i 2 π u cos ⁡ θ d l ) ∑ m = − M M exp ⁡ ( i 2 π d m p ) rect ( z ′ − m Δ Δ ) .
U 3 ( u ) = A u ∫ − ∞ ∞ ∫ − ∞ ∞ U 2 ( u ) exp ⁡ [ − i 2 π u ( sin ⁡ α z ′ − sin ⁡ θ i x ′ ) ] d x ′ d z ′ ,
U 3 ( u ) = Δ ABU 0 ( u ) u η 2 ( u ) sinc ( u Δ sin α ) exp ⁡ ( i 2 π u cos ⁡ θ d l )
× ∑ m = − M M exp ⁡ ( i 2 π d m p ) exp ⁡ [ − i 2 π m u Δ sin ⁡ α ] ,
u n = n ( 2 M + 1 ) Δ sin ⁡ α + l Δ sin ⁡ α ,
ϕ m = 2 π d m p ,
U ( u ) = U 3 ( u ) Δ ABU 0 ( u ) u η 2 ( u ) sinc ( u Δ sin α ) .
I n = ∣ U ( u n ) ∣ 2 = ∣ ∑ m = − M M exp ⁡ ( i ϕ m ) exp ⁡ ( − i 2 π n m 2 M + 1 ) ∣ 2 = ∣ DFT { exp ⁡ ( i ϕ m ) } ∣ 2 ,
Δ = 1 ( u max − u min ) sin α ,
( 2 M + 1 ) Δ = 1 δ u sin ⁡ α .
∑ n = − M M ∣ U ( u n ) ∣ 2 = ( 2 M + 1 ) 2 .
exp ⁡ [ i 2 π u h ( x ′ − d m ) ] exp ⁡ [ i 2 π u f ( z ′ − m Δ ) ] = { ∑ k C k ( u ) exp ⁡ [ i 2 π k ( x ′ − d m ) p ] }
× { ∑ k D k ( u ) exp ⁡ [ i 2 π k ( z ′ − m Δ ) g ] } ,
U 3 ( u ) = Δ ABU 0 ( u ) u η 2 ( u ) ξ ( u ) sinc [ Δ ( u sin α − 1 g ) ] exp ⁡ ( i 2 π u cos θ d l )
× ∑ m = − M M exp ⁡ ( i 2 π d m p ) exp ⁡ [ − i 2 π m u Δ sin ⁡ α ] ,
U ( u ) = U 3 ( u ) Δ ABU 0 ( u ) u η 2 ( u ) ξ ( u ) sinc [ Δ ( u sin α − 1 g ) ] .
u 0 sin ⁡ α = 1 g ,
E = ∑ n = − M M ( I n d − γ I n ) 2 ,
γ = ∑ n = − M M I n I n d ∑ n = − M M I n 2 .
min ϕ m { E ( ϕ m ) } and 0 ≤ ϕ m < 2 π .
d E d ϕ m = 2 Im [ exp ⁡ ( − i ϕ m ) · ∑ k = − M M 2 γ ( γ I k − I k d ) U ( u k ) exp ⁡ ( i 2 π k m 2 M + 1 ) ] ,
ζ = I ( u p ) I 0 ( u p ) × 100 % ,

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