Abstract

Diffractive optical elements (DOEs) are often used in pattern formation for display purposes. Constructing these images from two or more colors greatly enhances their visual effect. To achieve this with DOEs is not simple, as they are inherently wavelength specific. We discuss an algorithm for designing quantized elements that produce distinct intensity patterns in the far field for two wavelengths. The benefits of applying bias phase to the dual-wavelength problem are investigated. The difference between the best and the worst choice of bias phase is shown to produce a variation of up to 2% in the efficiency. The mean square error can vary by up to a factor of 2 between the best and the worst case. It is also critically important to understand how the values of the two wavelengths affect the result. We present an analysis of how choosing different pairs of wavelengths in the design process affects the quality of our results.

© 2006 Optical Society of America

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References

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    [CrossRef]
  3. M. R. Taghizadeh and A. J. Waddie, "Micro-optical and optoelectronic components for optical interconnection applications" in Proceedings of the From Quantum Optics to Photonics Conference, Acta Phys. Pol.101, 175-187 (2002).
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    [CrossRef] [PubMed]
  6. J. R. Sze and M. H. Lu, "Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams," Opt. Eng. (Bellingham) 41, 3127-3135 (2002).
    [CrossRef]
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    [CrossRef]
  13. A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, "A novel algorithm for designing diffractive optical elements for 2 colour far-field pattern formation," J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
    [CrossRef]
  14. K. Balluder and M. R. Taghizadeh, "Optimized phase quantization for diffractive elements by use of bias phase," Opt. Lett. 24, 1756-1758 (1999).
    [CrossRef]
  15. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffractive plane pictures," Optik (Stuttgart) 35, 237-246 (1972).
  16. J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465 (2002).
    [CrossRef]
  17. M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
    [CrossRef]

2005

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, "A novel algorithm for designing diffractive optical elements for 2 colour far-field pattern formation," J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

2003

M. J. Thomson and M. R. Taghizadeh, "Diffractive elements for high-power fibre coupling applications," J. Mod. Opt. 50, 1691-1699 (2003).

H. Hua, Y. Ha, and J. P. Rolland, "Design of an ultralight and compact projection lens," Appl. Opt. 42, 97-107 (2003).
[CrossRef] [PubMed]

2002

J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465 (2002).
[CrossRef]

J. R. Sze and M. H. Lu, "Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams," Opt. Eng. (Bellingham) 41, 3127-3135 (2002).
[CrossRef]

2001

2000

M. Lo, B. Z. Dong, B. Y. Gu, and P. Meyrueis, "Non-periodic diffractive phase element for wavelength-division (de)multiplexing," Opt. Commun. 173, 217-221 (2000).
[CrossRef]

1999

1998

1997

M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
[CrossRef]

1995

1993

1978

1972

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffractive plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Balluder, K.

Barton, I. M.

I. M. Barton, P. Blair, and M. R. Taghizadeh, "Dual-wavelength operation diffractive phase elements for pattern formation," Opt. Express 1, 54-59 (1998).
[CrossRef]

M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
[CrossRef]

Bengtsson, J.

Blair, P.

I. M. Barton, P. Blair, and M. R. Taghizadeh, "Dual-wavelength operation diffractive phase elements for pattern formation," Opt. Express 1, 54-59 (1998).
[CrossRef]

M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
[CrossRef]

Caley, A. J.

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, "A novel algorithm for designing diffractive optical elements for 2 colour far-field pattern formation," J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

Dammann, H.

Dong, B. Z.

M. Lo, B. Z. Dong, B. Y. Gu, and P. Meyrueis, "Non-periodic diffractive phase element for wavelength-division (de)multiplexing," Opt. Commun. 173, 217-221 (2000).
[CrossRef]

Farn, M. W.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffractive plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Gu, B. Y.

M. Lo, B. Z. Dong, B. Y. Gu, and P. Meyrueis, "Non-periodic diffractive phase element for wavelength-division (de)multiplexing," Opt. Commun. 173, 217-221 (2000).
[CrossRef]

Ha, Y.

Hua, H.

Ichioka, Y.

Layet, B.

M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
[CrossRef]

Liu, J. S.

Lo, M.

M. Lo, B. Z. Dong, B. Y. Gu, and P. Meyrueis, "Non-periodic diffractive phase element for wavelength-division (de)multiplexing," Opt. Commun. 173, 217-221 (2000).
[CrossRef]

Lu, M. H.

J. R. Sze and M. H. Lu, "Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams," Opt. Eng. (Bellingham) 41, 3127-3135 (2002).
[CrossRef]

Medeiros, S. S.

Meyrueis, P.

M. Lo, B. Z. Dong, B. Y. Gu, and P. Meyrueis, "Non-periodic diffractive phase element for wavelength-division (de)multiplexing," Opt. Commun. 173, 217-221 (2000).
[CrossRef]

Ogura, Y.

Rolland, J. P.

Ross, N.

M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffractive plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Shirai, N.

Sommargren, Gary E.

Stern, M. B.

Sweeney, D. W.

Sze, J. R.

J. R. Sze and M. H. Lu, "Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams," Opt. Eng. (Bellingham) 41, 3127-3135 (2002).
[CrossRef]

Taghizadeh, M. R.

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, "A novel algorithm for designing diffractive optical elements for 2 colour far-field pattern formation," J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

M. J. Thomson and M. R. Taghizadeh, "Diffractive elements for high-power fibre coupling applications," J. Mod. Opt. 50, 1691-1699 (2003).

J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465 (2002).
[CrossRef]

K. Balluder and M. R. Taghizadeh, "Regenerative ring-laser design by use of an intracavity diffractive mode-selecting element," Appl. Opt. 38, 5768-5774 (1999).
[CrossRef]

K. Balluder and M. R. Taghizadeh, "Optimized phase quantization for diffractive elements by use of bias phase," Opt. Lett. 24, 1756-1758 (1999).
[CrossRef]

I. M. Barton, P. Blair, and M. R. Taghizadeh, "Dual-wavelength operation diffractive phase elements for pattern formation," Opt. Express 1, 54-59 (1998).
[CrossRef]

M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
[CrossRef]

M. R. Taghizadeh and A. J. Waddie, "Micro-optical and optoelectronic components for optical interconnection applications" in Proceedings of the From Quantum Optics to Photonics Conference, Acta Phys. Pol.101, 175-187 (2002).

Tanida, J.

Thomson, M. J.

M. J. Thomson and M. R. Taghizadeh, "Diffractive elements for high-power fibre coupling applications," J. Mod. Opt. 50, 1691-1699 (2003).

Veldkamp, W. B.

Waddie, A. J.

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, "A novel algorithm for designing diffractive optical elements for 2 colour far-field pattern formation," J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
[CrossRef]

M. R. Taghizadeh and A. J. Waddie, "Micro-optical and optoelectronic components for optical interconnection applications" in Proceedings of the From Quantum Optics to Photonics Conference, Acta Phys. Pol.101, 175-187 (2002).

Appl. Opt.

J. Mod. Opt.

M. J. Thomson and M. R. Taghizadeh, "Diffractive elements for high-power fibre coupling applications," J. Mod. Opt. 50, 1691-1699 (2003).

J. Opt. A, Pure Appl. Opt.

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, "A novel algorithm for designing diffractive optical elements for 2 colour far-field pattern formation," J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Microelectron. Eng.

M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, "Design and fabrication of diffractive optical elements," Microelectron. Eng. 34, 219-242 (1997).
[CrossRef]

Opt. Commun.

M. Lo, B. Z. Dong, B. Y. Gu, and P. Meyrueis, "Non-periodic diffractive phase element for wavelength-division (de)multiplexing," Opt. Commun. 173, 217-221 (2000).
[CrossRef]

Opt. Eng. (Bellingham)

J. R. Sze and M. H. Lu, "Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams," Opt. Eng. (Bellingham) 41, 3127-3135 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttgart)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffractive plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Other

M. R. Taghizadeh and A. J. Waddie, "Micro-optical and optoelectronic components for optical interconnection applications" in Proceedings of the From Quantum Optics to Photonics Conference, Acta Phys. Pol.101, 175-187 (2002).

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Figures (9)

Fig. 1
Fig. 1

DOE cross section with single-wavelength etch depths h. The option of adding h 1 to the etch depth, increasing the phase delay for λ 1 by 2 π , introduces a choice in the phase delay for λ 2 .

Fig. 2
Fig. 2

Two-color quantization. Numbers 1–8 show the available quantization levels. Open circles indicate etch depths that give a phase delay equivalent to Φ 1 . Solid circles indicate etch depths giving a phase delay equivalent to Φ 2 . Δ 1 and Δ 2 indicate the quantization error in terms of depth. The method tries to minimize the quantization error in terms of phase for both wavelengths.

Fig. 3
Fig. 3

Application of bias phase is equivalent to applying a shift of S 1 and S 2 to the original values of h 1 and h 2 , respectively. The original positions are indicated by the gray circles, the new positions by the open circles for λ 1 and black circles for λ 2 .

Fig. 4
Fig. 4

Desired intensity pattern.

Fig. 5
Fig. 5

The effect of applying different bulk shifts to the two desired phase profiles where the maximum λ 1 phase depth is 8 π . The four plots show variation in (a) efficiency for λ 1 , (b) efficiency for λ 2 , (c) MSE for λ 1 , and (d) MSE for λ 2 .

Fig. 6
Fig. 6

Effect of applying different bulk shifts to the two desired phase profiles where the maximum λ 1 phase depth is 6.8 π . The four plots show variation in (a) efficiency for λ 1 , (b) efficiency for λ 2 , (c) MSE for λ 1 , and (d) MSE for λ 2 .

Fig. 7
Fig. 7

Modeled results for optimizing the MSE.

Fig. 8
Fig. 8

Modeled results where the worst values of MSE are selected.

Fig. 9
Fig. 9

Results of applying the algorithm with no enhancement, bias phase, and depth bias on (a) nonuniformity for λ 1 , (b) nonuniformity for λ 2 , (c) efficiency for λ 1 , and (d) efficiency for λ 2 , for different values of α.

Equations (7)

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

Φ λ = 2 π ( n ( λ ) 1 ) h λ ,
2 π ( n ( λ 2 ) 1 ) λ 1 ( n ( λ 1 ) 1 ) λ 2 .
δ L , λ = ϕ L , λ e f f Φ λ ,
η = P F ( x , y ) E g ( X , Y ) ,
MSE = P G ( x , y ) F ( x , y ) 2 P F ( x , y ) 2 ,
Δ R = m a x F ( x , y ) G ( x , y ) G m a x ,
α = ( n ( λ 2 ) 1 ) λ 1 ( n ( λ 1 ) 1 ) λ 2 .

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