Abstract

We define the signal-relevant efficiency (SRE) of a diffractive optical element as a measure of the proportion of the incident field power that ends up in the desired output signal. An upper bound for SRE is determined in the presence of arbitrary constraints imposed on the element, such as phase-dependent loss due to absorption within the microstructure and quantization of the surface profile. We apply the theory to the important class of diffractive elements that contain only one desired diffraction order (such as diffractive lenses) and derive the surface profile that provides the highest efficiency allowed by the constraints.

© 2016 Optical Society of America

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References

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  1. F. Wyrowski, “Upper bound of the efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [Crossref]
  2. F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
    [Crossref]
  3. F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
    [Crossref]
  4. J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie-Verlag, 1997).
  5. U. Krackhardt, J. N. Mait, and N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
    [Crossref]
  6. G. Zhou, X. Yuan, P. Dowd, Y.-L. Lam, and Y.-C. Chan, “Efficient method for evaluation of the diffraction efficiency upper bound of diffractive phase elements,” Opt. Lett. 25, 1288–1290 (2000).
    [Crossref]
  7. L. A. Romero and F. M. Dickey, “Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings,” J. Opt. Soc. Am. A 24, 2280–2295 (2007).
    [Crossref]
  8. In Ref. [7] it is claimed that the upper bound formula presented in Ref. [1] is incorrect. However, this claim is wrong and the dimensions in Eq. (19) of Ref. [1] are correct. Moreover, the mathematical form of the definition given by Eq. (19) in Ref. [1] ensures that the upper bound of efficiency is always less than 100%.
  9. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003).
    [Crossref]
  10. H. Aagedal, “Simulation und Design paraxialer diffraktiver Systeme,” Ph.D. thesis (Universität Karlsruhe, 1998).
  11. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [Crossref]
  12. J. W. Goodman and A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
    [Crossref]

2007 (1)

2003 (1)

2000 (1)

1993 (1)

1992 (2)

1991 (1)

1970 (2)

J. W. Goodman and A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[Crossref]

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
[Crossref]

Aagedal, H.

H. Aagedal, “Simulation und Design paraxialer diffraktiver Systeme,” Ph.D. thesis (Universität Karlsruhe, 1998).

Chan, Y.-C.

Collins, S. A.

Dickey, F. M.

Dowd, P.

Goodman, J. W.

J. W. Goodman and A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[Crossref]

Honkanen, M.

Kettunen, V.

Krackhardt, U.

Lam, Y.-L.

Mait, J. N.

Romero, L. A.

Silvestri, A. M.

J. W. Goodman and A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[Crossref]

Streibl, N.

Tervo, J.

Turunen, J.

Wyrowski, F.

Yuan, X.

Zhou, G.

Appl. Opt. (1)

IBM J. Res. Dev. (1)

J. W. Goodman and A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
[Crossref]

Opt. Lett. (2)

Other (3)

In Ref. [7] it is claimed that the upper bound formula presented in Ref. [1] is incorrect. However, this claim is wrong and the dimensions in Eq. (19) of Ref. [1] are correct. Moreover, the mathematical form of the definition given by Eq. (19) in Ref. [1] ensures that the upper bound of efficiency is always less than 100%.

J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie-Verlag, 1997).

H. Aagedal, “Simulation und Design paraxialer diffraktiver Systeme,” Ph.D. thesis (Universität Karlsruhe, 1998).

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Figures (10)

Fig. 1.
Fig. 1. General setup under consideration. The field incident on the thin element with transmission function t ( x , y ) is labeled as U in ( x , y ) . The field U out ( x ˜ , y ˜ ) at the output plane of the system, where the desired field is defined only within a finite region W (the signal window), can be obtained through a linear operator L .
Fig. 2.
Fig. 2. Cross section in Hilbert space that contains the field after the grating and the desired field, represented as vectors. The projection of the output field onto the desired field is given by the normalized inner product between the two, and the size of this projection represents the SRE. The gray area shows all possible output fields, and of those the field U opt yields the largest projection upon U desired .
Fig. 3.
Fig. 3. Left: the point that gives the largest projection can be obtained by letting the length of U desired approach infinity and finding the point in the allowed distribution closest to it. Right: a property of the Hilbert space is that the projection is not influenced by applying a linear operation [Eq. (9)]. Finding the SRE upper bound is therefore equivalent to finding the U in t , t A c that lies closest to the inverse of the desired distribution located at lim r L 1 { r U desired } .
Fig. 4.
Fig. 4. Flowchart describes how to obtain the upper bound for the SRE.
Fig. 5.
Fig. 5. Representation of the transmission constraint A c for elements with complex refractive index n = 1 + Δ n + i κ with (a)  κ / Δ n = 0.1 and (b)  κ / Δ n = 0.8 . The allowed transmittance values are located on the spirals.
Fig. 6.
Fig. 6. Determination of t opt ( x ) for ( m ¯ , n ¯ ) = ( 1,0 ) . (a) The constraint A c under consideration with κ / Δ n = 0.2 . (b) Finding the value of t opt by search of the point closest to A c for a single point x = 0 in the circle r exp ( i 2 π x ) , r : the black line connects these two points. (c) Values of t opt ( x ) constructed for all x . (d) The resulting phase profile ϕ opt ( x ) used to define t opt ( x ) by inserting into Eq. (29).
Fig. 7.
Fig. 7. Designs with different types of constraints. Left: phase only. Middle: quantized phase. Right: binary-amplitude. Top row: retrieval of t opt . Bottom row: resulting phase/amplitude profiles.
Fig. 8.
Fig. 8. Phase-only grating with various quantized phase levels Z . The dots denote the efficiencies computed by the SRE theory for the indicated number of phase levels. The line is given by Eq. (36).
Fig. 9.
Fig. 9. Effect of absorption in the efficiency of gratings with one-point signal. Compares the computed profile obtained by SRE theory with the gratings with standard triangular surface profile and binary amplitude gratings.
Fig. 10.
Fig. 10. Phase and amplitude transmission profiles of DOE from SRE theory for selected levels of absorption.

Equations (36)

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U t ( x , y ) = t ( x , y ) U in ( x , y ) ,
t ( x , y ) = exp [ i 2 π ( n ^ 1 ) λ h ( x , y ) ] .
U out ( x ˜ , y ˜ ) = L { U t ( x , y ) } .
U 1 | U 2 := R 2 U 1 ( x ) U 2 ( x ) ¯ d x ,
U = U | U .
L 2 C ( R 2 ) = U desired H error H freedom
U out = α U desired + U error + U freedom ,
α = U out | U desired U desired | U desired
= U t | L 1 U desired U desired | U desired .
η SRE := α U desired 2 U in 2
= | U out | U desired | 2 U in 2 U desired 2 .
U out U in 2 = U desired U desired 2 .
η SRE max := | U opt | U desired | 2 U in 2 U desired 2
= | L { U in t opt } | U desired | 2 U in 2 U desired 2 .
lim r r U desired .
t opt := argmax t A c | L { U in t opt } | U desired | 2 U in 2 U desired 2
= argmin t A c lim r L { U in t } r U desired 2 .
t opt = argmin t A c lim r t r L 1 { U desired } U in 2 .
t ideal := L 1 { U desired } U in ,
t opt = argmin t A c lim r t r t ideal 2 .
t ideal = F 1 { U desired } , t opt = e i Arg ( t ideal ) .
η SRE max = | e i Arg ( t ideal ) | t ideal | 2 t ideal 2
= | | t ideal | | 2 t ideal 2 ,
n ^ ( x , y ) = { n ^ 0 z h ( x , y ) 1 else ,
n ^ = 1 + Δ n + i κ .
t ( x , y ) = | t ( x , y ) | exp [ i ϕ ( x , y ) ] ,
| t ( x , y ) | = exp [ ( 2 π / λ ) κ h ( x , y ) ] ,
ϕ ( x , y ) = ( 2 π / λ ) Δ n h ( x , y ) ,
A c ( ϕ ) = exp [ ( κ / Δ n ) ϕ ] .
U out ( m , n ) = t ( x , y ) e i 2 π ( m x + n y ) d x d y
U t ( x , y ) = t ( x , y ) = ( m , n ) = U out ( m , n ) e i 2 π ( m x + n y ) .
η ( m , n ) = | U out ( m , n ) | 2 .
U desired ( m , n ) = δ m m ¯ , n n ¯ ,
η SRE max = | t opt ( x , y ) | F 1 { δ m m ¯ , n n ¯ } | 2 δ m m ¯ , n n ¯ 2 = | t opt ( x , y ) | e i 2 π ( m ¯ x + n ¯ y ) | 2 .
t opt = argmin t A c lim r t r e i 2 π ( m ¯ x + n ¯ y ) 2 .
η quant sinc ( 1 / Z ) 2 × η ,

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