Abstract

The range of validity and the accuracy of scalar diffraction theory for periodic diffractive phase elements (DPE’s) is evaluated by a comparison of diffraction efficiencies predicted from scalar theory to exact results calculated with a rigorous electromagnetic theory. The effects of DPE parameters (depth, feature size, period, index of refraction, angle of incidence, fill factor, and number of binary levels) on the accuracy of scalar diffraction theory is determined. It is found that, in general, the error of scalar theory is significant ( > ±5%) when the feature size is less than 14 wavelengths (s < 14λ). The error is minimized when the fill factor approaches 50%, even for small feature sizes (s = 2λ); for elements with an overall fill factor of 50% the larger period of the DPE replaces the smaller feature size as the condition of validity for scalar diffraction theory. For an 8-level DPE of refractive index 1.5 analyzed at normal incidence the error of the scalar analysis is greater than ±5% when the period is less than 20 wavelengths (Λ < 20λ). The accuracy of the scalar treatment degrades as either the index of refraction, the depth, the number of binary levels, or the angle of incidence is increased. The conclusions are, in general, applicable to nonperiodic as well as other periodic (trapezoidal, two-dimensional) structures.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 402.
  2. D. A. Gremaux, N. C. Gallagher, “Limits of scalar diffraction theory for conducting gratings,” Appl. Opt. 32, 1948–1953 (1993).
    [CrossRef] [PubMed]
  3. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.
  5. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), p. 15.
  6. M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 184.
  7. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]

1993 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1982 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 402.

Gallagher, N. C.

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

Gremaux, D. A.

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 184.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 402.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (4)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 402.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), p. 15.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 184.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Single-level, asymmetric periodic diffractive phase element (DPE) with depth d, period Λ, fill factor f, and refractive index n2. A plane wave is incident at angle θ0.

Fig. 2
Fig. 2

Modeling a DPE by an infinitely thin, periodic phase mask. The phase difference between bins is characterized by Δϕ.

Fig. 3
Fig. 3

First-order diffraction efficiencies versus normalized depth at normal incidence with n2 = 1.5 and fixed feature size (s = 5λ). Rigorous diffraction efficiencies are for two complementary fill factors (f = 0.25 and 0.75). The vertical dashed line denotes the optimal depth dopt.

Fig. 4
Fig. 4

Error versus normalized depth for three refractive indices (n2 = 1.5, 2.0, and 4.0) at normal incidence, s = 5λ, and f = 0.25. The optimal depths for the refractive indices are denoted with the vertical dashed lines. The optimal depth decreases as the index of refraction increases.

Fig. 5
Fig. 5

% Error versus angle of incidence for a high refractive index (n2 = 4.0), complementary fill factors [f = 0.25 (open symbols) and f = 0.75 (solid symbols)], and varying feature sizes (s = 5λ and 15λ).

Fig. 6
Fig. 6

% Error versus angle of incidence for n2 = 1.5, complementary fill factors (f = 0.25 and 0.75), and varying feature sizes (s = 5λ and 15λ).

Fig. 7
Fig. 7

% Error versus fill factor at normal incidence with n2 = 1.5. The vertical dashed line represents a symmetric DPE (f = 0.5).

Fig. 8
Fig. 8

System geometry for a multilevel binary grating with depth d, period Λ, N binary levels, and refractive index n2. Note that the effective feature size of the DPE is Λ/N.

Fig. 9
Fig. 9

Decomposing an N-level profile into N − 1 complementary slices. All the slices have the identical properties (n1, n2, and Λ) of the composite multilevel DPE, except that the depth of each slice [d/(N − 1)] is reduced.

Fig. 10
Fig. 10

4-level DPE reduced to three, complementary slices.

Fig. 11
Fig. 11

% Error versus normalized feature size for the three single-level slices (marked with symbols) and for the 4-level profile that they approximate (dashed curve). The data are calculated at normal incidence with an index of refraction of n2 = 1.5.

Fig. 12
Fig. 12

% Error versus normalized grating period at normal incidence for n2 = 4.0 and a varying number of binary levels (N = 2, 4, 8, and 16).

Fig. 13
Fig. 13

% Error versus normalized grating period at normal incidence for n2 = 1.5 and a varying number of binary levels (N = 2, 4, 8, and 16).

Fig. 14
Fig. 14

% Error versus angle of incidence for n2 = 4.0 and a varying number of grating periods (Λ = 5λ, 10λ, 15λ, and 25λ).

Fig. 15
Fig. 15

% Error versus angle of incidence for n2 = 1.5 and a varying number of grating periods (Λ = 5λ, 10λ, 15λ, and 25λ)

Fig. 16
Fig. 16

System geometry for the derivation of the phase functions for general DPE’s of depth d and refractive index n2. In region I two parallel rays (A and B) are incident on the element at angle θ0. θ0 is the angle in the medium, and P1 and P2 are planes of equal phase in regions I and III, respectively.

Equations (16)

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

s = { ( 1 f ) Λ f Λ s 0.5 s < 0.5 .
Δ ϕ = k 0 d ( n 2 cos θ 0 n 1 cos θ 0 ) ,
η p = τ 2 ( θ 0 ) 4 f 2 Sinc 2 ( f p ) cos 2 ( 1 2 k 0 Δ ϕ + π 2 p ) ,
d opt = { 1 2 λ ( n 2 cos θ 0 n 1 cos θ 0 ) λ ( n 2 cos θ 0 n 1 cos θ 0 ) p odd p even .
η p = τ 2 ( θ 0 ) 4 f 2 Sinc 2 ( p f ) .
% Error = η RCWT η ST η RCWT × 100 % .
η p = τ 2 ( θ 0 ) Sinc 2 ( p N ) ,
d opt = ( N 1 N ) p λ 0 n 2 cos θ 0 n 1 cos θ 0
Δ ϕ = k 0 d ( n 2 cos θ 0 n 1 tan θ 0 sin θ 0 + n 1 cos θ 0 n 2 tan θ 0 sin θ 0 ) = k 0 d ( n 2 cos θ 0 n 1 cos θ 0 ) .
t uc ( x ) = τ ( θ 0 ) [ i = 1 2 rect ( x w i Λ ) exp ( j i Δ ϕ ) δ ( x s i Λ ) ] ,
t ( x ) = t uc ( x ) p = δ ( x p Λ ) .
T ( f x ) = τ ( θ 0 ) { i = 1 2 w i Λ Sinc ( w i Λ f x ) × exp [ j ( i Δ ϕ + 2 π s i Λ f x ) ] } 1 Λ p = δ ( f x p Λ ) .
U + ( f x ) = τ ( θ 0 ) { i = 1 2 w i Sinc ( w i p ) exp [ j ( i Δ ϕ + 2 π s i p ) ] } × p = δ ( f x + k 0 n 1 sin θ 0 P Λ ) .
η p = τ 2 ( θ 0 ) | i = 1 2 w i Sinc ( w i p ) × exp [ j i ( k 0 d ( n 1 cos θ 0 n 2 cos θ 0 ) + p π ] | 2 .
Δ ϕ = k 0 d N 1 ( n 2 cos θ 0 n 1 cos θ 0 ) .
η p = τ 2 ( θ 0 ) Sinc 2 ( p N ) 1 N 2 | i = 1 N exp { j i [ k 0 d N 1 ( n 1 cos θ 0 n 2 cos θ 0 ) + p 2 π N ] } | 2 .

Metrics