Abstract

In order to accurately analyze and design the transmittance characteristic of a diffraction phase grating, the validity of both the scalar diffraction theory and the effective medium theory is quantitatively evaluated by the comparison of diffraction efficiencies predicted from both simplified theories to exact results calculated by the rigorous vector electromagnetic theory. The effect of surface profile parameters, including the normalized period, the normalized depth, and the fill factor for the precision of the simplified methods is determined at normal incidence. It is found that, in general, when the normalized period is more than four wavelengths of the incident light, the scalar diffraction theory is useful to estimate the transmittance of the phase grating. When the fill factor approaches 0.5, the error of the scalar method is minimized, and the scalar theory is accurate even at the grating period of two wavelengths. The transmittance characteristic as a function of the normalized period is strongly influenced by the grating duty cycle, but the diffraction performance on the normalized depth is independent of the fill factor of the grating. Additionally, the effective medium theory is accurate for evaluating the diffraction efficiency within an error of less than around 1% when no higher-order diffraction waves appear and only the zero-order waves exist. The precision of the effective medium theory for calculating transmittance properties as a function of the normalized period, the normalized groove depth, and the polarization state of incident light is insensitive to the fill factor of the phase grating.

© 2011 Optical Society of America

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References

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2008 (2)

A. D. Papadopoulos and E. N. Glytsis, “Finite-difference-time-domain analysis of finite- number-of-periods holographic and surface-relief gratings,” Appl. Opt. 47, 1981–1994 (2008).
[CrossRef] [PubMed]

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

2003 (1)

2001 (1)

1997 (1)

1994 (2)

1993 (3)

1990 (1)

1982 (2)

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” J. Mod. Opt. 29, 289–312 (1982).
[CrossRef]

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

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Botten, L. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” J. Mod. Opt. 29, 289–312 (1982).
[CrossRef]

Brundrett, D. L.

Cowan, J. J.

Craig, M. S.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” J. Mod. Opt. 29, 289–312 (1982).
[CrossRef]

Fainman, Y.

Fan, Z. X.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Gallagher, N. C.

Gaylord, T. K.

Glytsis, E. N.

Grann, E. B.

Gremaux, D. A.

Haggans, C. W.

Jin, Y. X.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Li, L.

Li, L. F.

Liu, S. J.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Ma, J. Y.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (Institute of Physics, Bristol, 2001), pp. 41.

Maystre, D.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” J. Mod. Opt. 29, 289–312 (1982).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” J. Mod. Opt. 29, 289–312 (1982).
[CrossRef]

Moharam, M. G.

Nakagawa, W.

Nevière, M.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” J. Mod. Opt. 29, 289–312 (1982).
[CrossRef]

Papadopoulos, A. D.

Pommet, D. A.

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Shao, J. D.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Sun, P.-C.

Tyan, R.-C.

Xu, C.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Xu, F.

Appl. Opt. (3)

J. Mod. Opt. (1)

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” J. Mod. Opt. 29, 289–312 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (1)

H. A. Macleod, Thin Film Optical Filters (Institute of Physics, Bristol, 2001), pp. 41.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of diffraction phase grating. In the air region, two parallel rays (a and b) are incident on the element at an angle θ 0 . θ g is the diffraction angle. P 1 and P 2 are planes of equal phase in the regions of air and substrate, respectively. In our calculation of diffraction efficiencies across this paper, the incident angle is restricted to be zero.

Fig. 2
Fig. 2

Transmittances versus the normalized period at normal incidence for the duty cycle of 0.5 by using the FMM. (a) and (b): For TE polarization and TM polarization, respectively, with the normalized groove depth d / λ = 0.5 . (c) and (d): For TE polarization and TM polarization, respectively, with the normalized groove depth d / λ = 1.0 .

Fig. 3
Fig. 3

Comparison of diffraction efficiencies between the scalar method and FMM versus the normalized period for the varying fill factor. (a) and (b) represent the normalized groove depth, 0.5 λ , and (c) and (d) represent the normalized groove depth, 1.0 λ .

Fig. 4
Fig. 4

Comparison of transmittance characteristic between the scalar method and FMM for the zeroth order and ± 1 orders as a function of normalized groove depth at normal incidence. (a)–(c): For Λ / λ = 2.0 at the fill factors of 0.3, 0.5, and 0.7. (d)–(f): For Λ / λ = 5.0 at the fill factors of 0.3, 0.5, and 0.7. (g)–(i): For Λ / λ = 8.0 at the fill factors of 0.3, 0.5, and 0.7.

Fig. 5
Fig. 5

Comparison of transmittance characteristics of rigorous vector theory and the zeroth-order and second-order EMT with respect to the normalized period at normal incidence for the fill factors of 0.3, 0.5, and 0.7. (a)–(c): For TE polarization and the normalized groove depth 0.5. (d)–(f): For TM polarization and the normalized groove depth 0.5. (g)–(i): For TE polarization and the normalized groove depth 1.0. (j)–(l): For TM polarization and the normalized groove depth 1.0.

Fig. 6
Fig. 6

Transmittances of rigorous vector theory and the zeroth-order and second-order EMT as a function of normalized groove depth for the fill factor of 0.5. (a) and (b): For TE wave and TM wave, respectively.

Tables (1)

Tables Icon

Table 1 Three Kinds of Calculating Methods for Estimating the Diffraction Efficiency of the Phase Grating

Equations (11)

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t ( x ) = { ( n g cos θ g / n 0 cos θ 0 ) 1 / 2 τ ( θ 0 ) exp ( i φ a ) , 0 < x < f Λ ( n g cos θ g / n 0 cos θ 0 ) 1 / 2 τ ( θ 0 ) exp ( i φ b ) , f Λ < x < Λ ,
η ( λ ) = | 1 Λ 0 Λ t ( x ) exp ( 2 π i m x / Λ ) d x | 2 ,
η 0 = ( n g cos θ g / n 0 cos θ 0 ) τ 2 ( θ 0 ) [ 1 2 f ( 1 f ) ( 1 cos Δ φ ) ] ,
η m 0 = ( n g cos θ g / n 0 cos θ 0 ) τ 2 ( θ 0 ) [ ( 1 / m 2 π 2 ) ( 1 cos 2 m π f ) ( 1 cos Δ φ ) ] .
n TE ( 2 ) = [ ( n TE ( 0 ) ) 2 + 1 3 ( Λ λ ) 2 π 2 f 2 ( 1 f ) 2 ( n g 2 n 0 2 ) 2 ] 1 / 2 ,
n TM ( 2 ) = [ ( n TM ( 0 ) ) 2 + 1 3 ( Λ λ ) 2 π 2 f 2 ( 1 f ) 2 ( 1 n g 2 1 n 0 2 ) 2 ( n TM ( 0 ) ) 6 ( n TE ( 0 ) ) 2 ] 1 / 2 ,
n TE ( 0 ) = [ ( 1 f ) n 0 2 + f n g 2 ] 1 / 2 ,
n TM ( 0 ) = [ ( 1 f ) / n 0 2 + f / n g 2 ] 1 / 2 ,
[ B C ] = { q = 1 N [ cos δ q ( i sin δ q ) / η q i η q sin δ q cos δ q ] } [ 1 η s ] ,
n 0 sin θ 0 = n ( q ) sin θ q = n s sin θ s .
R = ( η 0 Y η 0 + Y ) ( η 0 Y η 0 + Y ) * ,

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