Abstract

Diffraction properties of dielectric surface-relief gratings were investigated by solving Maxwell’s equations numerically using the differential method. The diffraction efficiency of a grating with a groove depth about twice as deep as the grating period is comparable with the efficiency of a volume phase grating. Dielectric surface-relief gratings interferometrically recorded in photoresist can possess a high diffraction efficiency of up to 94% (throughput efficiency 85%). Calculated results were also found in good agreement with the experimental data.

© 1984 Optical Society of America

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References

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  1. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).
  2. M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 71, 811 (1981).
    [CrossRef]
  3. T. A. Shankoff, “Phase Holograms in Dichromated Gelatin,” Appl. Opt. 7, 2101 (1968).
    [CrossRef] [PubMed]
  4. T. Kubota, T. Ose, M. Sasaki, K. Honda, “Hologram Formation with Red Light in Methylene Blue Sensitized Dichromated Gelatin,” Appl. Opt. 15, 556 (1976).
    [CrossRef] [PubMed]
  5. A. Graube, “Advances in Bleaching Methods for Photographically Recorded Holograms,” Appl. Opt. 13, 2942 (1974).
    [CrossRef] [PubMed]
  6. R. Petit, Ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980).
    [CrossRef]
  7. M. G. Moharam, T. K. Gaylord, “Diffraction Analysis of Dielectric Surface-Relief Gratings,” J. Opt. Soc. Am. 72, 1385 (1982).
    [CrossRef]
  8. C. J. Kramer, “Holographic Laser Scanners for Nonimpact Printing,” Laser Focus 17, 70 (June1981).
  9. H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo-Opt. Instrum. Eng. 390, 174 (1983).
  10. R. C. Enger, S. K. Case, “High-Frequency Holographic Transmission Gratings in Photoresist,” J. Opt. Soc. Am. 73, 1113 (1983).
    [CrossRef]
  11. P. Vincent, in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer, Berlin, 1980), Chap. 4.
  12. M. A. Melkanoff, J. Raynal, T. Sawada, Methods in Computational Physics (Academic, New York, 1966).

1983

H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo-Opt. Instrum. Eng. 390, 174 (1983).

R. C. Enger, S. K. Case, “High-Frequency Holographic Transmission Gratings in Photoresist,” J. Opt. Soc. Am. 73, 1113 (1983).
[CrossRef]

1982

1981

M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 71, 811 (1981).
[CrossRef]

C. J. Kramer, “Holographic Laser Scanners for Nonimpact Printing,” Laser Focus 17, 70 (June1981).

1976

1974

1969

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

1968

Case, S. K.

Enger, R. C.

Funato, H.

H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo-Opt. Instrum. Eng. 390, 174 (1983).

Gaylord, T. K.

Graube, A.

Honda, K.

Kogelnik, H.

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

Kramer, C. J.

C. J. Kramer, “Holographic Laser Scanners for Nonimpact Printing,” Laser Focus 17, 70 (June1981).

Kubota, T.

Melkanoff, M. A.

M. A. Melkanoff, J. Raynal, T. Sawada, Methods in Computational Physics (Academic, New York, 1966).

Moharam, M. G.

Ose, T.

Raynal, J.

M. A. Melkanoff, J. Raynal, T. Sawada, Methods in Computational Physics (Academic, New York, 1966).

Sasaki, M.

Sawada, T.

M. A. Melkanoff, J. Raynal, T. Sawada, Methods in Computational Physics (Academic, New York, 1966).

Shankoff, T. A.

Vincent, P.

P. Vincent, in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer, Berlin, 1980), Chap. 4.

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

J. Opt. Soc. Am.

Laser Focus

C. J. Kramer, “Holographic Laser Scanners for Nonimpact Printing,” Laser Focus 17, 70 (June1981).

Proc. Soc. Photo-Opt. Instrum. Eng.

H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo-Opt. Instrum. Eng. 390, 174 (1983).

Other

P. Vincent, in Electromagnetic Theory of Gratings, R. Petit, Ed. (Springer, Berlin, 1980), Chap. 4.

M. A. Melkanoff, J. Raynal, T. Sawada, Methods in Computational Physics (Academic, New York, 1966).

R. Petit, Ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980).
[CrossRef]

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Figures (20)

Fig. 1
Fig. 1

Geometry of dielectric surface-relief grating. The power of the incident wave is P0, and that of the transmitted first-order diffracted wave is P1. The incident angle is θ.

Fig. 2
Fig. 2

Transmitted first-order diffraction efficiency curves as a function of λ/d for a sinusoidal surface-relief grating with n0 = 1.66 and h/d = 0.5 at the first Bragg angle: —, TE polarization; - - -, TM polarization.

Fig. 3
Fig. 3

Same as Fig. 2 except h/d = 1.0.

Fig. 4
Fig. 4

Same as Fig. 2 except h/d = 1.5.

Fig. 5
Fig. 5

Same as Fig. 2 except h/d = 2.0.

Fig. 6
Fig. 6

Diffraction efficiencies as a function of shape factor d1/d for triangular gratings with n0 = 1.66 and h/d = 1.0 at the first Bragg angle for TE polarization. λ/d = 0.7, 0.8, 0.9, 1.0, and 1.2.

Fig. 7
Fig. 7

Same as Fig. 6 except h/d = 1.5.

Fig. 8
Fig. 8

Effect of groove width w for rectangular surface-relief gratings in a TE polarization. Efficiency curves as a function of h/d for n0 = 1.66, λ/d = 1.414, and θ = 45° (first Bragg angle).

Fig. 9
Fig. 9

Effect of grating profile in TE polarization. First-order efficiency curves as a function of h/d are shown for isosceles triangular gratings —, sinusoidal gratings - - -, and rectangular gratings ·····, where n0 = 1.66, λ/d = 1.414, and θ = 45°.

Fig. 10
Fig. 10

Effect of the dielectric refractive index n0 for sinusoidal gratings in TE polarization. First-order efficiency curves as a function of h/d are shown for n0 = 1.66 —, 1.50 - - -, and 2.00 ····· where λ/d = 1.414 and θ = 45°.

Fig. 11
Fig. 11

Normalized diffraction efficiency as a function of angle of incidence for TE polarization. Sinusoidal surface relief grating with n0 = 1.66 and h/d = 1.9, —; sinusoidal volume phase gratings with the average refractive index 1.66 and the spatial modulation the refractive index n1 = 0.02, - - -, whewre λ/d = 1.414.

Fig. 12
Fig. 12

Normalized diffraction efficiency as a function of incident wavelength on a sinusoidal grating with d = 0.447 μm at the first Bragg angle and TE polarization. Surface-relief grating with n0 = 1.66 and h = 0.85 μm, —; volume phase grating with an average refractive index 1.66 and a spatial modulation of the refractive index n1 = 0.02, - - -.

Fig. 13
Fig. 13

Photoresist grating geometry on a glass plate. The incident power is I0, and the first-order transmitted power of the diffracted wave on a glass–air boundary is I1.

Fig. 14
Fig. 14

Electron micrograph of a cross section of a fabricated grating with d = 0.45 μm and h = 0.72 μm.

Fig. 15
Fig. 15

Top view of the same grating as in Fig. 14.

Fig. 16
Fig. 16

Theoretical and experimental curves of throughput efficiency as a function of incident angle and TE polarization. The grating profile consists of rectangular and semicircular sections, where d = 0.447 μm, h = 0.73 μm, λ = 0.633 μm, and n0 = 1.66.

Fig. 17
Fig. 17

Theoretical and experimental curves of throughput efficiency as a function of groove depth h and TE polarization, where d = 0.447 μm, λ = 0.633 μm, and n0 = 1.66.

Fig. 18
Fig. 18

Optimum angle of incidence for maximum first-order efficiency as a function of grating period d with two values of h/d and TE polarization. Grating profile is the same as in Fig. 16, where λ = 0.633 μm and n0 = 1.66. Theoretical curve is for h/d = 1.0, —, and 1.5, - - -, and compared with measured values ○. First Bragg angle locus, ·····, is also shown for comparison.

Fig. 19
Fig. 19

Diffraction efficiency of a sinusoidal surface-relief grating with h/d = 1.0 and d = 0.75 μm as a function of the incident angle and TE polarization, where λ = 0.633 μm and n0 = 1.66. First Bragg angle is 24.95°.

Fig. 20
Fig. 20

Comparison between theoretical and measured diffraction efficiency spectra of a sinusoidal grating with a 0.556-μm grating period and 0.8-μm groove depth at the first Bragg angle: —, TE polarization; - - -, TM polarization. Measured values: ●, TE polarization; ○, TM polarization.

Equations (15)

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Δ E z ( x , y ) + k 2 E z ( x , y ) = 0 , where k 2 = ( 2 π λ n ) 2 .
E z ( x , y ) = m = - E m ( y ) exp ( j α m X ) .
k 2 ( x , y ) = m = - ( k 2 ) m exp ( j m K x ) ,
d 2 d y 2 E m = α m 2 E m - p = - ( k 2 ) m - p · E p .
x [ 1 k 2 · H z ( x , y ) x ] + y [ 1 k 2 · H z ( x , y ) y ] + H z ( x , y ) = 0.
E ˜ ( x , y ) = 1 k 2 ( x , y ) · y H z ( x , y ) ,
y E ˜ ( x , y ) = - x [ 1 k 2 · x H z ( x , y ) ] - H z ( x , y ) .
1 k 2 ( x , y ) = m = - ( 1 k 2 ) m exp ( j m k x ) ,
H z ( x , y ) = m = - H m ( y ) exp ( j α m x ) ,
E ˜ ( x , y ) = m = - E ˜ m ( y ) exp ( j α m x ) .
d d y H m ( y ) = p = - ( k 2 ) m - p E p ( y ) ,
d d y E ˜ m ( y ) = α m p = - [ α p · ( 1 k 2 ) m - p · H p ( y ) ] - H m ( y ) .
η = ( diffracted wave power ) ( incident wave power ) .
θ B = sin - 1 ( λ / 2 d ) .
η t 1 = I 1 / I 0 ,

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