Abstract

We present a brief tutorial overview of the basic physics of periodically modulated index structures and how they affect an incident electromagnetic wave in various simple limits. Discussions of the wave-vector spectrum, coupled-amplitude equations, and finite grating-size effects are included. We also discuss ways of implementing modulated index structures.

© 1990 Optical Society of America

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References

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  1. A. W. Cook, “The reflection and transmission of light by any system of parallel isotropic films,” J. Opt. Soc. Am. 38, 954–964 (1948).
    [CrossRef]
  2. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955).
  3. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  4. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 8.
  5. T. Fukuzawa, M. Nakamura, “Mode coupling in thin-film chirped gratings,” Opt. Lett. 4, 343–345 (1979).
    [CrossRef] [PubMed]
  6. L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988), and references therein.
    [CrossRef]
  7. A. Sommerfeld, Optics (Academic, New York, 1964), pp. 180–183.
  8. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 19.
  9. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  10. D. Heitman, N. Kroo, C. Schultz, Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B 35, 2660–2666 (1987).
    [CrossRef]
  11. R. W. Gruhlke, D. G. Hall, “Band structure for dissimilar electromagnetic waves in a periodic structure,” Phys. Rev. B 40, 5367–5371 (1989).
    [CrossRef]
  12. V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, “k-gaps for surface polaritions on gratings,” Phys. Rev. B 37, 9089–9092 (1988).
    [CrossRef]
  13. T. G. Brown, University of Rochester, Rochester, New York 14627 (personal communication).
  14. W. R. Klein, W. D. Cook, “Unified approach to ultrasonic light deflection,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
    [CrossRef]
  15. N. S. N. Nath, “The diffraction of light by supersonic waves,” Proc. Indian Acad. Sci. Sect. A 8, 499–503 (1938).
  16. M. G. Moharam, L. Young, “Criterion for Bragg and Raman-Nath diffraction regimes,” Appl. Opt. 17, 1757–1759 (1978).
    [CrossRef] [PubMed]
  17. H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1985), p. 103.

1989 (1)

R. W. Gruhlke, D. G. Hall, “Band structure for dissimilar electromagnetic waves in a periodic structure,” Phys. Rev. B 40, 5367–5371 (1989).
[CrossRef]

1988 (2)

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, “k-gaps for surface polaritions on gratings,” Phys. Rev. B 37, 9089–9092 (1988).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988), and references therein.
[CrossRef]

1987 (1)

D. Heitman, N. Kroo, C. Schultz, Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B 35, 2660–2666 (1987).
[CrossRef]

1979 (1)

1978 (1)

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

1967 (1)

W. R. Klein, W. D. Cook, “Unified approach to ultrasonic light deflection,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

1948 (1)

1938 (1)

N. S. N. Nath, “The diffraction of light by supersonic waves,” Proc. Indian Acad. Sci. Sect. A 8, 499–503 (1938).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 8.

Brown, T. G.

T. G. Brown, University of Rochester, Rochester, New York 14627 (personal communication).

Celli, V.

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, “k-gaps for surface polaritions on gratings,” Phys. Rev. B 37, 9089–9092 (1988).
[CrossRef]

Cook, A. W.

Cook, W. D.

W. R. Klein, W. D. Cook, “Unified approach to ultrasonic light deflection,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Fukuzawa, T.

Gruhlke, R. W.

R. W. Gruhlke, D. G. Hall, “Band structure for dissimilar electromagnetic waves in a periodic structure,” Phys. Rev. B 40, 5367–5371 (1989).
[CrossRef]

Hall, D. G.

R. W. Gruhlke, D. G. Hall, “Band structure for dissimilar electromagnetic waves in a periodic structure,” Phys. Rev. B 40, 5367–5371 (1989).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988), and references therein.
[CrossRef]

Haruna, M.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1985), p. 103.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955).

Heitman, D.

D. Heitman, N. Kroo, C. Schultz, Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B 35, 2660–2666 (1987).
[CrossRef]

Klein, W. R.

W. R. Klein, W. D. Cook, “Unified approach to ultrasonic light deflection,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Kroo, N.

D. Heitman, N. Kroo, C. Schultz, Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B 35, 2660–2666 (1987).
[CrossRef]

Maradudin, A. A.

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, “k-gaps for surface polaritions on gratings,” Phys. Rev. B 37, 9089–9092 (1988).
[CrossRef]

Mills, D. L.

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, “k-gaps for surface polaritions on gratings,” Phys. Rev. B 37, 9089–9092 (1988).
[CrossRef]

Moharam, M. G.

Nakamura, M.

Nath, N. S. N.

N. S. N. Nath, “The diffraction of light by supersonic waves,” Proc. Indian Acad. Sci. Sect. A 8, 499–503 (1938).

Nishihara, H.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1985), p. 103.

Schultz, C.

D. Heitman, N. Kroo, C. Schultz, Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B 35, 2660–2666 (1987).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1964), pp. 180–183.

Suhara, T.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1985), p. 103.

Szentirmay, Z.

D. Heitman, N. Kroo, C. Schultz, Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B 35, 2660–2666 (1987).
[CrossRef]

Tran, P.

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, “k-gaps for surface polaritions on gratings,” Phys. Rev. B 37, 9089–9092 (1988).
[CrossRef]

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988), and references therein.
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 8.

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 19.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Young, L.

Appl. Opt. (1)

IEEE J. Lightwave Technol. (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988), and references therein.
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein, W. D. Cook, “Unified approach to ultrasonic light deflection,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Phys. Rev. B (3)

D. Heitman, N. Kroo, C. Schultz, Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B 35, 2660–2666 (1987).
[CrossRef]

R. W. Gruhlke, D. G. Hall, “Band structure for dissimilar electromagnetic waves in a periodic structure,” Phys. Rev. B 40, 5367–5371 (1989).
[CrossRef]

V. Celli, P. Tran, A. A. Maradudin, D. L. Mills, “k-gaps for surface polaritions on gratings,” Phys. Rev. B 37, 9089–9092 (1988).
[CrossRef]

Proc. Indian Acad. Sci. Sect. A (1)

N. S. N. Nath, “The diffraction of light by supersonic waves,” Proc. Indian Acad. Sci. Sect. A 8, 499–503 (1938).

Other (7)

A. Sommerfeld, Optics (Academic, New York, 1964), pp. 180–183.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 19.

H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1985), p. 103.

T. G. Brown, University of Rochester, Rochester, New York 14627 (personal communication).

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 8.

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Figures (19)

Fig. 1
Fig. 1

(a) An example of a modulated (along x) index stucture with average index n ¯ and peak-to-peak modulation 2Δn. The regions of high and low index are of equal length Λ/2, and there are a total of N periods. (b) Fourier spectrum of the modulated index structure that is shown in (a). (c) Wave vectors associated with first three odd orders of the modulated index structure. Note that the finite length of the modulated structure leads to an uncertainty in the length of the wave vectors.

Fig. 2
Fig. 2

Different geometries in which a plane wave is incident onto a modulated index structure.

Fig. 3
Fig. 3

Dispersion characteristics of plane waves of wave vector k propagating in an infinitely long, modulated index structure with wave vector κ1. Note the ω gap that occurs when counterpropagating waves are coupled by the grating wave vector κ1.

Fig. 4
Fig. 4

(a) Geometry in which a plane wave of amplitude Ei(0) is incident onto a grating structure of length L. Waves are both reflected from and transmitted through the structure. (b) Variation in the transmitted and reflected fluxes within the modulated index structure for incident and reflected wave vectors with κ 1 = 2 n ¯ k 0.

Fig. 5
Fig. 5

Reflection (solid curves) and transmission (dotted curves) coefficients for a grating versus detuning from the Bragg condition (Δk = 0) for various grating strengths ΓL.

Fig. 6
Fig. 6

Evolution of the forward [E+(x)] traveling and backward [E(x)] traveling fields in a distributed-feedback laser.

Fig. 7
Fig. 7

Periodic modulation introduced in the phase front of an incident plane wave on transmission through a modulated index structure for the case where Q = κ 1 2 L / ( n ¯ k 0 ) 1.

Fig. 8
Fig. 8

Evolution of the functions J0ϕ) (∝E0, the undiffracted field) and J1ϕ) (∝E1, the first-order diffracted field) as a function of Δϕ (∝ ΓL, the grating strength).

Fig. 9
Fig. 9

Wave-vector mismatch produced by scattering into the first order at an angle θ for light incident in a direction orthogonal to the grating wave vector. Note that Δ k κ 1 2 / n ¯ k 0.

Fig. 10
Fig. 10

Geometry for the analysis of scattering of a plane wave from a periodically modulated medium of infinite extent along the z direction and finite width L along the x direction. The incident wave vector with wave vector k ( n ¯ k 0 ) is incident onto the grating at angle θ; the scattered wave with wave vector q propagates at the angle θ.

Fig. 11
Fig. 11

Plot of the function sinc(u) = sin(u)/u.

Fig. 12
Fig. 12

Variation in percentage of the incident light diffracted into the various orders with increasing grating strength V = ΓL for normal incidence (k ·κ = 0) in the limit Q ≪ 1.

Fig. 13
Fig. 13

Percentage of the light diffracted into the zero (I0) and first (I+1) orders by a grating as a function of the normalized angle of incidence α for V = 2 and increasing values of the Raman–Nath parameter Q. Here α = n ¯ k 0 / κ 1 sin θ and V = 2ΓL. Note the narrowing of the angular distribution with increasing Q. The dashed lines correspond to the Bragg angle (α = 0.5). In (d) we also show diffraction into the −1 order (I−1). These data were taken from Ref. 14.

Fig. 14
Fig. 14

Geometry for the analysis of scattering from a finite grating element with dimensions Lx and Lz along the x and z directions, respectively.

Fig. 15
Fig. 15

Two examples of beam writing a modulated index structure: (a) interference between two optical beams produces a modulated index structure in a thin photoresist film, (b) scanned electron beam writes an index change into the surface region of an appropriate material.

Fig. 16
Fig. 16

Surface-relief grating produced by, for example, ion milling through a mask, in the surface of a material.

Fig. 17
Fig. 17

Periodic index structure produced in a thin slab of electro-optic material produced by periodic electrodes with alternating voltages applied to them.

Fig. 18
Fig. 18

Grating structures produced by the interference of two incident light beams in a material with an intensity-dependent refractive index. A total of three beams are incident onto the material. Taken in pairs, they produce gratings that are Bragg matched to the third beam.

Fig. 19
Fig. 19

Diffraction of light by a traveling acoustic wave of frequency Ω and wave vector κ1. The diffracted light is Doppler upshifted in frequency because the grating is moving.

Equations (41)

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n ( κ ) = 0 L n ( x ) exp ( j κ x ) d x ,
| n ( κ ) | 2 Δ n 2 κ 2 sin 2 ( κ Λ 4 ) sin 2 ( N κ Λ 2 ) sin 2 ( κ Λ 2 ) .
| n ( κ ) | 2 Δ n 2 κ p 2 sin 2 p π 2 N 2 ,
κ p = p 2 π Λ , p = 1 , 2 , 3 .
| n ( κ ) | 2 ( Δ n Λ N ) 2 ( 2 π p ) 2 , p = 1 , 3 , 5 , ,
n ( x ) = n ¯ + p Δ n p ( x ) = n ¯ + p Δ n p sin κ p x ,
E i ( x , t ) = 1 2 E i ( x ) exp [ j ( ω t n ¯ k 0 x ) ] + c . c . ,
E d ( x , t ) = 1 2 E d ( x ) exp [ j ( ω t n ¯ k 0 x ) ] + c . c . ,
2 E + n ( x ) 2 k 0 2 E = 0 ,
d d x E i ( x ) = Γ E d ( x ) exp ( j Δ k x ) ,
d d x E d ( x ) = Γ E i ( x ) exp ( j Δ k x ) ,
Γ = Δ n 1 k 0 2 , Δ k = 2 n ¯ k 0 κ 1 .
E i ( x ) = [ A 1 exp ( μ x ) + A 2 exp ( μ x ) ] exp ( j Δ k / 2 ) , E d ( x ) = [ B 1 exp ( μ x ) + B 2 exp ( μ x ) ] exp ( j Δ k / 2 )
μ = ± ( Γ 2 Δ k 2 / 4 ) 1 / 2 .
R = 4 Γ 2 sinh 2 ( μ L ) 4 μ 2 cosh 2 ( μ L ) + Δ k 2 sinh 2 ( μ L ) .
Γ = ± j μ sinh ( μ L ) = k 0 Δ n 1 2 + 1 2 j Δ α 1 ,
ϕ = ϕ 0 + p = 1 Δ ϕ p cos ( κ p x ) ,
ϕ 0 = 1 2 ( n 1 + n 2 ) k 0 L = n ¯ k 0 L , Δ ϕ 1 1 2 ( n 2 n 1 ) k 0 L = Γ L .
E = 1 2 E i exp { j ( ω t n ¯ k 0 z + ϕ 0 + Δ ϕ p cos [ ( κ p x ) ] } + c . c .
exp [ j Δ ϕ 1 cos ( κ 1 x ) ] = m = ( j ) m J m ( Δ ϕ 1 ) exp ( j m κ 1 x ) ,
E = 1 2 E i m = ( j ) m J m ( Δ ϕ 1 ) × exp [ j ( ω t n ¯ k 0 z + ϕ 0 m κ 1 x ) ] + c . c .
E ( x , z ) = y ˆ { E i exp [ j ( k x x + k z z ) ] + E d ( z ) exp [ j ( q x x ) ] } .
k x 2 + k z 2 = q x 2 + q z 2 = n ¯ 2 k 0 2 .
( 2 + n ¯ 2 k 0 2 ) E ( x , z ) = 2 n ¯ Δ n 1 ( x , z ) k 0 2 E ( x , z ) ,
( d 2 d z 2 + q z 2 ) E d ( z ) exp ( j q x x ) = 2 n ¯ Δ n 1 ( x , z ) k 0 2 E i exp [ j ( k x x + k z z ) ] .
( d 2 d z 2 + q z 2 ) E d ( z ) = n ¯ Δ n 1 E i W ( z ) exp ( j k z z ) δ ( k x q x ± κ 1 ) ,
W ( z ) = { 1 L / 2 < z < L / 2 0 otherwise ,
E d ( z ) = G ( z , z ) Q ( z ) d z ,
G ( z , z ) = 1 2 j q z exp ( j q z | z z | ) ,
E d ( z ) = j ( k 0 2 n ¯ Δ n 1 2 q z ) L E i exp ( j q z z ) × sinc [ ( k z q z ) L / 2 ] δ ( k x q x , ± κ 1 ) ,
sin θ ± m = ± m κ 1 n k 0 .
E ( x , z ) = y ˆ { E i exp [ j ( k x x + k z z ) ] + E d ( x , z ) } .
( 2 x 2 + 2 z 2 + n ¯ 2 k 0 2 ) E d ( x , z ) = K ( x , z ) ,
K ( x , z ) = 2 n ¯ Δ n 1 ( x , z ) k 0 2 E i exp [ j ( k x x + k z z ) ] .
E d ( x , z ) = L z / 2 L z / 2 L x / 2 L x / 2 G ( x , z ; x , z ) K ( x , z ) d x d z ,
G ( x , z ; x , z ) = j 4 H 0 ( 1 ) ( q | r r | ) ,
G ( x , z ; x , z ) = j 4 ( 2 π q ) 1 / 2 exp [ j ( q | r r | π / 4 ) ] ( | r r | ) 1 / 2 ,
E d ( x , z ) = j n ¯ Δ n 1 L x L z 2 E i ( k 0 4 2 π q ) 1 / 2 exp [ j ( k r π / 4 ) ] r 1 / 2 × sinc ( Δ k x L x 2 ) sinc ( Δ k z L z 2 ) ,
Δ n 1 p eff S sin ( Ω t κ 1 x ) ,
Δ n i j r i j k E k ,
n = n 0 + n 2 I ,

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