Abstract

Diffraction by thin refractive-index gratings of arbitrary periodic shape is treated. Three analytical approaches are indicated and are shown to be equivalent. Resultant expressions for the diffraction efficiencies are given for all diffracted orders.

© 1978 Optical Society of America

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References

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  1. M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  2. R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
    [CrossRef]
  3. R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [CrossRef]
  4. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  5. R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353–362 (1976).
    [CrossRef]
  6. K. L. Zankel and E. A. Hiedemann, “Diffraction of light by ultrasonic waves progressing with finite but moderate amplitudes in liquids,” J. Acoust. Soc. Am. 31, 44–54 (1959).
    [CrossRef]

1977 (1)

1976 (1)

1975 (1)

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[CrossRef]

1973 (1)

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

1959 (1)

K. L. Zankel and E. A. Hiedemann, “Diffraction of light by ultrasonic waves progressing with finite but moderate amplitudes in liquids,” J. Acoust. Soc. Am. 31, 44–54 (1959).
[CrossRef]

Alferness, R.

Cadilhac, M.

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Gaylord, T. K.

Hiedemann, E. A.

K. L. Zankel and E. A. Hiedemann, “Diffraction of light by ultrasonic waves progressing with finite but moderate amplitudes in liquids,” J. Acoust. Soc. Am. 31, 44–54 (1959).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Magnusson, R.

Neviere, M.

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Petit, R.

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[CrossRef]

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Zankel, K. L.

K. L. Zankel and E. A. Hiedemann, “Diffraction of light by ultrasonic waves progressing with finite but moderate amplitudes in liquids,” J. Acoust. Soc. Am. 31, 44–54 (1959).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

J. Acoust. Soc. Am. (1)

K. L. Zankel and E. A. Hiedemann, “Diffraction of light by ultrasonic waves progressing with finite but moderate amplitudes in liquids,” J. Acoust. Soc. Am. 31, 44–54 (1959).
[CrossRef]

J. Opt. Soc. Am. (2)

Nouv. Rev. Opt. (1)

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[CrossRef]

Opt. Commun. (1)

M. Neviere, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

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Figures (1)

FIG. 1
FIG. 1

The geometry of a thin, phase grating with an incident plane wave and multiple diffracted output waves. The spatial modulation of the refractive index is indicated by the line pattern.

Tables (1)

Tables Icon

TABLE I Example diffraction efficiencies of thin phase gratings.

Equations (19)

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Q = 2 π λ d / n 0 L 2 cos θ ,
S i / z + j [ i ( m - i ) / 2 ] Q S i + γ h = 1 ( f h S i - h - f h * S i + h ) = 0 ,
Δ n = h = 1 ( n c h cos h K x + n s h sin h K x ) ,
Δ n = n 1 h = 1 ( f c h 2 + f s h 2 ) 1 / 2 sin [ h K x + tan - 1 ( f c h / f s h ) ] ,
τ ( x , z ) = exp ( - j 2 π z d Δ n / λ cos θ ) .
τ ( x , z ) = i = - S i ( z ) exp ( j i K x ) .
S i ( z ) = L - 1 0 L τ ( x , z ) exp ( - j i K x ) d x .
S i ( z ) = ( 2 π ) - 1 0 2 π exp [ - j 2 γ z f ( ξ ) ] exp ( - j i ξ ) d ξ ,
S i / z + γ h = 1 ( f h S i - h - f h * S i + h ) = 0 ,
τ ( x , z ) = m = 1 exp ( - j a c m cos m K x ) × n = 1 exp ( - j a s n sin n K x ) ,
exp ( j b cos α ) = i = - j i J i ( b ) exp ( j i α )
exp ( j b sin α ) = i = - J i ( b ) exp ( j i α ) ,
τ ( x , z ) = i = - ( i c m , i s n m = 1 n = 1 ( - j ) i c m ( - 1 ) i s n × J i c m ( a c m ) J i s n ( a s n ) ) exp ( j i K x ) .
m = 1 i c m m + n = 1 i s n n = i ,
S i = i c m , i s n m = 1 n = 1 ( - j ) i c m ( - 1 ) i s n J i c m ( a c m ) J i s n ( a s n ) .
i = - S i S i * = 1.
i = - ( c i / c 0 ) S i S i * = 1 ,
η i = ( i c 2 = - i c 3 = - i c 4 = - i s 1 = - i s 2 = - i s 3 = - × J i - 2 i c 2 - 3 i c 3 - 4 i c 4 - i s 1 - 2 i s 2 - 3 i s 3 ( a c 1 ) × J i c 2 ( a c 2 ) J i c 3 ( a c 3 ) J i c 4 ( a c 4 ) × J i s 1 ( a s 1 ) J i s 2 ( a s 2 ) J i s 3 ( a s 3 ) ) 2 .
η i = ( n = 2 J 0 2 ( a s n ) ) { J i ( a s 1 ) + q = 1 n = 2 [ J q ( a s n ) / J 0 ( a s n ) ] [ J i - q n ( a s 1 ) + ( - 1 ) q J i + q n ( a s 1 ) ] + m = 1 q = 1 p = 1 n = 2 [ J q ( a s ( p + 1 ) ) / J 0 ( a s ( p + 1 ) ) ] [ J m ( a s ( n + p ) ) / J 0 ( a s ( n + p ) ) ] × [ J i - [ q ( p + 1 ) + m ( n + p ) ] ( a s 1 ) + ( - 1 ) m J i - [ q ( p + 1 ) - m ( n + p ) ] ( a s 1 ) + ( - 1 ) q J i + [ q ( p + 1 ) - m ( n + p ) ] ( a s 1 ) + ( - 1 ) m + q J i + [ q ( p + 1 ) + m ( n + p ) ] ( a s 1 ) ] + l = 1 m = 1 q = 1 r = 1 p = 1 n = 2 } 2 .