Abstract

A method for the calculation of the propagation of a light beam through an inhomogeneous medium is presented. A theoretical analysis of this beam-propagation method is given, and a set of conditions necessary for the accurate application of the method is derived. The method is illustrated by the study of a number of integrated-optic structures, such as thin-film waveguides and gratings.

© 1981 Optical Society of America

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References

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  1. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [Crossref]
  2. M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [Crossref] [PubMed]
  3. C. Yeh and et al., “Multimode inhomogeneous fiber couplers,” Appl. Opt. 18, 489–495 (1979).
    [Crossref] [PubMed]
  4. M. D. Feit and J. A. Fleck, “Calculation of dispersion in graded-index multimode fibers by a propagating-beam method,” Appl. Opt. 18, 2843–2851 (1979).
    [Crossref] [PubMed]
  5. J. van der Donk and P. E. Lagasse, “Calculation of light propagation in integrated optic multimode waveguide structures,” presented at the Optical Communication Conference, Amsterdam, The Netherlands, 1979.
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1969).
  7. J. Van Roey and P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423–429 (1981).
    [Crossref] [PubMed]

1981 (1)

1979 (2)

1978 (1)

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Feit, M. D.

Fleck, J. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1969).

Lagasse, P. E.

J. Van Roey and P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423–429 (1981).
[Crossref] [PubMed]

J. van der Donk and P. E. Lagasse, “Calculation of light propagation in integrated optic multimode waveguide structures,” presented at the Optical Communication Conference, Amsterdam, The Netherlands, 1979.

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

van der Donk, J.

J. van der Donk and P. E. Lagasse, “Calculation of light propagation in integrated optic multimode waveguide structures,” presented at the Optical Communication Conference, Amsterdam, The Netherlands, 1979.

Van Roey, J.

Yeh, C.

Appl. Opt. (4)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Other (2)

J. van der Donk and P. E. Lagasse, “Calculation of light propagation in integrated optic multimode waveguide structures,” presented at the Optical Communication Conference, Amsterdam, The Netherlands, 1979.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1969).

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Figures (7)

Fig. 1
Fig. 1

Percent error of the computed field amplitude as a function of propagation distance for different step sizes. Details about the waveguide geometry are given in the text.

Fig. 2
Fig. 2

Amplitude of the fields propagated in a bent thin-film waveguide.

Fig. 3
Fig. 3

Refractive-index profile showing a step.

Fig. 4
Fig. 4

Periodic extension of the refractive-index profile and absorbers used in the BPM.

Fig. 5
Fig. 5

Calculated field amplitude as a function of depth and propagation distance in slab waveguides.

Fig. 6
Fig. 6

Beams in a periodic medium: 1, incident beam; 2, reflected beam; 3, periodic medium; z, propagation direction for the BPM.

Fig. 7
Fig. 7

Gaussian beams incident upon a rectangular grating. The incident beam has no divergence in a and an angular spread of 3° in b.

Tables (1)

Tables Icon

Table 1 Number of Multiplications for One Propagation Step

Equations (82)

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2 ϕ + k 2 n 2 ( r ) ϕ = 0 ,
2 ϕ + k 2 n 0 2 ( r ) ϕ = - k 2 Δ n 2 ( r ) ϕ = p ( r ) ,
2 ψ + k 2 n 0 2 ( r ) ψ = 0 ,
ψ / z = â ψ ( x , y , z 0 ) ,
Ψ ( k x , k y , z ) = - + ψ ( x , y , z ) exp [ - j ( k x x + k y y ) ] d x d y .
Ψ z ( k x , k y , z ) = - j ( k 2 n 0 2 - k x 2 - k y 2 ) 1 / 2 Ψ ( k x , k y , z ) .
ϕ = ϕ 1 + ϕ 2 .
e 1 ( r r ) = { 0 for z < z 1 / 2 for z = z 1 for z > z .
ϕ 1 ( r ) = - + G ( r r ) e 1 ( r r ) p ( r ) d V .
ϕ 1 z = - + G z ( r r ) e 1 ( r r ) p ( r ) d V × - + G ( r r ) δ ( z - z ) p ( r ) d V .
â ϕ 1 .
- + G ( x , y , z x , y , z ) ( - k 2 ) Δ n 2 ( x , y , z ) ϕ ( x , y , z ) d x d y .
ϕ 1 z = â ϕ 1 + b ˆ ϕ .
ϕ 1 z = â ϕ 1 + b ˆ ϕ 1 .
b ˆ ϕ 1 ( r ) = - + exp ( - j k n 0 D ) 4 π D k 2 Δ n 2 ( x , y , z ) × ϕ 1 ( x , y , z ) d x d y ,
[ b ˆ ϕ 1 ] = k 2 { e x p [ - j k n 0 ( x 2 + y 2 ) ] 1 / 2 4 π ( x 2 + y 2 ) } × [ Δ n 2 ϕ 1 ] = - j k 2 2 1 ( k 2 n 0 2 - k x 2 - k y 2 ) 1 / 2 × [ Δ n 2 ϕ 1 ] ,
1 ( k 2 n 0 2 - k x 2 - k y 2 ) 1 / 2 = m = 0 + ( - 1 ) m ( 2 m ) ! [ - ( k x 2 + k y 2 ) ] m 2 2 m ( m ! ) 2 ( k n 0 ) 2 m + 1 .
b ˆ 1 = - - j k 2 2 m = 0 + ( - 1 ) m ( 2 m ) ! 2 2 m ( m ! ) 2 ( k n 0 ) 2 m + 1 × p = 0 m m ! p ! ( m - p ) ! 2 m ( Δ n 2 ϕ 1 ) 2 p x 2 ( m - p ) y .
b ˆ ϕ 1 = - j k 2 n 0 Δ n 2 ϕ 1 .
ϕ 1 z = â ϕ 1 - j k 2 n 0 Δ n 2 ϕ 1 .
/ z = â ,
ϕ 1 ( r ) = exp [ Γ ( r ) ] ( r ) ,
Γ z = - j k 2 n 0 Δ n 2 + â [ exp ( Γ ) ] - exp ( Γ ) â ( ) exp ( Γ ) .
Γ = - j k 2 n 0 Δ n 2 ( z - z 0 )             for ( z - z 0 ) small .
ϕ 1 ( x , y , z 0 + Δ z ) = ( x , y , z 0 + Δ z ) exp ( - j k 2 n 0 Δ n 2 Δ z ) .
( x , y , z 0 + Δ z ) = n A n ψ n ( x , y ) exp ( - j k n Δ z ) , where A n = - + ( x , y , z 0 ) ψ n ( x , y ) d x d y .
Γ = n B n ( x , y ) · Δ z n .
ϕ 1 = ξ = e Γ + γ ,
γ Γ .
Im γ π .
ξ z = b ˆ ( ϕ 1 ) + â ( ξ ) - ξ â ( ) .
ξ ( x ) = m = 0 1 m ! m ξ x m ( x - x ) m .
â ( ξ ) = m = 0 1 m ! m ξ x m â [ ( x - x ) m ( x ) ] ,
ξ z = b ˆ ( ϕ 1 ) ϕ 1 ξ + â [ ( x - x ) ( x ) ] ξ x .
e Γ z = - j k 2 n 0 Δ n 2 e Γ - j k Δ n e Γ .
γ z = j 4 k n 0 3 ( Δ n 2 1 ϕ 1 2 ϕ 1 x 2 + 2 Δ n 2 x 1 ϕ 1 ϕ 1 x + 2 Δ n 2 x 2 ) .
| Δ n 2 x | max < 2 π p Δ n 2 max , | 2 Δ n 2 x 2 | max < 4 π 2 p 2 Δ n 2 max ,
| 1 ϕ 1 ϕ 1 x | max < k n 0 sin α , | 1 ϕ 1 2 ϕ 1 x 2 | max < k 2 n 0 2 sin 2 α .
| γ z | < Δ n 2 max 4 k n 0 3 ( 2 π p + k n 0 sin α ) 2
γ < Δ n 2 max 4 k n 0 3 ( 2 π p + k n 0 sin α ) 2 Δ z .
γ < k Δ n 2 max sin 2 α 4 n 0 Δ z .
Δ n max sin 2 α Δ z λ 1 ,
γ < 8 π 2 Δ n max k n 0 2 p 2 Δ z .
p λ n 0 2
p 1 n 0 ( 2 π λ Δ n max Δ z ) 1 / 2 .
ξ z = - j k 2 n 0 Δ n 2 ξ + a [ ( x - x ) ( x ) ] ξ x .
γ z = â [ ( x - x ) ( x ) ] ξ ( Γ x + γ x ) .
γ = - j k 2 â [ ( x - x ) ( x ) ] Δ n x ( Δ z ) 2 .
π p | â [ ( x - x ) ( x ) ] | max Δ z 1 ,
{ 2 π Δ n max p λ | â [ ( x - x ) ( x ) ] | max } 1 / 2 Δ z 1.
â ϕ = h * ϕ ,
â [ ( x - x ) ( x ) ] = - ( x h ) * .
[ x h ] = - k x ( k 2 n 0 2 - k x 2 ) 1 / 2 ,
| ( x h ) * | max tan α .
π tan α p Δ z 1
( 2 π p λ tan α Δ n max ) 1 / 2 Δ z 1.
Δ z 45 μ m ,
Δ z < 40 μ m .
n 0 ( x ) = n 1 x < 0 = n 2 x > 0 ,
| n ( x ) - n 0 ( x ) n 0 ( x ) | 1             for all x .
ψ m and ψ m x continuous at x = 0.
ψ m and 1 n 0 2 ψ m x continuous at x = 0.
c 1 cos ( γ 1 x ) - x 1 < x < x 1 c 2 cos [ γ 2 ( x - x 2 ) ] x 1 < x < X - x 1
γ 1 tan ( γ 1 x 1 ) = γ 2 tan [ γ 2 ( x 1 - X 2 ) ] ;
c 3 sin ( γ 1 x ) - x 1 < x < x 1 c 4 sin [ γ 2 ( x - X 2 ) ] x 1 < x < X - x 1
1 γ 1 tan ( γ 1 x 1 ) = 1 γ 2 tan [ γ 2 ( x 1 - X 2 ) ] ,
( x , z 0 ) = ϕ 1 ( x , z 0 ) = m = 1 M A m ( z 0 ) ψ m ( x ) ,
A m = - x 1 X - x 1 ϕ 1 ( x , z 0 ) ψ m * ( x ) d x ,
S ( x ; K ) = k = - + sin [ π ( x - k K ) ] π ( x - k K ) = 1 K sin ( π x ) sin ( π x K )             for K even = 1 K sin ( π x ) tan ( π x K )             for K odd ,
( x , z 0 ) = ϕ 1 ( x , z 0 ) = k = 0 K - 1 S ( x - x k Δ x ; K ) ϕ 1 ( x k , z 0 )             for - x 1 < x < x - x 1 ,
A m = k = 0 K - 1 ϕ 1 ( x k , z 0 ) I k m ,
I k m = - x 1 X - x 1 S ( x - x k Δ x ; K ) ψ m * ( x ) d x .
( x , z 0 + Δ z ) = m = 1 M A m ψ m ( x ) exp ( j β m Δ z ) .
ϕ = ϕ I + ϕ R ,
ϕ I z + ϕ R z = â ϕ I + â ϕ R + b ˆ ϕ I + b ˆ ϕ R .
ϕ I z = â ϕ I + b ˆ ϕ R , ϕ R z = â ϕ R + b ˆ ϕ I ,
ϕ i = ψ i exp ( - j k n o u ¯ i · r ¯ ) ,             i = I , R ,
Δ n 2 = E cos U = E e j U + e - j U 2 ,
ψ I z = exp ( j k u ¯ I · r ¯ ) â [ exp ( - j k u ¯ I · r ¯ ) ψ I ] + j k u ¯ I · u ¯ 2 r ¯ ψ I + exp ( j k u ¯ I · r ¯ ) - + d x j 4 H 0 ( 2 ) ( k n o x - x ) × ( - k 2 E ) e j U + e - j U 2 exp ( - j k u ¯ R · r ¯ ) ψ R ( x ) = â I ψ I - j k 2 8 - + d x E ( x ) ψ R ( x ) exp ( j k u ¯ I · r ¯ - j k u ¯ R · r ¯ - j U ) × H 0 ( 2 ) ( k n 0 x - x ) exp [ j k u ¯ I · ( r ¯ - r ¯ ) ] ,
k u ¯ I = k u ¯ R + grad U + grad μ
ψ I z = â I ψ I - j k 2 8 - + d x E ( x ) ψ R ( x ) exp [ j μ ( x ) ] × H 0 ( 2 ) ( k n 0 x - x ) exp [ j k u ¯ I · ( r ¯ - r ¯ ) ] .
2 x 2 y [ ϕ ] + 2 z 2 y [ ϕ ] + [ k 2 n 0 2 ( x ) - k y 2 ] y [ ϕ ] = 0 ,