Abstract

The propagation of electromagnetic waves in the presence of a locally two-dimensional deformed plane waveguide is considered. We are concerned with the conversion of an incident beam into a guided beam as well as with the interaction of a guided beam with a local deformation. We look for a rigorous solution of Maxwell’s equations, i.e., a solution in which the errors depend only on the numerical methods used for evaluation. We outline the mathematical aspect of this rather formidable problem and emphasize the numerical difficulties that we have to overcome. An example is presented to give an idea of the capabilities of our computer program.

© 1981 Optical Society of America

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References

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  1. J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360 (1977).
    [CrossRef]
  2. J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a locally deformed plane waveguide,” Opt. Commun. 22, 221 (1977).
    [CrossRef]
  3. M. C. Hutley and et al., “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
    [CrossRef]
  4. R. Petit, Electromagnetic Theory of Gratings, Petit, ed. (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  5. Throughout the paper, when a is real and negative, a means i-a.
  6. Since Fourier transforms are defined as in distribution theory, the integral notation is purely formal.
  7. If, for example, the incident field is a plane wave in normal incidence [Ei= exp(−iy)], EDl(ζ) contains a term in δ(ζ).
  8. Let us recall that ζT= 1 implies that T= P(1/ζ) + aδ(ζ), where a is an arbitrary constant.
  9. P[iu(ζ)/g(ζ)] is the distribution that, to any test function ϕ(ζ) assignslim∊→0∫l∊iu(ζ)g(ζ)ϕ(ζ)dζ,where I∊ is the set defined by |ζ− ζG| > ∊ and |ζ+ ζG| > ∊.
  10. The Fourier transform of H(x) is 1/2δ(ζ) + (1/2πi)P(1/ζ), where P(1/ζ) is the principal value distribution.
  11. P. Vincent, “Singularity expansions for cylinders of finite conductivity,” Appl. Phys. 17, 239–248 (1978).
    [CrossRef]
  12. We do not set in boldface letters that, like C, represent elements of an abstract vector space.
  13. There is no possible confusion between the operator D and the rectangular domain that appears in Fig. 1.
  14. The letters topped by the inverted-wedge sign are used for data (functions or constants) related to a given practical problem.
  15. J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 252.
  16. P. Vincent and M. Nevière, “The reciprocity theorm for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
    [CrossRef]
  17. J. P. Hugonin, “On the numerical study of the deformed dielectric waveguide,” presented at the international Union Radio Scientifique Internationale Symposium, Munich, 1980.
  18. More generally, the use of an approximation of L−1is likely to increase the speed of convergence of the process.

1979 (1)

P. Vincent and M. Nevière, “The reciprocity theorm for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
[CrossRef]

1978 (1)

P. Vincent, “Singularity expansions for cylinders of finite conductivity,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

1977 (2)

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360 (1977).
[CrossRef]

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a locally deformed plane waveguide,” Opt. Commun. 22, 221 (1977).
[CrossRef]

1975 (1)

M. C. Hutley and et al., “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Hugonin, J. P.

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a locally deformed plane waveguide,” Opt. Commun. 22, 221 (1977).
[CrossRef]

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360 (1977).
[CrossRef]

J. P. Hugonin, “On the numerical study of the deformed dielectric waveguide,” presented at the international Union Radio Scientifique Internationale Symposium, Munich, 1980.

Hutley, M. C.

M. C. Hutley and et al., “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Nevière, M.

P. Vincent and M. Nevière, “The reciprocity theorm for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
[CrossRef]

Petit, R.

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a locally deformed plane waveguide,” Opt. Commun. 22, 221 (1977).
[CrossRef]

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360 (1977).
[CrossRef]

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

Van Bladel, J.

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 252.

Vincent, P.

P. Vincent and M. Nevière, “The reciprocity theorm for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
[CrossRef]

P. Vincent, “Singularity expansions for cylinders of finite conductivity,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

Appl. Phys. (1)

P. Vincent, “Singularity expansions for cylinders of finite conductivity,” Appl. Phys. 17, 239–248 (1978).
[CrossRef]

Nouv. Rev. Opt. (1)

M. C. Hutley and et al., “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Opt. Acta (1)

P. Vincent and M. Nevière, “The reciprocity theorm for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
[CrossRef]

Opt. Commun. (2)

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360 (1977).
[CrossRef]

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a locally deformed plane waveguide,” Opt. Commun. 22, 221 (1977).
[CrossRef]

Other (13)

J. P. Hugonin, “On the numerical study of the deformed dielectric waveguide,” presented at the international Union Radio Scientifique Internationale Symposium, Munich, 1980.

More generally, the use of an approximation of L−1is likely to increase the speed of convergence of the process.

We do not set in boldface letters that, like C, represent elements of an abstract vector space.

There is no possible confusion between the operator D and the rectangular domain that appears in Fig. 1.

The letters topped by the inverted-wedge sign are used for data (functions or constants) related to a given practical problem.

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), p. 252.

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

Throughout the paper, when a is real and negative, a means i-a.

Since Fourier transforms are defined as in distribution theory, the integral notation is purely formal.

If, for example, the incident field is a plane wave in normal incidence [Ei= exp(−iy)], EDl(ζ) contains a term in δ(ζ).

Let us recall that ζT= 1 implies that T= P(1/ζ) + aδ(ζ), where a is an arbitrary constant.

P[iu(ζ)/g(ζ)] is the distribution that, to any test function ϕ(ζ) assignslim∊→0∫l∊iu(ζ)g(ζ)ϕ(ζ)dζ,where I∊ is the set defined by |ζ− ζG| > ∊ and |ζ+ ζG| > ∊.

The Fourier transform of H(x) is 1/2δ(ζ) + (1/2πi)P(1/ζ), where P(1/ζ) is the principal value distribution.

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Figures (9)

Fig. 1
Fig. 1

Some practical problems that have suggested this theoretical study.

Fig. 2
Fig. 2

The grating description. We denote by yu the ordinate associated with the top of the groove [yu = max f(x)].

Fig. 3
Fig. 3

The locally deformed plane surface.

Fig. 4
Fig. 4

The locally deformed slab waveguide.

Fig. 5
Fig. 5

Examples of configurations studied.

Fig. 6
Fig. 6

The locally deformed plane waveguide and the schematic description of the associated fields. All the examples of Fig. 5 are particular cases of this configuration. Here y1 = yu but y0yl. The incident fields (Iu, Il) and the diffracted fields (Du, Dl) are represented by black arrows lying on a cone. Large and white arrows schematize incident and diffracted modes (IG, DG)

Fig. 7
Fig. 7

The coordinate transformation. Do not confuse block capitals XYZ and lower-case letters xyz.

Fig. 8
Fig. 8

Physical meaning of the diffraction patterns. R1 and R2 are detectors for the diffracted field.

Fig. 9
Fig. 9

Schematic representation of results obtained for γ = nusin(45°). The waveguide can support only one TE mode and one TH mode. Actually these modes are propagating obliquely with respect to Oz. The obstacle is illuminated by the TE mode coming from the left. The incident energy is taken as unity. The diffraction patterns are drawn using a continuous line for TE polarization and a dashed line for TH.

Equations (80)

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Δ E + n 2 ( x , y ) E = 0 ,
N ( y ) = { 1 if y 0 n l if y < 0 ,
D ( x , y ) = n 2 ( x , y ) - N 2 ( y ) .
E ( x , y ) = m = - Ê m ( y ) exp ( i ζ m x ) ,
D ( x , y ) = m = - D ˆ m ( y ) exp ( i ζ m x ) ,
d 2 Ê m ( y ) d y 2 = [ ζ m 2 - N 2 ( y ) ] Ê m ( y ) - p = - Ê p ( y ) D ˆ m - p ( y ) .
Ê m ( y ) = { E I u ( m ) exp [ - i χ u ( m ) y ] + E D u ( m ) exp [ i χ u ( m ) y ] if y y u , E D l ( m ) exp [ - i χ l ( m ) y ] if y < 0
E D u ( m ) = 1 2 exp [ - i χ u ( m ) y u ] × [ Ê m ( y u ) - i χ u ( m ) d Ê m d y ( y u ) ] ,
E I u ( m ) = 1 2 exp [ i χ u ( m ) y u ] × [ Ê m ( y u ) + i χ u ( m ) d Ê m d y ( y u ) ] .
E I u = E D l ,             E D u = N E D l ,
E D l = ( 1 , 0 0 ) E D l = ( 0 , 1 0 ) E D l = ( 0 , 0 , 1 ) .
N ( y ) = { 1 if y 0 n l if y < 0 ,
D ( x , y ) = n 2 ( x , y ) - N 2 ( y ) .
E ( x , y ) = - Ê ( ζ , y ) exp ( i ζ x ) d ζ , D ( x , y ) = - D ˆ ( ζ , y ) exp ( i ζ x ) d ζ .
2 Ê ( ζ , y ) y 2 = [ ζ 2 - N 2 ( y ) ] Ê ( ζ , y ) - D ˆ ( ζ , y ) * Ê ( ζ , y ) ,
Ê ( ζ , y ) = { E I u ( ζ ) exp ( - i χ u y ) + E D u ( ζ ) exp ( i χ u y ) if y y u E D l ( ζ ) exp ( - i χ l y ) if y 0 ,
J Ê ( ζ , y ) = 1 2 exp ( i χ u y u ) [ χ u ( ζ ) Ê ( ζ , y u ) + i Ê y ( ζ , y u ) ] .
Ê 0 ( ζ , y ) = { E D l ( ζ ) { 1 T ( ζ ) exp [ - i χ u ( ζ ) y ] + R ( ζ ) T ( ζ ) exp [ i χ u ( ζ ) y ] } if y > 0 E D l ( ζ ) exp [ - i χ l ( ζ ) y ] if y 0 ,
Ê ( ζ , y ) = Ê 0 ( ζ , y ) + e ( ζ , y ) ,
2 Ê 0 y 2 = [ ζ 2 - N 2 ( y ) ] Ê 0 ,
2 e y 2 = ( ζ 2 - N 2 ) e + D ˆ ( ζ , y ) * [ Ê 0 ( ζ , y ) + e ( ζ , y ) ] .
i u ( ζ ) = J e ( ζ , y ) .
i u = T E D l .
χ u ( ζ ) E I u ( ζ ) = g ( ζ ) E D l ( ζ ) + i u ( ζ ) ,
g ( ζ ) = χ u ( ζ ) T ( ζ ) = 1 2 [ χ u ( ζ ) + χ l ( ζ ) ] .
E D l ( ζ ) = 1 g ( ζ ) [ δ ( ζ ) - i u ( ζ ) ]
i u ( ζ ) = T { 1 g ( ζ ) [ δ ( ζ ) - i u ( ζ ) ] } .
g ( ζ ) E D l ( ζ ) = - i u ( ζ ) ,
E D l ( ζ ) = - P [ i u ( ζ ) g ( ζ ) ] + a + δ ( ζ - ζ G ) + a - δ ( ζ + ζ G ) ,
i u ( ζ ) = T { - P [ i u ( ζ ) g ( ζ ) ] + a + δ ( ζ - ζ G ) + a - δ ( ζ + ζ G ) } .
Ê ( ζ , y ) = [ e + P ( 1 ζ - ζ G ) + e - P ( 1 ζ + ζ G ) + a + δ ( ζ - ζ G ) + a - δ ( ζ + ζ G ) ] exp [ - i χ l ( ζ G ) y ] + U ( ζ , y ) ,
E ( x , y ) = exp [ - i χ l ( ζ G ) y ] { [ I l exp ( i ζ G x ) + D l exp ( - i ζ G x ) ] H ( - x ) + [ I r exp ( - i ζ G x ) + D r exp ( - i ζ G x ) ] H ( x ) } + E ˜ ( x , y ) ,
π i e + = - I l + D r , π i e - = I r - D l , 2 a + = I l + D r , 2 a - = I r + D l .
( I l I r ) = A ( a + a - ) ,             ( D l D r ) = ( a + a - ) .
( a + a - ) = A - 1 ( 1 0 ) ,             ( D l D r ) = A - 1 ( 1 0 ) .
curl E = i ω μ 0 H , curl H = - i ω 0 n 2 E ,
E ( x , y , z ) = E ( x , y ) exp ( i γ z ) / ω 0 , H ( x , y , z ) = H ( x , y ) exp ( i γ z ) / ω μ 0 ,
E z y = i ( 1 - γ 2 n 2 ) H x + γ n 2 H z x , H x y = i n 2 E z + x ( i E z x + γ E x ) , H z y = i ( n 2 - γ 2 ) E x - γ E z x , E x y = - i H z - x [ 1 n 2 ( i H z x + γ H x ) ] ,
{ E y = - i n 2 H z x - γ n 2 H x H y = i E z x + γ E x .
n 2 ( x , y ) = N 2 ( y ) + D + ( x , y ) , 1 n 2 ( x , y ) = 1 N 2 ( y ) + D - ( x , y ) ,
E ( x , y ) = - Ê ( ζ , y ) exp ( i ζ x ) d ζ , H ( x , y ) = - Ĥ ( ζ , y ) exp ( i ζ x ) d ζ , D ± ( x , y ) = - D ˆ ± ( ζ , y ) exp ( i ζ x ) d ζ ,
β = ζ 2 + γ 2 , χ u ( ζ ) = n u 2 - β 2 , χ l ( ζ ) = n l 2 - β 2 .
F y = W β F + P y E ,
W β F = W β [ Ê Z Ĥ X Ĥ Z Ê X ] = [ i Ĥ X i ( N 2 - β 2 ) Ê Z - i N 2 Ê X - i ( 1 - β 2 N 2 ) Ĥ Z ] ,
P y F = P y [ Ê Z Ĥ X Ĥ Z Ê X ] = [ 0 i ζ β ( D ˆ + * Ê z ) - i γ β ( D ˆ + * Ê x ) - i γ β ( D ˆ + * Ê z ) - i ζ β ( D ˆ + * Ê x ) i β [ D ˆ - * ( β Ĥ Z ) ] ] ,
Ê x = ζ / β Ê X - γ / β Ê Z , Ê z = γ / β Ê X + ζ / β Ê Z ,
F ( ζ , y ) = ( E I u ( ζ ) - χ u E I u ( ζ ) H I u ( ζ ) χ u / n u 2 H I u ( ζ ) ) exp ( - i χ u y ) + ( E D u ( ζ ) χ u E D u ( ζ ) H D u ( ζ ) - χ u / n u 2 H D u ( ζ ) ) exp ( i χ u y ) ,
F ( ζ , y ) = ( E D l ( ζ ) - χ l E D l ( ζ ) H D l ( ζ ) χ l / n l 2 H D l ( ζ ) ) exp ( - i χ l y ) + ( E I l ( ζ ) χ l E I l ( ζ ) H I l ( ζ ) - χ l / n l 2 H I l ( ζ ) ) exp ( i χ l y ) .
I u ( ζ ) = χ u ( ζ ) ( E I u ( ζ ) H I u ( ζ ) ) , D u ( ζ ) = χ u ( ζ ) ( E D u ( ζ ) H D u ( ζ ) ) , I l ( ζ ) = ( E I l ( ζ ) H I l ( ζ ) ) , D l ( ζ ) = ( E D l ( ζ ) H D l ( ζ ) ) .
F ( ζ , y l ) = [ I l ( ζ ) , D l ( ζ ) ] , I u ( ζ ) = J [ F ( ζ , y u ) ] , D u ( ζ ) = D [ F ( ζ , y u ) ] .
F y = W β F .
G [ D l ( ζ ) ] = I u ( ζ ) = ( χ u / T E ( ζ ) 0 0 χ u / T H ( ζ ) ) D l ( ζ ) .
E = ϕ E j ( y ) exp ( ± i ζ E j x ) e ( ± ζ E j ) , j = 1 , , m E , H = ϕ H j ( y ) exp ( ± i ζ H j x ) e ( ± ζ H j ) , j = 1 , , m H ,
- ϕ E j ( y ) 2 d y = 1 ,             - ϕ H j ( y ) 2 / N 2 ( y ) d y = 1.
m = m E + m H , ζ G j = { ζ E j if j m E ζ H j - m E if m E < j m .
C G ( x , y ) = j = 1 m [ I l j ψ + j ( y ) exp ( i ζ G j x ) + D l j ψ - j ( y ) exp ( - ζ G j x ) ] H ( - x ) + j = 1 m [ I r j ψ - j ( y ) exp ( - i ζ G j x ) + D r j ψ + j ( y ) exp ( i ζ G j x ) ] H ( x ) ,
F ( ζ , y ) = j = 1 m { [ D r j + I l j 2 δ ( ζ - ζ G j ) + D r j - I l j 2 π i P ( 1 ζ - ζ G j ) ] ψ + j ( y ) + [ I r j + D l j 2 δ ( ζ + ζ G j ) + I r j - D l j 2 π i P ( 1 ζ + ζ G j ) ] ψ - j ( y ) } + U ( ζ , y ) ,
F ( ζ , y ) = F 0 ( ζ , y ) + F 1 ( ζ , y ) + f ( ζ , y ) ,
f y = W β f + P y ( F 0 + F 1 + f ) ,
f ( ζ , y 0 ) = 0.
i u ( ζ ) = J [ f ( ζ , y u ) ] ,
I u ( ζ ) = G [ D l ( ζ ) ] + I u 1 ( ζ ) + i u ( ζ ) ,
I u ( ζ ) = J [ F ( ζ , y u ) ] ,             I u 1 ( ζ ) = J [ F 1 ( ζ , y u ) ] ,
J [ F 0 ( ζ , y u ) ] = G [ D l ( ζ ) ] .
( 1 ) i u ( ζ ) = T [ I l ( ζ ) , D l ( ζ ) ] , ( 2 ) D u ( ζ ) = D [ F ( ζ , y u ) ] , ( 3 ) the vector I G and D G
D l ( ζ ) = P { G - 1 [ I u ( ζ ) - I u 1 ( ζ ) - i u ( ζ ) ] } + j = 1 m [ a + j δ + j ( ζ ) + a - j δ - j ( ζ ) ] ,
δ ± j ( ζ ) = [ δ ( ζ ζ G j ) 0 ] if j m E ( TE modes ) , δ ± j ( ζ ) = [ 0 δ ( ζ ζ G j ) ] if m E < j m ( TH modes ) ,
i u = i u C ,
i u C = T { I l ( ζ ) , P [ G - 1 ( I u - I u 1 ) ] + j = 1 m [ a + j δ + j ( ζ ) + a - j δ - j ( ζ ) ] }
i u = i u + T ( 0 , P { G - 1 [ i u ( ζ ) ] } ) .
I u = I ˇ u , I l = I ˇ l , A = ( 0 , , 0 ) , I u = 0 , I l = 0 , A = ( 1 , , 0 ) , I u = 0 , I l = 0 , A = ( 0 , , 1 ) .
I u = I ˇ u , I l = I ˇ l , I G = I ˇ G .
L X = B ,
n u 2 n u 2 - γ 2 χ u E D u ( ζ ) 2
1 n u 2 - γ 2 χ u H D u ( ζ ) 2
n l 2 n l 2 - γ 2 χ l E D l ( ζ ) 2
1 n l 2 - γ 2 χ l H D l ( ζ ) 2
B = n = 1 p ( B , Ŷ n ) Ŷ n + R p .
X p = n = 1 p ( B , Ŷ n ) X ˆ n ,
lim 0 l i u ( ζ ) g ( ζ ) ϕ ( ζ ) d ζ ,