Abstract

A boundary variation method for the forward modeling of multilayered diffraction optics is presented. The approach permits fast and high-order accurate modeling of periodic transmission optics consisting of an arbitrary number of materials and interfaces of general shape subject to plane-wave illumination or, by solving a sequence of problems, illumination by beams. The key elements of the algorithm are discussed, as are details of an efficient implementation. Numerous comparisons with exact solutions and highly accurate direct solutions confirm the accuracy, the versatility, and the efficiency of the proposed method.

© 2004 Optical Society of America

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References

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  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  2. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
    [CrossRef]
  3. K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
    [CrossRef]
  4. D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1141 (1999).
    [CrossRef]
  5. P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudospectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 16, 1124–1130 (1999).
    [CrossRef]
  6. J. S. Hesthaven, P. G. Dinesen, J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
    [CrossRef]
  7. J. S. Hesthaven, “High-order accurate methods in time-domain computational electromagnetics: a review,” Adv. Electron. Electron Phys. 127, 59–123 (2003).
  8. O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
    [CrossRef]
  9. O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
    [CrossRef]
  10. O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  11. P. G. Dinesen, J. S. Hesthaven, “A fast and accurate boundary variation method for diffractive gratings,” J. Opt. Soc. Am. A 17, 1565–1572 (2000).
    [CrossRef]
  12. P. G. Dinesen, J. S. Hesthaven, “A fast and accurate boundary variation method for diffractive gratings. II.The three-dimensional vectorial case,” J. Opt. Soc. Am. A 18, 2876–2885 (2001).
    [CrossRef]
  13. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 1–52.
    [CrossRef]
  14. O. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh, Sect. A 122, 317–340 (1992).
    [CrossRef]
  15. G. A. Baker, P. Graves-Morris, Padé Approximants, 2nd ed., Vol. 59 of Encyclopedia of Mathematics and Its Applications (Cambridge U. Press, Cambridge, UK.1996).
  16. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

2003 (1)

J. S. Hesthaven, “High-order accurate methods in time-domain computational electromagnetics: a review,” Adv. Electron. Electron Phys. 127, 59–123 (2003).

2001 (1)

2000 (1)

1999 (3)

1996 (1)

1994 (1)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
[CrossRef]

1993 (3)

1992 (1)

O. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh, Sect. A 122, 317–340 (1992).
[CrossRef]

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Baker, G. A.

G. A. Baker, P. Graves-Morris, Padé Approximants, 2nd ed., Vol. 59 of Encyclopedia of Mathematics and Its Applications (Cambridge U. Press, Cambridge, UK.1996).

Bruno, O.

Dinesen, P. G.

Gallagher, N. C.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
[CrossRef]

Gaylord, T. K.

K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Glytsis, E. N.

Graves-Morris, P.

G. A. Baker, P. Graves-Morris, Padé Approximants, 2nd ed., Vol. 59 of Encyclopedia of Mathematics and Its Applications (Cambridge U. Press, Cambridge, UK.1996).

Hesthaven, J. S.

Hirayama, K.

Lading, L.

Lichtenberg, B.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
[CrossRef]

Lynov, J. P.

J. S. Hesthaven, P. G. Dinesen, J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[CrossRef]

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudospectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 16, 1124–1130 (1999).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Petit, R.

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 1–52.
[CrossRef]

Prather, D. W.

Reitich, F.

Shi, S.

Wilson, D. W.

Yeh, P.

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

Adv. Electron. Electron Phys. (1)

J. S. Hesthaven, “High-order accurate methods in time-domain computational electromagnetics: a review,” Adv. Electron. Electron Phys. 127, 59–123 (2003).

J. Comput. Phys. (1)

J. S. Hesthaven, P. G. Dinesen, J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[CrossRef]

J. Opt. Soc. Am. A (8)

O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
[CrossRef]

O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
[CrossRef]

O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

P. G. Dinesen, J. S. Hesthaven, “A fast and accurate boundary variation method for diffractive gratings,” J. Opt. Soc. Am. A 17, 1565–1572 (2000).
[CrossRef]

P. G. Dinesen, J. S. Hesthaven, “A fast and accurate boundary variation method for diffractive gratings. II.The three-dimensional vectorial case,” J. Opt. Soc. Am. A 18, 2876–2885 (2001).
[CrossRef]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1141 (1999).
[CrossRef]

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudospectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 16, 1124–1130 (1999).
[CrossRef]

Opt. Eng. (1)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Proc. R. Soc. Edinburgh, Sect. A (1)

O. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh, Sect. A 122, 317–340 (1992).
[CrossRef]

Other (3)

G. A. Baker, P. Graves-Morris, Padé Approximants, 2nd ed., Vol. 59 of Encyclopedia of Mathematics and Its Applications (Cambridge U. Press, Cambridge, UK.1996).

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

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 1–52.
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Generic setup for scattering by a problem with multiple interfaces.

Fig. 2
Fig. 2

Specific example to illustrate scheme for a problem with two interfaces.

Fig. 3
Fig. 3

(a) Problem setup for the test with one plane layer, (b) decay of error in field amplitudes as a function of internal reflections or bounces.

Fig. 4
Fig. 4

(a) Problem setup for the test with a stack of plane layers, (b) decay of error in field amplitudes as a function of internal reflections or bounces.

Fig. 5
Fig. 5

Problem specification for a single layer with a shallow curved interface.

Fig. 6
Fig. 6

E z computed at different heights y, with y = 0 corresponding to the vertical position of the shallow curved interface: (a) y = - 3 , (b) y = - 1 , (c) y = - 0.5 , (d) y = 3 . Illumination is TE polarized at normal incidence. Results are shown for a fixed number of internal bounces and compared with a highly accurate spectral solution, illustrating the importance of accounting for the multiple internal reflections.

Fig. 7
Fig. 7

E z computed at different heights y, with y = 0 corresponding to the vertical position of the shallow curved interface: (a) y = - 3 , (b) y = - 1 , (c) y = - 0.5 , (d) y = 3 . Illumination is TE polarized at normal incidence. Results are shown for converged solutions in terms of internal reflections and compared with a highly accurate spectral solution.

Fig. 8
Fig. 8

Problem specification for a single layer with a deep curved interface.

Fig. 9
Fig. 9

E z computed at different heights y, with y = 0 corresponding to the vertical position of the deep curved interface: (a) y = - 3 , (b) y = - 1.5 , (c) y = - 0.5 , (d) y = 3 . Illumination is TE polarized at normal incidence. Results are shown for converged solutions in terms of internal reflections and compared with a highly accurate spectral solution. E e = 1 is a solution that includes the two evanescent modes from each single-interface solve with the smallest |β|.

Fig. 10
Fig. 10

Problem specification for a single layer with an integrated lens.

Fig. 11
Fig. 11

E z computed at different heights y, with y = 0 corresponding to the vertical position of the integrated lens: (a) y = - 2 , (b) y = - 1 , (c) y = 1 , (d) y = 3 . Illumination is TE polarized at normal incidence. Results are shown for converged solutions in terms of internal reflections and compared with a highly accurate spectral solution.

Fig. 12
Fig. 12

Problem specification for a single layer with two curved interfaces.

Fig. 13
Fig. 13

E z computed at different heights y, with y = 0 corresponding to the vertical position of the slowly varying interface: (a) y = - 3 , (b) y = - 1.5 , (c) y = - 0.5 , (d) y = 3 . Illumination is TE polarized at normal incidence. Results are shown for converged solutions in terms of internal reflections and compared with a highly accurate spectral solution. E e = 1 is a solution that includes the two evanescent modes from each single-interface solve with the smallest |β|.

Tables (2)

Tables Icon

Table 1 Convergence of Scattering Efficiencies and Relation to the Threshold Value w Used in the Iterative Approach

Tables Icon

Table 2 Time and Space Used in the Multilayer Boundary Variation Method Computations Presented in the Paper

Equations (37)

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E inc ( x ,   t ) H inc ( x ,   t ) = A E A H exp [ i ( k inc     x - ω t ) ] ,
k inc = 2 π λ v   0 k ^ inc , ω = 2 π   c v λ v .
× E j = i ω H j , × H j = - i ω j E j
    E j =     H j = 0 , j { 0 ,   1 , , N } .
n ^ j × E j - 1 + δ j - 1 , 0 E inc H j - 1 + δ j - 1 , 0 H inc = n ^ j × E j H j ,
n ^ j × E j - 1 = - n ^ j × δ j - 1 , 0 E inc .
n ^ j × × ( H j - 1 + δ j - 1 , 0 H inc )
= - n ^ j     ( H j - 1 + δ j - 1 , 0 H inc ) = - ( H j - 1 + δ j - 1 , 0 H inc ) n ^ j = 0
Δ u j + | k j | 2 u j = 0 , j = 0 , , N ,
u j - 1 ( x ,   f j ( x ) ) - u j ( x ,   f j ( x ) ) = - δ j - 1 , 0 u inc ( x ,   f j ( x ) ) ,
n ^ j   u j - 1 ( x ,   f j ( x ) ) - C j   n ^ j   u j ( x ,   f j ( x ) )
= - n ^ j   δ j - 1 , 0 u inc ( x ,   f j ( x ) )
C j = 1 , TE polarization j - 1 / j , TM polarization .
u j - 1 ( x ,   f j ( x ) ) = - δ j - 1 , 0 E z inc ( x ,   f j ( x ) ) ,
n ^ j   u j - 1 ( x ,   f j ( x ) ) = - n ^ j   δ j - 1 , 0 H z inc ( x ,   f j ( x ) )
u ± ( x + L ,   y ) = exp ( ik x inc L ) u ± ( x ,   y ) .
K = 2 π L , α n = k x inc + nK , α n 2 + ( β n ± ) 2 = | k ± | 2 .
u ± ( x ,   y ) = n = - B n ±   exp [ i ( α n x β n ± y ) ] .
n Π +   β n + | B n + | 2 +   C 0 n Π -   β n - | B n - | 2   = k y inc ,
f δ ( x ) = δ f ( x ,   L ) ;
B n ± ( δ ) = k = 0   1 k ! d k B n ± ( δ ) d δ k δ = 0 δ k = k = 0 d k , n δ k ;
u + ( x ,   f δ ( x ) ) = - exp [ ik x inc x + ik y inc δ f ( x ,   L ) ] .
1 k ! k u + δ k δ = 0 = n = - d k , n   exp [ i ( α n x - β n + y ) ] .
1 r ! r u + δ r y , δ = 0 = - ( ik y inc ) r   f r r !   exp ( ik x inc x ) - k = 0 r - 1   f r - k ( r - k ) !   r - k y r - k   1 k !   k u + δ k y , δ = 0 .
[ f ( x ,   L ) ] r r ! = l = - rF rF C r , l   exp ( iKlx ) .
d k , n = - ( ik y inc ) k C k , n - r = 0 k - 1 q = max [ - rF , n - ( k - r ) F ] min [ rF , n + ( k - r ) F ] C k - r , n - q ( - i β q ) k - r d r , q .
B n ( δ ) = k = 0 D d k , n δ k ,
[ L / M ] = a 0 + a 1 δ + + a L δ L 1 + b 1 δ + + b M δ M ,
u ± ( x ,   y ) = n = - p ± - E e q ± + E e B n ±   exp [ i ( α n x β n ± y ) ] ,
E ± ( x ,   y ) = n = - 1 1 E n ± ,
E n ± = B n ±   exp [ i ( α n x β n ± y ) ] , n { - 1 ,   0 ,   1 } .
α 2 + β 2 = | k j | 2 .
u ± ( x ,   y ) = n = - p ± - E e q ± + E e E n ± ,
E n ± = B n ±   exp [ i ( α n x β n ± y ) ]
n { - E e - p ± ,   - E e - p ± + 1 , , q ± + E e } .
= l = 1 s 0 e l 0 + 0 N   l = 1 s N e l N - 1 ,
u j = A 1 j   exp [ i ( α 1 j x - β 1 j y ) ] + A 2 j   exp [ i ( α 2 j x + β 2 j y ) ]

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