Abstract

We study the far-field reflected diffraction pattern of an index discontinuity in a thin one-dimensional slab illuminated by a plane wave and show that a time-saving modeling technique based on plane wave expansion approaches fairly well the Maxwell-based rigorous models. This method is simple to implement, and it fur thermore allows a good understanding of the optical phenomena involved in the propagation of light through the slab.

© 2011 Optical Society of America

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    [CrossRef]

2010 (1)

2009 (1)

2008 (1)

2005 (1)

2002 (1)

2001 (1)

2000 (1)

1999 (1)

1998 (1)

1994 (1)

1990 (1)

1981 (1)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am 71, 811-818 (1981).
[CrossRef]

1976 (1)

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

Aagedal, H.

Bosch, S.

Campos, J.

Carcolé, E.

Chavel, P.

Cottrell, D. M.

Davis, J. A.

Drauschke, A.

Fang, Z.

Gaylord, T. K.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am 71, 811-818 (1981).
[CrossRef]

Glytsis, E. N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Sec. 3.10, pp. 55-60.

Goudail, F.

Hedman, T. R.

Hugonin, J. P.

Kettunen, V.

Kuang, D.

Kuittinen, M.

Lalanne, P.

Lalor, E.

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

Lilly, R. A.

Moharam, M. G.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am 71, 811-818 (1981).
[CrossRef]

Moulin, G.

Peloux, M.

Pfeil, A. v.

Sherman, G. C.

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

Singer, W.

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

Swanson, G. J.

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Technical Report 914 (MIT Lexington Lincoln Lab, 1991).

Taboury, J.

Testorf, M.

Tiziani, H.

Turunen, J.

Wang, H.

Wyrowski, F.

Appl. Opt. (5)

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760-776 (1976).
[CrossRef]

J. Opt. Soc. Am (1)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am 71, 811-818 (1981).
[CrossRef]

J. Opt. Soc. Am. A (6)

Other (2)

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Technical Report 914 (MIT Lexington Lincoln Lab, 1991).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Sec. 3.10, pp. 55-60.

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Figures (11)

Fig. 1
Fig. 1

(a) Partial waves 1 ( x < 0 ) and 2 ( x > 0 ) and (b) partial waves 3 [ x > 2 H tan ( θ n 02 ) ] and 4 ( x < 0 ). Partial waves 1 and 4 have semi-infinite pupils on the left, and partial waves 2 and 3 have semi-infinite pupils on the right.

Fig. 2
Fig. 2

(a) Partial waves 5 [ 2 H tan ( θ n 01 ) < x < H tan ( θ n 01 ) ] and 5 [ H tan ( θ n 12 ) < x < 2 H tan ( θ n 12 ) ] and (b) partial waves 6 [ H tan ( θ n 01 ) < x < 0 ] and 6 [ 0 < x < H tan ( θ n 12 ) ]. Partial waves 5, 6, 5 , and 6 are limited by the vertical index discontinuity in the slab and therefore have finite-size pupils.

Fig. 3
Fig. 3

Unfolded scheme associated to the study of partial wave 3. The thick vertical dashed lines stand for the n 1 n 2 discontinuity, and the horizontal dashed lines stand for the interface between the slab and the substrate.

Fig. 4
Fig. 4

TE mode ( n 1 = 1 , n 2 = 1.5 , n 3 = 1.5 , H = 4 μm , θ = 30 ° ). Central peak associated to cases 1 to 4 and peak associated to cases 5 and 6. The specular peak grows to infinity at θ = 30 ° , as should be expected, and it has therefore been truncated.

Fig. 5
Fig. 5

TE mode ( n 1 = 1.4 , n 2 = 1.5 , n 3 = 1 , H = 20 μm , θ = 10 ° ). Left (near 34.4 ° ): peak associated to cases 5 and 6 , middle (near 10 ° ): central peak associated to cases 1 to 4 and right (near 10 ° ): peak associated to cases 5 and 6. The central peak grows to infinity at θ = 10 ° , as should be expected, and it has therefore been truncated.

Fig. 6
Fig. 6

TE mode ( n 1 = 1.5 , n 2 = 1.4 , n 3 = 1 , H = 20 μm , θ = 10 ° ). Left (near 10 ° ): central peak associated to cases 1 to 4 and right (near 10 ° ): peak associated to cases 5 and 6. The central peak grows to infinity at θ = 10 ° , as should be expected, and it has therefore been truncated.

Fig. 7
Fig. 7

TE mode ( n 1 = 1.5 , n 2 = 1.4 , n 3 = 1 , H = 20 μm , θ = 10 ° ). Peak associated to partial waves 5 and 6, calculated using our FOM, considering case 5 only (thick orange curve), case 6 only (brown, thick, dashed curve), cases 1 to 4 (magenta dashed–dotted curve), and considering cases 1, 2, 3, 4, 5, and 6 (blue solid curve).

Fig. 8
Fig. 8

Unfolded scheme associated to the study of partial wave 5.

Fig. 9
Fig. 9

Unfolded scheme associated to the study of partial wave 6.

Fig. 10
Fig. 10

Unfolded scheme associated to the study of partial wave 5 .

Fig. 11
Fig. 11

Unfolded scheme associated to the study of partial wave 6 .

Equations (31)

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u ( α ) 1 α 2 λ u ˜ ( α λ ) .
u 3 ( x , 2 H + ) = He ( x ) e i 2 π λ n 0 x sin θ t 02 ( θ ) ,
t 02 TE ( θ ) = 2 n 0 cos ( θ ) n 0 cos ( θ ) + n 2 cos ( θ n 02 ) .
u ˜ 3 , 2 H ( μ ) = t 02 ( θ ) 2 π i ( μ n 0 sin θ / λ ) .
u 3 , r ( x , z ) = u ˜ 3 , 2 H ( α n λ ) e i 2 π λ ( 2 H + z ) n 2 2 α n 2 e i 2 π λ x α n d α n λ ,
u 3 , R ( x , z ) = r 23 ( α n ) u ˜ 3 , 2 H ( α n λ ) e i 2 π λ ( 2 H + z ) n 2 2 α n 2 e i 2 π λ x α n d α n λ ,
r 23 , TE ( α n ) = n 2 cos θ n n 3 cos θ n , 3 n 2 cos θ n + n 3 cos θ n , 3 = n 2 2 α n 2 n 3 2 α n 2 n 2 2 α n 2 + n 3 2 α n 2 .
u 3 , R T ( x , z ) = t 20 ( α n ) r 23 ( α n ) u ˜ 3 , 2 H ( α n λ ) e i 2 π λ 2 H n 2 2 α n 2 e i 2 π λ z n 0 2 α n 2 e i 2 π λ x α n d α n λ ,
t 20 , TE ( α n ) = 2 n 2 cos θ n n 2 cos θ n + n 0 cos θ n , 0 = 2 n 2 2 α n 2 n 2 2 α n 2 + n 0 2 α n 2 .
u 3 , ( α ) 1 α 2 λ t 20 ( α ) r 23 ( α ) u ˜ 3 , 2 H ( α λ ) e i 2 π λ 2 H n 2 2 α 2 .
u 1 ( x , 0 + ) = He ( x ) e i 2 π λ n 0 x sin θ t 01 ( θ ) ,
u 1 , ( α ) 1 α 2 λ u ˜ 1 , 0 ( α λ ) .
u 2 ( x , 0 + ) = He ( x ) e i 2 π λ n 0 x sin θ t 02 ( θ ) ,
u 2 , ( α ) 1 α 2 λ u ˜ 2 , 0 ( α λ ) .
u 4 ( x , 0 + ) = He ( x ) e i 2 π λ n 0 x sin θ t 01 ( θ ) r 13 ( θ ) t 10 ( θ ) e i 2 π λ × 2 n 1 H cos θ n 01 ,
u 4 , ( α ) 1 α 2 λ u ˜ 4 , 0 ( α λ ) .
u 5 ( 0 , z ) = Π H ( z + 3 H 2 ) e i 2 π λ n 1 ( 2 H + z ) cos θ n 01 t 01 ( θ ) r 12 ( θ ) ,
u ˜ 5 , 0 ( μ ) = t 01 ( θ ) r 12 ( θ ) e i 2 π λ 2 H n 1 cos θ n 01 H sinc [ H ( μ n 1 cos θ n 01 λ ) ] e i 3 π H ( μ n 1 cos θ n 01 λ ) .
u 5 , r ( x , z ) = u ˜ 5 , 0 ( α n λ ) e i 2 π λ x n 1 2 α n 2 e i 2 π λ z α n d α n λ .
r 13 , TE ( α n ) = n 1 cos θ n n 3 cos θ n , 3 n 1 cos θ n + n 3 cos θ n , 3 = α n n 3 2 n 1 2 + α n 2 α n + n 3 2 n 1 2 + α n 2 .
u 5 , R T ( x , z ) = t 10 ( α n ) r 13 ( α n ) u ˜ 5 , 0 ( α n λ ) e i 2 π λ z n 0 2 ( n 1 2 α n 2 ) e i 2 π λ x n 1 2 α n 2 d α n λ .
t 10 , TE ( α n ) = 2 n 1 cos θ n n 1 cos θ n + n 0 cos θ n , 0 = 2 α n α n + n 0 2 n 1 2 + α n 2 .
u 5 , ( α ) 1 α 2 λ t 10 ( n 1 2 α 2 ) r 13 ( n 1 2 α 2 ) u ˜ 5 , 0 ( n 1 2 α 2 λ ) α n 1 2 α 2 .
u 6 ( 0 , z ) = Π H ( z + H 2 ) e i 2 π λ n 1 2 H cos θ n 01 e i 2 π λ n 1 z cos θ n 01 t 01 ( θ ) r 13 ( θ ) r 12 ( θ )
u 6 , ( α ) 1 α 2 λ t 10 ( n 1 2 α 2 ) u ˜ 6 , 0 ( n 1 2 α 2 λ ) α n 1 2 α 2 .
u 5 ( 0 + , z ) = Π H ( z + 3 H 2 ) e i 2 π λ n 1 ( 2 H + z ) cos θ n 01 t 01 ( θ ) r 12 ( θ ) .
u 5 , ( α ) 1 α 2 λ t 20 ( n 2 2 α 2 ) r 23 ( n 2 2 α 2 ) u ˜ 5 , 0 ( n 2 2 α 2 λ ) α n 2 2 α 2 ,
t 20 , TE ( α n ) = 2 α n α n + n 0 2 n 2 2 + α n 2 ,
r 23 , TE ( α n ) = α n n 3 2 n 2 2 + α n 2 α n + n 3 2 n 2 2 + α n 2 .
u 6 ( 0 + , z ) = Π H ( z + H 2 ) e i 2 π λ n 1 2 H cos θ n 01 e i 2 π λ n 1 z cos θ n 01 t 01 ( θ ) r 13 ( θ ) t 12 ( θ ) .
u 6 , ( α ) 1 α 2 λ t 20 ( n 2 2 α 2 ) u ˜ 6 , 0 ( n 2 2 α 2 λ ) α n 2 2 α 2 .

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