1Department of Physics II and Institute of Optical Research, The Royal Institute of Technology, S-100 44, Stockholm, Sweden
2Laboratoire d’Optique Electromagnétique, Equipe de Recherche Associée au Centre National de la Recherche Scientifique No. 597, Faculté des Sciences et Techniques, Centre de Saint-Jérôme, F-13397 Marseille Cedex 4, France
M. Breidne and D. Maystre, "Variational theory of diffraction gratings and its application to the study of ghosts," J. Opt. Soc. Am. 72, 499-506 (1982)
A variational theory is presented that describes the diffraction properties of perfectly conducting gratings in TE polarization. By using this formalism we solve the problem of the energy distribution of light diffracted by gratings with periodic errors.
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TE Efficiency in the −1 Order Obtained with Different Functions φ and Different Variational Formulas for Sinusoidal Gratings with Different Normalized Depths
EφPavageau
EφMirror
d
E(0)
E(1)
E(2)
E(0)
E(1)
E(2)
EφIntegral
0.05
0.03728
0.02386
0.02589
0.00745
0.02981
0.2 × 106
0.02400
0.10
0.13852
0.09050
0.09734
0.02914
0.11728
–
0.09260
0.20
0.40881
0.29574
0.29963
0.10945
0.43571
–
0.32431
Table 2
TE Efficiency in the −1 Order Obtained from Different Trial Functions ϕ(m) and Compared with the Values given by the Integral Formalisma
h/d
NIP
NTG
φ(0)
φ(1)
φ(2)
φ(3)
φ(5)
φIntegral
0.05
35
31
0.02414
0.02400
0.02400
0.02400
0.02400
0.02400
17
15
0.02414
0.02400
0.02400
0.02400
0.02400
0.02400
0.10
35
31
0.09467
0.09261
0.09260
0.09260
0.09260
0.09260
17
15
0.09467
0.09261
0.09260
0.09260
0.09260
0.09260
0.40
35
31
0.72880
0.83579
0.83705
0.83629
0.83630
0.83630
17
15
0.72888
0.83615
0.83748
0.83672
0.83671
0.83671
0.70
35
31
0.23216
0.68525
0.84461
0.86761
0.86763
0.86763
The efficiencies have been calculated for different normalized depths, number of integration points (NIP), and number of terms in the series giving G(x, ξ, θ) (NTG).
Table 3
TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya
Efficiency
Order
Variational
Integral
−2
0.34339
0.33998
−1 (ghost)
0.00441
0.00438
0
0.31345
0.31127
+1 (ghost)
0.00441
0.00438
+2
0.34341
0.33998
∑
1.00907
0.99999
The grating has the profile given by formula (22); h1/d = 0.1, h2/d = 0.09, θ = 0°. The variational formalism values have been calculated from Eq. (8).
Table 4
TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya
Efficiency
Order
Variational
Variational (After Renormalization)
Integral
−2
0.28086
0.24494
0.24234
−1 (ghost)
0.07377
0.06434
0.06740
0
0.43737
0.38145
0.38051
+1 (ghost)
0.07376
0.06433
0.06741
+2
0.28086
0.24494
0.24234
∑
1.14661
1.00000
1.00000
The grating has the profile given by formula (22); h1/d = 0.1, h2/d = 0.06, θ = 0°. The variational formalism values have been calculated from Eq. (8).
Table 5
TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya
Efficiency
Order
Variational
Variational (After Renormalization)
Integral
3 (ghost)
0.00579
0.00565
0.00521
2
0.50370
0.49158
0.49193
1 (ghost)
0.00994
0.00970
0.00983
0
0.49690
0.48494
0.48491
1 (ghost)
0.00833
0.00813
0.00816
∑
1.02466
1.00000
1.00004
The grating has the profile given by formula (22); h1/d = 0.1, h2/d = 0.085, θ = 20°.
Tables (5)
Table 1
TE Efficiency in the −1 Order Obtained with Different Functions φ and Different Variational Formulas for Sinusoidal Gratings with Different Normalized Depths
EφPavageau
EφMirror
d
E(0)
E(1)
E(2)
E(0)
E(1)
E(2)
EφIntegral
0.05
0.03728
0.02386
0.02589
0.00745
0.02981
0.2 × 106
0.02400
0.10
0.13852
0.09050
0.09734
0.02914
0.11728
–
0.09260
0.20
0.40881
0.29574
0.29963
0.10945
0.43571
–
0.32431
Table 2
TE Efficiency in the −1 Order Obtained from Different Trial Functions ϕ(m) and Compared with the Values given by the Integral Formalisma
h/d
NIP
NTG
φ(0)
φ(1)
φ(2)
φ(3)
φ(5)
φIntegral
0.05
35
31
0.02414
0.02400
0.02400
0.02400
0.02400
0.02400
17
15
0.02414
0.02400
0.02400
0.02400
0.02400
0.02400
0.10
35
31
0.09467
0.09261
0.09260
0.09260
0.09260
0.09260
17
15
0.09467
0.09261
0.09260
0.09260
0.09260
0.09260
0.40
35
31
0.72880
0.83579
0.83705
0.83629
0.83630
0.83630
17
15
0.72888
0.83615
0.83748
0.83672
0.83671
0.83671
0.70
35
31
0.23216
0.68525
0.84461
0.86761
0.86763
0.86763
The efficiencies have been calculated for different normalized depths, number of integration points (NIP), and number of terms in the series giving G(x, ξ, θ) (NTG).
Table 3
TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya
Efficiency
Order
Variational
Integral
−2
0.34339
0.33998
−1 (ghost)
0.00441
0.00438
0
0.31345
0.31127
+1 (ghost)
0.00441
0.00438
+2
0.34341
0.33998
∑
1.00907
0.99999
The grating has the profile given by formula (22); h1/d = 0.1, h2/d = 0.09, θ = 0°. The variational formalism values have been calculated from Eq. (8).
Table 4
TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya
Efficiency
Order
Variational
Variational (After Renormalization)
Integral
−2
0.28086
0.24494
0.24234
−1 (ghost)
0.07377
0.06434
0.06740
0
0.43737
0.38145
0.38051
+1 (ghost)
0.07376
0.06433
0.06741
+2
0.28086
0.24494
0.24234
∑
1.14661
1.00000
1.00000
The grating has the profile given by formula (22); h1/d = 0.1, h2/d = 0.06, θ = 0°. The variational formalism values have been calculated from Eq. (8).
Table 5
TE Efficiency in the Different Propagating Orders as Obtained from the Variational and Integral Formalisms, Respectivelya
Efficiency
Order
Variational
Variational (After Renormalization)
Integral
3 (ghost)
0.00579
0.00565
0.00521
2
0.50370
0.49158
0.49193
1 (ghost)
0.00994
0.00970
0.00983
0
0.49690
0.48494
0.48491
1 (ghost)
0.00833
0.00813
0.00816
∑
1.02466
1.00000
1.00004
The grating has the profile given by formula (22); h1/d = 0.1, h2/d = 0.085, θ = 20°.