Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina
We present a rigorous electromagnetic approach to wave diffraction by corrugated gratings made of uniaxial crystals. The optic axis of the anisotropic medium is assumed to lie on the mean surface of the grating, inclined at an arbitrary angle with respect to the grooves. The diffraction problem is exactly analyzed as a two-medium boundary-value problem. We simplify the fully vectorial treatment by first writing the fields everywhere in terms of the components of the electric and magnetic fields along the groove direction. Then a coordinate transformation mapping the corrugated interface into a plane is used, and the transformed propagation equations are solved by means of a differential method. The theory is exemplified numerically for the case of gratings made of sodium nitrate, and the results are compared against those obtained with a simplified formalism invoking the Rayleigh hypothesis.
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Comparison between Efficiencies Calculated with the Present Method (PM) and the Rayleigh Method12 (RM) for Zero Reflected, Transmitted Ordinary and Transmitted Extraordinary Orders, Different Values of h/d, and TE Incident-Wave Polarizationa
h/d
r110
r120
t120
t1E0
PC
0.2
RM
2.349940 × 10−2
9.291237 × 10−4
0.5003882
0.3817185
0.999999
PM
2.349713 × 10−2
9.308289 × 10−4
0.5003998
0.3817256
1.000570
0.28
RM
1.824551 × 10−2
7.281403 × 10−4
0.4819141
0.3734616
0.999988
PM
1.829426 × 10−2
7.309628 × 10−4
0.4819555
0.3734602
1.000964
0.3
RM
1.779921 × 10−2
7.131458 × 10−4
0.4802781
0.3727218
0.999976
PM
1.786616 × 10−2
7.165323 × 10−4
0.4803594
0.3727146
1.001070
N⊥ = 1.584, N|| = 1.336, λ/d = 1, τ = 40°, and θ0 = 5°. The column labeled PC gives the sum of the diffracted efficiencies.
Table 2
Comparison between Efficiencies Calculated with the Present Method (PM) and the Rayleigh Method12 (RM) for Zero Reflected, Transmitted Ordinary and Transmitted Extraordinary Orders, Different Values of h/d, and TM Incident-Wave Polarizationa
h/d
r210
r220
t2O0
t2E0
PC
0.2
RM
9.291416 × 10−4
1.502894 × 10−2
0.3808179
0.5782137
0.999999
PM
9.283676 × 10−4
1.503069 × 10−2
0.3809162
0.5783028
1.000396
0.28
RM
7.284584 × 10−4
1.178003 × 10−2
0.3766673
0.5794421
0.999920
PM
7.295183 × 10−4
1.178330 × 10−2
0.3769412
0.5796687
1.000627
0.3
RM
7.135574 × 10−4
1.158635 × 10−2
0.3762650
0.5795436
0.999846
PM
7.158222 × 10−4
1.159385 × 10−2
0.3766379
0.5798405
1.000687
N⊥ = 1.584, N|| = 1.336, λ/d = 1, τ = 40°, and θ0 = 5° The column labeled PC gives the sum of the diffracted efficiencies.
Table 3
Comparison between Efficiencies Calculated with the Present Method (PM) and the Rayleigh Method12 (RM) for Zero Reflected, Transmitted Ordinary and Transmitted Extraordinary Orders, Different Values of h/d, and TE Incident-Wave Polarizationa
h/d
r110
r120
t1O0
t1E0
PC
0.2
RM
3.257077 × 10−3
1.554082 × 10−4
0.4173479
0.3615221
0.999970
PM
3.237676 × 10−3
1.542516 × 10−4
0.4184191
0.3619553
1.000705
0.28
RM
5.917718 × 10−3
2.678355 × 10−4
0.3592572
0.3411313
1.001449
PM
6.152642 × 10−3
2.619493 × 10−4
0.3580000
0.3403898
1.001244
0.3
RM
5.960516 × 10−3
2.915793 × 10−4
0.3552869
0.3394751
1.003220
PM
6.423553 × 10−3
2.689754 × 10−4
0.3503602
0.3374002
1.000046
N⊥ = 1.584, N|| = 1.336, λ/d = 0.5, τ = 40°, and θ0 = 5°. The column labeled PC gives the sum of the diffracted efficiencies.
Table 4
Comparison between Efficiencies Calculated with the Present Method (PM) and the Rayleigh Method12 (RM) for Zero Reflected, Transmitted Ordinary and Transmitted Extraordinary Orders, Different Values of h/d, and TM Incident-Wave Polarizationa
h/d
r210
r220
t2O0
t2E
PC
0.2
RM
1.574516 × 10−4
3.261164 × 10−3
0.3218745
0.5471473
0.999865
PM
1.545082 × 10−4
3.246229 × 10−3
0.3225694
0.5479696
1.000503
0.28
RM
2.556615 × 10−4
4.322938 × 10−3
0.2970085
0.5385833
1.001903
PM
2.649408 × 10−4
4.878990 × 10−3
0.2951340
0.5382009
1.000850
0.3
RM
2.338947 × 10−4
5.393274 × 10−3
0.2928591
0.5313452
0.992259
PM
2.734019 × 10−4
4.955189 × 10−3
0.2922081
0.5376271
1.001000
N⊥ = 1.584, N|| = 1.336, λ/d = 0.5, τ = 40°, and θ0 = 5°. The column labeled PC gives the sum of the diffracted efficiencies.
Tables (4)
Table 1
Comparison between Efficiencies Calculated with the Present Method (PM) and the Rayleigh Method12 (RM) for Zero Reflected, Transmitted Ordinary and Transmitted Extraordinary Orders, Different Values of h/d, and TE Incident-Wave Polarizationa
h/d
r110
r120
t120
t1E0
PC
0.2
RM
2.349940 × 10−2
9.291237 × 10−4
0.5003882
0.3817185
0.999999
PM
2.349713 × 10−2
9.308289 × 10−4
0.5003998
0.3817256
1.000570
0.28
RM
1.824551 × 10−2
7.281403 × 10−4
0.4819141
0.3734616
0.999988
PM
1.829426 × 10−2
7.309628 × 10−4
0.4819555
0.3734602
1.000964
0.3
RM
1.779921 × 10−2
7.131458 × 10−4
0.4802781
0.3727218
0.999976
PM
1.786616 × 10−2
7.165323 × 10−4
0.4803594
0.3727146
1.001070
N⊥ = 1.584, N|| = 1.336, λ/d = 1, τ = 40°, and θ0 = 5°. The column labeled PC gives the sum of the diffracted efficiencies.
Table 2
Comparison between Efficiencies Calculated with the Present Method (PM) and the Rayleigh Method12 (RM) for Zero Reflected, Transmitted Ordinary and Transmitted Extraordinary Orders, Different Values of h/d, and TM Incident-Wave Polarizationa
h/d
r210
r220
t2O0
t2E0
PC
0.2
RM
9.291416 × 10−4
1.502894 × 10−2
0.3808179
0.5782137
0.999999
PM
9.283676 × 10−4
1.503069 × 10−2
0.3809162
0.5783028
1.000396
0.28
RM
7.284584 × 10−4
1.178003 × 10−2
0.3766673
0.5794421
0.999920
PM
7.295183 × 10−4
1.178330 × 10−2
0.3769412
0.5796687
1.000627
0.3
RM
7.135574 × 10−4
1.158635 × 10−2
0.3762650
0.5795436
0.999846
PM
7.158222 × 10−4
1.159385 × 10−2
0.3766379
0.5798405
1.000687
N⊥ = 1.584, N|| = 1.336, λ/d = 1, τ = 40°, and θ0 = 5° The column labeled PC gives the sum of the diffracted efficiencies.
Table 3
Comparison between Efficiencies Calculated with the Present Method (PM) and the Rayleigh Method12 (RM) for Zero Reflected, Transmitted Ordinary and Transmitted Extraordinary Orders, Different Values of h/d, and TE Incident-Wave Polarizationa
h/d
r110
r120
t1O0
t1E0
PC
0.2
RM
3.257077 × 10−3
1.554082 × 10−4
0.4173479
0.3615221
0.999970
PM
3.237676 × 10−3
1.542516 × 10−4
0.4184191
0.3619553
1.000705
0.28
RM
5.917718 × 10−3
2.678355 × 10−4
0.3592572
0.3411313
1.001449
PM
6.152642 × 10−3
2.619493 × 10−4
0.3580000
0.3403898
1.001244
0.3
RM
5.960516 × 10−3
2.915793 × 10−4
0.3552869
0.3394751
1.003220
PM
6.423553 × 10−3
2.689754 × 10−4
0.3503602
0.3374002
1.000046
N⊥ = 1.584, N|| = 1.336, λ/d = 0.5, τ = 40°, and θ0 = 5°. The column labeled PC gives the sum of the diffracted efficiencies.
Table 4
Comparison between Efficiencies Calculated with the Present Method (PM) and the Rayleigh Method12 (RM) for Zero Reflected, Transmitted Ordinary and Transmitted Extraordinary Orders, Different Values of h/d, and TM Incident-Wave Polarizationa
h/d
r210
r220
t2O0
t2E
PC
0.2
RM
1.574516 × 10−4
3.261164 × 10−3
0.3218745
0.5471473
0.999865
PM
1.545082 × 10−4
3.246229 × 10−3
0.3225694
0.5479696
1.000503
0.28
RM
2.556615 × 10−4
4.322938 × 10−3
0.2970085
0.5385833
1.001903
PM
2.649408 × 10−4
4.878990 × 10−3
0.2951340
0.5382009
1.000850
0.3
RM
2.338947 × 10−4
5.393274 × 10−3
0.2928591
0.5313452
0.992259
PM
2.734019 × 10−4
4.955189 × 10−3
0.2922081
0.5376271
1.001000
N⊥ = 1.584, N|| = 1.336, λ/d = 0.5, τ = 40°, and θ0 = 5°. The column labeled PC gives the sum of the diffracted efficiencies.