A design procedure is described to determine the thicknesses of single-layer coatings of a given dielectric on a given metallic substrate so that any input polarization of light is preserved after two reflections at the same angle of incidence from a parallel-mirror beam displacer or an axicon. This is achieved by equalizing the net complex p and s reflection coefficients (also called the radial and azimuthal eigenvalues of an axicon) after two reflections. The net polarization-independent reflectance (insertion loss) of the device is computed and found to exceed the net minimum parallel reflectance of the uncoated device for incidence above a certain angle. Thus the dielectric films serve (1) protective, (2) polarization-corrective, and (3) reflectance-enhancement functions simultaneously. The sensitivity of the designs (deviation from the condition of polarization preservation) to small film-thickness and angle-of-incidence errors is examined. Results are presented graphically and in tables of applications of the method to beam displacers and axicons for He–Ne laser light, λ = 0.6328 μm, using Al2O3 (or Si2O3)/Al and MgF2/Al film–substrate systems, and for CO2 laser light, λ = 10.6 μm, using ZnS or ThF4 films on an Ag substrate.
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ϕ is the angle of incidence in degrees; (ζ1,ζ2) is the normalized film-thickness solution pair; Dϕ is the film-thickness period in microns;
are the net reflectances of the uncoated device for the p and s polarizations; and ℛc is the net reflectance of the coated (polarization-corrected) device.
Table II
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1,Δd2) = (+1, ±1) nm, (II) Angle-of-Incidence Errors (Δϕ1,Δϕ2) = (+0.5, ±0.5°) to the Beam Displacer and Axicon Designs of Table I
Error type
Angle of incidence (ϕ) (deg)
Magnitude error (|ρn| − 1) × 104
Phase error (Δn) (deg)
+,−
+,+
+,−
+,+
I
30
7.717
−0.6488
0.3349
0.5126
45
20.35
3.418
0.8871
0.9740
A
42.19
17.63
1.8600
1.3343
B
27.02
16.68
2.9275
1.3713
60
C
−10.69
−12.34
3.3389
1.3617
II
30
4.533
1.131
−0.5056
0.3790
45
9.996
3.608
−1.09
0.1117
A
31.95
7.492
−1.5111
0.2194
B
17.32
7.222
−2.2037
0.2136
60
C
42.05
7.492
0.1109
0.2287
Table III
Summary of Design Results for MgF2/Al Polarization-Preserving Beam Displacers and Axicons at Three Angles of Incidence a
ϕ (deg)
ζ1
ζ2
Dϕ(μm)
ℛc
35
0.481728
0.510810
0.25208
0.7906
0.8546
0.7048
45
0.430213
0.564371
0.26693
0.7627
0.8733
0.7133
60
0.387927
0.611417
0.29451
0.6896
0.9089
0.7352
See text (or footnote to Table I) for explanation of notation.
Table IV
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1,Δd2) = (+1, ±1) nm, (II) Angle-of-Incidence Errors (Δϕ1,Δϕ2) = (+0.5, ±0.5°) to the Beam Displacer and Axicon Designs of Table III
Error type
Angle of incidence (ϕ) (deg)
Magnitude error (|ρn| − 1) × 103
Phase error (Δn) (deg)
+,−
+,+
+,−
+,+
I
35
0.1055
−0.4605
0.0496
0.7412
45
0.8767
0.4743
0.3369
1.0856
60
2.528
0.0568
0.9317
1.4810
II
35
0.0694
0.2115
0.0890
0.0718
45
0.4418
0.4489
−0.6309
0.1507
60
1.1113
0.9599
−1.9322
0.30722
Table V
Summary of Design Results for ZnS/Ag Polarization-Preserving Beam Displacers and Axicons at Three Angles of Incidence. a
ϕ (deg)
ζ1
ζ2
Dϕ(μm)
ℛc
A
0.248008
0.563386
2.47382
0.9839
0.9879
0.9606
30
B
0.389366
0.610939
2.47382
0.9839
0.9879
0.9545
C
0.418178
0.710221
2.47382
0.9839
0.9879
0.9593
A
0.259708
0.568079
2.54378
0.9804
0.9901
0.9611
45
B
0.385571
0.614810
2.54378
0.9804
0.9901
0.9568
C
0.412711
0.699789
2.54378
0.9804
0.9901
0.9601
A
0.272138
0.573447
2.62064
0.9724
0.9930
0.9641
60
B
0.381525
0.618975
2.62064
0.9724
0.9930
0.9627
C
0.406417
0.688617
2.62064
0.9724
0.9930
0.9637
See text (or footnote of Table I) for explanation of notation.
Table VI
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1,Δd2) = (+1, ±1) nm, (II) Angle-of-Incidence Errors (Δϕ1, Δϕ2) = (+0.5, ±0.5°) to the Beam Displacer and Axicon Designs of Table V
Type of error
Angle of incidence ϕ (deg)
Magnitude error (|ρn| − 1) × 104
Phase error (deg) Δn × 10
+,−
+,+
+,−
+,+
I
A
0.1472
−0.1454
−0.5402
0.2000
30
B
0.2051
0.02712
0.0919
0.2338
C
0.1552
0.1406
0.5156
0.2094
A
0.3619
−0.3335
−1.16
0.3939
45
B
0.4555
0.0633
0.2046
0.4430
C
0.3780
0.3156
1.08
0.4107
A
0.6909
−0.5617
−1.8122
0.5447
60
B
0.7210
0.1077
0.3284
0.5711
C
0.6909
0.5121
1.6234
0.5600
II
A
−0.8746
0.0503
−7.597
0.0224
30
B
0.1575
0.0597
−9.327
0.0259
C
0.8465
0.0537
−8.021
0.0230
A
−1.3839
0.1429
−14.33
0.0645
45
B
0.2457
0.1627
−14.94
0.0717
C
1.2966
0.1528
−16.66
0.0660
A
−1.3388
0.2299
−25.42
0.1034
60
B
0.2078
0.2433
−27.13
0.1064
C
1.1549
0.2440
−25.93
0.1036
Table VII
Summary of Design Data for ThF4-Ag Polarization-Preserving Beam Displacers and Axicons at Three Angles of Incidence a
ϕ (deg)
ζ1
ζ2
Dϕ(μm)
ℛ
30
0.482391
0.517601
4.15523
0.9839
0.9879
0.9741
45
0.426483
0.573512
4.51509
0.9804
0.9901
0.9753
60
0.389570
0.610428
4.99245
0.9724
0.9930
0.9778
See text (or footnote of Table I) for explanation of notation.
Table VIII
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1, Δd2) = (+1, ±1) nm, (II) Angle-of-Incidence Errors (Δϕ1, Δϕ2) = (0.5, ±0.5°) to Beam Displacer and Axicon Designs of Table VII
Type of error
Angle of incidence (ϕ) (deg)
Magnitude error (|ρn| − 1) × 106
Phase error (Δn in deg)
+,−
+,+
+,−
+,+
I
30
0.5513
−0.1490
0.0002
0.0338
45
5.1550
−0.1639
0.0022
0.0645
60
12.25
0.0894
0.0054
0.0870
II
30
0.0440
9.8190
−0.0941
0.0043
45
4.6190
33.190
−0.7465
0.0145
60
9.089
67.70
−2.0816
0.0298
Tables (8)
Table I
Summary of Design Results for Al2O3 (or Si2O3)/Al Polarization-Preserving Beam Dlsplacers and Axicons at Three Angles of Incidence
ϕ is the angle of incidence in degrees; (ζ1,ζ2) is the normalized film-thickness solution pair; Dϕ is the film-thickness period in microns;
are the net reflectances of the uncoated device for the p and s polarizations; and ℛc is the net reflectance of the coated (polarization-corrected) device.
Table II
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1,Δd2) = (+1, ±1) nm, (II) Angle-of-Incidence Errors (Δϕ1,Δϕ2) = (+0.5, ±0.5°) to the Beam Displacer and Axicon Designs of Table I
Error type
Angle of incidence (ϕ) (deg)
Magnitude error (|ρn| − 1) × 104
Phase error (Δn) (deg)
+,−
+,+
+,−
+,+
I
30
7.717
−0.6488
0.3349
0.5126
45
20.35
3.418
0.8871
0.9740
A
42.19
17.63
1.8600
1.3343
B
27.02
16.68
2.9275
1.3713
60
C
−10.69
−12.34
3.3389
1.3617
II
30
4.533
1.131
−0.5056
0.3790
45
9.996
3.608
−1.09
0.1117
A
31.95
7.492
−1.5111
0.2194
B
17.32
7.222
−2.2037
0.2136
60
C
42.05
7.492
0.1109
0.2287
Table III
Summary of Design Results for MgF2/Al Polarization-Preserving Beam Displacers and Axicons at Three Angles of Incidence a
ϕ (deg)
ζ1
ζ2
Dϕ(μm)
ℛc
35
0.481728
0.510810
0.25208
0.7906
0.8546
0.7048
45
0.430213
0.564371
0.26693
0.7627
0.8733
0.7133
60
0.387927
0.611417
0.29451
0.6896
0.9089
0.7352
See text (or footnote to Table I) for explanation of notation.
Table IV
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1,Δd2) = (+1, ±1) nm, (II) Angle-of-Incidence Errors (Δϕ1,Δϕ2) = (+0.5, ±0.5°) to the Beam Displacer and Axicon Designs of Table III
Error type
Angle of incidence (ϕ) (deg)
Magnitude error (|ρn| − 1) × 103
Phase error (Δn) (deg)
+,−
+,+
+,−
+,+
I
35
0.1055
−0.4605
0.0496
0.7412
45
0.8767
0.4743
0.3369
1.0856
60
2.528
0.0568
0.9317
1.4810
II
35
0.0694
0.2115
0.0890
0.0718
45
0.4418
0.4489
−0.6309
0.1507
60
1.1113
0.9599
−1.9322
0.30722
Table V
Summary of Design Results for ZnS/Ag Polarization-Preserving Beam Displacers and Axicons at Three Angles of Incidence. a
ϕ (deg)
ζ1
ζ2
Dϕ(μm)
ℛc
A
0.248008
0.563386
2.47382
0.9839
0.9879
0.9606
30
B
0.389366
0.610939
2.47382
0.9839
0.9879
0.9545
C
0.418178
0.710221
2.47382
0.9839
0.9879
0.9593
A
0.259708
0.568079
2.54378
0.9804
0.9901
0.9611
45
B
0.385571
0.614810
2.54378
0.9804
0.9901
0.9568
C
0.412711
0.699789
2.54378
0.9804
0.9901
0.9601
A
0.272138
0.573447
2.62064
0.9724
0.9930
0.9641
60
B
0.381525
0.618975
2.62064
0.9724
0.9930
0.9627
C
0.406417
0.688617
2.62064
0.9724
0.9930
0.9637
See text (or footnote of Table I) for explanation of notation.
Table VI
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1,Δd2) = (+1, ±1) nm, (II) Angle-of-Incidence Errors (Δϕ1, Δϕ2) = (+0.5, ±0.5°) to the Beam Displacer and Axicon Designs of Table V
Type of error
Angle of incidence ϕ (deg)
Magnitude error (|ρn| − 1) × 104
Phase error (deg) Δn × 10
+,−
+,+
+,−
+,+
I
A
0.1472
−0.1454
−0.5402
0.2000
30
B
0.2051
0.02712
0.0919
0.2338
C
0.1552
0.1406
0.5156
0.2094
A
0.3619
−0.3335
−1.16
0.3939
45
B
0.4555
0.0633
0.2046
0.4430
C
0.3780
0.3156
1.08
0.4107
A
0.6909
−0.5617
−1.8122
0.5447
60
B
0.7210
0.1077
0.3284
0.5711
C
0.6909
0.5121
1.6234
0.5600
II
A
−0.8746
0.0503
−7.597
0.0224
30
B
0.1575
0.0597
−9.327
0.0259
C
0.8465
0.0537
−8.021
0.0230
A
−1.3839
0.1429
−14.33
0.0645
45
B
0.2457
0.1627
−14.94
0.0717
C
1.2966
0.1528
−16.66
0.0660
A
−1.3388
0.2299
−25.42
0.1034
60
B
0.2078
0.2433
−27.13
0.1064
C
1.1549
0.2440
−25.93
0.1036
Table VII
Summary of Design Data for ThF4-Ag Polarization-Preserving Beam Displacers and Axicons at Three Angles of Incidence a
ϕ (deg)
ζ1
ζ2
Dϕ(μm)
ℛ
30
0.482391
0.517601
4.15523
0.9839
0.9879
0.9741
45
0.426483
0.573512
4.51509
0.9804
0.9901
0.9753
60
0.389570
0.610428
4.99245
0.9724
0.9930
0.9778
See text (or footnote of Table I) for explanation of notation.
Table VIII
Magnitude and Phase Errors Caused by Introducing (I) Film-Thickness Errors (Δd1, Δd2) = (+1, ±1) nm, (II) Angle-of-Incidence Errors (Δϕ1, Δϕ2) = (0.5, ±0.5°) to Beam Displacer and Axicon Designs of Table VII