When this research was performed the author was with Plummer Precision Optics, 601 Montgomery Avenue, Pennsburg, Pennsylvania 18073; the author is now with Barr Associates, Inc., 2 Lyberty Way, P.O. Box 557, Westford, Massachusetts 01886-3616
The principal aspects of rugate filter theory are reviewed and expanded to show how the Fourier-transform technique can be used to design rugate filters that fulfill many optical coating functions.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Comparison between the Reflectances Obtained by the Exact (RA Exact) and Approximate (RA Approx.) Optical Thickness Calculation Methodsa
RA(%) Desired
xA (nm) Exact
n+Exact
n− Exact
RA(%) Exact
xA(nm) Approx.
n+ Approx.
n− Approx.
RA(%) Approx.
99.00
5962.0
1.6395
1.7689
99.00
5925.5
1.5838
1.8311
98.93
90.00
3551.7
1.6177
1.7926
90.01
3599.9
1.5495
1.8716
90.71
50.00
1691.3
1.4658
1.9785
50.07
1744.8
1.4501
1.9998
50.29
4.00
448.9
1.7915
1.6188
4.00
401.3
1.8697
1.5510
2.67
The layer matches the outer media. The truncated and the original phase factors are zero in all cases. nM = 2.0 and nm = 1.45.
Table 2
Influence of β on the Continuous Fourier Component of the Kaiser Apodization Windowa
β
0.0
1.0
2.0
3.0
4.0
5.0
6.0
W(0)
1.0
0.9282
0.7955
0.6842
0.6037
0.5448
0.5
xA (nm)
3599.9
3878.4
4525.3
5261.5
5963.1
6607.7
7199.8
The optical thicknesses are those required to achieve 90% reflectance at 1 μm with an optical admittance profile truncated by means of the corresponding Kaiser function. We assume no admittance contrast at the outer boundaries of the rugate mirror and a phase factor equal to 0°.
Table 3
Optical Thickness of Several Truncated Single-Line Rugate Mirrors Designed to have Minimum Optical Thicknessa
Phase Factors and Minimum and Maximum Optical Admittances for Three Quarter-Wave Rugate Antireflection Coatings
m
ΦA (deg)
ηm
ηM
1
90
1.45
1.8419
2
−90
1.45
1.6200
3
90
1.45
1.5584
Table 6
Results Obtained in the Case of Narrow-Band Filtersa
ΦA
−hAσa
σq/σA
Resonance
(2n + 1)π/2
p/2
q/p
Resonance at σA
(2n + p)π
p/2
(2q + 1)/2p
No resonance at σA
nπ
(2p + 1)/4
(2q + 1)/(2p + 1)
Resonance at σA
nπ
(2p + n)/2
(2q + 1)/(2p + n)
No resonance at σA
n, p, and q are integers. The wave numbers σq give the positions of the transmitting peaks.
Tables (6)
Table 1
Comparison between the Reflectances Obtained by the Exact (RA Exact) and Approximate (RA Approx.) Optical Thickness Calculation Methodsa
RA(%) Desired
xA (nm) Exact
n+Exact
n− Exact
RA(%) Exact
xA(nm) Approx.
n+ Approx.
n− Approx.
RA(%) Approx.
99.00
5962.0
1.6395
1.7689
99.00
5925.5
1.5838
1.8311
98.93
90.00
3551.7
1.6177
1.7926
90.01
3599.9
1.5495
1.8716
90.71
50.00
1691.3
1.4658
1.9785
50.07
1744.8
1.4501
1.9998
50.29
4.00
448.9
1.7915
1.6188
4.00
401.3
1.8697
1.5510
2.67
The layer matches the outer media. The truncated and the original phase factors are zero in all cases. nM = 2.0 and nm = 1.45.
Table 2
Influence of β on the Continuous Fourier Component of the Kaiser Apodization Windowa
β
0.0
1.0
2.0
3.0
4.0
5.0
6.0
W(0)
1.0
0.9282
0.7955
0.6842
0.6037
0.5448
0.5
xA (nm)
3599.9
3878.4
4525.3
5261.5
5963.1
6607.7
7199.8
The optical thicknesses are those required to achieve 90% reflectance at 1 μm with an optical admittance profile truncated by means of the corresponding Kaiser function. We assume no admittance contrast at the outer boundaries of the rugate mirror and a phase factor equal to 0°.
Table 3
Optical Thickness of Several Truncated Single-Line Rugate Mirrors Designed to have Minimum Optical Thicknessa