Abstract

A periodic multilayer reflector composed of two nonabsorbing materials is found to have a phase retardance that is a function of the refractive indices of the materials, the periodic design, and the incidence angle. The addition of a reflection enhancing periodic multilayer to a mirror causes the phase retardance to change to a unique value that depends on the properties of the added layers and the incidence angle. After reaching this limiting value the retardance is not changed by the further addition of reflection enhancing layers with the same periodic structure. Equations for the limiting phase retardance of a periodic multilayer reflector are provided.

© 1982 Optical Society of America

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References

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  1. J. H. Apfel, Appl. Opt. 20, 1024 (1981).
    [CrossRef] [PubMed]
  2. H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969), pp. 114–115.
  3. C. K. Carniglia, J. H. Apfel, J. Opt. Soc. Am. 70, 523 (1980).
    [CrossRef]
  4. In the previous work the phase thickness of the film was represented by ϕ. Here the symbol β is used for phase thickness of a layer to conform to a common representation for periodic multilayers.

1981 (1)

1980 (1)

Apfel, J. H.

Carniglia, C. K.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969), pp. 114–115.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Other (2)

In the previous work the phase thickness of the film was represented by ϕ. Here the symbol β is used for phase thickness of a layer to conform to a common representation for periodic multilayers.

H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969), pp. 114–115.

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Figures (8)

Fig. 1
Fig. 1

Boundaries of the high reflectance zones for s- and p-polarization of a periodic multilayer as a function of the phase thicknesses of the layers. The incidence angle is 45°, and refractive indices are 1.9 and 3.4 in this example. For two-layer periods of the form ab the ordinate and abscissa represent phase thicknesses of the layers. For three-layer two-material symmetric periods of the form a/2 b a/2, the ordinate represents combined phase thicknesses of the outer layers.

Fig. 2
Fig. 2

Reflectance of p-polarized radiation by a multilayer mirror at a fixed wavelength as a function of the thicknesses of the two nonabsorbing materials. This plot represents an eleven-layer reflector with six high-index (na = 3.4) and five low-index (nb = 1.9) layers illuminated at 45°. The vertical axis is logarithm of the standing wave ratio (SWR).

Fig. 3
Fig. 3

Phase retardance D vs average phase shift A for a film of index 3.4 on top of a high reflectance substructure at an incidence angle of 45° in air or vacuum. Increasing layer thickness results in movement from right to left. Tick marks indicate intervals of one-tenth of one quarterwave optical thickness.

Fig. 4
Fig. 4

DA plot for several periodic reflectors composed of layers of index 1.9 and 3.4 at a 4.0-μm wavelength and 45° incidence angle. P1 is a period design 0.55L1.10H.55L and P2 is a period design 0.5H1.0L.5H, where L and H represent quarterwave optically thick layers of the low- and high-index materials. The DA values for one to six periods of each design on a silver (ñ = 2.3 − 24.3i) are plotted. The DA point for silver (Ag) is also shown.

Fig. 5
Fig. 5

Diagram indicating the variables used to define the reflectance at the interfaces of three layers on a high reflectance substructure.

Fig. 6
Fig. 6

DA plot for the limiting phase properties of periodic reflectors with symmetric periods of the type a/2 b a/2. The labels on the curves correspond to the similarly labeled ticked lines on Fig. 1.

Fig. 7
Fig. 7

DA plot for the limiting phase properties of two-layer period reflectors of the type ababaab with na = 3.4 and nb = 1.9. The ticked curves and labels correspond to the similarly labeled ticked lines in Fig. 1.

Fig. 8
Fig. 8

DA plot for the limiting phase properties of two-layer period reflectors of the type ababaab with na = 1.9 and nb = 3.4. The ticked curves and labels correspond to the similarly ticked lines in Fig. 1.

Tables (2)

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Table I Properties of silver mirror enhanced with P1 = 0.55L1.10H.55L

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Table II Properties of silver mirror enhanced with P2 = 0.5H1.0L.5H

Equations (16)

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cos β a cos β b - γ sin β a sin β b = - 1 ,
sin v = ± ( n a - n b n a + n b ) sin u ,
a 2 b a 2
tan 1 2 δ 0 = n tan ( 1 2 δ 1 - β 1 ) .
tan ½ δ 0 = ( n 1 / n 0 ) tan ½ δ 1 .
tan ½ δ s = ( n 3 / n 0 ) tan ½ δ 3 ,
tan ½ δ 2 = ( n 3 / n 0 ) tan ( ½ δ 3 - β 3 ) = ( n 2 / n 0 ) tan ½ δ 2 ,
tan ½ δ 1 = ( n 2 / n 0 ) tan ( ½ δ 2 - β 2 ) = ( n 1 / n 0 ) tan ½ δ 1 ,
tan ½ δ 0 = ( n 1 / n 0 ) tan ( ½ δ 1 - β 1 ) .
( n 2 / n 1 ) tan ½ δ 2 = tan ( ½ δ 0 - β 1 ) ,
tan ½ δ 0 = tan ( ½ δ 1 - β 1 ) ,
tan ½ δ 1 = ( n 2 / n 1 ) tan ( ½ δ 2 - β 2 ) .
T 1 = ( T 0 + tan β 1 ) / ( 1 - T 0 tan β 1 ) ,
T 1 = n T 0 ( 1 - n tan β 1 tan β 2 ) - n ( tan β 1 + n tan β 2 ) T 0 ( n tan β 1 + tan β 2 ) + ( n - tan β 1 tan β 2 ) ,
T 0 2 = - 2 n tan β 1 - tan β 2 ( n 2 - tan 2 β 1 ) 2 n tan β 1 + tan β 2 ( 1 - n 2 tan 2 β 1 ) .
A T 0 2 + B T 0 + C = 0 ,

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