Abstract

A new design technique for multilayer reflectors is presented It is useful when slight absorption by one or both of the coating materials limits the performance of the reflector. The basic procedure is to add layers to a given substrate or multilayer system one pair at a time. The thickness of each layer is chosen to give the maximum increase in reflectance for each pair of layers added. In general, the optical thicknesses of the layers in each pair are not quarter waves, but depend on the optical constants of the materials, as well as the starting reflectance of the subsystem. By using such optimized pairs, it is possible to exceed the reflectance limit usually imposed on quarter-wave stack reflectors by absorption. Expressions for the optimum design and the ultimate reflectance for a high reflector made with a given set of coating materials are given. Other design techniques found in the literature require more layers to achieve the same level of reflectance as the present method.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Koppelmann, “On the theory of multilayers consisting of weakly absorbing materials and their use as interferometer mirrors,” (in German) Ann. Phys. (Leipzig) 5, 388–396 (1960).
  2. P. Baumeister and O. Arnon, “Use of hafnium dioxide in multilayer dielectric reflectors for the near UV,” Appl. Opt. 16, 439–444 (1977).
    [Crossref] [PubMed]
  3. M. Sparks and M. Flannery, “Simplified description of multilayer dielectric reflectors,” J. Opt. Soc. Am. 69, 993–1006 (1979).
    [Crossref]
  4. P. H. Lissberger, “The ultimate reflectance of multilayer dielectric mirrors,” Opt. Acta 25, 291–298 (1978).
    [Crossref]
  5. G. W. DeBell, “The design and measurement of low absorptance optical interference coatings,” Ph.D. thesis, Institute of Optics, University of Rochester, 1972 (unpublished).
  6. Joseph H. Apfel, “Optical coating design with reduced electric field intensity,” Appl. Opt. 16, 1880–1885 (1977);Eberhard Spiller, “Reflective multilayer coatings for the far uv region,” Appl. Opt. 15, 2333–2338 (1976).
    [Crossref] [PubMed]
  7. Leo Young, “Prediction of absorption loss in multilayer interference filters,” J. Opt. Soc. Am. 52, 753–761 (1962).
    [Crossref]
  8. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).
  9. When a term in δij occurs with a coefficient of the form, it is considered to be of order higher than k/n. This is strictly true only as rj→ 1. In Eq. (27) the coefficient is and is not small.

1979 (1)

1978 (1)

P. H. Lissberger, “The ultimate reflectance of multilayer dielectric mirrors,” Opt. Acta 25, 291–298 (1978).
[Crossref]

1977 (2)

1962 (1)

1960 (1)

G. Koppelmann, “On the theory of multilayers consisting of weakly absorbing materials and their use as interferometer mirrors,” (in German) Ann. Phys. (Leipzig) 5, 388–396 (1960).

Apfel, Joseph H.

Arnon, O.

Baumeister, P.

DeBell, G. W.

G. W. DeBell, “The design and measurement of low absorptance optical interference coatings,” Ph.D. thesis, Institute of Optics, University of Rochester, 1972 (unpublished).

Flannery, M.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

Koppelmann, G.

G. Koppelmann, “On the theory of multilayers consisting of weakly absorbing materials and their use as interferometer mirrors,” (in German) Ann. Phys. (Leipzig) 5, 388–396 (1960).

Lissberger, P. H.

P. H. Lissberger, “The ultimate reflectance of multilayer dielectric mirrors,” Opt. Acta 25, 291–298 (1978).
[Crossref]

Sparks, M.

Young, Leo

Ann. Phys. (Leipzig) (1)

G. Koppelmann, “On the theory of multilayers consisting of weakly absorbing materials and their use as interferometer mirrors,” (in German) Ann. Phys. (Leipzig) 5, 388–396 (1960).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Opt. Acta (1)

P. H. Lissberger, “The ultimate reflectance of multilayer dielectric mirrors,” Opt. Acta 25, 291–298 (1978).
[Crossref]

Other (3)

G. W. DeBell, “The design and measurement of low absorptance optical interference coatings,” Ph.D. thesis, Institute of Optics, University of Rochester, 1972 (unpublished).

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

When a term in δij occurs with a coefficient of the form, it is considered to be of order higher than k/n. This is strictly true only as rj→ 1. In Eq. (27) the coefficient is and is not small.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

FIG. 1
FIG. 1

SWR and reflectance of a quarter-wave stack versus number of layers for two different values of k1 for the high-index material. The substrate has ns = 1.51.

FIG. 2
FIG. 2

SWR and reflectance versus optical thickness of the low- and high-index layers for a single pair of layers added to a quarter-wave stack. The line at SWR = 2355 represents the maximum attainable by the quarter-wave stack. This illustrates the maximization procedure for determining the optimum pair.

FIG. 3
FIG. 3

SWR and reflectance versus optical thickness of the low-index layer for several pairs of layers added to a single quarter-wave high-index layer on a substrate with ns = 1.51. The starting SWR for each pair is taken to be the peak value from the previous pair, indicated by a tic mark.

FIG. 4
FIG. 4

Diagram illustrating the variables used to determine the optimum pair.

FIG. 5
FIG. 5

Left scale: The asymptotic optimum value of the optical thickness of the low-index layer in quarter waves versus k1/k2 with n1/n2 as a parameter. Right scale: The corresponding improvement in the performance limit Δ l of the optimum-pair stack as compared to the Koppelmann limit Δk for the quarter-wave stack. Note that R = 1 − 2Δ.

FIG. 6
FIG. 6

SWR and reflectance versus number of layers for an optimum-pair stack, an optimized-fixed-low stack and a quarter-wave stack with absorption in both materials. The designs are given in Table I.

FIG. 7
FIG. 7

Absorptance versus number layers for the three designs in Fig. 6.

FIG. 8
FIG. 8

SWR and reflectance versus number of layers for an optimum-pair stack, a quarter-wave stack and three periodic stacks denoted by the optical thickness of the low-index layer. The designs are given in Table II. Only the high-index material is absorbing.

FIG. 9
FIG. 9

Absorptance versus number of layers for the five designs in Fig. 8.

FIG. 10
FIG. 10

Reflectance versus wavelength for four of the designs in Fig. 8. Each reflector consists of 27 layers. All but the optimum-pair stack have reached their ultimate reflectance after this number of layers.

Tables (2)

Tables Icon

TABLE I Multilayer dielectric mirror: both materials absorbing. ns = 1.51, nH = 2.35 − i 0.001, nL = 1.47 − i0.0001

Tables Icon

TABLE II Multilayer dielectric mirror: one material absorbing. ns = 1.51, nH = 2.35 − i0.001, nL = 1.35

Equations (53)

Equations on this page are rendered with MathJax. Learn more.

n H = n 1 i k 1 ,
n L = n 2 i k 2 ,
r = ( n o n e ) / ( n o + n e ) ,
n e = ( n H ) 1 + l ( n L ) 1 l n s 1 ,
SWR = ( 1 + R ) / ( 1 R ) .
SWR = n e / n o = ( n H ) 1 + l ( n L ) 1 l ( n o n s ) 1 .
R k = 1 2 Δ k ,
Δ k = π n o ( k 1 + k 2 ) / ( n 1 2 n 2 2 ) .
r s = r s e i δ s ,
r i j ( n i i k i ) ( n j i k j ) ( n i i k i ) + ( n j i k j ) , i , j = 0 , 1 , 2 .
r i j = r i j e i δ i j r i j ( 1 i δ i j ) ,
r i j ( n i n j ) / ( n i + n j )
δ i j [ ( k i / n i ) ( k j / n j ) ] ( 1 r i j 2 ) / 2 r i j .
r j = r j e i δ j , j = 0 , 1 , 2 ,
r 2 = r 20 + r s 1 + r 20 r s ,
ϕ j ( 2 π / λ 0 ) n j d j , j = 1 , 2 ,
α j ( 2 π / λ 0 ) k j d j = ( k j / n j ) ϕ j , j = 1 , 2 .
r i = r i j + r j e 2 α j e i 2 ϕ j 1 + r i j r j e 2 α j e i 2 ϕ j .
2 ϕ 1 + δ 1 = π ,
r 1 e i δ 1 = r 12 e i δ 12 + r 2 e 2 α 2 e i ( δ 2 + 2 ϕ 2 ) 1 + r 12 r 2 e 2 α 2 e i ( δ 12 + δ 2 + 2 ϕ 2 ) ,
tan δ 1 r 2 e 2 α 2 ( 1 r 12 2 ) sin ( δ 2 + 2 ϕ 2 ) r 12 ( 1 + r 2 2 e 4 α 2 ) + r 2 e 2 α 2 ( 1 + r 12 2 ) cos ( δ 2 + 2 ϕ 2 ) ,
d r 0 / d ϕ 2 = 0 .
r 0 r 0 = r 01 r 1 e 2 α 1 1 r 01 r 1 e 2 α 1 ,
1 r 1 d r 1 d ϕ 2 = 2 k 1 n 1 d ϕ 1 d ϕ 2 .
1 r 1 d r 1 d ϕ 2 = 2 i ( 1 i k 1 / n 1 ) d ϕ 1 d ϕ 2 .
1 r 1 d r 1 d ϕ 2 = 2 i r 2 ( 1 i k 2 / n 2 ) ( 1 r 12 2 ) e 2 α 2 e i 2 ϕ 2 ( 1 + r 12 r 2 e 2 α 2 e i 2 ϕ 2 ) ( r 12 + r 2 e 2 α 2 e i 2 ϕ 2 ) .
r 2 ( 1 r 12 2 ) 1 i k 2 / n 2 1 i k 1 / n 1 = { r 2 ( 1 + r 12 2 ) + r 12 [ e 2 α 2 e i ( δ 2 + 2 ϕ 2 ) + r 2 2 e 2 α 2 e i ( δ 2 + 2 ϕ 2 ) ] } d ϕ 1 d ϕ 2 .
δ 12 ( 1 + r 12 ) 1 r 12 r 2 ( 1 r 12 2 ) δ 12 + r 12 ( 1 r 2 2 e 4 α 2 ) sin ( δ 2 + 2 ϕ 2 ) r 2 ( 1 + r 12 2 ) + r 12 ( 1 + r 2 2 ) cos ( δ 2 + 2 ϕ 2 ) ,
a = 4 n 1 2 r 2 [ ( k 1 / n 1 ) ( k 2 / n 2 ) ] ,
b = a ( 1 + r 2 2 ) / 2 r 2 ,
c = ( n 1 2 n 2 2 ) ( 1 r 2 2 e 4 α 2 ) .
2 ϕ 2 + δ 2 = β + γ ,
sin β = a / ( b 2 + c 2 ) 1 / 2 ,
tan γ = b / c .
ϕ 2 = π / 2 + ( β + γ ) / 2 .
r j = ( 1 Δ j ) , for j = 0 , 2 , s ,
Δ 2 Δ s ( n 2 / n o ) .
r 1 = ( 1 Δ 1 ) e i δ 1 .
Δ 1 r 12 e 2 i ϕ 2 ( Δ 2 + 2 α 2 + i δ 12 ) 1 r 12 e 2 i ϕ 2 e 2 i ϕ 2 ( Δ 2 + 2 α 2 ) i r 12 δ 12 r 12 e 2 i ϕ 2
e i δ 1 = r 12 e i 2 ϕ 2 1 r 12 e i 2 ϕ 2 .
tan 2 ϕ 1 = ( 1 r 12 2 ) sin 2 ϕ 2 2 r 12 ( 1 + r 12 2 ) cos 2 ϕ 2 .
tan ( π ϕ 1 ) = ( n 1 / n 2 ) tan ϕ 2
Δ 1 ( 1 r 12 2 ) [ Δ s n 2 / n o + 2 α 2 + ( k 1 / n 1 k 2 / n 2 ) sin 2 ϕ 2 ] 1 + r 12 2 2 r 12 cos 2 ϕ 2 .
Δ 1 Δ 0 n 1 / n o 2 α 1 .
Δ l 2 n o ( n 1 2 n 2 2 ) [ k 1 ( 2 ϕ 1 sin 2 ϕ 1 ) 1 cos 2 ϕ 1 + k 2 ( 2 ϕ 2 sin 2 ϕ 2 ) 1 cos 2 ϕ 2 ] .
tan ϕ 2 n 2 = ( k 1 ϕ 1 / n 1 2 ) + ( k 2 ϕ 2 / n 2 2 ) ( k 1 / n 1 ) ( k 2 / n 2 ) .
2 ϕ 1 + δ 1 = π .
r 1 = ( 1 Δ 1 ) e i δ 1 ,
r 0 = ( 1 Δ 0 ) e i δ 0 ,
Δ 0 ( 1 r 01 2 ) Δ 1 + 2 α 1 + ( k 1 / n 1 ) sin ( 2 ϕ 1 + δ 1 ) 1 + r 01 2 + 2 r 01 cos ( 2 ϕ 1 + δ 1 ) .
2 ϕ 1 + δ 1 = π + ɛ
Δ 0 ( 1 r 01 2 ) Δ 1 + ( k 1 / n 1 ) ( π δ 1 ) + ( k 1 / n 1 ) ɛ 3 / 6 ( 1 r 01 ) 2 + r 01 ɛ 2 .
e i δ 0 = r 01 + e i ( 2 ϕ 1 + δ 1 ) 1 + r 01 e i ( 2 ϕ 1 + δ 1 ) .