Abstract

Ohmer's algorithm for the complete exact synthesis of three-layer equivalent periods has been generalized for oblique incidence, and a theory was developed for periods with unsplit equivalent admittances, observing simultaneously the corresponding equivalent phase thicknesses. In a parallel development general conditions were deduced for the depolarization of outer media by quarterwave extensions of the central system. In combination, these building blocks enable the design of depolarized partial reflectors. A number of concrete designs are presented on various reflection levels.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Herpin, C. R. Acad. Sci. 225, 182 (1947).
  2. L. I. Epstein, J. Opt. Soc. Am. 42, 806 (1952).
    [CrossRef]
  3. A. Thelen, “Design of Multilayer Interference Filters,” in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), Vol. 5.
  4. A. Thelen, J. Opt. Soc. Am. 61, 365 (1971).
    [CrossRef]
  5. B. V. Landau, P. H. Lissberger, J. Opt. Soc. Am. 62, 1258 (1972).
    [CrossRef]
  6. K. Rabinovitch, A. Pagis, Opt. Acta 21, 963 (1974).
    [CrossRef]
  7. H. A. MacLeod, Opt. Acta 25, 93 (1978).
    [CrossRef]
  8. P. H. Lissberger, D. R. Gibson, Opt. Acta 27, 1295 (1980).
    [CrossRef]
  9. P. W. Baumeister, C. Ufford, J. Opt. Soc. Am. 64, 239 (1974).
  10. Z. Knittl, P. Pokorný, Jemná Mech. Opt. 23, 205 (1978).
  11. M. C. Ohmer, J. Opt. Soc. Am. 68, 137 (1978).
    [CrossRef]
  12. In a less effective but more general way a depolarized semireflector could be constructed even if we do not succeed in keeping the ϕe at 90°, but the deviation from the λ/4 condition should only be moderate. Anyway, in this detuned case the phase jumps get out of control and it is only the amplitudes that can be depolarized.
  13. V. R. Costich, Appl. Opt. 9, 866 (1970).
    [CrossRef] [PubMed]
  14. Z. Knittl, Optics of Thin Films (Wiley, London, 1976), p. 361.
  15. Denoting tanϕ1 = y, tanϕ2 = x, the condition for the intersectional point in a three layer is Px2 + Qx + F = 0, where P = Ay2 + B/y2 + C, Q = Dy + E/y, and A, B, C, D, E, and F are algebraic functions of the nν and Yνp, Yνs, ν = 1,2. Choosing a set of y, one obtains the corresponding set of x (if real). The intersectional levels Ye and ϕe are determined afterward.
  16. No analytical proof has been attempted to prove that M and N are true straight lines, even when they appear so graphically.
  17. It being known that approximately ϕe = ∑ϕν, it may appear surprising that, with ϕ2 in the range (180°, 360°), the equivalent thicknesses ϕe may have the small values indicated in Fig. 3. The explanation is in the accompanying values ϕ1, not shown for brevity. Thus, e.g., in the case of ϕe = 90°, for ϕ2 > 90° they turn out to be negative and the expected sum total is maintained: ϕ2 = 180° implies ϕ1 = −45°, etc. For realistic purposes the negative ϕ1 may be converted positive by adding 180° to each (compare other examples of conversion in the main text), thus increasing the total phase thickness by 360°. Thus the ϕe -labeling in Fig. 3 may also be understood modulo 360°. Alternatively it could also go by the values sinϕe which would be more universal. Figure 3 is only meant as an additional illustration of the limited appearance of the intersections not as a design nomogram.

1980

P. H. Lissberger, D. R. Gibson, Opt. Acta 27, 1295 (1980).
[CrossRef]

1978

H. A. MacLeod, Opt. Acta 25, 93 (1978).
[CrossRef]

Z. Knittl, P. Pokorný, Jemná Mech. Opt. 23, 205 (1978).

M. C. Ohmer, J. Opt. Soc. Am. 68, 137 (1978).
[CrossRef]

1974

K. Rabinovitch, A. Pagis, Opt. Acta 21, 963 (1974).
[CrossRef]

P. W. Baumeister, C. Ufford, J. Opt. Soc. Am. 64, 239 (1974).

1972

1971

1970

1952

1947

A. Herpin, C. R. Acad. Sci. 225, 182 (1947).

Baumeister, P. W.

P. W. Baumeister, C. Ufford, J. Opt. Soc. Am. 64, 239 (1974).

Costich, V. R.

Epstein, L. I.

Gibson, D. R.

P. H. Lissberger, D. R. Gibson, Opt. Acta 27, 1295 (1980).
[CrossRef]

Herpin, A.

A. Herpin, C. R. Acad. Sci. 225, 182 (1947).

Knittl, Z.

Z. Knittl, P. Pokorný, Jemná Mech. Opt. 23, 205 (1978).

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), p. 361.

Landau, B. V.

Lissberger, P. H.

P. H. Lissberger, D. R. Gibson, Opt. Acta 27, 1295 (1980).
[CrossRef]

B. V. Landau, P. H. Lissberger, J. Opt. Soc. Am. 62, 1258 (1972).
[CrossRef]

MacLeod, H. A.

H. A. MacLeod, Opt. Acta 25, 93 (1978).
[CrossRef]

Ohmer, M. C.

Pagis, A.

K. Rabinovitch, A. Pagis, Opt. Acta 21, 963 (1974).
[CrossRef]

Pokorný, P.

Z. Knittl, P. Pokorný, Jemná Mech. Opt. 23, 205 (1978).

Rabinovitch, K.

K. Rabinovitch, A. Pagis, Opt. Acta 21, 963 (1974).
[CrossRef]

Thelen, A.

A. Thelen, J. Opt. Soc. Am. 61, 365 (1971).
[CrossRef]

A. Thelen, “Design of Multilayer Interference Filters,” in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), Vol. 5.

Ufford, C.

P. W. Baumeister, C. Ufford, J. Opt. Soc. Am. 64, 239 (1974).

Appl. Opt.

C. R. Acad. Sci.

A. Herpin, C. R. Acad. Sci. 225, 182 (1947).

J. Opt. Soc. Am.

Jemná Mech. Opt.

Z. Knittl, P. Pokorný, Jemná Mech. Opt. 23, 205 (1978).

Opt. Acta

K. Rabinovitch, A. Pagis, Opt. Acta 21, 963 (1974).
[CrossRef]

H. A. MacLeod, Opt. Acta 25, 93 (1978).
[CrossRef]

P. H. Lissberger, D. R. Gibson, Opt. Acta 27, 1295 (1980).
[CrossRef]

Other

A. Thelen, “Design of Multilayer Interference Filters,” in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), Vol. 5.

In a less effective but more general way a depolarized semireflector could be constructed even if we do not succeed in keeping the ϕe at 90°, but the deviation from the λ/4 condition should only be moderate. Anyway, in this detuned case the phase jumps get out of control and it is only the amplitudes that can be depolarized.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), p. 361.

Denoting tanϕ1 = y, tanϕ2 = x, the condition for the intersectional point in a three layer is Px2 + Qx + F = 0, where P = Ay2 + B/y2 + C, Q = Dy + E/y, and A, B, C, D, E, and F are algebraic functions of the nν and Yνp, Yνs, ν = 1,2. Choosing a set of y, one obtains the corresponding set of x (if real). The intersectional levels Ye and ϕe are determined afterward.

No analytical proof has been attempted to prove that M and N are true straight lines, even when they appear so graphically.

It being known that approximately ϕe = ∑ϕν, it may appear surprising that, with ϕ2 in the range (180°, 360°), the equivalent thicknesses ϕe may have the small values indicated in Fig. 3. The explanation is in the accompanying values ϕ1, not shown for brevity. Thus, e.g., in the case of ϕe = 90°, for ϕ2 > 90° they turn out to be negative and the expected sum total is maintained: ϕ2 = 180° implies ϕ1 = −45°, etc. For realistic purposes the negative ϕ1 may be converted positive by adding 180° to each (compare other examples of conversion in the main text), thus increasing the total phase thickness by 360°. Thus the ϕe -labeling in Fig. 3 may also be understood modulo 360°. Alternatively it could also go by the values sinϕe which would be more universal. Figure 3 is only meant as an additional illustration of the limited appearance of the intersections not as a design nomogram.

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 (12)

Fig. 1
Fig. 1

Function H (ne) for index pair 2.35/1.38 in the HLH and LHL configurations. Cardinal points for H = ±1 are indicated.

Fig. 2
Fig. 2

Intersectional values Ye and h = Hs = Hp as functions of n1 for various n2. Line MM separates the LHL and HLH cases. To the right of line NN negative values with |h| < 1 are secured for the HLH cases.

Fig. 3
Fig. 3

Oblique equivalent admittances Yes, Yep as functions of ϕ2 for various prescribed ϕe (in the admissible ranges). Index pair 2.22/1.49 in the configurations: (a) HLH and (b) LHL. The constant intersectional levels are 2.94 and 3.23, respectively. (The accompanying ϕ1 are not plotted.)17

Fig. 4
Fig. 4

Additional dispersions due to conversion of negative ϕ values for the HLH cases from Table V. Index combinations and basic nonconverted relative tunings: (a) 2.0–1.38–2.0; ±0.7854: ∓0.5490:±0.7854, (b) 2.6–1.49–2.6; ±0.6421:∓0.2486:±0.6421, (c) 2.6–1.87–2.6; ±0.9600:∓0.2154:±0.9600. These have common dispersion curves ➀ with minimum ϕ dependence. Curves ➁ correspond to conversion by 2 of the central negative values in the uppersign schemes, curves ➂ to that of the outer negative values in the lower-sign schemes.

Fig. 5
Fig. 5

The S-20a (five layers): (a) index scheme 1/1.3–2.6–1.87– 2.6–2.09/1.52; tuning 1:0.96:1.0844:0.96:1; (b) compromise on n1 = 1.38

Fig. 6
Fig. 6

The S-30a,b (nine layers): index scheme 1/(1.7–2.175)2– 1.7–2.6–1.49–2.6–2.09/1.52; tuning: (a) (1)5:0.6422:1.7511:0.6422:1; (b) (1)5:1.3577:0.2488:1.3577:1.

Fig. 7
Fig. 7

The S-33b (nine layers): (a) index scheme 1/1.49–2.31–1.7–2.6–1.49–2.6–1.49–1.345–1.7/1.52; tuning (1)3:1.3577:0.2488:1.3577:(1)3; (b) n2 = 2.35, n8 = 1.38; (c) n2 = 2.5, n8 = 1.38; (d) n2 = n4 = n6 = 2.5, n8 = 1.38.

Fig. 8
Fig. 8

(a) S-38a (nine layers), index scheme 1/(1.7–2.175)2–1.7–2.6–1.87–2.6–2.09/1.53, tuning (1)5:0.96:1.0844:0.96:1; (b) S-47a (nine layers), index scheme 1/(1.7–2.5)2–2.0–2.6–1.87–2.6–2.09/1.52, tuning (1)5:0.96:1.0844:0.96:1.

Fig. 9
Fig. 9

The S-44 (eleven layers), index scheme 1/1.3–2.6–1.87– 2.6–2.0–1.38–2.0–2.6–1.87–2.6–2.09/1.52, tuning of outer layers 1:1, tuning of inner layers: (a) S-44aaa, 0.96:1.084:0.96:0.7855:1.451: 0.7855:0.96:1.084:0.96; (b) S-44bbb, 1.04:0.9155:1.04:1.2144:0.5488: 1.2144:1.04:0.9155:1.04.

Fig. 10
Fig. 10

The S-48 (fifteen layers), index scheme 1/(1.49–2.31–1.7)–2.6–1.49–2.6–2.0–1.38–2.0–2.6–1.49–2.6–(1.49–1.345–1.7)/1.52. Tuning of kernel: if 0.6422:1.7511:0.6422 ≡ A, 1.3577:0.2488:1.3577 ≡ B, 0.7855:1.4511:0.7855 ≡ A′; 1.2144:0.5488:1.2144 ≡ B′; then (a) S-48aaa, AA′A; (b) S-48bab, BA′B; (c) S-48bbb, BB′B; (d) S-48aba, AB′A; (e) compromise on S-48bbb: n2 = 2.35, n14 = 1.38, n4 = n6 = n10 = n12 = 2.5; (f) additional compromise on n2 = 2.5.

Fig. 11
Fig. 11

The S-53bbb (fifteen layers); (a) index scheme 1/(1.7–2.5)2–2.0–2.6–1.49–2.6–2.0–1.38–2.0–2.6–1.49–2.6–2.09/1.52, tuning (1)5:B:B′:B:(1) (for B, B′, see Fig. 10 caption); (b) compromise n6 = n8 = n12 = n14 = 2.5.

Fig. 12
Fig. 12

The S-62 (fifteen layers), index scheme 1/1.49–2.31–1.7–2.6–1.87–2.6–2.0–1.38–2.0–2.6–1.87–2.6–1.49–1.345–1.7/1.52, tuning of outer layers (1)3:(1)3; tuning of inner layers: (a) S-62aaa, 0.96:1.0844:0.96:0.7855:1.4511:0.7855:0.96:1.0844:0.96; (b) S-62bbb, 1.04:0.9155:1.04:1.2144:0.5488:1.2144:1.04:0.9155:1.04.

Tables (9)

Tables Icon

Table I Design of Equivalent Period in Oblique Incidence (45°/Air) by Parallel Demands: Pairs (Yep, Yes) Compatible by the Equality Hp = Hs = ; nH = 2.35, nL = 1.38

Tables Icon

Table II Design of Depolarized Periods with Refractive Indices 2.0–1.38–2.0, at 45° from the Air, with Intersectional Level 2.659; Postulated ϕe: (a) 45°, (b) −45°; Two Solutions A and B in ϕ2 for Each Case with ϕ1 Split and Partially Spurious

Tables Icon

Table III Checking the ϕ 1 Compromise on Selected Index Combinations and Various ϕe

Tables Icon

Table IV Reduced Form of Table II After Elimination of Spurious Results, ϕ Compromise, and Conversion to Positive Thicknesses

Tables Icon

Table V Reduced Form of Results for 2.0–1.38–2.0 When ϕe = ±90°

Tables Icon

Table VI Final Tunings of Selected Depolarized Periods of the λ/4 Type; Each Value ϕ° Also Expressed as a Multiple of 90°

Tables Icon

Table VII Complete Schematic Designs for an Array of Depolarized Partial Reflectors with Outer Matchings from the Appendix

Tables Icon

Table VIII Refractive Indices for Three-Layer Depolarizing Outer-Left (Upper Lines) and Outer-Right (Lower Lines) λ/4 Extensions (Unless Specified, the Refractive Indices are Copied from Preceding Lines)

Tables Icon

Table IX Refractive Indices for Five-Layer Depolarizing Outer-Left (Upper Lines) and Outer-Right (Lower Lines) λ/4 Extensions (Unless Specified, the Refractive Indices are Copied from Preceding Lines)

Equations (49)

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

ν ( cos ϕ ν j / n ν sin ϕ ν j n ν sin ϕ ν cos ϕ ν ) = ( α j β j γ α )
α = cos 2 ϕ 1 cos ϕ 2 p sin 2 ϕ 1 sin ϕ 2 ,
β = 1 n 1 ( A + B ) , γ = n 1 ( A B ) ,
A = sin 2 ϕ 1 cos ϕ 2 + p cos 2 ϕ 1 sin ϕ 2 ,
B = q sin ϕ 2 ,
p = 1 2 ( n 1 n 2 + n 2 n 1 ) , q = 1 2 ( n 1 n 2 n 2 n 1 ) .
n 1 β γ n 1 = 2 B .
cos ϕ e = α , β = sin ϕ e n e , γ = n e sin ϕ e ,
n e = ( γ β ) 1 / 2 .
sin ϕ 2 = H sin ϕ e , where H = n 1 n e 1 n e n 1 1 n 1 n 2 1 n 2 n 1 1 = n 1 2 n e 2 n 1 2 n 2 2 · n 2 n e .
cos ϕ 2 = R sin ϵ , p sin ϕ 2 = R cos ϵ ,
R sin ( ϵ 2 ϕ 1 ) = α ,
| H sin ϕ e | 1 .
H ( n 1 2 / n e ) = H ( n e ) ,
H ( n 1 2 / n 2 ) = 1 .
for HLH : 2.35 = n H < n e < n H 2 / n L = 4.01 ,
for LHL : 0.81 = n L 2 / n H < n e < n L = 1.38 ,
ϕ 1 A = ϵ ψ 2 ; ϕ 1 A = ϵ ψ 2 = ϵ + ψ 2 90 ° ,
ϕ 1 B = ϵ ψ 2 = ( 90 ° + ϕ 1 A ) ; ϕ 1 B = ϵ + ψ 2 = ( 90 ° + ϕ 1 A ) .
ϕ 2 A = 30.8 ° ; ϕ 1 A = 13.5 ° ; ϕ 1 A = 47.8 ° ; ( 132.2 ) n e / ϕ e 1.7 / 60 ° 3.248 / 60 ° ϕ 2 B = 149.2 ° ; ϕ 1 B = 42.2 ° ; ( 137.8 ° ) ϕ 1 B = 103.5 ° . ( 76.5 ° )
sin ϕ 2 = H s ( Y es ) · sin ϕ e p , sin ϕ 2 = H p ( Y ep ) · sin ϕ e p .
sin ϕ 2 = sin ϕ e .
H s ( Y es ) = H p ( Y ep ) ,
Y es 2 + Y 1 p 2 Y ep 2 Ω Y ep Y es Y 1 s 2 = 0 , with Ω = Y 1 p 2 Y 2 p 2 Y 1 s 2 Y 2 s 2 Y 2 s Y 2 p ,
Y e = ( Y 1 p 2 Ω Y 1 s 2 1 Ω ) 1 / 2 ,
sin ϕ 2 = h sin ϕ e ,
Y e = r 2 + ( r 2 ) 2 + Y 1 2 ,
r = ( Y 1 2 Y 2 2 ) sin ϕ 2 Y 2 sin ϕ e ( taken for p or s ) .
0.7854 0.549 0.7854 .
0.7854 1.451 0.7854
0.7854 0.549 0.7854
1.2146 0.549 1.2146 .
refractivities 2.6 1.49 2.6 Y e = 3.245 relative tunings ( a ) 0.6422 1.7511 0.6422 ( b ) 1.3577 0.2488 1.3577 ,
( 1.7 2.175 ) 2 1.7 Y e = 1.077 ,
( 1.077 × 2.85 3.245 2 1.077 × 2.85 + 3.245 2 ) = 30.0 % .
n n } = n 2 cos ( 1 2 θ ) ,
n = 1.3 , Y p = Y s = 1.69 ; n = 2.09 , Y p = Y s = 2.85 ;
n 0 | n | | n | G Y | | Y .
1 ( n A n B ) m n C ( n A n B ) m n C / G .
m 11 = m 22 = 0 , m 21 = 1 m 12 = ( 1 ) m ( Y A Y B ) m Y Cj ,
1 2 ( 1 1 / Y 0 1 1 / Y 0 ) V 0 1 ( 0 m 12 m 21 0 ) = m 21 2 Y 0 ( 1 1 / Y eff 1 1 / Y eff ) .
( 0 m 12 m 21 0 ) ( 1 1 Y G Y G ) V G = m 12 Y G ( 1 1 Y eff , Y eff ) .
Y eff = m 21 m 12 1 Y = 1 Y ( Y A Y B ) 2 m Y C ,
Y p Y s = n 2 , Y s Y p = cos 2 Θ , cos 2 Θ = 1 a 2 n 2 ,
Y p Y s = 1 n 2 ( n A n B ) 4 m n C 4 ,
Y s Y p = cos 2 Θ ( cos Θ A cos Θ B ) 4 m cos 4 θ C .
Y = n C 2 n ( n A n B ) 2 m ,
cos 2 Θ B = cos 2 Θ A · W m , W m = ( cos 2 Θ C cos Θ ) 1 / m .
1 n B 2 = W m n A 2 + 1 W m a 2 .

Metrics