Abstract

An approximate mathematical description of tunable metal film reflector filters for the medium and far ir is derived. The absorptive film is characterized by two easily measured quantities, and a mirror reflectivity smaller than one is taken into account. The dependence of reflectivity, phase, and transmissivity on wavelength is studied. Expressions for the filter bandwidth and a lower limit of the bandwidth are obtained. Minimum losses depend primarily on the reflection coefficient of the mirror and not on the film properties. Two interesting configurations can be distinguished: one with high loss and small bandwidth (I) and one with low loss and large bandwidth (II). Measurements of bandwidth and minimum losses at 10 gm using Ge and ZnSe film substrates and gold and aluminum films show very good agreement with theory. Typical experimental results for configuration I yielded a 4.6% bandwidth (given as fraction of the free spectral range) and 2.9% additional reflection losses (caused by the insertion of the film). Smaller losses (≈0.5%) were obtained in configuration II in connection with broader bandwidths. Possible applications include low loss longitudinal mode suppression and rotational line selection in gas lasers.

© 1976 Optical Society of America

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References

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  1. R. Abrams, Appl. Phys. Lett. 25, 304 (1974).
    [CrossRef]
  2. J. J. Degnan, J. Appl. Phys. 45, 257 (1974).
    [CrossRef]
  3. P. W. Smith, Proc. IEEE 60, 422 (1972).
    [CrossRef]
  4. Yu. V. Troitskii, J. Appl. Spectrosc. 12, 325 (1970).
    [CrossRef]
  5. W. R. Leeb, Appl. Phys. 6, 267 (1975).
    [CrossRef]
  6. W. Wiesemann, Appl. Opt. 12, 2909 (1973).
    [CrossRef] [PubMed]
  7. H. Eichler, G. Schick, W. Wiesemann, IEEE J. Quantum Electron, QE-11, 168 (1975).
    [CrossRef]
  8. Yu. V. Troitskii, Opt. Spectrosc. 25, 309 (1968).
  9. Yu. V. Troitskii, Radio Eng. Electron. Phys. 14, 1423 (1969).
  10. N. D. Goldina, M. I. Zakharov, Yu. V. Troitskii, J. Appl. Spectrosc. 10, 29 (1969).
    [CrossRef]
  11. V. I. Donin, Yu. V. Troitskii, N. D. Goldina, Opt. Spectrosc. 26, 64 (1969).
  12. P. W. Smith, M. V. Schneider, H. G. Danielmeyer, Bell Syst. Tech. J. 48, 1405 (1969). [There is a sign error in Eqs. (19)–(23), which results in a phase shift of π/2 in β.]
  13. I. M. Keterov, Yu. M. Kirin, B. Ya. Yurshin, Opt. Commun. 13, 238 (1975).
    [CrossRef]
  14. V. P. Avtonomov, E. T. Antropov, N. N. Sobolev, Yu. V. Troitskii, Sov. J. Quantum Electron. 2, 300 (1972).
    [CrossRef]
  15. A. A. Kovalev, A. S. Provorov, A. V. Shishaev, Sov. J. Quantum Electron. 5, 85 (1975).
    [CrossRef]
  16. L. N. Hadley, D. M. Dennison, J. Opt. Soc. Am. 37, 451 (1947).
    [CrossRef] [PubMed]
  17. L. N. Hadley, D. M. Dennison, J. Opt. Soc. Am. 38, 483 (1948).
    [CrossRef] [PubMed]
  18. H. Wolter, “Optik dünner Schichten,” in Encyclopedia of Physics (Springer, Berlin, 1956), Vol. 24.
  19. G. Hass, L. Hadley, in American Institute of Physics Handbook, D. E. Grey, Ed. (McGraw-Hill, New York, 1972), p. 6-124–6-155.
  20. D. Kelsall, Appl. Opt. 9, 85 (1970).
    [CrossRef] [PubMed]
  21. W. R. Leeb, unpublished.
  22. W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
    [CrossRef]

1975 (4)

H. Eichler, G. Schick, W. Wiesemann, IEEE J. Quantum Electron, QE-11, 168 (1975).
[CrossRef]

I. M. Keterov, Yu. M. Kirin, B. Ya. Yurshin, Opt. Commun. 13, 238 (1975).
[CrossRef]

A. A. Kovalev, A. S. Provorov, A. V. Shishaev, Sov. J. Quantum Electron. 5, 85 (1975).
[CrossRef]

W. R. Leeb, Appl. Phys. 6, 267 (1975).
[CrossRef]

1974 (2)

R. Abrams, Appl. Phys. Lett. 25, 304 (1974).
[CrossRef]

J. J. Degnan, J. Appl. Phys. 45, 257 (1974).
[CrossRef]

1973 (1)

1972 (2)

P. W. Smith, Proc. IEEE 60, 422 (1972).
[CrossRef]

V. P. Avtonomov, E. T. Antropov, N. N. Sobolev, Yu. V. Troitskii, Sov. J. Quantum Electron. 2, 300 (1972).
[CrossRef]

1970 (2)

Yu. V. Troitskii, J. Appl. Spectrosc. 12, 325 (1970).
[CrossRef]

D. Kelsall, Appl. Opt. 9, 85 (1970).
[CrossRef] [PubMed]

1969 (4)

Yu. V. Troitskii, Radio Eng. Electron. Phys. 14, 1423 (1969).

N. D. Goldina, M. I. Zakharov, Yu. V. Troitskii, J. Appl. Spectrosc. 10, 29 (1969).
[CrossRef]

V. I. Donin, Yu. V. Troitskii, N. D. Goldina, Opt. Spectrosc. 26, 64 (1969).

P. W. Smith, M. V. Schneider, H. G. Danielmeyer, Bell Syst. Tech. J. 48, 1405 (1969). [There is a sign error in Eqs. (19)–(23), which results in a phase shift of π/2 in β.]

1968 (1)

Yu. V. Troitskii, Opt. Spectrosc. 25, 309 (1968).

1965 (1)

W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

1948 (1)

1947 (1)

Abrams, R.

R. Abrams, Appl. Phys. Lett. 25, 304 (1974).
[CrossRef]

Antropov, E. T.

V. P. Avtonomov, E. T. Antropov, N. N. Sobolev, Yu. V. Troitskii, Sov. J. Quantum Electron. 2, 300 (1972).
[CrossRef]

Avtonomov, V. P.

V. P. Avtonomov, E. T. Antropov, N. N. Sobolev, Yu. V. Troitskii, Sov. J. Quantum Electron. 2, 300 (1972).
[CrossRef]

Danielmeyer, H. G.

P. W. Smith, M. V. Schneider, H. G. Danielmeyer, Bell Syst. Tech. J. 48, 1405 (1969). [There is a sign error in Eqs. (19)–(23), which results in a phase shift of π/2 in β.]

Degnan, J. J.

J. J. Degnan, J. Appl. Phys. 45, 257 (1974).
[CrossRef]

Dennison, D. M.

Donin, V. I.

V. I. Donin, Yu. V. Troitskii, N. D. Goldina, Opt. Spectrosc. 26, 64 (1969).

Eichler, H.

H. Eichler, G. Schick, W. Wiesemann, IEEE J. Quantum Electron, QE-11, 168 (1975).
[CrossRef]

Goldina, N. D.

N. D. Goldina, M. I. Zakharov, Yu. V. Troitskii, J. Appl. Spectrosc. 10, 29 (1969).
[CrossRef]

V. I. Donin, Yu. V. Troitskii, N. D. Goldina, Opt. Spectrosc. 26, 64 (1969).

Hadley, L.

G. Hass, L. Hadley, in American Institute of Physics Handbook, D. E. Grey, Ed. (McGraw-Hill, New York, 1972), p. 6-124–6-155.

Hadley, L. N.

Hass, G.

G. Hass, L. Hadley, in American Institute of Physics Handbook, D. E. Grey, Ed. (McGraw-Hill, New York, 1972), p. 6-124–6-155.

Kelsall, D.

Keterov, I. M.

I. M. Keterov, Yu. M. Kirin, B. Ya. Yurshin, Opt. Commun. 13, 238 (1975).
[CrossRef]

Kirin, Yu. M.

I. M. Keterov, Yu. M. Kirin, B. Ya. Yurshin, Opt. Commun. 13, 238 (1975).
[CrossRef]

Kovalev, A. A.

A. A. Kovalev, A. S. Provorov, A. V. Shishaev, Sov. J. Quantum Electron. 5, 85 (1975).
[CrossRef]

Leeb, W. R.

W. R. Leeb, Appl. Phys. 6, 267 (1975).
[CrossRef]

W. R. Leeb, unpublished.

Provorov, A. S.

A. A. Kovalev, A. S. Provorov, A. V. Shishaev, Sov. J. Quantum Electron. 5, 85 (1975).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

Schick, G.

H. Eichler, G. Schick, W. Wiesemann, IEEE J. Quantum Electron, QE-11, 168 (1975).
[CrossRef]

Schneider, M. V.

P. W. Smith, M. V. Schneider, H. G. Danielmeyer, Bell Syst. Tech. J. 48, 1405 (1969). [There is a sign error in Eqs. (19)–(23), which results in a phase shift of π/2 in β.]

Shishaev, A. V.

A. A. Kovalev, A. S. Provorov, A. V. Shishaev, Sov. J. Quantum Electron. 5, 85 (1975).
[CrossRef]

Smith, P. W.

P. W. Smith, Proc. IEEE 60, 422 (1972).
[CrossRef]

P. W. Smith, M. V. Schneider, H. G. Danielmeyer, Bell Syst. Tech. J. 48, 1405 (1969). [There is a sign error in Eqs. (19)–(23), which results in a phase shift of π/2 in β.]

Sobolev, N. N.

V. P. Avtonomov, E. T. Antropov, N. N. Sobolev, Yu. V. Troitskii, Sov. J. Quantum Electron. 2, 300 (1972).
[CrossRef]

Troitskii, Yu. V.

V. P. Avtonomov, E. T. Antropov, N. N. Sobolev, Yu. V. Troitskii, Sov. J. Quantum Electron. 2, 300 (1972).
[CrossRef]

Yu. V. Troitskii, J. Appl. Spectrosc. 12, 325 (1970).
[CrossRef]

N. D. Goldina, M. I. Zakharov, Yu. V. Troitskii, J. Appl. Spectrosc. 10, 29 (1969).
[CrossRef]

Yu. V. Troitskii, Radio Eng. Electron. Phys. 14, 1423 (1969).

V. I. Donin, Yu. V. Troitskii, N. D. Goldina, Opt. Spectrosc. 26, 64 (1969).

Yu. V. Troitskii, Opt. Spectrosc. 25, 309 (1968).

Wiesemann, W.

H. Eichler, G. Schick, W. Wiesemann, IEEE J. Quantum Electron, QE-11, 168 (1975).
[CrossRef]

W. Wiesemann, Appl. Opt. 12, 2909 (1973).
[CrossRef] [PubMed]

Wolter, H.

H. Wolter, “Optik dünner Schichten,” in Encyclopedia of Physics (Springer, Berlin, 1956), Vol. 24.

Yurshin, B. Ya.

I. M. Keterov, Yu. M. Kirin, B. Ya. Yurshin, Opt. Commun. 13, 238 (1975).
[CrossRef]

Zakharov, M. I.

N. D. Goldina, M. I. Zakharov, Yu. V. Troitskii, J. Appl. Spectrosc. 10, 29 (1969).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. (1)

W. R. Leeb, Appl. Phys. 6, 267 (1975).
[CrossRef]

Appl. Phys. Lett. (1)

R. Abrams, Appl. Phys. Lett. 25, 304 (1974).
[CrossRef]

Bell Syst. Tech. J. (1)

P. W. Smith, M. V. Schneider, H. G. Danielmeyer, Bell Syst. Tech. J. 48, 1405 (1969). [There is a sign error in Eqs. (19)–(23), which results in a phase shift of π/2 in β.]

IEEE J. Quantum Electron (1)

H. Eichler, G. Schick, W. Wiesemann, IEEE J. Quantum Electron, QE-11, 168 (1975).
[CrossRef]

J. Appl. Phys. (2)

J. J. Degnan, J. Appl. Phys. 45, 257 (1974).
[CrossRef]

W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

J. Appl. Spectrosc. (2)

N. D. Goldina, M. I. Zakharov, Yu. V. Troitskii, J. Appl. Spectrosc. 10, 29 (1969).
[CrossRef]

Yu. V. Troitskii, J. Appl. Spectrosc. 12, 325 (1970).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

I. M. Keterov, Yu. M. Kirin, B. Ya. Yurshin, Opt. Commun. 13, 238 (1975).
[CrossRef]

Opt. Spectrosc. (2)

Yu. V. Troitskii, Opt. Spectrosc. 25, 309 (1968).

V. I. Donin, Yu. V. Troitskii, N. D. Goldina, Opt. Spectrosc. 26, 64 (1969).

Proc. IEEE (1)

P. W. Smith, Proc. IEEE 60, 422 (1972).
[CrossRef]

Radio Eng. Electron. Phys. (1)

Yu. V. Troitskii, Radio Eng. Electron. Phys. 14, 1423 (1969).

Sov. J. Quantum Electron. (2)

V. P. Avtonomov, E. T. Antropov, N. N. Sobolev, Yu. V. Troitskii, Sov. J. Quantum Electron. 2, 300 (1972).
[CrossRef]

A. A. Kovalev, A. S. Provorov, A. V. Shishaev, Sov. J. Quantum Electron. 5, 85 (1975).
[CrossRef]

Other (3)

H. Wolter, “Optik dünner Schichten,” in Encyclopedia of Physics (Springer, Berlin, 1956), Vol. 24.

G. Hass, L. Hadley, in American Institute of Physics Handbook, D. E. Grey, Ed. (McGraw-Hill, New York, 1972), p. 6-124–6-155.

W. R. Leeb, unpublished.

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Figures (9)

Fig. 1
Fig. 1

(a) and (b) Filter geometry and equivalent four layer system.

Fig. 2
Fig. 2

Reflectivity and transmissivity as a function of β for u = 8 and v = −6. Full lines are for configuration I (N1 = 1, N3 = 4), dashed lines for configuration II (N1 = 4, N3 = 1). In both cases Rm = 0.994, corresponding to = 1.5 × 10−3.

Fig. 3
Fig. 3

As Fig. 2, but u = 2, v = −0.5.

Fig. 4
Fig. 4

Bandwidth Bp as a function of parameter u for different values of p. Full lines are for N3 = 2.4 (ZnSe), broken lines for N3 = 4 (Ge); N1 = 1, v = 0.

Fig. 5
Fig. 5

Intensity distribution within the filter. A minimum loss situation is shown, where the absorptive film is centered at an intensity minimum (distorted vertical scale, exaggerated thickness D). The distance S is approximately equal to an integer (m) multiple of half-wavelength within the medium N1 (corresponding β ≈ 0). The inset indicates the way of calculating L3/2, where β has been assumed to be exactly zero (intensity minimum at N2N1 boundary).

Fig. 6
Fig. 6

Phase of the reflected wave for the filter of Fig. 2. Full line is for configuration I, dashed line for configuration II.

Fig. 7
Fig. 7

Phase of reflected wave for the filter of Fig. 3. Full line is for configuration I, dashed line for configuration II.

Fig. 8
Fig. 8

Measured R(β) curve (full line) for a gold film of Dn = 80 Å on a ZnSe substrate in configuration I. The dashed line gives the calculated curve for u = 9.0, v = −6.4, N1 = 1, N3 = 2.4, and Rm = 0.987. The scales for measured and calculated curves have been adjusted to give the same Rmax.

Fig. 9
Fig. 9

As Fig. 8, but for a chromium film of Dn = 110 Å on a germanium substrate: u = 1.98; = −0.5; N3 = 4.

Tables (2)

Tables Icon

Table I Measured and Calculated Bandwith B0.2 in Configuration I

Tables Icon

Table II Measured and Calculated Minimum Additional Losses

Equations (38)

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r r * = R ,             r m r m * = R m .
r m = ( N 1 - N 0 ) / ( N 1 + N 0 ) .
r = ( N 3 - N 2 ) [ ( N 2 + N 1 ) exp ( j β ) + r m ( N 2 - N 1 ) exp ( - j β ) ] exp ( j d N 2 ) + ( N 3 + N 2 ) ( N 3 + N 2 ) [ ( N 2 + N 1 ) exp ( j β ) + r m ( N 2 - N 1 ) exp ( - j β ) ] exp ( j d N 2 ) + ( N 3 - N 2 ) [ ( N 2 - N 1 ) exp ( j β ) + r m ( N 2 + N 1 ) exp ( - j β ) ] exp ( - j d N 2 ) [ ( N 2 - N 1 ) exp ( j β ) + r m ( N 2 + N 1 ) exp ( - j β ) ] exp ( - j d N 2 ) .
β = 2 π N 1 S / λ
d = 2 π D / λ ,
T = N 0 N 3 | 8 N 1 N 2 N 3 ( N 1 + N 0 ) D e n 3 | 2 ,
2 π D n / λ 1 ,             2 π D k / λ 1 ,
N 1 N 3 ( n 2 + k 2 ) .
r = { [ ( N 3 - u ) - N 1 ] cos β + v sin β } + j [ ( N 3 - N 1 - u ) sin β - v cos β ] { [ ( N 3 + u ) + N 1 ] cos β - v sin β } + j [ ( N 3 + N 1 + u ) sin β + v cos β ] ,
u = 2 k d n ,
v = d ( n 2 - k 2 ) ,
= ( 1 + r m ) / ( 1 - r m ) = [ 1 - ( R m ) 1 / 2 ] / [ 1 + ( R m ) 1 / 2 ] .
R = { [ ( N 3 - u ) - N 1 ] cos β + v sin β } 2 + [ ( N 3 - N 1 - u ) sin β - v cos β ] 2 { [ ( N 3 + u ) + N 1 ] cos β - v sin β } 2 + [ ( N 3 - N 1 + u ) sin β + v cos β ] 2 .
T = ( 4 N 1 N 3 ) / ( D e n 13 ) ,
B p = β 1 - β 2 / π ,
1 - p = ( v β 1 , 2 - N 1 ) 2 + [ ( N 3 - u ) β 1 , 2 ] 2 ( v β 1 , 2 - N 1 ) 2 + [ ( N 3 + u ) β 1 , 2 ] 2 .
B p = 2 N 1 [ 2 N 3 u ( 2 - p ) / p - N 3 2 - u 2 ] 1 / 2 π [ 2 N 3 u ( 2 - p ) / p - N 3 2 - u 2 - v 2 ] .
v β 1 , 2 N 1 and N 3 2 u ( 2 - p ) / p
B p = 2 N 1 π [ 2 N 3 u ( 2 - p ) / p - u 2 ] 1 / 2 .
B p min = 2 p N 1 / [ π N 3 ( 2 - p ) ] .
B p = N 1 π ( p N 3 u ) 1 / 2 .
( B p / B p ) = ( N 3 / N 1 ) 3 / 2 ,
D I d z ,
L = L 1 + L 2 + L 3
L = 1 - R max ,
L 1 = ( 1 - R m ) ( N 3 / N 1 ) ,
L 2 = 1 - R max - T - L 3 .
L 2 = [ ( u N 3 ) / ( 4 N 1 2 ) ] ( 1 - R m ) 2 .
L 3 = 1 - R ° max ,
r = - [ 1 - ( N 3 / N 2 ) j tan ( d N 2 / 2 ) ] / [ 1 + ( N 3 / N 2 ) j tan ( d N 2 / 2 ) ] ,
r r * = - 1 - 2 Re [ 2 j ( N 3 / N 2 ) tan ( d N 2 / 2 ) ] .
1 - r r * = ( N 3 / 3 ) n k d 3
L 3 = ( 2 N 3 / 3 ) n k d 3 .
L = N 3 N 1 ( 1 - R m ) + u N 3 4 N 1 2 ( 1 - R m ) 2 + N 3 3 u d 2 ,
A = L 2 + L 3 .
φ = arctan 2 N 3 [ N 1 ( 1 - 2 ) sin β cos β - v sin 2 β + 2 v cos 2 β ] N 3 2 ( 2 cos 2 β + sin 2 β ) - [ ( u + N 1 ) cos β - v sin β ] 2 - [ ( N 1 + u ) sin β + v cos β ] 2 .
L S = ± Δ R + 2 L AR + 2 α D 3
L S = 2 R A + 2 L A R + 2 α D 3

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