Abstract

The possibility of making multilayer reflecting coatings by using materials whose dielectric susceptibility is very near to unity at the frequency under consideration is discussed. For the pair of substances with a given Re and Im the explicit expressions are obtained for the value of reflection coefficient and for the optimum thickness of layers. The published data on Re and Im allowed us to choose the pairs of substances for which one may hope to create (in principle) the mirrors for normal incidence with R ~ 50% at λ ≳ 30 Å.

© 1977 Optical Society of America

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References

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  1. A. V. Vinogradov, I. I. Sobelman, E. A. Yukov, Kvant. Elektron. 2, 105 (1975) [Sov. J. Quantum Electron. 5(1), 59 (1975)].
  2. R. C. Elton, Appl. Opt. 14, 97 (1975).
    [CrossRef] [PubMed]
  3. A. N. Zherikhin, K. N. Koshelev, V. S. Letokhov, Kvant. Elektron.3, No. 1 (1976) [Sov. J. Quantum Electron. (in press)].
  4. A. V. Vinogradov, I. Yu. Skobelev, I. I. Sobelman, E. A. Yukov, Kvant. Elektron.2, 2100 (1975) [Sov. J. Quantum Electron. (in press)].
  5. A. P. Lukirsky, E. P. Savinov, O. A. Ershov, Yu. F. Shepelev, Opt. Spektrosk. 16, 310 (1964) [Opt. Spectrosc. 16, 168 (1964)].
  6. O. A. Ershov, I. A. Brytov, A. P. Lukirsky, Opt. Spektrosk. 22, 127 (1967) [Opt. Spectrosc. 22, 66 (1967)].
  7. J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1968).
  8. J. T. Cox, G. Hass, J. B. Ramsey, W. R. Hunter, J. Opt. Soc. Am. 64, 423 (1974).
    [CrossRef]
  9. V. Daneu, Appl. Opt. 14, 962 (1975).
    [PubMed]
  10. B. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).
  12. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  13. A. V. Antonov, A. I. Isakov, V. I. Makarov, S. A. Startsev, JETP Pis. Red. 20, 632 (1974) [JETP Lett. 20, 289 (1974)].
  14. E. Spiller, in Proceedings of 9th International Congress of the International Commission for Optics, B. J. Thompson, R. R. Shannon, Eds. (National Academy of Sciences, Washington, D.C.1974).

1975 (3)

A. V. Vinogradov, I. I. Sobelman, E. A. Yukov, Kvant. Elektron. 2, 105 (1975) [Sov. J. Quantum Electron. 5(1), 59 (1975)].

R. C. Elton, Appl. Opt. 14, 97 (1975).
[CrossRef] [PubMed]

V. Daneu, Appl. Opt. 14, 962 (1975).
[PubMed]

1974 (2)

J. T. Cox, G. Hass, J. B. Ramsey, W. R. Hunter, J. Opt. Soc. Am. 64, 423 (1974).
[CrossRef]

A. V. Antonov, A. I. Isakov, V. I. Makarov, S. A. Startsev, JETP Pis. Red. 20, 632 (1974) [JETP Lett. 20, 289 (1974)].

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

1967 (1)

O. A. Ershov, I. A. Brytov, A. P. Lukirsky, Opt. Spektrosk. 22, 127 (1967) [Opt. Spectrosc. 22, 66 (1967)].

1964 (2)

B. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
[CrossRef]

A. P. Lukirsky, E. P. Savinov, O. A. Ershov, Yu. F. Shepelev, Opt. Spektrosk. 16, 310 (1964) [Opt. Spectrosc. 16, 168 (1964)].

Antonov, A. V.

A. V. Antonov, A. I. Isakov, V. I. Makarov, S. A. Startsev, JETP Pis. Red. 20, 632 (1974) [JETP Lett. 20, 289 (1974)].

Batterman, B.

B. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Brytov, I. A.

O. A. Ershov, I. A. Brytov, A. P. Lukirsky, Opt. Spektrosk. 22, 127 (1967) [Opt. Spectrosc. 22, 66 (1967)].

Cole, H.

B. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
[CrossRef]

Cox, J. T.

Daneu, V.

Elton, R. C.

Ershov, O. A.

O. A. Ershov, I. A. Brytov, A. P. Lukirsky, Opt. Spektrosk. 22, 127 (1967) [Opt. Spectrosc. 22, 66 (1967)].

A. P. Lukirsky, E. P. Savinov, O. A. Ershov, Yu. F. Shepelev, Opt. Spektrosk. 16, 310 (1964) [Opt. Spectrosc. 16, 168 (1964)].

Hass, G.

Hunter, W. R.

Isakov, A. I.

A. V. Antonov, A. I. Isakov, V. I. Makarov, S. A. Startsev, JETP Pis. Red. 20, 632 (1974) [JETP Lett. 20, 289 (1974)].

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Koshelev, K. N.

A. N. Zherikhin, K. N. Koshelev, V. S. Letokhov, Kvant. Elektron.3, No. 1 (1976) [Sov. J. Quantum Electron. (in press)].

Letokhov, V. S.

A. N. Zherikhin, K. N. Koshelev, V. S. Letokhov, Kvant. Elektron.3, No. 1 (1976) [Sov. J. Quantum Electron. (in press)].

Lukirsky, A. P.

O. A. Ershov, I. A. Brytov, A. P. Lukirsky, Opt. Spektrosk. 22, 127 (1967) [Opt. Spectrosc. 22, 66 (1967)].

A. P. Lukirsky, E. P. Savinov, O. A. Ershov, Yu. F. Shepelev, Opt. Spektrosk. 16, 310 (1964) [Opt. Spectrosc. 16, 168 (1964)].

Makarov, V. I.

A. V. Antonov, A. I. Isakov, V. I. Makarov, S. A. Startsev, JETP Pis. Red. 20, 632 (1974) [JETP Lett. 20, 289 (1974)].

Ramsey, J. B.

Samson, J. A. R.

J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1968).

Savinov, E. P.

A. P. Lukirsky, E. P. Savinov, O. A. Ershov, Yu. F. Shepelev, Opt. Spektrosk. 16, 310 (1964) [Opt. Spectrosc. 16, 168 (1964)].

Shepelev, Yu. F.

A. P. Lukirsky, E. P. Savinov, O. A. Ershov, Yu. F. Shepelev, Opt. Spektrosk. 16, 310 (1964) [Opt. Spectrosc. 16, 168 (1964)].

Skobelev, I. Yu.

A. V. Vinogradov, I. Yu. Skobelev, I. I. Sobelman, E. A. Yukov, Kvant. Elektron.2, 2100 (1975) [Sov. J. Quantum Electron. (in press)].

Sobelman, I. I.

A. V. Vinogradov, I. I. Sobelman, E. A. Yukov, Kvant. Elektron. 2, 105 (1975) [Sov. J. Quantum Electron. 5(1), 59 (1975)].

A. V. Vinogradov, I. Yu. Skobelev, I. I. Sobelman, E. A. Yukov, Kvant. Elektron.2, 2100 (1975) [Sov. J. Quantum Electron. (in press)].

Spiller, E.

E. Spiller, in Proceedings of 9th International Congress of the International Commission for Optics, B. J. Thompson, R. R. Shannon, Eds. (National Academy of Sciences, Washington, D.C.1974).

Startsev, S. A.

A. V. Antonov, A. I. Isakov, V. I. Makarov, S. A. Startsev, JETP Pis. Red. 20, 632 (1974) [JETP Lett. 20, 289 (1974)].

Vinogradov, A. V.

A. V. Vinogradov, I. I. Sobelman, E. A. Yukov, Kvant. Elektron. 2, 105 (1975) [Sov. J. Quantum Electron. 5(1), 59 (1975)].

A. V. Vinogradov, I. Yu. Skobelev, I. I. Sobelman, E. A. Yukov, Kvant. Elektron.2, 2100 (1975) [Sov. J. Quantum Electron. (in press)].

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Yukov, E. A.

A. V. Vinogradov, I. I. Sobelman, E. A. Yukov, Kvant. Elektron. 2, 105 (1975) [Sov. J. Quantum Electron. 5(1), 59 (1975)].

A. V. Vinogradov, I. Yu. Skobelev, I. I. Sobelman, E. A. Yukov, Kvant. Elektron.2, 2100 (1975) [Sov. J. Quantum Electron. (in press)].

Zherikhin, A. N.

A. N. Zherikhin, K. N. Koshelev, V. S. Letokhov, Kvant. Elektron.3, No. 1 (1976) [Sov. J. Quantum Electron. (in press)].

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

J. Opt. Soc. Am. (1)

JETP Pis. Red. (1)

A. V. Antonov, A. I. Isakov, V. I. Makarov, S. A. Startsev, JETP Pis. Red. 20, 632 (1974) [JETP Lett. 20, 289 (1974)].

Kvant. Elektron. (1)

A. V. Vinogradov, I. I. Sobelman, E. A. Yukov, Kvant. Elektron. 2, 105 (1975) [Sov. J. Quantum Electron. 5(1), 59 (1975)].

Opt. Spektrosk. (2)

A. P. Lukirsky, E. P. Savinov, O. A. Ershov, Yu. F. Shepelev, Opt. Spektrosk. 16, 310 (1964) [Opt. Spectrosc. 16, 168 (1964)].

O. A. Ershov, I. A. Brytov, A. P. Lukirsky, Opt. Spektrosk. 22, 127 (1967) [Opt. Spectrosc. 22, 66 (1967)].

Rev. Mod. Phys. (1)

B. Batterman, H. Cole, Rev. Mod. Phys. 36, 681 (1964).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

E. Spiller, in Proceedings of 9th International Congress of the International Commission for Optics, B. J. Thompson, R. R. Shannon, Eds. (National Academy of Sciences, Washington, D.C.1974).

J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1968).

A. N. Zherikhin, K. N. Koshelev, V. S. Letokhov, Kvant. Elektron.3, No. 1 (1976) [Sov. J. Quantum Electron. (in press)].

A. V. Vinogradov, I. Yu. Skobelev, I. I. Sobelman, E. A. Yukov, Kvant. Elektron.2, 2100 (1975) [Sov. J. Quantum Electron. (in press)].

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Figures (2)

Fig. 1
Fig. 1

Dependence of Re (or Im) on the coordinate z in the periodic structure.

Fig. 2
Fig. 2

Solution opt for Eq. (17) for the optimum relative thickness (dashed curve) and optimum value of reflection coefficient R for the case Re1 = Re2 and L = ∞ [Eq. (18), solid curve] vs parameter Im2/Im(12).

Equations (37)

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( z ) Re ( z ) + Im ( z ) 1 + δ 0 + δ + exp ( 2 i q z ) + δ - exp ( - 2 i q z ) .
δ 0 , ± = l - 1 0 l [ 1 , exp ( 2 i q z ) ] [ ( z ) - 1 ] d z ,
E ( z , t ) = exp ( - i ω t ) [ E + ( z ) exp ( i q z ) + E - ( z ) exp ( - i q z ) ] .
1 q d E + d z - i ( 2 l - λ λ + 1 2 δ 0 ) E + - i δ + 2 E - = 0 , - 1 q d E - d z - i ( 2 l - λ λ + 1 2 δ 0 ) E - - i δ - 2 E + = 0.
E + ( z = 0 ) = 1 , E - ( z = L ) = 0.
E + ( z ) = E 1 + exp ( A z ) + E 2 + exp ( - A z ) , E - ( z ) = - E 1 + α - 1 - ( 1 - α + α - ) 1 / 2 exp ( A z ) - E 2 + α - 1 + ( 1 + α + α - ) 1 / 2 exp ( - A z )
A = - i q 1 2 ( δ 0 + 2 2 l - λ λ ) ( 1 - α + α - ) 1 / 2 ,
α ± = δ ± δ 0 + 2 [ ( 2 l + λ ) / λ ] .
r = - α - [ 1 - exp ( - 2 A L ) ] [ 1 + ( 1 - α + α - ) 1 / 2 ] - exp ( - 2 A L ) [ 1 - ( 1 - α + α - ) 1 / 2 ] .
r ( δ ± 0 ) = - i q L δ - 2 · 1 - exp ( + i q L { δ 0 + 2 [ ( 2 l - λ ) ] } ) i q L { δ 0 + 2 [ ( 2 l - λ ) / λ ] } .
r ( L 0 ) = i q L δ - 2 i π 2 L l δ - .
r ( L ) = α - 1 + ( 1 + α + α - ) 1 / 2 .
δ 0 = δ 1 β + δ 2 ( 1 - β ) ,
δ + = δ - = ( δ 1 - δ 2 ) sin π β π ,
α + = α - = α = sin π β π δ 1 - δ 2 δ 1 β + δ 2 ( 1 - β ) + 2 [ ( 2 l - λ ) / λ ] .
r = - α · 1 - exp { i q L [ ( sin π β ) / π β ] ( δ 1 - δ 2 ) [ ( 1 - α 2 ) 1 / 2 / α ] } [ 1 + ( 1 - α 2 ) 1 / 2 ] - [ 1 - ( 1 - α 2 ) 1 / 2 ] exp { i q L [ ( sin π β ) / π β ] ( δ 1 - δ 2 ) [ ( 1 - α 2 ) 1 / 2 / α ] }
R ( L 0 , β = ½ ) = ¼ δ 1 - δ 2 2 ( L / l ) 2 .
N ( R ~ 1 ) 2 δ 1 - δ 2 - 1 .
l opt = λ 2 ( 1 - δ 1 + δ 2 4 ) ,
R = { tanh [ ( δ 1 - δ 2 ) ( L / 2 l ) ] } 2 .
λ 0 - Δ λ 2 < λ < λ 0 + Δ λ 2 , Δ λ λ 0 = δ 1 - δ 2 π , λ 0 = 2 l ( 1 + δ 1 + δ 2 4 ) .
α = a ( β ) = sin ( π β ) π [ β + Im δ 2 / Im ( δ 1 - δ 2 ) ] .
tan ( π β opt ) = π [ β opt + Im 2 Im ( 1 - 2 ) ] .
R = { a ( β opt ) 1 + [ 1 - a 2 ( β opt ) ] 1 / 2 } 2 [ cos ( π β opt ) 1 + sin ( π β opt ) ] 2 .
β 1 2 , R | 1 2 π Im ( 1 - 2 ) Im 2 | 2 1.
β opt 1 π ( 3 π Im 2 Im 1 ) 1 / 3 1 ;     R 1 - 2 ( 3 π Im 2 Im 1 ) 1 / 3 .
N = L l ( Im 1 ) - 1 ( 3 π Im 2 Im 1 ) - 1 / 3 ~ 2 ( Im 1 ) - 1 ( 1 - R ) - 1 .
R = | α 1 + ( 1 - α 2 ) 1 / 2 | 2 ,
α = 1 - i t 1 - i ρ a 1 - i t 1 - i ρ · sin ( π β ) π [ β + Im 2 / Im ( 1 - 2 ) ] ,
t = Re ( 1 - 2 ) Im ( 1 - 2 ) , ρ ( β , l ) = 2 - λ + 2 { 1 + ¼ λ [ β Re δ 1 + ( 1 - β ) Re δ 2 ] } λ [ β Im 1 + ( 1 - β ) Im 2 ]
( R β ) ρ = d a d β · R · 2 Re [ 1 ( 1 - α 2 ) 1 / 2 ] = 0 ,
( R ρ ) β = R ( 1 - i ρ ) ( 1 - α 2 ) 1 / 2 · 2 Im [ ( 1 - i ρ ) ( 1 - α 2 ) 1 / 2 ] = 0.
ρ opt = a 2 ( β opt ) · t , l opt = λ 2 [ 1 - β 2 Re ( 1 - 2 ) ( 1 - sin 2 π β 2 π β ) - Re δ 2 2 ] .
R opt = a 2 ( 1 + t 2 ) a 4 t 2 + { 1 + [ ( 1 - a 2 ) ( 1 + a 2 t 2 ) ] 1 / 2 } 2 .
N = 1 A l = 2 a ( ( 1 - a 2 ) { [ Re ( 1 - 2 ) ] 2 a 2 + [ Im ( 1 - 2 ) ] 2 } ) 1 / 2 .
chromium , δ 1 = ( 39.7 + 17.9 i ) · 10 - 3 ; carbon , δ 2 = ( 20.8 + 3.75 i ) · 10 - 3 .
nickel , δ 1 = ( 60 + 15 i ) · 10 - 4 ; titanium , δ 2 = ( 18.5 + 4.06 i ) · 10 - 4 .

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