Abstract

Successive evaporation of two correctly chosen elements now make possible highly efficient X–UV mirrors. However, good results depend on two points: First we must know the manufacturing tolerances; these depend strongly on the spectral range and the refractive index of materials used for the coating. The other important parameter is the roughness of each interface, which can result in significant losses in the reflectivity of the mirrors. This paper examines both points. Specular reflectance measurements at 0.159 nm are very sensitive to the roughness of each layer. To demonstrate this sensitivity we develop a method of calculation of this reflectance taking into account the roughness of each interface of the multilayer.

© 1984 Optical Society of America

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References

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  1. W. Deubner, Ann. Phys. Leipzig 5, 261 (1930).
    [CrossRef]
  2. T. W. Barbee, in Proceedings, Topical Conference on Low Energy X-Ray Diagnostics (American Institute of Physics, New York, 1980).
  3. J. P. Henry, E. Spiller, M. Weisskopf, Proc. Soc. Photo-Opt. Instrum. Eng. 316, 166 (1981).
  4. A. V. Vinogradov, B. Ya. Zeldovich, Appl. Opt. 16, 89 (1977).
    [CrossRef] [PubMed]
  5. P. Croce, J. Opt. (Paris) 12, 163 (1981).
    [CrossRef]
  6. Y. Lepetre, L. Nevot, B. Vidal, in Proceedings, Fourth International Conference on Plasmas, CIP 82, Nice, France13–17 Sept. 1982.
  7. L. G. Parratt, Phys. Rev. 95, 359 (1954).
    [CrossRef]
  8. A. E. Rosenbluth, P. Lee, Appl. Phys. Lett. 40, 466 (1982).
    [CrossRef]
  9. R. Marmoret, J. M. Andre, Appl. Opt. 22, 17 (1983).
    [CrossRef] [PubMed]
  10. B. L. Henke et al.. At. Data Nucl. Data Tables 27, 1 (1982).
    [CrossRef]
  11. J. C. Rife, J. F. Osantowski, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 103 (1981).
  12. B. Vidal, A. Fornier, E. Pelletier, Appl. Opt. 17, 1038 (1978).
    [CrossRef] [PubMed]
  13. P. Beckmann, A. Spizzichino, The scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  14. J. M. Eastman, “Scattering by ALL Dielectric Multilayer Band-Pass Filters and Mirrors for Lasers,” Phys. Thin Films 10, 000 (1978).
  15. C. K. Carniglia, Opt. Eng. 18, 104 (March–April1979).
    [CrossRef]
  16. P. Croce, J. Opt. (Paris) 10, 141 (1979).
    [CrossRef]
  17. P. Croce, L. Prod’homme, Nouv. Rev. Opt. 7, 121 (1976).
    [CrossRef]
  18. L. Nevot, P. Croce, Rev. Phys. Appl. 15, 761 (1980).
    [CrossRef]
  19. P. Bousquet, F. Flory, P. Roche, J. Opt. Soc. Am. 71, 1115 (1981).
    [CrossRef]
  20. A. Roger, Opt. Acta 30, 575 (1983).
    [CrossRef]
  21. J. H. Apfel, Appl. Opt. 15, 2339 (1976).
    [CrossRef] [PubMed]
  22. A. K. Hagenlocher, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 118 (1981).

1983

1982

A. E. Rosenbluth, P. Lee, Appl. Phys. Lett. 40, 466 (1982).
[CrossRef]

B. L. Henke et al.. At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

1981

J. C. Rife, J. F. Osantowski, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 103 (1981).

P. Croce, J. Opt. (Paris) 12, 163 (1981).
[CrossRef]

P. Bousquet, F. Flory, P. Roche, J. Opt. Soc. Am. 71, 1115 (1981).
[CrossRef]

A. K. Hagenlocher, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 118 (1981).

J. P. Henry, E. Spiller, M. Weisskopf, Proc. Soc. Photo-Opt. Instrum. Eng. 316, 166 (1981).

1980

L. Nevot, P. Croce, Rev. Phys. Appl. 15, 761 (1980).
[CrossRef]

1979

C. K. Carniglia, Opt. Eng. 18, 104 (March–April1979).
[CrossRef]

P. Croce, J. Opt. (Paris) 10, 141 (1979).
[CrossRef]

1978

J. M. Eastman, “Scattering by ALL Dielectric Multilayer Band-Pass Filters and Mirrors for Lasers,” Phys. Thin Films 10, 000 (1978).

B. Vidal, A. Fornier, E. Pelletier, Appl. Opt. 17, 1038 (1978).
[CrossRef] [PubMed]

1977

1976

J. H. Apfel, Appl. Opt. 15, 2339 (1976).
[CrossRef] [PubMed]

P. Croce, L. Prod’homme, Nouv. Rev. Opt. 7, 121 (1976).
[CrossRef]

1954

L. G. Parratt, Phys. Rev. 95, 359 (1954).
[CrossRef]

1930

W. Deubner, Ann. Phys. Leipzig 5, 261 (1930).
[CrossRef]

Andre, J. M.

Apfel, J. H.

Barbee, T. W.

T. W. Barbee, in Proceedings, Topical Conference on Low Energy X-Ray Diagnostics (American Institute of Physics, New York, 1980).

Beckmann, P.

P. Beckmann, A. Spizzichino, The scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Bousquet, P.

Carniglia, C. K.

C. K. Carniglia, Opt. Eng. 18, 104 (March–April1979).
[CrossRef]

Croce, P.

P. Croce, J. Opt. (Paris) 12, 163 (1981).
[CrossRef]

L. Nevot, P. Croce, Rev. Phys. Appl. 15, 761 (1980).
[CrossRef]

P. Croce, J. Opt. (Paris) 10, 141 (1979).
[CrossRef]

P. Croce, L. Prod’homme, Nouv. Rev. Opt. 7, 121 (1976).
[CrossRef]

Deubner, W.

W. Deubner, Ann. Phys. Leipzig 5, 261 (1930).
[CrossRef]

Eastman, J. M.

J. M. Eastman, “Scattering by ALL Dielectric Multilayer Band-Pass Filters and Mirrors for Lasers,” Phys. Thin Films 10, 000 (1978).

Flory, F.

Fornier, A.

Hagenlocher, A. K.

A. K. Hagenlocher, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 118 (1981).

Henke, B. L.

B. L. Henke et al.. At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

Henry, J. P.

J. P. Henry, E. Spiller, M. Weisskopf, Proc. Soc. Photo-Opt. Instrum. Eng. 316, 166 (1981).

Lee, P.

A. E. Rosenbluth, P. Lee, Appl. Phys. Lett. 40, 466 (1982).
[CrossRef]

Lepetre, Y.

Y. Lepetre, L. Nevot, B. Vidal, in Proceedings, Fourth International Conference on Plasmas, CIP 82, Nice, France13–17 Sept. 1982.

Marmoret, R.

Nevot, L.

L. Nevot, P. Croce, Rev. Phys. Appl. 15, 761 (1980).
[CrossRef]

Y. Lepetre, L. Nevot, B. Vidal, in Proceedings, Fourth International Conference on Plasmas, CIP 82, Nice, France13–17 Sept. 1982.

Osantowski, J. F.

J. C. Rife, J. F. Osantowski, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 103 (1981).

Parratt, L. G.

L. G. Parratt, Phys. Rev. 95, 359 (1954).
[CrossRef]

Pelletier, E.

Prod’homme, L.

P. Croce, L. Prod’homme, Nouv. Rev. Opt. 7, 121 (1976).
[CrossRef]

Rife, J. C.

J. C. Rife, J. F. Osantowski, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 103 (1981).

Roche, P.

Roger, A.

A. Roger, Opt. Acta 30, 575 (1983).
[CrossRef]

Rosenbluth, A. E.

A. E. Rosenbluth, P. Lee, Appl. Phys. Lett. 40, 466 (1982).
[CrossRef]

Spiller, E.

J. P. Henry, E. Spiller, M. Weisskopf, Proc. Soc. Photo-Opt. Instrum. Eng. 316, 166 (1981).

Spizzichino, A.

P. Beckmann, A. Spizzichino, The scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Vidal, B.

B. Vidal, A. Fornier, E. Pelletier, Appl. Opt. 17, 1038 (1978).
[CrossRef] [PubMed]

Y. Lepetre, L. Nevot, B. Vidal, in Proceedings, Fourth International Conference on Plasmas, CIP 82, Nice, France13–17 Sept. 1982.

Vinogradov, A. V.

Weisskopf, M.

J. P. Henry, E. Spiller, M. Weisskopf, Proc. Soc. Photo-Opt. Instrum. Eng. 316, 166 (1981).

Zeldovich, B. Ya.

Ann. Phys. Leipzig

W. Deubner, Ann. Phys. Leipzig 5, 261 (1930).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

A. E. Rosenbluth, P. Lee, Appl. Phys. Lett. 40, 466 (1982).
[CrossRef]

At. Data Nucl. Data Tables

B. L. Henke et al.. At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

J. Opt. (Paris)

P. Croce, J. Opt. (Paris) 10, 141 (1979).
[CrossRef]

P. Croce, J. Opt. (Paris) 12, 163 (1981).
[CrossRef]

J. Opt. Soc. Am.

Nouv. Rev. Opt.

P. Croce, L. Prod’homme, Nouv. Rev. Opt. 7, 121 (1976).
[CrossRef]

Opt. Acta

A. Roger, Opt. Acta 30, 575 (1983).
[CrossRef]

Opt. Eng.

C. K. Carniglia, Opt. Eng. 18, 104 (March–April1979).
[CrossRef]

Phys. Rev.

L. G. Parratt, Phys. Rev. 95, 359 (1954).
[CrossRef]

Phys. Thin Films

J. M. Eastman, “Scattering by ALL Dielectric Multilayer Band-Pass Filters and Mirrors for Lasers,” Phys. Thin Films 10, 000 (1978).

Proc. Soc. Photo-Opt. Instrum. Eng.

A. K. Hagenlocher, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 118 (1981).

J. C. Rife, J. F. Osantowski, Proc. Soc. Photo-Opt. Instrum. Eng. 315, 103 (1981).

J. P. Henry, E. Spiller, M. Weisskopf, Proc. Soc. Photo-Opt. Instrum. Eng. 316, 166 (1981).

Rev. Phys. Appl.

L. Nevot, P. Croce, Rev. Phys. Appl. 15, 761 (1980).
[CrossRef]

Other

P. Beckmann, A. Spizzichino, The scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Y. Lepetre, L. Nevot, B. Vidal, in Proceedings, Fourth International Conference on Plasmas, CIP 82, Nice, France13–17 Sept. 1982.

T. W. Barbee, in Proceedings, Topical Conference on Low Energy X-Ray Diagnostics (American Institute of Physics, New York, 1980).

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Figures (16)

Fig. 1
Fig. 1

Schematic representation of a multilayer thin film. The coordinate axis used in calculating the properties of the structure is shown.

Fig. 2
Fig. 2

Schematic representation of waves involved at a perfect interface between two media of refractive indices ñ1 and ñ2.

Fig. 3
Fig. 3

Theoretical reflectance at 0.154 nm for a 23-metallic layer stack of design: silicon substrate [2 dL − 11(dHdL) − dL] air, where dH is an 0.8-nm thick tungsten layer and dL is a 2.58-nm thick carbon layer.11 For tungsten δH = 0.457 × 10−4, βH = 0.4 × 10−5; for carbon δL = 0.66 × 10−5, βL = 0.11 × 10−7; for silicon substrate δS = 0.756 × 10−5, βS = 0.170 × 10−6.

Fig. 4
Fig. 4

Theoretical performances vs wavelength of the stack described in Fig. 3 at γ = 45°.

Fig. 5
Fig. 5

Theoretical performance vs wavelength of 99 metallic layers of design: silicon substrate [49 (dLdH) − dL] vacuum; — S polarization at γ = 45°. The maximum of reflectance for the P polarization is ~10−4 where dH,dL,δH,δL,βH,βL,δS,δS are the same as in Fig. 3.

Fig. 6
Fig. 6

Envelopes showing at 0.154 nm the logarithm of the reflectance found with random errors of 3% standard deviation: — perfect filter of Fig. 3; — filters with random errors.

Fig. 7
Fig. 7

Envelopes showing at 0.154 nm the logarithm of reflectance found with random errors of 1% standard deviation: — perfect filter of Fig. 3; — filters with random errors.

Fig. 8
Fig. 8

Envelopes showing the reflectance found in the soft x-ray range with random errors of 1% standard deviation: — perfect filter of Fig. 4; — filters with random errors.

Fig. 9
Fig. 9

Envelopes showing the reflectance found in the soft x-ray range with random errors of 1% standard deviation: — perfect filter of 99 layers of Fig. 5; — filters with random errors.

Fig. 10
Fig. 10

Schematic contour around the roughness of the interface where the Green theorem is used.

Fig. 11
Fig. 11

Schematic calculation of the Wronskian with a middle stage S′. From W S S ¯ = 0, we can deduce reciprocity relations.

Fig. 12
Fig. 12

(a) Profile of |E|2 for the S polarization case inside of the stack: silicon substrate [dL, 11(dLdH)] vacuum; λ = 0.475 nm; γ = 45°. (b) Same stack as (a) but we assume a perturbation of 0.4 nm on each interface of the stack. (c) Enlarged view of the three first layers of the stack.

Fig. 13
Fig. 13

Losses in reflectance values at 0.154 nm when we assume a 0.4-nm perturbation at each interface of the stack described in Fig. 3: — same filter as Fig. 3; — filter with roughness.

Fig. 14
Fig. 14

Theoretical reflectance in the soft x-ray range for a stack silicon substrate [dH − 49(dLdH)] vacuum where dH, dL, and all δ and β are those used in Fig. 3: — theoretical reflectance of the same stack as (a) assuming 0.8-nm roughness on the tungsten layer and 0.3-nm roughness on the carbon layer.

Fig. 15
Fig. 15

Evolution of the diffraction curve (logR) at 0.154 nm when the perturbation induced by the roughness increases from 0 to 0.8 nm for dH and to 0.3 nm for dL. The stack is defined in Fig. 3.

Fig. 16
Fig. 16

(a) Evolution of diffraction curve when perturbation induced by roughness increases from zero to σ1σ12 = 0 and σ13σ24 = 0.4 nm. (b) Same calculation as (a) but σ1, … σ12,σ13 = 0 and σ14σ24 = 0.4 nm. σ1 is the perturbation induced by the substrate roughness.

Equations (34)

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m λ = 2 d sin γ m ,             m = 1 , 2 , .
E ( x , y ) = [ A ˜ exp ( - i β 2 y ) + B ˜ exp ( + i β 2 y ) ] exp ( i α x ) .
E ( x , y ) = [ C ˜ exp ( - i β 1 y ) + D ˜ exp ( + i β 1 y ) ] exp ( i α x ) ,
k 0 = 2 π λ 0 ,
α = k 0 sin θ
θ = π 2 - γ ,
β 1 = ( k 0 n ˜ 1 ) 2 - α 2 ,             β 2 = ( k 0 n ˜ 2 ) 2 - α 2 .
| A ˜ B ˜ | = 1 2 | p 11 p 12 p 21 p 22 | · | C ˜ D ˜ | ,
p 11 = a 1 exp [ - i ( β 1 - β 2 ) y ] , p 12 = a 2 exp [ + i ( β 1 + β 2 ) y ] , p 21 = a 2 exp [ - i ( β 1 + β 2 ) y ] , p 22 = a 1 exp [ + i ( β 1 - β 2 ) y ] ,
a 1 = 1 + β 1 β 2 ,             a 2 = 1 - β 1 β 2 .
R = | p 21 p 11 | 2 .
β 1 β 2 × ɛ 2 ɛ 1 ,
Δ E + k 2 E = 0 ,
Δ E ˜ + k ˜ 2 E = 0 with k ˜ 2 = { k 1 2 = ω 2 ɛ 1 μ 0 if y < y 0 k 2 2 = ω 2 ɛ 2 μ 0 if y > y 0 .
S ( E ˜ Δ E - E Δ E ˜ ) d x d y = M N O P ( E ˜ d E d n - E d E ˜ d n ) d l ,
1 L M N ( E ˜ E y - E E ˜ y ) d x - 1 L O P ( E ˜ E y - E E ˜ y ) d x = y 1 y 2 ( k 2 - k ˜ 2 ) E E ˜ d y x ,
W S S ˜ = 2 i β 2 ( A B ˜ - A ˜ B ) - 2 i β 1 ( C D ˜ - C ˜ D ) = y 1 y 2 ( k 2 - k ˜ 2 ) E E ˜ d y x .
β 2 A ˜ B = β 1 C ˜ D ;
W S S = 2 i β 2 A B - 2 i β 1 C D .
W S S = 2 i β 2 ( A - A ˜ ) B ,
A - A ˜ = 1 2 i β 2 B y 1 y 2 ( k 2 - k ˜ 2 ) E E d y x .
E ( x , y ) = exp ( i α x - i β 1 y )
E ( x , y ) = B exp ( - i α x + i β 2 y )             if y > y 0
A - A ˜ = 1 2 i β 2 ( k 1 2 - k 2 2 ) y 0 h ( x ) exp [ i ( β 2 - β 1 ) y ] d y x .
A - A ˜ = β 1 + β 2 2 β 2 ( exp [ i ( β 2 - β 1 ) h ( x ) ] x - 1 ) .
f ( h ( x ) ) x = - + f ( y ) w ( y ) d y ,
w ( y ) = ( 2 π σ 2 ) - 1 / 2 exp [ ( y - y 0 ) 2 / ( 2 σ 2 ) ] .
A - A ˜ = β 1 + β 2 2 β 2 { - + w ( y ) exp [ i ( β 2 - β 1 ) y ] d y - 1 } = β 1 + β 2 2 β 2 exp [ - i ( β 1 - β 2 ) y 0 ] { exp [ - ( β 2 - β 1 ) 2 σ 2 2 ] - 1 } ,
A = A ˜ exp [ - ( β 2 - β 1 ) 2 σ 2 2 ] .
[ A B ] = [ p 11 exp [ - ( β 1 - β 2 ) 2 σ 2 2 ] p 12 exp [ - ( β 1 + β 2 ) 2 σ 2 2 ] p 21 exp [ - ( β 1 + β 2 ) 2 σ 2 2 ] p 22 exp [ - ( β 1 - β 2 ) 2 σ 2 2 ] ] × [ C D ] .
· ( 1 k 2 H ) + H = 0 ,
1 L M N O P ( 1 k 2 H ˜ d H d n - 1 k 2 H d H ˜ d n ) d l = y 1 y 2 ( 1 k 2 - 1 k ˜ 2 ) H · H ˜ d y x .
W S S ˜ = 2 i β 2 k 2 2 ( A B ˜ - A ˜ B ) - 2 i β 1 k 1 2 ( C D ˜ - C ˜ D ) .
R = R ˜ exp ( - 16 π 2 σ 2 λ 2 sin 2 γ ) ,

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