Abstract

The phase shift upon transmission of a nonabsorbing multilayer is shown to be a monotonically decreasing function of the wave number, with an average slope proportional to the optical thickness of the coating. Two limiting situations of the phase shift upon reflection are examined: In one the phase monotonically increases with wavelength, and in the other the phase oscillates. The phase shift upon reflection is derivable from Kramers–Kronig-type relationships, provided the radiant reflectance and the Blaschke factors are known. Characteristic features of the refractive-index profile related to these factors are discussed.

© 1997 Optical Society of America

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References

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  1. K. Ferencz, R. Szipocs, “Recent developments of laser optical coatings in Hungary,” Opt. Eng. 32, 2525–2538 (1993).
    [CrossRef]
  2. F. Lemarquis, E. Pelletier, “Optical coating without phase dispersion for a Fabry–Perot interferometer,” Appl. Opt. 35, 4987–4992 (1996).
    [CrossRef] [PubMed]
  3. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1958).
  4. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).
  5. F. Stern, “Elementary theory of the optical property of solids,” in Solid State Physics, F. Seitz, D. Turnbull, eds. (Academic, New York, 1963), Vol. 15, pp. 299–420.
  6. W. Lichten, “Precise wavelength measurements and optical phase shifts. I. General theory,” J. Opt. Soc. Am. A 2, 189–1876 (1985).
  7. P. Grosse, V. Offermann, “Analysis of reflectance data using the Kramers–Kronig relation,” Appl. Phys. A 52, 138–144 (1991).
    [CrossRef]
  8. M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Prob. 11, 1–28 (1995).
    [CrossRef]
  9. A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their application,” Appl. Opt. 32, 5417–5426 (1993).
    [CrossRef] [PubMed]
  10. I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” USSR Comput. Math. Math. Phys. 33, 387–395 (1993).
  11. A. V. Tikhonravov, “Amplitude-phase properties of the spectral coefficients of laminar media,” USSR Comput. Math. Math. Phys. 25, 77–83 (1985).
    [CrossRef]
  12. S. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Eq. (1.3.26).
  13. H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986), p. 51.
    [CrossRef]
  14. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989).
  15. Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
    [CrossRef]
  16. A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin film optics,” Inverse Prob. 11, 251–270 (1995).
    [CrossRef]
  17. P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallin, 1971) (in Russian).
  18. H. Pohlack, “Problem of reducing reflectance of optical glass at nonnormal incidence,” in Jenaer Jahrbuch (VEB Carl Zeiss, Jena, 1952) (in German).
  19. E. Delano, R. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 69–137.
  20. B. Levin, Entire Functions Roots Distribution (Gostekhteorizdat, Moscow, 1956) (in Russian).

1996

1995

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin film optics,” Inverse Prob. 11, 251–270 (1995).
[CrossRef]

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Prob. 11, 1–28 (1995).
[CrossRef]

1993

I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” USSR Comput. Math. Math. Phys. 33, 387–395 (1993).

K. Ferencz, R. Szipocs, “Recent developments of laser optical coatings in Hungary,” Opt. Eng. 32, 2525–2538 (1993).
[CrossRef]

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their application,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

1991

P. Grosse, V. Offermann, “Analysis of reflectance data using the Kramers–Kronig relation,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

1989

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

1985

W. Lichten, “Precise wavelength measurements and optical phase shifts. I. General theory,” J. Opt. Soc. Am. A 2, 189–1876 (1985).

A. V. Tikhonravov, “Amplitude-phase properties of the spectral coefficients of laminar media,” USSR Comput. Math. Math. Phys. 25, 77–83 (1985).
[CrossRef]

Delano, E.

E. Delano, R. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 69–137.

Ferencz, K.

K. Ferencz, R. Szipocs, “Recent developments of laser optical coatings in Hungary,” Opt. Eng. 32, 2525–2538 (1993).
[CrossRef]

Furman, S. A.

S. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Eq. (1.3.26).

Grosse, P.

P. Grosse, V. Offermann, “Analysis of reflectance data using the Kramers–Kronig relation,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

Kard, P. G.

P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallin, 1971) (in Russian).

Klibanov, M. V.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Prob. 11, 1–28 (1995).
[CrossRef]

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin film optics,” Inverse Prob. 11, 251–270 (1995).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1958).

Lemarquis, F.

Levin, B.

B. Levin, Entire Functions Roots Distribution (Gostekhteorizdat, Moscow, 1956) (in Russian).

Lichten, W.

W. Lichten, “Precise wavelength measurements and optical phase shifts. I. General theory,” J. Opt. Soc. Am. A 2, 189–1876 (1985).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1958).

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986), p. 51.
[CrossRef]

Mandel, L.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).

Offermann, V.

P. Grosse, V. Offermann, “Analysis of reflectance data using the Kramers–Kronig relation,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

Pegis, R.

E. Delano, R. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 69–137.

Pelletier, E.

Pohlack, H.

H. Pohlack, “Problem of reducing reflectance of optical glass at nonnormal incidence,” in Jenaer Jahrbuch (VEB Carl Zeiss, Jena, 1952) (in German).

Sacks, P. E.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Prob. 11, 1–28 (1995).
[CrossRef]

Stern, F.

F. Stern, “Elementary theory of the optical property of solids,” in Solid State Physics, F. Seitz, D. Turnbull, eds. (Academic, New York, 1963), Vol. 15, pp. 299–420.

Szipocs, R.

K. Ferencz, R. Szipocs, “Recent developments of laser optical coatings in Hungary,” Opt. Eng. 32, 2525–2538 (1993).
[CrossRef]

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989).

Tikhonravov, A. V.

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin film optics,” Inverse Prob. 11, 251–270 (1995).
[CrossRef]

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Prob. 11, 1–28 (1995).
[CrossRef]

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their application,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” USSR Comput. Math. Math. Phys. 33, 387–395 (1993).

A. V. Tikhonravov, “Amplitude-phase properties of the spectral coefficients of laminar media,” USSR Comput. Math. Math. Phys. 25, 77–83 (1985).
[CrossRef]

S. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Eq. (1.3.26).

Zuev, I. V.

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin film optics,” Inverse Prob. 11, 251–270 (1995).
[CrossRef]

I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” USSR Comput. Math. Math. Phys. 33, 387–395 (1993).

Am. J. Phys.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[CrossRef]

Appl. Opt.

Appl. Phys.

P. Grosse, V. Offermann, “Analysis of reflectance data using the Kramers–Kronig relation,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

Inverse Prob.

M. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Prob. 11, 1–28 (1995).
[CrossRef]

A. V. Tikhonravov, M. V. Klibanov, I. V. Zuev, “Numerical study of the phaseless inverse scattering problem in thin film optics,” Inverse Prob. 11, 251–270 (1995).
[CrossRef]

J. Opt. Soc. Am.

W. Lichten, “Precise wavelength measurements and optical phase shifts. I. General theory,” J. Opt. Soc. Am. A 2, 189–1876 (1985).

Opt. Eng.

K. Ferencz, R. Szipocs, “Recent developments of laser optical coatings in Hungary,” Opt. Eng. 32, 2525–2538 (1993).
[CrossRef]

USSR Comput. Math. Math. Phys.

I. V. Zuev, A. V. Tikhonravov, “The uniqueness of the determination of the parameters of a stratified medium from the energy reflectance,” USSR Comput. Math. Math. Phys. 33, 387–395 (1993).

A. V. Tikhonravov, “Amplitude-phase properties of the spectral coefficients of laminar media,” USSR Comput. Math. Math. Phys. 25, 77–83 (1985).
[CrossRef]

Other

S. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Eq. (1.3.26).

H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986), p. 51.
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989).

P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallin, 1971) (in Russian).

H. Pohlack, “Problem of reducing reflectance of optical glass at nonnormal incidence,” in Jenaer Jahrbuch (VEB Carl Zeiss, Jena, 1952) (in German).

E. Delano, R. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 69–137.

B. Levin, Entire Functions Roots Distribution (Gostekhteorizdat, Moscow, 1956) (in Russian).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1958).

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).

F. Stern, “Elementary theory of the optical property of solids,” in Solid State Physics, F. Seitz, D. Turnbull, eds. (Academic, New York, 1963), Vol. 15, pp. 299–420.

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Figures (11)

Fig. 1
Fig. 1

Complex zeros (dots) and poles (crosses) of the amplitude reflection coefficient of an eight-layer stack of alternating high-index and low-index layers. Each layer is λ0/4 in optical thickness at λ0 of 500 nm. The high, low, substrate, and ambient medium indices are nH = 2.35, nL = 1.45, ns = 1.52, and na = 1.00, respectively. A high-index layer is adjacent to the substrate.

Fig. 2
Fig. 2

Graphic interpretation of arg(μj - k) in Eq. (10), relating to the phase shift upon transmission.

Fig. 3
Fig. 3

Explanation of the behavior of arg[t(k)] when k is increasing and passing near one of the f1(ν) zeros in the complex wave-number plane.

Fig. 4
Fig. 4

Phase shift upon transmission for two multilayers: (a) An eight-layer quarter-wave stack S| (H L)4 |air with nH = 2.35, nL = 1.45. (b) A three-layer antireflection coating S| M HH L |air with nM = 1.63, nH = 2.1, nL = 1.38. In both coatings ns = 1.52 and na = 1.00. The reference wavelength for the quarter-wave is 500 nm.

Fig. 5
Fig. 5

(a) Multiple reflections of beams in a two-layer coating. r0, r1, and r2 are the amplitude reflection coefficients at the lower, inner, and upper interfaces, respectively. (b) Phase difference PD [see Eq. (14)] and the radiant reflectance for the quarter-wave stack captioned in Fig. 4(a).

Fig. 6
Fig. 6

Explanation of the behavior of arg[r(k)] when all zeros of the amplitude reflection coefficient are in the lower half-plane.

Fig. 7
Fig. 7

Single-layer coatings of 0.5-µm optical thickness illustrating the limiting situations with the location of the amplitude reflection coefficient zeros. The refractive index of the first is 2.20 [solid line in (a)], and the refractive index of the second is 1.59 [dashed line in (a)]. The refractive indices of the substrate and the ambient medium are ns = 3.50 and na = 1.0. The zeros of the amplitude reflection coefficient of the first and the second coatings are shown in (b) and (c), respectively. The corresponding phase shifts upon reflection are presented in (d) and (e). (f) Radiant reflectance versus wave number, which is the same for both coatings.

Fig. 8
Fig. 8

Intermediate situation with the locations of the zeros of the amplitude reflection coefficient: (a) Refractive-index profile of the eight-layer quarter-wave stack captioned in Fig. 4; (b) first nine amplitude reflection coefficient zeros; (c), (d) phase shift upon reflection and the radiant reflectance of this multilayer. For the ordinate of (b) the physical units of σ are 104 cm-1.

Fig. 9
Fig. 9

Transfer of amplitude reflection coefficient zeros from one complex half-plane to another one as described by Eq. (21).

Fig. 10
Fig. 10

Refractive-index profiles of three inhomogeneous systems obtained from the single-layer coating shown in (a). [This coating is the same as the solid line in Fig. 7(a).] (d) Transfer of the first pair of zeros that is closest to the imaginary axis. (g), (j) Transfer of the two nearest and three nearest pairs of zeros, respectively. (b), (e), (h), (k) Amplitude reflection coefficient zeros for the aforementioned systems. (c), (f), (i), (l) Phase shifts upon reflection of these systems, respectively. The radiant reflectance curves of all the above systems are identical and appear in Fig. 7(f). For the ordinates of (b), (e), (h), and (k) the physical units of σ are 104 cm-1.

Fig. 11
Fig. 11

Transformation of the refractive-index profile corresponding to the transfer of a large number of zeros (see the text for detail). For the ordinates of (b) and (e), the physical units of σ are 104 cm-1.

Equations (31)

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k2π/λ.
Rk=rk2,  Tk=nsna-1τk2.
Tk=tk2.
rk2+tk2=1.
νk+iσ,
f1ν1tν,  f2νrνtν.
f1ν=1tν=12 na+nsns-1/2 na-1/2×Πj=11-νμj-1,
f2ν=rνtν=12 na-nsns-1/2 na-1/2×Πj=11-ννj-1.
f1k=1tk=12 na+nsns-1/2 na-1/2×Πj=1 μj-kμj-1.
argtk=-j=1 argμj-k+j=1 argμj.
ddk argtkaverage=-xa.
τ=t0t1 exp-jϕ11+r0r2 exp-j2ϕ1,
τ=t0t1t2 exp-jϕ1-jϕ21+r0r1 exp-j2ϕ1+r1r2 exp-j2ϕ2+r0r2 exp-j2ϕ1-j2ϕ2,
PD=argτk-xak,
arg(f2k=argrk-argtk=j=1 argνj-k-j=1 νj.
argrk=argf2k+argtk,
argrk=2argtk-argrk+π.
nsnansns, nnansn, nanansna.
argrk=argtk.
argrk=argtk±π/2.
r˜ν=rν νj*-ν νj+ννj-ν νj*+ν,
r˜k=rk.
arg|r˜k]=argrk+2 arctan2σj kk2-Vj2.
f1k=1tk,  f2k=rktk
f1k=j=1M aj expikxj,  f2k=j=1M bj expikxj,
-xa=x1<x2<<xM-1<xM=xa,  a10, aM0, b10, bM0.
f1ν=f10 Πj 1-νμj-1,
f2ν=f20 Πj 1-ννj-1,
f10=1/t0,  f20=r0/t0.
r0=na-nsna+ns-1,  t0=2nans1/2na+ns-1,
f10=12 na+nsna-1/2 ns-1/2,  f20=12 na-nsna-1/2 ns-1/2.

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