Abstract

The method of equivalent layers is a commonly used technique for designing optical multilayer interference coatings. Herpin’s theorem [C. R. Acad. Sci. 225, 182 (1947)] states that every symmetrical multilayer structure is equivalent, at one arbitrary wavelength, to a single homogeneous layer. The Herpin equivalent layer is described by two design parameters, the equivalent index and the equivalent thickness. Alternatively, we recently developed an exact coupled-mode analysis for the description of multilayer interference coatings composed of a symmetrical combination of layers. The design parameters of the coupled-mode theory are the exact coupling coefficient and the exact detuning coefficient. Recently we used this method in the design of chirped mirrors for dispersion compensation. We prove that the two methods are equivalent and derive relations that link the design parameters of both formalisms. By use of these relations it is possible to translate between the coupled-mode formalism and the method of equivalent layers. The simultaneous availability of both design methods gives a new perspective on the analytical design of optical interference coatings with challenging spectral response characteristics.

© 2000 Optical Society of America

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References

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  1. A. Herpin, “Calcul du pouvoir réflecteur d’un système stratifiè quelconque,” C. R. Acad. Sci. 225, 182–183 (1947).
  2. L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. 42, 807–810 (1952).
    [CrossRef]
  3. P. H. Berning, “Use of equivalent films in the design of infrared multilayer antireflection coatings,” J. Opt. Soc. Am. 52, 431–436 (1962).
    [CrossRef]
  4. A. Thelen, “Equivalent layers in multilayer filters,” J. Opt. Soc. Am. 56, 1533–1538 (1966).
    [CrossRef]
  5. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Adam Hilger, Bristol, UK, 1985).
  6. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989).
  7. C. J. van der Laan, H. J. Frankena, “Equivalent layers: another way to look at them,” Appl. Opt. 34, 681–687 (1995).
    [CrossRef] [PubMed]
  8. J. R. Pierce, “Coupling of modes propagation,” J. Appl. Phys. 25, 179–183 (1954).
    [CrossRef]
  9. M. Matsuhara, K. O. Hill, A. Watanabe, “Optical-waveguide filters: synthesis,” J. Opt. Soc. Am. 65, 804–809 (1975).
    [CrossRef]
  10. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides,” Opt. Lett. 12, 847–849 (1987).
    [CrossRef] [PubMed]
  11. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  12. N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
    [CrossRef]
  13. F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
    [CrossRef] [PubMed]
  14. N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron 4, 197–208 (1998).
    [CrossRef]
  15. In Eqs. (7)–(12) we define the elements of the transfer matrix slightly differently from the original reference (Ref. 12). In this paper we investigate the multilayer coating with respect to passband regions as the standard case. Hence the elements of the transfer matrix are expressed by trigonometric functions, unlike in Ref. 12, where the multilayer structure was investigated with respect to the fundamental stop band as the standard case. The formulas of this paper are obtained from Ref. 12 when the substitution γ → iγ is made.
  16. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984).
  17. R. Szipöcs, K. Ferencz, C. Spielmann, F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19, 201–203 (1994).
    [CrossRef] [PubMed]
  18. N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
    [CrossRef]

1999

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

1998

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron 4, 197–208 (1998).
[CrossRef]

1997

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

1995

1994

1987

1975

1972

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

1966

1962

1954

J. R. Pierce, “Coupling of modes propagation,” J. Appl. Phys. 25, 179–183 (1954).
[CrossRef]

1952

L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. 42, 807–810 (1952).
[CrossRef]

1947

A. Herpin, “Calcul du pouvoir réflecteur d’un système stratifiè quelconque,” C. R. Acad. Sci. 225, 182–183 (1947).

Berning, P. H.

Epstein, L. I.

L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. 42, 807–810 (1952).
[CrossRef]

Ferencz, K.

Frankena, H. J.

Haus, H. A.

Heine, C.

Herpin, A.

A. Herpin, “Calcul du pouvoir réflecteur d’un système stratifiè quelconque,” C. R. Acad. Sci. 225, 182–183 (1947).

Hill, K. O.

Kärtner, F. X.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron 4, 197–208 (1998).
[CrossRef]

F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

Keller, U.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron 4, 197–208 (1998).
[CrossRef]

F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Krausz, F.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Adam Hilger, Bristol, UK, 1985).

Matsuhara, M.

Matuschek, N.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron 4, 197–208 (1998).
[CrossRef]

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
[CrossRef] [PubMed]

Morf, R.

Ouellette, F.

Pierce, J. R.

J. R. Pierce, “Coupling of modes propagation,” J. Appl. Phys. 25, 179–183 (1954).
[CrossRef]

Scheuer, V.

Schibli, T.

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Spielmann, C.

Szipöcs, R.

Thelen, A.

A. Thelen, “Equivalent layers in multilayer filters,” J. Opt. Soc. Am. 56, 1533–1538 (1966).
[CrossRef]

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

Tilsch, M.

Tschudi, T.

van der Laan, C. J.

Watanabe, A.

Appl. Opt.

C. R. Acad. Sci.

A. Herpin, “Calcul du pouvoir réflecteur d’un système stratifiè quelconque,” C. R. Acad. Sci. 225, 182–183 (1947).

IEEE J. Quantum Electron.

N. Matuschek, F. X. Kärtner, U. Keller, “Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations,” IEEE J. Quantum Electron. 33, 295–302 (1997).
[CrossRef]

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron 4, 197–208 (1998).
[CrossRef]

J. Appl. Phys.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. R. Pierce, “Coupling of modes propagation,” J. Appl. Phys. 25, 179–183 (1954).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Other

In Eqs. (7)–(12) we define the elements of the transfer matrix slightly differently from the original reference (Ref. 12). In this paper we investigate the multilayer coating with respect to passband regions as the standard case. Hence the elements of the transfer matrix are expressed by trigonometric functions, unlike in Ref. 12, where the multilayer structure was investigated with respect to the fundamental stop band as the standard case. The formulas of this paper are obtained from Ref. 12 when the substitution γ → iγ is made.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984).

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Adam Hilger, Bristol, UK, 1985).

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

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Figures (2)

Fig. 1
Fig. 1

Refractive-index profile of a symmetrical combination of three homogeneous layers. The optical phase shift of each outer layer (medium 1) is ϕ1/2, with ϕ1 = kn 1 d 1, and the optical phase shift of the center layer (medium 2) is ϕ2 = kn 2 d 2. Here k is the vacuum wave number, n 1/2 is the refractive index of medium 1/2, and d 1/2 is the corresponding physical layer thickness (Λ = d 1 + d 2). The sum of the phase variables, ϕ = ϕ1 + ϕ2, defines the total optical phase shift of the three-layer combination; the difference of the phase variables, Δϕ = ϕ2 - ϕ1, is a measure of the duty cycle. The amplitudes E l/ r and H l/ r denote the electric and the magnetic fields, respectively, and A l/ r and B l/ r represent the traveling waves in the right and left directions on the left and right ends, respectively, of the multilayer structure.

Fig. 2
Fig. 2

Contour plots of (a) the exact coupling coefficient κ, (b) the exact detuning coefficient δ, (c) the normalized equivalent index N e /n 1, and (d) the equivalent thickness Γ e as functions of the phase variables ϕ and Δϕ. The phase variables as well as the detuning coefficient and the equivalent thickness are given in units of π. Shaded regions indicate real values of the parameter; white regions indicate complex parameter ranges. For the equivalent-layer parameters shown in (c) and (d), shaded and unshaded regions separate passband regions from stop-band regions. Note that only regions that fulfill the condition ϕ ≥ |Δϕ| are of physical relevance because the phase shifts, ϕ1/2, are positive quantities. The dashed curves in (a) are contour lines with κ = -2r = -0.5, the value from standard coupled-mode theory. The dotted lines in (a) and (c) are the zero contour lines of the coupling coefficient and the unity contour lines of the equivalent index, respectively. For these contour lines, medium 2 vanishes or acts as an absentee layer, as described in the text.

Equations (27)

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ElHl/Y0=M11iM12iM21M11ErHr/Y0,
Me=cosΓeiNesinΓeiNe sinΓecosΓe.
Ne=M21/M12,
cosΓe=M11.
AlBl=TArBr.
T=FG*GF*=FR+iFIG*GFR-iFI,
T=-cosγ+i δγsinγi κγsinγ-i κγsinγcosγ-i δγsinγ.
γ=δ2-κ2.
κ=-iαG,
δ=-αFI,
α=γsinγ,
γ=-i ln-FR+FR2-1FR<-1arctan1-FR2-FR-1FR0-arctan1-FR2FR+π0<FR+1-i lnFR-FR2-1+πFR>+1.
FR=M11,
FI=12n1M12+M21n1G=i2n1M12-M21n1n1M12=FI-iG,M21nI=FI+iG.
Ne=n1FI+iGFI-iG=n1δ+κδ-κ,
cosΓe=FR=cosγ+π.
Z=δ-κδ+κ.
Γe=±γ+2j-1π,  j.
Γe=-1ϕ/π+1γ+2j-1π, ϕ[2πj-1, 2πj[.
FR=11-r2cosϕ-r2 cosΔϕ,
F1=11-r2sinϕ+r2 sinΔϕ,
G=-2ir1-r2sin12ϕ+Δϕ,
ϕ=ϕ1+ϕ2,
Δϕ=ϕ2-ϕ1,
r=n2-n1n2+n1.
Nen1=sinϕ+r2 sinΔϕ+2r sinϕ+Δϕ/2sinϕ+r2 sinΔϕ-2r sinϕ+Δϕ/21/2.
Γe=arccos11-r2cosϕ-r2 cosΔϕ.

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