Abstract

A novel analytical thin-film design method is presented that is based on electrical engineering communication theory. The proposed thickness modulation describes the thickness modulation of discrete, homogeneous thin-film layers of a multilayer coating. One modulation scheme, amplitude modulation, is presented in which analytical equations determine individual layer thicknesses for a given modulation amplitude, carrier frequency (f c), direct-current bias, as well as several layers and refractive indices. The spectral performance (especially stop bands) of multilayer coatings with alternating layers of two refractive indices is presented for different carrier frequencies and modulation amplitudes. For f c ≤ 1, the ratio of the center frequencies of the first-order (f 1) and the next present stop band (f 2) is determined analytically from the modulation frequency for which f 2/f 1 = 2f c + 1. Particular cases of the carrier frequency produce virtual stop bands below the spectral frequency of the first-order stop band as well as high-frequency harmonics. Degenerate and other cases of thickness-modulated designs are presented, along with other modulation methods.

© 1998 Optical Society of America

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References

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  1. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989), Chap. 1, pp. 2–3.
  2. H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, 1981), Chap. 2.4, pp. 46–47.
  3. J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photon. News 8(6), 25 (1997).
  4. W. H. Southwell, “Use of gradient index for spectral filters,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. SPIE464, 110–114 (1984).
    [CrossRef]
  5. W. H. Southwell, “Spectral response calculations of rugate filters using coupled-wave theory,” J. Opt. Soc. Am. A 5, 1558–1564 (1988).
    [CrossRef]
  6. B. G. Bovard, “Derivation of a matrix describing a rugate dielectric thin film,” Appl. Opt. 27, 1998–2005 (1988).
    [CrossRef] [PubMed]
  7. W. H. Southwell, “Extended-bandwidth reflector designs by using wavelets,” Appl. Opt. 36, 314–318 (1997).
    [CrossRef] [PubMed]
  8. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  9. P. Baumeister, Optical Coating Technology, Short Course, University of California at Los Angeles, 13–17 January 1997 (P. Baumeister, Sebastopol, Calif., 1997).
  10. See, for example, E. Cojocaru, “Analytical solutions for zero-phase-shift transparent coatings on metallic reflectors at 10.6 μm,” Appl. Opt. 32, 4843–4845 (1993).
  11. Ref. 2, pp. 55–73.
  12. S. Haykin, Communication Systems (Wiley, New York, 1978), Chap. 3, p. 244.
  13. Ref. 1, p. 25.
  14. J. Millman, Microelectronics (McGraw-Hill, New York, 1979), Chap. 10-5, pp. 343–348.
  15. R. M. A. Azzam, B. E. Perilloux, “Polarized-light techniques for generating electrical signals of controllable waveform,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. SPIE464, 88–95 (1984).
    [CrossRef]
  16. M. Thomsen, Z. L. Wu, “Polarizing and reflective coatings based on half-wave layer pairs,” Appl. Opt. 36, 307–313 (1997).
    [CrossRef] [PubMed]

1997 (3)

1993 (1)

1988 (2)

1978 (1)

Azzam, R. M. A.

R. M. A. Azzam, B. E. Perilloux, “Polarized-light techniques for generating electrical signals of controllable waveform,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. SPIE464, 88–95 (1984).
[CrossRef]

Baumeister, P.

P. Baumeister, Optical Coating Technology, Short Course, University of California at Los Angeles, 13–17 January 1997 (P. Baumeister, Sebastopol, Calif., 1997).

Bovard, B. G.

Cojocaru, E.

Dobrowolski, J. A.

Haykin, S.

S. Haykin, Communication Systems (Wiley, New York, 1978), Chap. 3, p. 244.

Liddell, H. M.

H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, 1981), Chap. 2.4, pp. 46–47.

Lowe, D.

Millman, J.

J. Millman, Microelectronics (McGraw-Hill, New York, 1979), Chap. 10-5, pp. 343–348.

Perilloux, B. E.

R. M. A. Azzam, B. E. Perilloux, “Polarized-light techniques for generating electrical signals of controllable waveform,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. SPIE464, 88–95 (1984).
[CrossRef]

Southwell, W. H.

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989), Chap. 1, pp. 2–3.

Thomsen, M.

Wu, Z. L.

Appl. Opt. (5)

J. Opt. Soc. Am. A (1)

Opt. Photon. News (1)

J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photon. News 8(6), 25 (1997).

Other (9)

W. H. Southwell, “Use of gradient index for spectral filters,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. SPIE464, 110–114 (1984).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989), Chap. 1, pp. 2–3.

H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, 1981), Chap. 2.4, pp. 46–47.

P. Baumeister, Optical Coating Technology, Short Course, University of California at Los Angeles, 13–17 January 1997 (P. Baumeister, Sebastopol, Calif., 1997).

Ref. 2, pp. 55–73.

S. Haykin, Communication Systems (Wiley, New York, 1978), Chap. 3, p. 244.

Ref. 1, p. 25.

J. Millman, Microelectronics (McGraw-Hill, New York, 1979), Chap. 10-5, pp. 343–348.

R. M. A. Azzam, B. E. Perilloux, “Polarized-light techniques for generating electrical signals of controllable waveform,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. SPIE464, 88–95 (1984).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

QWOT of layers for a 36-layer TMD with arbitrarily selected parameters of k a = 0.5, T c = 1 μm, T base = 9 layers, and n = 36. The actual period repeats every 18 layers since the base period is an odd integer.

Fig. 2
Fig. 2

Theoretical spectral transmittance of 10 TMD structures from Table 1. The refractive index of the substrate is 1.52 for all designs, n L = 1.46 and n H = 2.25. Parts (a)–(j) correspond to the TMD structures as referenced on the bottom row of Table 1.

Fig. 3
Fig. 3

Same as Fig. 2 except for (a), T base = 2.5 (actual period = 10 layers; 3 periods = 30 layers) and (b), T base = 4.5 (actual period = 18 layers; 2 periods = 36 layers).

Fig. 4
Fig. 4

Ratio (f B /f A ) of the center spectral frequency of the primary and the next present stop bands (A and B) for transmittances of TMD structures shown in Figs. 2 and 3. Also included is the same ratio for a similar TMD structure except for T base = 1.5 (actual period = 6 layers; 4 periods = 24 layers). Between the base periods of 2 and 10, the ratio decreases from 2 to approximately 1.2.

Fig. 5
Fig. 5

Reflectivity of three stop bands versus k a . This TMD structure has T base = 4 layers (actual period = 4 layers; 5 periods = 20 layers) and T s = 1 μm, and k a varies from 0 to 1. The stop bands are at wave numbers 0.5, 1.0, and 1.5.

Tables (2)

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Table 1 Layer Thickness for 10 TMD Structuresa

Tables Icon

Table 2 Spectral Positions of Some Stop Bands for Each TMD Structure in Table 1

Equations (7)

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s t = A c 1 + k a m t cos 2 π f c t ,
t L = T c 1 + k a   cos 2 π f c L ,     L = 1 ,   2 , ,   n .
T base = 1 / f c ,
R = f B / f A = 2 / T base + 1 ,
R = 2 f c + 1 ,
T base = 2 / R - 1 .
n x = n a 1 + b 1   sin 2 π f 1 x + b 2   sin 2 π f 2 x ,

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