Abstract

The spectral spacing of multiple, high-reflectance stopbands is determined analytically for discrete thin-film layer-thickness-modulated designs. Discrete, sinusoidal modulation of layer thickness produces unique multilayer designs for fundamental modulation frequencies within the range 0 < f m≤ 0.5. Thickness-modulated designs typically have several high-reflectance stopbands at spectral frequencies that are a function of modulation frequency. All stopbands were verified by the trace of the characteristic matrix for each thickness-modulated design. A universal stopband equation is presented that predicts the relative spectral spacing of all possible stopbands for thickness-modulated designs. Two harmonic components of thickness-modulated designs are defined. The spectral performance (stopband position) is shown graphically for modulation amplitudes 0.25, 0.5, 0.75, 1.0, and 0 < f m ≤ 0.5.

© 1999 Optical Society of America

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References

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  1. B. E. Perilloux, “Discrete thin-film layer thickness modulation,” Appl. Opt. 37, 3527–3532 (1998).
    [CrossRef]
  2. G. Tempea, F. Krausz, C. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
    [CrossRef]
  3. P. Baumeister, Optical Coating Technology, Short Course, University of California at Los Angeles, 13–17 January1997 (P. Baumeister, Sebastopol, Calif., 1997).
  4. Ref. 3, “Fundamentals,” Chap. 2, pp. 50–51.
  5. Ref. 3, “Fundamentals,” Chap. 2, pg. 63.
  6. Ref. 3, “Reflectors, edge filters and periodic structures,” Chap. 5, pp. 6–7.
  7. H. M. Liddell, “Design of filters by analytical techniques,” in Computer-Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 2, pp. 46–47.

1998

B. E. Perilloux, “Discrete thin-film layer thickness modulation,” Appl. Opt. 37, 3527–3532 (1998).
[CrossRef]

G. Tempea, F. Krausz, C. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

Baumeister, P.

P. Baumeister, Optical Coating Technology, Short Course, University of California at Los Angeles, 13–17 January1997 (P. Baumeister, Sebastopol, Calif., 1997).

Ferencz, K.

G. Tempea, F. Krausz, C. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

Krausz, F.

G. Tempea, F. Krausz, C. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

Liddell, H. M.

H. M. Liddell, “Design of filters by analytical techniques,” in Computer-Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 2, pp. 46–47.

Perilloux, B. E.

Spielmann, C.

G. Tempea, F. Krausz, C. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

Tempea, G.

G. Tempea, F. Krausz, C. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

Appl. Opt.

IEEE J. Sel. Top. Quantum Electron.

G. Tempea, F. Krausz, C. Spielmann, K. Ferencz, “Dispersion control over 150 THz with chirped dielectric mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 193–196 (1998).
[CrossRef]

Other

P. Baumeister, Optical Coating Technology, Short Course, University of California at Los Angeles, 13–17 January1997 (P. Baumeister, Sebastopol, Calif., 1997).

Ref. 3, “Fundamentals,” Chap. 2, pp. 50–51.

Ref. 3, “Fundamentals,” Chap. 2, pg. 63.

Ref. 3, “Reflectors, edge filters and periodic structures,” Chap. 5, pp. 6–7.

H. M. Liddell, “Design of filters by analytical techniques,” in Computer-Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, UK, 1981), Chap. 2, pp. 46–47.

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Figures (5)

Fig. 1
Fig. 1

Typical spectral reflectance of a TMD, with T base = 6, k m = 0.5, and n = 30 layers. The average layer thickness for this TMD is 1-µm QWOT.

Fig. 2
Fig. 2

TMD stopbands (short, horizontal lines or dashes) as a function of spectral and modulation frequencies. For all TMD’s, k m = 0.5. The lines that start at f m = 0.0 and spectral frequencies 1, 3, 5, 9, etc., are determined by use of USE-TMD equation (7). Only the nearest ±10 orders to the zero order of the TMD (N) are shown. The average layer thickness for all TMD’s is 1-µm QWOT.

Fig. 3
Fig. 3

Same as Fig. 2, except k m = 0.25.

Fig. 4
Fig. 4

Same as Fig. 2, except k m = 0.75.

Fig. 5
Fig. 5

Same as Fig. 2, except k m = 1.0.

Tables (1)

Tables Icon

Table 1 Modulation Frequency, Base Period, and Number of Layers for Each Thickness-Modulated Design

Equations (9)

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tL=T01+km cos2πfmL,  L=1, 2,n,
Tbase=1/fm,
|x|>2.
cos2πfmL=cos2π1-fmL.
0<fm0.5,
RBA=σB/σA=2fm+1.
σM,N=σ02Nfm+2M-1,
-<M<,
-N.

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