Abstract

Optical interference coatings have been designed by using combinations of wavelets (fully apodized sine-wave refractive index groups). Like a single-line rugate filter, a single wavelet produces a stop band without harmonics and, in addition, has no sidelobes. The rules for combining wavelet refractive index structures have been derived for extended-bandwidth reflectors.

© 1997 Optical Society of America

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References

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  1. H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), p. 46.
  2. W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–96 (1993).
    [CrossRef]
  3. W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 46–59 (1993).
    [CrossRef]
  4. Small harmonics appear for large amplitude sine-wave index structures. To eliminate these, one should strictly use what are called exponential sine waves, in which the log of the refractive index is sinusoidal.
  5. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983).
    [CrossRef] [PubMed]
  6. W. H. Southwell, “Using apodization functions to reduce side-lobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989).
    [CrossRef] [PubMed]
  7. W. H. Southwell, “Extended bandwidth reflector designs using wavelets,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 11–13.

1989 (1)

1983 (1)

Crane, R. L.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–96 (1993).
[CrossRef]

Gunning, W. J.

W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 46–59 (1993).
[CrossRef]

Hall, R. L.

W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 46–59 (1993).
[CrossRef]

Johnson, W. E.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–96 (1993).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), p. 46.

Southwell, W. H.

W. H. Southwell, “Using apodization functions to reduce side-lobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989).
[CrossRef] [PubMed]

W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983).
[CrossRef] [PubMed]

W. H. Southwell, “Extended bandwidth reflector designs using wavelets,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 11–13.

W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 46–59 (1993).
[CrossRef]

Appl. Opt. (1)

Opt. Lett. (1)

Other (5)

H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), p. 46.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–96 (1993).
[CrossRef]

W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 46–59 (1993).
[CrossRef]

Small harmonics appear for large amplitude sine-wave index structures. To eliminate these, one should strictly use what are called exponential sine waves, in which the log of the refractive index is sinusoidal.

W. H. Southwell, “Extended bandwidth reflector designs using wavelets,” in Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 11–13.

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Figures (9)

Fig. 1
Fig. 1

Quarter-wave stack, with low refractive index material nL = 1.5 and high refractive index material nH = 2, produces a fundamental stop band surrounded by sidelobes and harmonic stop bands located at odd fractional wavelengths.

Fig. 2
Fig. 2

Rugate filter, as a sine-wave refractive index with peak-to-peak amplitude np = 0.5, produces a fundamental stop band surrounded by sidelobes but no harmonic stop bands. (The substrate and incident medium are the same as the average of the rugate refractive index.)

Fig. 3
Fig. 3

Refractive index wavelet, with the same amplitude and thickness as the rugate filter refractive index structure shown in Fig. 2. It produces no sidelobes and no harmonic stop bands. (The substrate and incident medium are the same as the average refractive index of the wavelet.)

Fig. 4
Fig. 4

Refractive index wavelet appended by a quintic matching region to connect smoothly to the substrate of refractive index ns = 1.52 on the right, and down to a low refractive index of 1.47 on the left, which is at the air interface.

Fig. 5
Fig. 5

Two half-overlapping wavelets used to extend the width of the stop band over that of the single wavelet shown in Fig. 4. The goal is to maintain an optical density over 2 (99% reflectivity) throughout the stop band.

Fig. 6
Fig. 6

Seven wavelets used to produce an extended bandwidth of the stop band.

Fig. 7
Fig. 7

Optimum phase differences (×’s) between adjacent wavelets (as a function of the number of cycles in the wavelet) to ensure high reflectance in the extended stop band. The dashed curve is a best-fit straight line.

Fig. 8
Fig. 8

Optimum phase differences (×’s) between adjacent wavelets (as a function of the wavelet amplitude np) to ensure high reflectance in the extended stop band. The dashed curve is a best-fit quadratic function.

Fig. 9
Fig. 9

Refractive index wavelet solution to the hot-mirror problem presented at the 1995 Optical Interference Coatings Conference.7 Although this solution wouldn’t qualify according to the rules of that contest (unless the gradient-index structure is considered as one layer), it does illustrate a means to obtain excellent performance without optimization.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

nx=na+0.5npAxsin4πx/λ+ϕ,
Ax=10t3-15t4+6t5,
t=2x/T  for x<T/2,  =2T-x/T  for x>T/2,
T=Ncλ/2,
λ+=1.07+0.3np-0.4
ϕ=90+1050np-0.6-24.5Nc-40+20np-0.6Nc-40,

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