Abstract

The scattering behavior of the all-dielectric twin-cavity narrow-band interference filter is studied both in theory and in experiment in two cases, l1 = l2 and l1l2, where l1 and l2 are the optical thicknesses of the two cavities. It has been shown that the scattering properties are determined mainly by the spacers in which the electric-field intensities are large because of the presence of large standing-wave fields. The scattered light cones are found on both sides of the filter illuminated by a monochromatic light of which the wavelength (λL) is shorter than the peak wavelength (λ0′) of the filter. The scattering angle of each cone is equal to the tilted angle of the filter when the peak wavelength of the filter shifts to the illumination wavelength. For the case l1l2, the distributions of the scattered light on both sides of the filter are quite different. The analytical calculations are in good agreement with experimental results. The possible applications of scattering in the twin-cavity filter in determining the bandwidth of the peak transmittance and the optical thicknesses of two spacers are addressed.

© 1997 Optical Society of America

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References

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  1. P. Giacomo, “Les couches réflechissantes multidiélectriques appliquées à l’interféromètrie de Fabry–Perot. Etude théorique et expérimentale des couches réelles,” Rev. Opt. Theor. Instrum. 35, 317–354 (1956).
  2. G. Koppelmann, “Irregularities in evaporated interference films and their connection with light scattering,” Optik 17, 416–425 (1960).
  3. J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D dissertation (University of Rochester, Rochester, N.Y., 1974).
  4. S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
    [CrossRef]
  5. J. Ebert, H. Pannhorst, H. Küster, H. Welling, “Scatter losses of broadband interference coatings,” Appl. Opt. 18, 818–822 (1979).
    [CrossRef] [PubMed]
  6. J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of the theory and experiment,” Appl. Opt. 19, 669–679 (1980).
    [CrossRef] [PubMed]
  7. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  8. J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties,” Appl. Opt. 22, 3207–3219 (1983).
    [CrossRef] [PubMed]
  9. C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
    [CrossRef] [PubMed]
  10. C. Amra, C. Grèzes-Besset, L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32, 5492–5503 (1993).
    [CrossRef] [PubMed]
  11. A. Duparre, S. Kassam, “Relation between light scattering and the microstructure of optical thin film,” Appl. Opt. 32, 5475–5480 (1993).
    [CrossRef]
  12. C. Amra, “From light scattering to the microstructure of the thin-film multilayers,” Appl. Opt. 32, 5481–5491 (1993).
    [CrossRef] [PubMed]
  13. C. Amra, “Light scattering from multilayer optics. I. Tools of investigation,” J. Opt. Soc. Am. A 11, 197–210 (1994).
    [CrossRef]
  14. C. Amra, “Light scattering from multilayer optics. II. Application to experiment,” J. Opt. Soc. Am. A 11, 211–226 (1994).
    [CrossRef]
  15. J. R. Gee, I. J. Hodgkinson, H. A. Macleod, “Moisture-dependent anisotropic effects in optical coatings,” Appl. Opt. 24, 3188–3192 (1985).
    [CrossRef] [PubMed]
  16. J. R. Gee, I. J. Hodgkinson, P. W. Wilson, “Scatter from fluid patches in optical thin-film coatings,” Appl. Opt. 25, 2688–2694 (1986).
    [CrossRef] [PubMed]
  17. N. C. Craft, “Highly cascadable optically bistable device for large fan-out optical computing uses,” Appl. Opt. 27, 1764–1768 (1988).
    [CrossRef] [PubMed]
  18. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 60.

1994 (2)

1993 (3)

1992 (1)

1988 (1)

1986 (1)

1985 (1)

1983 (1)

1981 (1)

1980 (1)

1979 (2)

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[CrossRef]

J. Ebert, H. Pannhorst, H. Küster, H. Welling, “Scatter losses of broadband interference coatings,” Appl. Opt. 18, 818–822 (1979).
[CrossRef] [PubMed]

1960 (1)

G. Koppelmann, “Irregularities in evaporated interference films and their connection with light scattering,” Optik 17, 416–425 (1960).

1956 (1)

P. Giacomo, “Les couches réflechissantes multidiélectriques appliquées à l’interféromètrie de Fabry–Perot. Etude théorique et expérimentale des couches réelles,” Rev. Opt. Theor. Instrum. 35, 317–354 (1956).

Amra, C.

Apfel, J. H.

Bennett, J. M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 60.

Bousquet, P.

Bruel, L.

Craft, N. C.

Duparre, A.

Eastman, J. M.

J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D dissertation (University of Rochester, Rochester, N.Y., 1974).

Ebert, J.

Elson, J. M.

Flory, F.

Gee, J. R.

Giacomo, P.

P. Giacomo, “Les couches réflechissantes multidiélectriques appliquées à l’interféromètrie de Fabry–Perot. Etude théorique et expérimentale des couches réelles,” Rev. Opt. Theor. Instrum. 35, 317–354 (1956).

Gourley, S. J.

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[CrossRef]

Grèzes-Besset, C.

Hodgkinson, I. J.

Kassam, S.

Koppelmann, G.

G. Koppelmann, “Irregularities in evaporated interference films and their connection with light scattering,” Optik 17, 416–425 (1960).

Küster, H.

Lissberger, P. H.

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[CrossRef]

Macleod, H. A.

Pannhorst, H.

Pelletier, E.

Rahn, J. P.

Roche, P.

Welling, H.

Wilson, P. W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 60.

Appl. Opt. (10)

J. Ebert, H. Pannhorst, H. Küster, H. Welling, “Scatter losses of broadband interference coatings,” Appl. Opt. 18, 818–822 (1979).
[CrossRef] [PubMed]

J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of the theory and experiment,” Appl. Opt. 19, 669–679 (1980).
[CrossRef] [PubMed]

J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties,” Appl. Opt. 22, 3207–3219 (1983).
[CrossRef] [PubMed]

J. R. Gee, I. J. Hodgkinson, H. A. Macleod, “Moisture-dependent anisotropic effects in optical coatings,” Appl. Opt. 24, 3188–3192 (1985).
[CrossRef] [PubMed]

J. R. Gee, I. J. Hodgkinson, P. W. Wilson, “Scatter from fluid patches in optical thin-film coatings,” Appl. Opt. 25, 2688–2694 (1986).
[CrossRef] [PubMed]

N. C. Craft, “Highly cascadable optically bistable device for large fan-out optical computing uses,” Appl. Opt. 27, 1764–1768 (1988).
[CrossRef] [PubMed]

C. Amra, J. H. Apfel, E. Pelletier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
[CrossRef] [PubMed]

C. Amra, “From light scattering to the microstructure of the thin-film multilayers,” Appl. Opt. 32, 5481–5491 (1993).
[CrossRef] [PubMed]

A. Duparre, S. Kassam, “Relation between light scattering and the microstructure of optical thin film,” Appl. Opt. 32, 5475–5480 (1993).
[CrossRef]

C. Amra, C. Grèzes-Besset, L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32, 5492–5503 (1993).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

S. J. Gourley, P. H. Lissberger, “Optical scattering in multilayer thin films,” Opt. Acta 26, 117–143 (1979).
[CrossRef]

Optik (1)

G. Koppelmann, “Irregularities in evaporated interference films and their connection with light scattering,” Optik 17, 416–425 (1960).

Rev. Opt. Theor. Instrum. (1)

P. Giacomo, “Les couches réflechissantes multidiélectriques appliquées à l’interféromètrie de Fabry–Perot. Etude théorique et expérimentale des couches réelles,” Rev. Opt. Theor. Instrum. 35, 317–354 (1956).

Other (2)

J. M. Eastman, “Surface scattering in optical interference coatings,” Ph.D dissertation (University of Rochester, Rochester, N.Y., 1974).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 60.

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Figures (9)

Fig. 1
Fig. 1

Experimental setup: θ, scattering angle. The λ/2 plate is used to change the illumination polarization.

Fig. 2
Fig. 2

Spatial distributions of the forward-scattered light in the twin-cavity filter G(HL)44H(LH)4L(HL)44H(LH)4 with λ0′ = 524.9 nm. The scattering angles are 16°, 17°, and 58.3°, respectively, when a p-polarized 514.5-nm laser illuminates the filter. The dark regions appear on the upper and the lower parts of the big ring. (a) The directly transmitted spot is on one of the two small brighter rings. (b) The directly transmitted spot is on the big ring. (c) The two small scattered rings are magnified.

Fig. 3
Fig. 3

Spatial distributions of the forward-scattered light in twin-cavity filter G(HL)44H(LH)4L(HL)43.9H(LH)4 with λ0′ = 526.0 nm. The directly transmitted light spot is blocked. The s-polarized 501.7-nm laser beam radiates the filter from coating side (a) and from glass (b). The scattering angles are 24°, 32°, and 66°, respectively.

Fig. 4
Fig. 4

Thin slice of material of phase thickness dφ in a general dielectric coating. A, B are the characteristic matrices; n0 and ns are the refractive indices of the air and the substrate, respectively.

Fig. 5
Fig. 5

Twin-cavity interference filter. φ12 - φ11, φ22 - φ21 are the phase thicknesses of spacers 1 and 2, respectively.

Fig. 6
Fig. 6

Single-cavity interference filter: φI, phase thickness of the spacer; R1, R2, characteristic matrices of both reflective stacks.

Fig. 7
Fig. 7

Calculated values of (a) the scattering parameter |Δt|2 and (b) the experimental result for the single-cavity filter G(HL)44H(LH)4 with λ0 = 512.5 nm, where nH = 2.35, nL = 1.35, ns = 1.52, n0 = 1.00, Δn = 0.05. The wavelength of the incidence laser beam is 514.5 nm. The calculated and the measured scattering angles are 14.7°, 14.5°, respectively. The directly transmitted light spot is blocked.

Fig. 8
Fig. 8

Calculated values of the scattering parameter |Δt|2 for the twin-cavity filter G(HL)44H(LH)4L(HL)44H(LH)4 with λ0 = 514.0 nm, where nH = 2.35, nL = 1.35, ns = 1.52, n0 = 1.00, Δn = 0.05 for both cavities. The scattering angles are 16.5°, 17.4°, and 60.5° when the filter is illuminated with a 514.5-nm (a) p-polarized and (b) s-polarized laser beam. No peak corresponding to a large angle can be observed in (b) that agrees with the darkness of the upper and the lower parts of the largest ring of Figs. 2(a) and 2(b).

Fig. 9
Fig. 9

Calculated values of the scattering parameters |Δt|2 for the twin-cavity filter G(HL)44H(LH)4L(HL)43.9H(LH)4 with λ0 = 515.0 nm, where nH = 2.35, nL = 1.35, ns = 1.52, n0 = 1.00, Δn = 0.05 for both cavities. The s-polarized 501.7-nm laser beam radiates the filter from (a) the coating side and (b) the glass. The scattering angles are 24.4°, 31.3°, and 65.9°.

Equations (18)

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Pincoh=kapy, z2dydz×Iin.
dt=stΔndφ,
dr=srΔndφ.
t=2n0n0m11+nsm12+m21+nsm22,
r=n0m11+nsm12-m21+nsm22n0m11+nsm12+m21+nsm22
st=-it2a22+n0a12b11+nsb12/n0,
sr=-it2b11+nsb122/n0,
st=-ita22+n0a12E/E0+/n0,
sr=-iE/E0+2/n0,
Δt=StφΔnφdφ,
Δr=SrφΔnφdφ.
Δt=0φStφΔnφdφ=φ11φ12St1φΔn1φdφ+φ21φ22St2φΔn2φdφ,
Δr=0φSrφΔnφdφ=φ11φ12Sr1φΔn1φdφ+φ21φ22Sr2φΔn2φdφ,
St=-it22η0V2Q1-V1Q2/ηI2cos2φ-φI+V2Q1+V1Q2/η12cos φI+iV1Q1-V2Q2sin2φ-φI+V1Q1+V2Q2sin φI/ηI,
Sr=-it2η0Q1 cosφI-φ-iQ2 sinφI-φ/ηI2,
ηj=nj cos θj,s polarizationnj/cos θj,p polarization  j=0, s, I,  θj is the angle of light beam in jth media.
ΔtI=0φIStφΔnφdφ=-iΔnt2η0V2Q1-V1Q2ηI2sin φI+V2Q1+V1Q2ηI2cos φIφI+iφIηIV1Q1+V2Q2sin φ1,
ΔrI=0φISrφΔnφdφ=-iΔnt22η0Q12-Q22ηI2φI+Q122+Q222ηI2sin 2φI+iQ1Q2η11-cos 2φI.

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