Abstract

The majority of the layers of a multilayer optical coating typically have optical thicknesses equal to one quarter of the appropriate wavelength of radiation. Replacing this constraint with the stipulation that a pair of adjacent layers should have a total optical thickness of one half of a wavelength introduces a significant new component of design flexibility while having a minimal impact upon the desired optical properties of the film. Taking a matrix approach, we derive a general expression for the reflectance of a periodic thin-film structure that is based on layers of two different materials of arbitrary thickness. This result is applied to highly reflective coatings at normal incidence and to off-normal polarizing coatings. Specific results involving HfO2/SiO2 films and TiO2/SiO2 films are displayed. We discuss how the thickness of the high-index layers may be reduced to increase damage thresholds. We also show a mirror design that is effective not only at λ = 1.06 µm but also at the frequency-doubled wavelength.

© 1997 Optical Society of America

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References

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  1. See, for example, J. H. Apfel, “Optical coating design with reduced electric field intensity,” Appl. Opt. 16, 1880–1885 (1977); D. H. Gill, B. E. Newman, J. McLeod, “Use of non-quarter-wave designs to increase the damage resistance of reflectors at 532 and 1064 manometers,” Natl. Bur. Stand. (U.S.) Spec. Publ. 509, 260–270 (1977); I. Y. Milev, S. S. Dimov, D. V. Terziev, J. I. Iordanova, L. B. Todorova, A. B. Gelkova, “Laser-induced damage threshold measurements of optical dielectric coatings at λ = 1.06 µm,” J. Appl. Phys. 70, 4057–4060 (1991); B. E. Newnam, S. R. Foltyn, L. J. Jolin, “Multiple-shot laser damage resistance of nonquarterwave reflector designs for 248 nm,” Natl. Bur. Stand. (U.S.) Spec. Publ. 638, 363–379 (1981).
  2. Z. Fan, Z. Wu, X. Tang, “Laser induced damage in optical coatings,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 459–468 (1992).
    [CrossRef]
  3. P. H. Lissberger, “The ultimate reflectance of multilayer dielectric mirrors,” Opt. Acta 25, 291–298 (1978).
    [CrossRef]
  4. B. V. Landau, P. H. Lissberger, “Theory of induced-transmission filters in terms of the concept of equivalent layers,” J. Opt. Soc. Am. 62, 1258–1264 (1972).
    [CrossRef]
  5. J. F. DeFord, M. R. Kozlowski, “Modeling of electric-field enhancements at nodular defects in dielectric mirror coatings,” in Laser-Induced Damage in Optical Materials, Proc. SPIE1848, 455–469 (1992).
  6. R. J. Tench, R. Chow, M. R. Kozlowski, “Characterization of defect geometries in multilayer optical coatings,” J. Vac. Sci. Technol. A12, 2808–2813 (1994); “Investigation of the microstructure of coatings for high power lasers by non-optical techniques,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 596–602 (1994).
    [CrossRef]
  7. D. G. Cahill, T. H. Allen, “Thermal conductivity of sputtered and evaporated SiO2 and TiO2 optical coatings,” Appl. Phys. Lett. 65, 309–311 (1994); Z. L. Wu, P. K. Kuo, L. Wei, S. L. Gu, R. L. Thomas, “Photothermal characterization of optical thin films,” Thin Solid Films 236, 191–198 (1993); D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986); J. C. Lambropoulos, M. R. Jolly, C. A. Amsden, S. E. Gillman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
    [CrossRef] [PubMed]
  8. E. Spiller, “Low-loss reflection coatings using absorbing materials,” Appl. Phys. Lett. 20, 365–367 (1972).
    [CrossRef]
  9. H. A. Macleod, Thin-Film Optical Filters (Elsevier, New York, 1969).
  10. J. C. Monga, “Multilayer thin-film polarizers with reduced electric-field intensity,” J. Mod. Opt. 36, 769–784 (1989).
    [CrossRef]

1994

D. G. Cahill, T. H. Allen, “Thermal conductivity of sputtered and evaporated SiO2 and TiO2 optical coatings,” Appl. Phys. Lett. 65, 309–311 (1994); Z. L. Wu, P. K. Kuo, L. Wei, S. L. Gu, R. L. Thomas, “Photothermal characterization of optical thin films,” Thin Solid Films 236, 191–198 (1993); D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986); J. C. Lambropoulos, M. R. Jolly, C. A. Amsden, S. E. Gillman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef] [PubMed]

1989

J. C. Monga, “Multilayer thin-film polarizers with reduced electric-field intensity,” J. Mod. Opt. 36, 769–784 (1989).
[CrossRef]

1978

P. H. Lissberger, “The ultimate reflectance of multilayer dielectric mirrors,” Opt. Acta 25, 291–298 (1978).
[CrossRef]

1977

1972

Allen, T. H.

D. G. Cahill, T. H. Allen, “Thermal conductivity of sputtered and evaporated SiO2 and TiO2 optical coatings,” Appl. Phys. Lett. 65, 309–311 (1994); Z. L. Wu, P. K. Kuo, L. Wei, S. L. Gu, R. L. Thomas, “Photothermal characterization of optical thin films,” Thin Solid Films 236, 191–198 (1993); D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986); J. C. Lambropoulos, M. R. Jolly, C. A. Amsden, S. E. Gillman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef] [PubMed]

Apfel, J. H.

Cahill, D. G.

D. G. Cahill, T. H. Allen, “Thermal conductivity of sputtered and evaporated SiO2 and TiO2 optical coatings,” Appl. Phys. Lett. 65, 309–311 (1994); Z. L. Wu, P. K. Kuo, L. Wei, S. L. Gu, R. L. Thomas, “Photothermal characterization of optical thin films,” Thin Solid Films 236, 191–198 (1993); D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986); J. C. Lambropoulos, M. R. Jolly, C. A. Amsden, S. E. Gillman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef] [PubMed]

Chow, R.

R. J. Tench, R. Chow, M. R. Kozlowski, “Characterization of defect geometries in multilayer optical coatings,” J. Vac. Sci. Technol. A12, 2808–2813 (1994); “Investigation of the microstructure of coatings for high power lasers by non-optical techniques,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 596–602 (1994).
[CrossRef]

DeFord, J. F.

J. F. DeFord, M. R. Kozlowski, “Modeling of electric-field enhancements at nodular defects in dielectric mirror coatings,” in Laser-Induced Damage in Optical Materials, Proc. SPIE1848, 455–469 (1992).

Fan, Z.

Z. Fan, Z. Wu, X. Tang, “Laser induced damage in optical coatings,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 459–468 (1992).
[CrossRef]

Kozlowski, M. R.

R. J. Tench, R. Chow, M. R. Kozlowski, “Characterization of defect geometries in multilayer optical coatings,” J. Vac. Sci. Technol. A12, 2808–2813 (1994); “Investigation of the microstructure of coatings for high power lasers by non-optical techniques,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 596–602 (1994).
[CrossRef]

J. F. DeFord, M. R. Kozlowski, “Modeling of electric-field enhancements at nodular defects in dielectric mirror coatings,” in Laser-Induced Damage in Optical Materials, Proc. SPIE1848, 455–469 (1992).

Landau, B. V.

Lissberger, P. H.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Elsevier, New York, 1969).

Monga, J. C.

J. C. Monga, “Multilayer thin-film polarizers with reduced electric-field intensity,” J. Mod. Opt. 36, 769–784 (1989).
[CrossRef]

Spiller, E.

E. Spiller, “Low-loss reflection coatings using absorbing materials,” Appl. Phys. Lett. 20, 365–367 (1972).
[CrossRef]

Tang, X.

Z. Fan, Z. Wu, X. Tang, “Laser induced damage in optical coatings,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 459–468 (1992).
[CrossRef]

Tench, R. J.

R. J. Tench, R. Chow, M. R. Kozlowski, “Characterization of defect geometries in multilayer optical coatings,” J. Vac. Sci. Technol. A12, 2808–2813 (1994); “Investigation of the microstructure of coatings for high power lasers by non-optical techniques,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 596–602 (1994).
[CrossRef]

Wu, Z.

Z. Fan, Z. Wu, X. Tang, “Laser induced damage in optical coatings,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 459–468 (1992).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. G. Cahill, T. H. Allen, “Thermal conductivity of sputtered and evaporated SiO2 and TiO2 optical coatings,” Appl. Phys. Lett. 65, 309–311 (1994); Z. L. Wu, P. K. Kuo, L. Wei, S. L. Gu, R. L. Thomas, “Photothermal characterization of optical thin films,” Thin Solid Films 236, 191–198 (1993); D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986); J. C. Lambropoulos, M. R. Jolly, C. A. Amsden, S. E. Gillman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef] [PubMed]

E. Spiller, “Low-loss reflection coatings using absorbing materials,” Appl. Phys. Lett. 20, 365–367 (1972).
[CrossRef]

J. Mod. Opt.

J. C. Monga, “Multilayer thin-film polarizers with reduced electric-field intensity,” J. Mod. Opt. 36, 769–784 (1989).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

P. H. Lissberger, “The ultimate reflectance of multilayer dielectric mirrors,” Opt. Acta 25, 291–298 (1978).
[CrossRef]

Other

Z. Fan, Z. Wu, X. Tang, “Laser induced damage in optical coatings,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 459–468 (1992).
[CrossRef]

H. A. Macleod, Thin-Film Optical Filters (Elsevier, New York, 1969).

J. F. DeFord, M. R. Kozlowski, “Modeling of electric-field enhancements at nodular defects in dielectric mirror coatings,” in Laser-Induced Damage in Optical Materials, Proc. SPIE1848, 455–469 (1992).

R. J. Tench, R. Chow, M. R. Kozlowski, “Characterization of defect geometries in multilayer optical coatings,” J. Vac. Sci. Technol. A12, 2808–2813 (1994); “Investigation of the microstructure of coatings for high power lasers by non-optical techniques,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 596–602 (1994).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Optical coating, consisting of alternating high and low index of refraction layers.

Fig. 2
Fig. 2

Reflectance as a function of fraction x of the half-wave layer pair associated with hafnia; a film of 23 total layers with 1.06-µm normally incident light is depicted.

Fig. 3
Fig. 3

Variations in reflectances as a function of hafnia fraction x, shown for films of different numbers of layer pairs. Light with a wavelength of 1.06 µm is normally incident on the films.

Fig. 4
Fig. 4

Reflectance as a function of incident wavelength for normally incident radiation, comparing the traditional quarter-wave design to a selected half-wave design.

Fig. 5
Fig. 5

Reflectances of 1.06-µm radiation incident at 56.5° on a TiO2–SiO2 polarizing film with seven layer pairs, as a function of TiO2 fraction x.

Fig. 6
Fig. 6

Reflectances of 1.06-µm radiation incident at 56.5° on a hafnia–silica polarizing film with 11 layer pairs, as a function of hafnia fraction x.

Tables (2)

Tables Icon

Table 1 Comparison of the Optical Properties of Polarizing Films Based on TiO2–SiO2 Layers

Tables Icon

Table 2 Comparison of the Optical Properties of Polarizing Films Based on Hafnia–Silica Layers

Equations (30)

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R=η0B-C2η0B+C2,
BC=r=1ncos δri sin δr/ηriηr sin δrcos δr1ηn+1,
δr=2πNrdr cos θr/λ, ηr=Nr cos θr TE waves, =Nr/cos θr TM waves,
N0 sin θ0=Nr sin θr.
d1=0.5-ελc2N1 cos θ1, d2=0.5+ελc2N2 cos θ2.
A11=121+η2η1cosπλcλ+121-η2η1cos2πελcλ, A12=i21η2-1η1sin2πελcλ+i21η2+1η1sinπλcλ, A21=i2η2-η1sin2πελcλ+i2η2+η1sinπλcλ, A22=121+η1η2cosπλcλ+121-η1η2cos2πελcλ.
λ1,2=s±s2-42,
s=122+η1η2+η2η1cosπλcλ+122-η1η2-η2η1cos2πελcλ.
S=A12A12λ1-A11A22-λ1, S-1=c0A22-λ1-A12A11-λ1A12,
S-1AS=λ100λ2.
Ap=c0A12A22-λ1λ1p-λ1-A11λ2pA122λ2p-λ1pλ1-A11A22-λ11p-λ2pA12A22-λ1λ2p-λ1-A11λ1p.
M2p+1=cos δ1i sin δ1η1iη1 sin δ1cos δ1,
δ1=π2-πελcλ.
R=η0BR-CR2+η0BI-CI2η0BR+CR2+η0BI+CI2,
B=BR+iBI, C=CR+iCI.
BR=-ηn+1η1A12iA22-λ1λ1p-λ1-A11λ2psin δ1+ηn+1A122λ2p-λ1pcos δ1, BI=A12iA22-λ1λ1p-λ1-A11λ2pcos δ1+η1A122λ2p-λ1psin δ1, CR=λ1-A11A22-λ1λ1p-λ2pcos δ1-η1A12i×A22-λ1λ2p-λ1-A11λ1psin δ1, CI=ηn+1η1λ1-A11A22-λ1λ1p-λ2psin δ1+ηn+1A12iA22-λ1λ2p-λ1-A11λ1pcos δ1.
λ1=expiϕ, λ2=exp-iϕ,
λ1+λ2=s,
cos ϕ=s2.
BR=2 sin ϕ cos pϕ-A22-A11sin pϕcos δ1-2A12iη1 sin pϕ sin δ1, BI=2 sin ϕ cos pϕ-A22-A11sin pϕηn+1η1sin δ1+2A12iηn+1 sin pϕ cos δ1, CR=-2A21iηn+1η1sin pϕ sin δ1+A22-A11sin pϕ+2 sin ϕ cos pϕηn+1 cos δ1, CI=2A21isin pϕ cos δ1+A22-A11sin pϕ+2 sin ϕ cos pϕη1 sin δ1,
R1-2η0B*C+BC*η02B2+C2.
B*C+BC*=-2ηn+1A122λ1-λ22.
η02B2+C2Kλ22p,
1-Rλ2-2p.
A11=-121+η2η1+121-η2η1cos2πε, A12=i21η2-1η1sin2πε, A21=i2η2-η1sin2πε, A22=-121+η1η2+121-η1η2cos2πε.
s=-122+η1η2+η2η1+122-η1η2-η2η1cos2πε.
δ1=π2-πε,
BR=-ηn+1η1A12iA22-λ1λ1p-λ1-A11λ2pcos πε+ηn+1A122λ2p-λ1psin πε, BI=A12iA22-λ1λ1p-λ1-A11λ2psin πε+η1A122λ2p-λ1pcos πε, CR=λ1-A11A22-λ1λ1p-λ2psin πε-η1A12i×A22-λ1λ2p-λ1-A11λ1pcos πε, CI=ηn+1η1λ1-A11A22-λ1λ1p-λ2pcos πε+ηn+1A12iA22-λ1λ2p-λ1-A11λ1psin πε.
R=BR-CR2+BI-CI2BR+CR2+BI+CI2.
BC=η12p+1η22pηn+1.

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