Abstract

We present a thorough description of high-performance thin-film optical filters with high flatness. These components can combine several tens or hundreds of layers and are manufactured using plasma-assisted reactive magnetron sputtering. Stress compensation is achieved using dual-side coatings with appropriate spectral function. Examples are described of highly reflecting mirrors at 515 nm with 15 nm flatness peak-to-valley up to and over 75 mm diameter aperture, narrow bandpass filters, and filters with broadband controlled transmission.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  8. M.-M. de Denus-Baillargeon, T. Schmitt, S. Larouche, and L. Martinu, “Design and fabrication of stress-compensated optical coatings: Fabry-Perot filters for astronomical applications,” Appl. Opt. 53, 2616–2624 (2014).
    [Crossref]
  9. T. Begou, H. Krol, D. Stojcevski, F. Lemarchand, M. Lequime, C. Grezes-Besset, and J. Lumeau, “Complex optical interference filters with stress compensation for space applications,” CEAS Space J. 9, 441–449 (2017).
    [Crossref]
  10. H. Liu, L. Jensen, P. Ma, and D. Ristau, “Stress compensated anti-reflection coating for high power laser deposited with IBS SiO2 and ALD Al2O3,” Appl. Surf. Sci. 476, 521–527 (2019).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  17. L. Abel-Tiberini, F. Lemarquis, and M. Lequime, “Dedicated spectrophotometer for localized transmittance and reflectance measurements,” Appl. Opt. 45, 1386–1391 (2006).
    [Crossref]
  18. https://www.eli-np.ro/ .
  19. C. A. Ur, “Gamma beam system at ELI-NP,” AIP Conf. Proc. 1645, 237 (2015).
    [Crossref]
  20. K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.
  21. A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
    [Crossref]
  22. P. Kupinski and H. A. Macleod, “Advances in optical manufacturing: measurement considerations when specifying optical coatings,” Laser Focus World 51, 1645–1655 (2015).
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    [Crossref]

2019 (2)

H. Liu, L. Jensen, P. Ma, and D. Ristau, “Stress compensated anti-reflection coating for high power laser deposited with IBS SiO2 and ALD Al2O3,” Appl. Surf. Sci. 476, 521–527 (2019).
[Crossref]

M. Vignaux, F. Lemarch, T. Begou, C. Grèzes-Besset, and J. Lumeau, “Automated method for the determination of the all-optical monitoring strategy of complex thin-film filters,” Opt. Express 27, 12373–12390 (2019).
[Crossref]

2018 (1)

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

2017 (3)

T. Begou, H. Krol, D. Stojcevski, F. Lemarchand, M. Lequime, C. Grezes-Besset, and J. Lumeau, “Complex optical interference filters with stress compensation for space applications,” CEAS Space J. 9, 441–449 (2017).
[Crossref]

T. Begou and J. Lumeau, “Accurate analysis of mechanical stress in dielectric multilayers,” Opt. Lett. 42, 3217–3220 (2017).
[Crossref]

A. Piegari and A. Sytchkova, “Phase distortion and thickness variation in the design of optical coatings,” Proc. SPIE 10562, 105621H (2017).
[Crossref]

2015 (2)

P. Kupinski and H. A. Macleod, “Advances in optical manufacturing: measurement considerations when specifying optical coatings,” Laser Focus World 51, 1645–1655 (2015).

C. A. Ur, “Gamma beam system at ELI-NP,” AIP Conf. Proc. 1645, 237 (2015).
[Crossref]

2014 (3)

2012 (1)

2008 (1)

2006 (1)

2004 (1)

The VIRGO Collaboration, “The VIRGO large mirrors: a challenge for low loss coatings,” Classical Quantum Gravity 21, S935 (2004).
[Crossref]

1996 (1)

1993 (1)

1988 (1)

A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
[Crossref]

1909 (1)

G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. London Ser. A 82, 172 (1909).
[Crossref]

Abadias, G.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Abel-Tiberini, L.

Barthel, E.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Beaugerard, N.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Begou, T.

Bischoff, M.

Cassou, K.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Chason, E.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Chen, W.-L.

de Denus-Baillargeon, M.-M.

Deacon, D. A. G.

A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
[Crossref]

Dobrowolski, J. A.

Doll, G. L.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Douillet, D.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Dupraz, K.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Falcoz, F.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Grezes-Besset, C.

T. Begou, H. Krol, D. Stojcevski, F. Lemarchand, M. Lequime, C. Grezes-Besset, and J. Lumeau, “Complex optical interference filters with stress compensation for space applications,” CEAS Space J. 9, 441–449 (2017).
[Crossref]

Grèzes-Besset, C.

Gu, P.-F.

Hand, R. D.

Jensen, L.

H. Liu, L. Jensen, P. Ma, and D. Ristau, “Stress compensated anti-reflection coating for high power laser deposited with IBS SiO2 and ALD Al2O3,” Appl. Surf. Sci. 476, 521–527 (2019).
[Crossref]

Keckes, J.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Kozlov, A.

Krol, H.

T. Begou, H. Krol, D. Stojcevski, F. Lemarchand, M. Lequime, C. Grezes-Besset, and J. Lumeau, “Complex optical interference filters with stress compensation for space applications,” CEAS Space J. 9, 441–449 (2017).
[Crossref]

Kupinski, P.

Lambropoulos, J. C.

Larouche, S.

Le barillec, T.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Lemarch, F.

Lemarchand, F.

T. Begou, H. Krol, D. Stojcevski, F. Lemarchand, M. Lequime, C. Grezes-Besset, and J. Lumeau, “Complex optical interference filters with stress compensation for space applications,” CEAS Space J. 9, 441–449 (2017).
[Crossref]

Lemarquis, F.

Lequime, M.

T. Begou, H. Krol, D. Stojcevski, F. Lemarchand, M. Lequime, C. Grezes-Besset, and J. Lumeau, “Complex optical interference filters with stress compensation for space applications,” CEAS Space J. 9, 441–449 (2017).
[Crossref]

L. Abel-Tiberini, F. Lemarquis, and M. Lequime, “Dedicated spectrophotometer for localized transmittance and reflectance measurements,” Appl. Opt. 45, 1386–1391 (2006).
[Crossref]

Liu, H.

H. Liu, L. Jensen, P. Ma, and D. Ristau, “Stress compensated anti-reflection coating for high power laser deposited with IBS SiO2 and ALD Al2O3,” Appl. Surf. Sci. 476, 521–527 (2019).
[Crossref]

Liu, X.

Lumeau, J.

Ma, P.

H. Liu, L. Jensen, P. Ma, and D. Ristau, “Stress compensated anti-reflection coating for high power laser deposited with IBS SiO2 and ALD Al2O3,” Appl. Surf. Sci. 476, 521–527 (2019).
[Crossref]

Macleod, H. A.

P. Kupinski and H. A. Macleod, “Advances in optical manufacturing: measurement considerations when specifying optical coatings,” Laser Focus World 51, 1645–1655 (2015).

Martens, A.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Martinu, L.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

M.-M. de Denus-Baillargeon, T. Schmitt, S. Larouche, and L. Martinu, “Design and fabrication of stress-compensated optical coatings: Fabry-Perot filters for astronomical applications,” Appl. Opt. 53, 2616–2624 (2014).
[Crossref]

Murray, C. E.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Ndiaye, C. F.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Nowitzki, T.

O’Keefe, A.

A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
[Crossref]

Oliver, J. B.

Papernov, S.

Peinaud, Y.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Piegari, A.

A. Piegari and A. Sytchkova, “Phase distortion and thickness variation in the design of optical coatings,” Proc. SPIE 10562, 105621H (2017).
[Crossref]

Rigatti, A. L.

Ristau, D.

H. Liu, L. Jensen, P. Ma, and D. Ristau, “Stress compensated anti-reflection coating for high power laser deposited with IBS SiO2 and ALD Al2O3,” Appl. Surf. Sci. 476, 521–527 (2019).
[Crossref]

Rocipon, H.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Schmid, A. W.

Schmitt, T.

Sebastiani, M.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Shen, W.

Smith, C.

Stenzel, O.

Stoessel, C. H.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Stojcevski, D.

T. Begou, H. Krol, D. Stojcevski, F. Lemarchand, M. Lequime, C. Grezes-Besset, and J. Lumeau, “Complex optical interference filters with stress compensation for space applications,” CEAS Space J. 9, 441–449 (2017).
[Crossref]

Stoney, G.

G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. London Ser. A 82, 172 (1909).
[Crossref]

Strauss, G. N.

G. N. Strauss, “Mechanical stress in optical coatings,” in Optical Interference Coatings, N. Kaiser and H. K. Pulker, eds., Vol. 88 of Springer Series in Optical Sciences (Springer, 2003).

Sullivan, B. T.

Sytchkova, A.

A. Piegari and A. Sytchkova, “Phase distortion and thickness variation in the design of optical coatings,” Proc. SPIE 10562, 105621H (2017).
[Crossref]

Thompson, G. B.

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Tikhonravov, A. V.

Ur, C. A.

C. A. Ur, “Gamma beam system at ELI-NP,” AIP Conf. Proc. 1645, 237 (2015).
[Crossref]

Variola, A.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

Vignaux, M.

Voß, O.

Wang, Y.

Wilbrandt, S.

Zhang, Y.-G.

Zomer, F.

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

AIP Conf. Proc. (1)

C. A. Ur, “Gamma beam system at ELI-NP,” AIP Conf. Proc. 1645, 237 (2015).
[Crossref]

Appl. Opt. (7)

Appl. Surf. Sci. (1)

H. Liu, L. Jensen, P. Ma, and D. Ristau, “Stress compensated anti-reflection coating for high power laser deposited with IBS SiO2 and ALD Al2O3,” Appl. Surf. Sci. 476, 521–527 (2019).
[Crossref]

CEAS Space J. (1)

T. Begou, H. Krol, D. Stojcevski, F. Lemarchand, M. Lequime, C. Grezes-Besset, and J. Lumeau, “Complex optical interference filters with stress compensation for space applications,” CEAS Space J. 9, 441–449 (2017).
[Crossref]

Classical Quantum Gravity (1)

The VIRGO Collaboration, “The VIRGO large mirrors: a challenge for low loss coatings,” Classical Quantum Gravity 21, S935 (2004).
[Crossref]

J. Vac. Sci. Technol. (1)

G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G. B. Thompson, E. Barthel, G. L. Doll, C. E. Murray, C. H. Stoessel, and L. Martinu, “Stress in thin films and coatings: current status, challenges, and prospects,” J. Vac. Sci. Technol. A36, 020801 (2018).
[Crossref]

Laser Focus World (1)

P. Kupinski and H. A. Macleod, “Advances in optical manufacturing: measurement considerations when specifying optical coatings,” Laser Focus World 51, 1645–1655 (2015).

Opt. Express (2)

Opt. Lett. (1)

Proc. R. Soc. London Ser. A (1)

G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. London Ser. A 82, 172 (1909).
[Crossref]

Proc. SPIE (1)

A. Piegari and A. Sytchkova, “Phase distortion and thickness variation in the design of optical coatings,” Proc. SPIE 10562, 105621H (2017).
[Crossref]

Rev. Sci. Instrum. (1)

A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544 (1988).
[Crossref]

Other (4)

K. Cassou, C. F. Ndiaye, N. Beaugerard, K. Dupraz, F. Falcoz, D. Douillet, T. Le barillec, A. Martens, Y. Peinaud, H. Rocipon, A. Variola, and F. Zomer, “Laser beam circulator for the generation of a high brilliance gamma beam at ELI-NP,” in High-Brightness Sources and Light-driven Interactions, OSA Technical Digest (online) (Optical Society of America, 2018), paper HW4A.7.

G. N. Strauss, “Mechanical stress in optical coatings,” in Optical Interference Coatings, N. Kaiser and H. K. Pulker, eds., Vol. 88 of Springer Series in Optical Sciences (Springer, 2003).

https://www.optilayer.com/ .

https://www.eli-np.ro/ .

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Figures (7)

Fig. 1.
Fig. 1. Spectral dependence of reflection at 22.5° angle of incidence for $s$ polarization (theory in orange and experimental in red) and $p$ polarization (theory in blue and experimental in green) in case of single-side coated mirrors.
Fig. 2.
Fig. 2. Typical flatness measured on an ELI-NP mirror after stress compensation.
Fig. 3.
Fig. 3. Typical structure of each face of the BPF1, top front face (FF), and bottom rear face (RF). S1 to S10 represent each of the basic quarter-wave structures composing the filter.
Fig. 4.
Fig. 4. Spectral dependence of the transmission of BPF1 in linear and log scales. Theory is in orange, and experimental data are in blue.
Fig. 5.
Fig. 5. Spectral dependence of the transmission of BPF2 in linear and log scales. Theory is in orange, and experimental data are in blue.
Fig. 6.
Fig. 6. Measured flatness on a witness glass coated with BPF2-FF (left), BPF2-RF (center), and BPF2-FF+RF (left).
Fig. 7.
Fig. 7. Spectral dependence of the transmission of EF1 (left) and EF2 (right). Theory is in orange, and experimental data are in blue.

Tables (8)

Tables Icon

Table 1. Specifications of the ELI-NP Laser Beam Circulator Mirrors

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Table 2. Typical Parameters of the Designed Mirrors

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Table 3. Theoretical Reflection Coefficients of the Different Mirrors for Different Incidence Wavelengths and Incidence Angles

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Table 4. Estimated Performance at 515 nm Deduced from Measurements at 532 nm

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Table 5. Average Flatness Performance of the Large-Aperture Mirrors

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Table 6. Typical Parameters of the Designed Bandpass Filters

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Table 7. Typical Wavelengths Used for Monitoring the Two Faces (Front Face, FF, and Rear Face, RF) of the Bandpass Filters

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Table 8. Typical Parameters of the Designed Filters with Broadband Controlled Transmission

Equations (4)

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σ = E S d S 2 6 t f ( 1 υ S ) ( 1 R S + f 1 R S ) ,
R S + f = d S 2 K S R S d S 2 K S + R S ( i = 1 N H σ H ( d H i ) d H i + j = 1 N L σ L ( d L j ) d L j ) ,
K S = E S 6 ( 1 υ S ) ,
( Σ i = 1 N H σ H ( d H i ) d H i + Σ j = 1 N L σ L ( d L j ) d L j ) F a c e A = ( Σ i = 1 N H σ H ( d H i ) d H i + Σ j = 1 N L σ L ( d L j ) d L j ) F a c e B .

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