Abstract

Mathematical expressions are developed for the phase-shift derivative with respect to the wavelength, in the case of nonquarter-wave, two-material, high-reflectance, periodic mirrors. These expressions are applied to the case of oblique incidence, and a condition relating the layer indices, which provides identical phase dispersion curves for both the P and S polarizations, is derived. The use of such mirrors in Fabry–Perot filters results in a common peak wavelength for both polarizations when the filter is used at oblique incidence, instead of the two separate spectral peaks usually observed. This property, which is illustrated by several numerical examples, is of great interest for the design of tunable filters, in which the angle of incidence is frequently used as a tuning parameter.

© 2013 Optical Society of America

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References

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  1. S. Kraft, B. Carnicero Dominguez, M. Drusch, J. L. Bézy, and R. Meynart, “FIMAS—feasibility study of a fluorescence imaging spectrometer to be flown on a small platform in tandem with sentinel 3,” presented at International Conference on Space Optics ICSO 2010, Rhodes, Greece, 4–8 October 2010.
  2. P. Gu and Z. Zheng, “Design of non-polarizing thin film edge filters,” J. Zhejiang Univ. Sci. A 7, 1037–1040 (2006).
    [CrossRef]
  3. “Semrock VersaChrome tunable bandpass filters,” http://www.semrock.com/semrock-versachrome-tunable-bandpass- filters. aspx .
  4. A. Thelen, “Nonpolarizing interference films inside a glass cube,” Appl. Opt. 15, 2983–2985 (1976).
    [CrossRef]
  5. P. Baumeister, “Bandpass design—application to nonnormal incidence,” Appl. Opt. 31, 504–512 (1992).
    [CrossRef]
  6. D. Cushing, “Thin film interference filter for 45° angle of incidence inside a glass prsim with extremely low polarization dependence,” in 43rd Annual Technical Conference Proceedings (SVC, 2000), pp. 252–257.
  7. H. Qi, R. Hong, K. Yi, J. Shao, and Z. Fan, “Nonpolarizing and polarizing filter design,” Appl. Opt. 44, 2343–2348 (2005).
    [CrossRef]
  8. D. Cushing, “Multilayer thin film dielectric bandpass filter,” U.S. patent 5,926,317 (20July1999).
  9. D. Cushing, “Bandpass filter for forty five degree angle with low polarization properties,” in Optical Interference Coatings, Vol. 9, OSA Technical Digest (Optical Society of America, 1998), pp. 226–228.
  10. P. Gu, H. Chen, Y. Zhang, H. Li, and X. Liu, “Wavelength-division multiplexed thin-film filters used in tilted incident angles of light,” Appl. Opt. 43, 2066–2070 (2004).
    [CrossRef]
  11. J. Birge and F. Kärtner, “Efficient analytic computation of dispersion from multilayer structures,” Appl. Opt. 45, 1478–1483 (2006).
    [CrossRef]
  12. J. Birge and F. Kärtner, “Efficient optimization of multilayer coatings for ultrafast optics using analytic gradients of dispersion,” Appl. Opt. 46, 2656–2662 (2007).
    [CrossRef]
  13. V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
    [CrossRef]
  14. A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, “Phase properties of multilayers,” Appl. Opt. 36, 4382–4392 (1997).
    [CrossRef]
  15. F. Abeles, “Remarque sur l’influence de la dispersion dans les systèmes de couches minces diélectriques,” J. Physique et le Radium 19, 327–334 (1958).
  16. P. W. Baumeister, Optical Coating Technology (SPIE, 2004), pp. 2–75.
  17. E. Garmire, “Theory of quarter-wave-stack dielectric mirrors used in a thin Fabry–Perot filter,” Appl. Opt. 42, 5442–5449 (2003).
    [CrossRef]
  18. W. Shen, X. Liu, B. Huang, Y. Zhu, and P. Gu, “The effect of reflection phase shift on the optical properties of a micro-opto-electo-mechanical system Fabry–Perot tunable filter,” J. Opt. A 6, 853–858 (2004).
    [CrossRef]
  19. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1988), p. 16.
  20. P. D. Atherton and N. K. Reay, “A narrow-gap, servo-controlled tunable Fabry–Perot filter for astronomy,” Mon. Not. R. Astron. Soc. 197, 507–511 (1981).
  21. A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
    [CrossRef]

2007 (2)

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
[CrossRef]

J. Birge and F. Kärtner, “Efficient optimization of multilayer coatings for ultrafast optics using analytic gradients of dispersion,” Appl. Opt. 46, 2656–2662 (2007).
[CrossRef]

2006 (3)

J. Birge and F. Kärtner, “Efficient analytic computation of dispersion from multilayer structures,” Appl. Opt. 45, 1478–1483 (2006).
[CrossRef]

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

P. Gu and Z. Zheng, “Design of non-polarizing thin film edge filters,” J. Zhejiang Univ. Sci. A 7, 1037–1040 (2006).
[CrossRef]

2005 (1)

2004 (2)

P. Gu, H. Chen, Y. Zhang, H. Li, and X. Liu, “Wavelength-division multiplexed thin-film filters used in tilted incident angles of light,” Appl. Opt. 43, 2066–2070 (2004).
[CrossRef]

W. Shen, X. Liu, B. Huang, Y. Zhu, and P. Gu, “The effect of reflection phase shift on the optical properties of a micro-opto-electo-mechanical system Fabry–Perot tunable filter,” J. Opt. A 6, 853–858 (2004).
[CrossRef]

2003 (1)

1997 (1)

1992 (1)

1981 (1)

P. D. Atherton and N. K. Reay, “A narrow-gap, servo-controlled tunable Fabry–Perot filter for astronomy,” Mon. Not. R. Astron. Soc. 197, 507–511 (1981).

1976 (1)

1958 (1)

F. Abeles, “Remarque sur l’influence de la dispersion dans les systèmes de couches minces diélectriques,” J. Physique et le Radium 19, 327–334 (1958).

Abeles, F.

F. Abeles, “Remarque sur l’influence de la dispersion dans les systèmes de couches minces diélectriques,” J. Physique et le Radium 19, 327–334 (1958).

Abraham, R.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Apolonski, A.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
[CrossRef]

Atherton, P. D.

P. D. Atherton and N. K. Reay, “A narrow-gap, servo-controlled tunable Fabry–Perot filter for astronomy,” Mon. Not. R. Astron. Soc. 197, 507–511 (1981).

Barton, E.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Baumeister, P.

Baumeister, P. W.

Bershady, M.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Bézy, J. L.

S. Kraft, B. Carnicero Dominguez, M. Drusch, J. L. Bézy, and R. Meynart, “FIMAS—feasibility study of a fluorescence imaging spectrometer to be flown on a small platform in tandem with sentinel 3,” presented at International Conference on Space Optics ICSO 2010, Rhodes, Greece, 4–8 October 2010.

Birge, J.

Bland-Hawthorn, J.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Carnicero Dominguez, B.

S. Kraft, B. Carnicero Dominguez, M. Drusch, J. L. Bézy, and R. Meynart, “FIMAS—feasibility study of a fluorescence imaging spectrometer to be flown on a small platform in tandem with sentinel 3,” presented at International Conference on Space Optics ICSO 2010, Rhodes, Greece, 4–8 October 2010.

Chen, H.

Crampton, D.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Cushing, D.

D. Cushing, “Bandpass filter for forty five degree angle with low polarization properties,” in Optical Interference Coatings, Vol. 9, OSA Technical Digest (Optical Society of America, 1998), pp. 226–228.

D. Cushing, “Multilayer thin film dielectric bandpass filter,” U.S. patent 5,926,317 (20July1999).

D. Cushing, “Thin film interference filter for 45° angle of incidence inside a glass prsim with extremely low polarization dependence,” in 43rd Annual Technical Conference Proceedings (SVC, 2000), pp. 252–257.

Doyon, R.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Drusch, M.

S. Kraft, B. Carnicero Dominguez, M. Drusch, J. L. Bézy, and R. Meynart, “FIMAS—feasibility study of a fluorescence imaging spectrometer to be flown on a small platform in tandem with sentinel 3,” presented at International Conference on Space Optics ICSO 2010, Rhodes, Greece, 4–8 October 2010.

Eikenberry, S.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Fan, Z.

Garmire, E.

Gu, P.

P. Gu and Z. Zheng, “Design of non-polarizing thin film edge filters,” J. Zhejiang Univ. Sci. A 7, 1037–1040 (2006).
[CrossRef]

P. Gu, H. Chen, Y. Zhang, H. Li, and X. Liu, “Wavelength-division multiplexed thin-film filters used in tilted incident angles of light,” Appl. Opt. 43, 2066–2070 (2004).
[CrossRef]

W. Shen, X. Liu, B. Huang, Y. Zhu, and P. Gu, “The effect of reflection phase shift on the optical properties of a micro-opto-electo-mechanical system Fabry–Perot tunable filter,” J. Opt. A 6, 853–858 (2004).
[CrossRef]

Hong, R.

Huang, B.

W. Shen, X. Liu, B. Huang, Y. Zhu, and P. Gu, “The effect of reflection phase shift on the optical properties of a micro-opto-electo-mechanical system Fabry–Perot tunable filter,” J. Opt. A 6, 853–858 (2004).
[CrossRef]

Javed, M.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Julian, J.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Julian, R.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Kärtner, F.

Kneib, J.-P.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Kraft, S.

S. Kraft, B. Carnicero Dominguez, M. Drusch, J. L. Bézy, and R. Meynart, “FIMAS—feasibility study of a fluorescence imaging spectrometer to be flown on a small platform in tandem with sentinel 3,” presented at International Conference on Space Optics ICSO 2010, Rhodes, Greece, 4–8 October 2010.

Krausz, F.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
[CrossRef]

Li, H.

Liu, X.

P. Gu, H. Chen, Y. Zhang, H. Li, and X. Liu, “Wavelength-division multiplexed thin-film filters used in tilted incident angles of light,” Appl. Opt. 43, 2066–2070 (2004).
[CrossRef]

W. Shen, X. Liu, B. Huang, Y. Zhu, and P. Gu, “The effect of reflection phase shift on the optical properties of a micro-opto-electo-mechanical system Fabry–Perot tunable filter,” J. Opt. A 6, 853–858 (2004).
[CrossRef]

Loop, D.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Meynart, R.

S. Kraft, B. Carnicero Dominguez, M. Drusch, J. L. Bézy, and R. Meynart, “FIMAS—feasibility study of a fluorescence imaging spectrometer to be flown on a small platform in tandem with sentinel 3,” presented at International Conference on Space Optics ICSO 2010, Rhodes, Greece, 4–8 October 2010.

Naumov, S.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
[CrossRef]

Pervak, V.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
[CrossRef]

Popov, K. V.

Qi, H.

Raines, N.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Reay, N. K.

P. D. Atherton and N. K. Reay, “A narrow-gap, servo-controlled tunable Fabry–Perot filter for astronomy,” Mon. Not. R. Astron. Soc. 197, 507–511 (1981).

Rowlands, N.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Scott, A.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Shao, J.

Shen, W.

W. Shen, X. Liu, B. Huang, Y. Zhu, and P. Gu, “The effect of reflection phase shift on the optical properties of a micro-opto-electo-mechanical system Fabry–Perot tunable filter,” J. Opt. A 6, 853–858 (2004).
[CrossRef]

Smith, J. D.

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Thelen, A.

A. Thelen, “Nonpolarizing interference films inside a glass cube,” Appl. Opt. 15, 2983–2985 (1976).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1988), p. 16.

Tikhonravov, A. V.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
[CrossRef]

A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, “Phase properties of multilayers,” Appl. Opt. 36, 4382–4392 (1997).
[CrossRef]

Trubetskov, M. K.

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
[CrossRef]

Yi, K.

Zhang, Y.

Zheng, Z.

P. Gu and Z. Zheng, “Design of non-polarizing thin film edge filters,” J. Zhejiang Univ. Sci. A 7, 1037–1040 (2006).
[CrossRef]

Zhu, Y.

W. Shen, X. Liu, B. Huang, Y. Zhu, and P. Gu, “The effect of reflection phase shift on the optical properties of a micro-opto-electo-mechanical system Fabry–Perot tunable filter,” J. Opt. A 6, 853–858 (2004).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. B (1)

V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87, 5–12(2007).
[CrossRef]

J. Opt. A (1)

W. Shen, X. Liu, B. Huang, Y. Zhu, and P. Gu, “The effect of reflection phase shift on the optical properties of a micro-opto-electo-mechanical system Fabry–Perot tunable filter,” J. Opt. A 6, 853–858 (2004).
[CrossRef]

J. Physique et le Radium (1)

F. Abeles, “Remarque sur l’influence de la dispersion dans les systèmes de couches minces diélectriques,” J. Physique et le Radium 19, 327–334 (1958).

J. Zhejiang Univ. Sci. A (1)

P. Gu and Z. Zheng, “Design of non-polarizing thin film edge filters,” J. Zhejiang Univ. Sci. A 7, 1037–1040 (2006).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

P. D. Atherton and N. K. Reay, “A narrow-gap, servo-controlled tunable Fabry–Perot filter for astronomy,” Mon. Not. R. Astron. Soc. 197, 507–511 (1981).

Proc. SPIE (1)

A. Scott, M. Javed, R. Abraham, S. Eikenberry, E. Barton, M. Bershady, J. Bland-Hawthorn, D. Crampton, R. Doyon, J. Julian, R. Julian, J.-P. Kneib, D. Loop, N. Raines, N. Rowlands, and J. D. Smith, “Performance of F2T2 tandem tunable etalon,” Proc. SPIE 6269, 62695J (2006).
[CrossRef]

Other (7)

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1988), p. 16.

“Semrock VersaChrome tunable bandpass filters,” http://www.semrock.com/semrock-versachrome-tunable-bandpass- filters. aspx .

D. Cushing, “Multilayer thin film dielectric bandpass filter,” U.S. patent 5,926,317 (20July1999).

D. Cushing, “Bandpass filter for forty five degree angle with low polarization properties,” in Optical Interference Coatings, Vol. 9, OSA Technical Digest (Optical Society of America, 1998), pp. 226–228.

D. Cushing, “Thin film interference filter for 45° angle of incidence inside a glass prsim with extremely low polarization dependence,” in 43rd Annual Technical Conference Proceedings (SVC, 2000), pp. 252–257.

P. W. Baumeister, Optical Coating Technology (SPIE, 2004), pp. 2–75.

S. Kraft, B. Carnicero Dominguez, M. Drusch, J. L. Bézy, and R. Meynart, “FIMAS—feasibility study of a fluorescence imaging spectrometer to be flown on a small platform in tandem with sentinel 3,” presented at International Conference on Space Optics ICSO 2010, Rhodes, Greece, 4–8 October 2010.

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Figures (10)

Fig. 1.
Fig. 1.

Schematic representation of a p-layer coating with propagating electric fields.

Fig. 2.
Fig. 2.

Illustration of a two-material, high-reflectance, periodic mirror with a reflectance factor equal to 1 and a reflection phase shift that remains unchanged when a new pair of layers is added.

Fig. 3.
Fig. 3.

Variations of the refractive index of the required high-index material as a function of the low-index material for the two particular cases corresponding to an air (n0=1) or a silica or equivalent (n0=1.47) incident medium. The two realistic solutions (n0=1, nL=1.47, nH=2.14) and (n0=1.47, nL=2.05, nH=3.45) are indicated. The dashed line corresponds to the lower limit for nH defined by nH=nL.

Fig. 4.
Fig. 4.

Phase dispersion for a high-reflectance quarter-wave mirror: Glass(1.52)/HLHLHL/Air, (nL=1.47, nH=2.14, formed with 23 or 24 layers, central wavelength=1000nm). All of the layers are matched for a 30° angle of incidence in air, and the calculations have been made at this particular incidence. When the mirror ends with a low-index layer, the phase dispersion curves are identical for both polarizations.

Fig. 5.
Fig. 5.

Phase-shift variations as a function of angle of incidence for a high-reflectance quarter-wave mirror: Glass(1.52)/HLHLHL/Air, (nL=1.47, nH=2.14, formed with 23 or 24 layers, central wavelength=1000nm). All of the layers are matched for a 30° angle of incidence in air, and the calculations have been made for a wavelength of 1000 nm. When the mirror ends with a low-index layer, the phase dispersion curves are identical for both polarizations.

Fig. 6.
Fig. 6.

Transmittance profiles at normal incidence for two single-cavity designs: Glass(1.52)/(HL)92A(LH)9/Glass(thick line), Glass(1.52)/(HL)8H2AH(LH)8/Glass(thin line), where 2A represents a half-wave layer of air. The layers have a quarter-wave thickness at normal incidence, with central wavelength=1000nm, nH=2.14, nL=1.47.

Fig. 7.
Fig. 7.

Transmittance profiles for a 30° angle of incidence (in air) for two single-cavity designs: Glass(1.52)/(HL)92A(LH)9/Glass(thick lines), Glass(1.52)/(HL)8H2AH(LH)8/Glass(thin lines). The layers have a quarter-wave thickness at normal incidence, central wavelength=1000nm, nH=2.14, nL=1.47, where 2A represents a half-wave layer of air. When the cavity mirrors end with a low-index layer, the peak wavelengths are identical for both polarizations.

Fig. 8.
Fig. 8.

Transmittance profiles for a 20° angle of incidence (in air) for two single-cavity designs: Glass(1.52)/(HL)950A(LH)9/Glass(thick line), Glass(1.52)/(HL)8H50AH(LH)8/Glass(thin line). The layers have a quarter-wave thickness at normal incidence, with central wavelength=1000nm, nH=2.14, nL=1.47, where 50A represents a spacer layer of air that has a 50 times quarter-wave thickness. When the cavity mirrors end with a low-index layer, the peak wavelengths are identical for both polarizations, and this property remains valid when the cavity thickness is varied in order to tune the peak wavelength.

Fig. 9.
Fig. 9.

Transmittance profiles at normal incidence for three-cavity designs: Glass(1.52)/(HL)64S(LH)6S(HL)64S(LH)6S(HL)64S(LH)6/Glass (thick line), Glass(1.52)/(LS)7L2SL(SL)7S(LS)7L2SL(SL)7S(LS)7L2SL(SL)7/Glass(thin line), Glass(1.52)/(HS)3H2SH(SH)3S(HS)3H2SH(SH)3S(HS)3H2SH(SH)3/Glass (dashed line). The layers have a quarter-wave thickness at normal incidence, with central wavelength=1700nm, nH=3.45, nL=2.15, where 2S represents a half-wave layer with an index equal to 1.47.

Fig. 10.
Fig. 10.

Transmittance profiles at oblique incidence for three-cavity designs: Glass(1.52)/(HL)64S(LH)6S(HL)64S(LH)6S(HL)64S(LH)6/Glass (thick lines,incidence of43°in the air) Glass(1.52)/(LS)7L2SL(SL)7S(LS)7L2SL(SL)7S(LS)7L2SL(SL)7/Glass (thin lines,incidence of 35°in the air) Glass(1.52)/(HS)3H2SH(SH)3S(HS)3H2SH(SH)3S(HS)3H2SH(SH)3/Glass(thin lines,incidence of 34°in the air). The layers have a quarter-wave thickness at normal incidence, with central wavelength=1700nm, nH=3.45, nL=2.15, where 2S represents a half-wave layer with an index equal to 1.47. When three materials are used, the peak wavelengths are very similar, whereas in the case of classical designs they are clearly separated.

Equations (52)

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(E0E0+)=(M11M12M21M22)·(ESES+)=M(ESES+).
M=I0·L1·I1·L2Lp·Ip.
Ii=(Ni+Ni+12NiNiNi+12NiNiNi+12NiNi+Ni+12Ni);Li=(ejδi00ejδi),
δi=2π·ni·cos(θi)·diλ.
r=E0E0+=M12M22.
M11=M22*,M12=M21*.
r=ejφr.
(M11M12M21M22)=(m11m12m21m22)·(M11M12M21M22),
r=M12M22=m11·M12+m12·M22m21·M12+m22·M22=m11·r+m12m21·r+m22.
m21·e2j·φr+(m22m11)·ej·φrm12=0.
(m21σ+2j·m21·φrσ)·e2j·φr+(m22σm11σ+j·(m22m11)·φrσ)·ej·φrm12σ=0.
(φrσ)λπ=j(m22σ)λπ(m11σ)λπ+(m12σ)λπ(m21σ)λπ2m21+m11m22.
(φrσ)λ0=j(m11σ)λ0(m22σ)λ0+(m12σ)λ0(m21σ)λ02m21+m22m11.
λc=2·(nH·dH·cos(θH)+nL·dL·cos(θL)),
ε=2·(nH·dH·cos(θH)nL·dL·cos(θL))λc.
m21(m22m11)m12=0.
Im(m21)Im(m22)=0.
(m11m21m12m22)=(N0+NH2N0N0NH2N0N0NH2N0N0+NH2N0)·(ejδH00ejδH)·(NH+NL2NHNHNL2NHNHNL2NHNH+NL2NH)·(ejδL00ejδL)·(NL+N02NLNLN02NLNLN02NLNL+N02NL).
δH=2π·nH·cos(θH)·dHλ=2πλ·(1+ε)·λc4=(1+ε)·π2·λcλ.
δL=(1ε)·π2·λcλ.
Im(m21)Im(m22)=jN02·NH·NL((NH+NL)·sin(π·λcλπ)(NHNL)·sin(ε·π·λcλπ))=0.
λπ=λc·(1+εNHNLNH+NL).
(m22σ)σπ(m11σ)σπ+(m12σ)σπ(m21σ)σπ=(2·Im(m22)σ)σπ+(2·Im(m12)σ)σπ.
2m21+m11m22=m21+m12=2·Re(m21).
(φrσ)σπ=N0·(NH+NL)·π·λc·cos(π·λc·σπ)N0·(NHNL)·ε·π·λc·cos(ε·π·λc·σπ)(NH+NL)·(NHNL)2·(cos(ε·π·λc·σπ)cos(π·λc·σπ)).
(φrσ)σπ=N0·(NH+NL)·π·λcN0·(NHNL)·ε·π·λc(NH+NL)·(NHNL).
(φrλ)λπ=πλc·N0(NHNL)ε.πλc·N0(NH+NL).
ε=2·(nL·dL·cos(θL)nH·dH·cos(θH))λc.
λ0=λc·(1+εNHNLNH+NL)
(φrλ)λ0=πλc·NH·NLN0·(NHNL)ε·πλc·NH·NLN0·(NH+NL).
ϕ=4·π·n.d.cos(θ)λpeak+2·φr(θ,λpeak)=2kπ,
nH·dH·cos(θH)=nL·dL·cos(θL)=λpeak4.
ϕ=[4.π·n.d.cos(θ)λpeak(d,θ)+2·φrP(θ,λpeak(d,θ))]polP=[4.π·n.d.cos(θ)λpeak(d,θ)+2·φrS(θ,λpeak(d,θ))]polS=2kπ.
[φrP(θ,λpeak(d,θ))]polP=[φrS(θ,λpeak(d,θ))]polS.
[φrλ]S=[φrλ]P.
[φrλ]S=[φrλ]Pand[φrθ]S=[φrθ]P.
n0·cos(θ0)nH·cos(θH)nL·cos(θL)=n0/cos(θ0)nH/cos(θH)nL/cos(θL).
nH·cos(θH)·nL·cos(θL)n0·cos(θ0)·(nH·cos(θH)nL·cos(θL))=(nH/cos(θH))·(nL/cos(θL))(n0/cos(θ0))·(nH/cos(θH)nL/cos(θL)).
nH3·(nL2n02)·cos(θH)=nL3·(nH2n02)·cos(θL).
n02=nH3·nL2nH2·nL3nH3nL3=nH2·nL2nH2+nH·nL+nL2.
nH=n0·nL·(n0+4·nL23·n02)2·(nL2n02).
φr(λ)=φr(λ0)+(φλ)λ0·(λλ0)=(φλ)λ0·(λλ0).
φr(λ)=[πλc·NH·NLN0·(NHNL)ε·πλc·NH·NLN0·(NH+NL)]·[λλc·(1+ε·NHNLNH+NL)].
(ϕrθ)λc,θ0=[πλc·NH·NLN0·(NHNL)]·θ[λc·(1+ε·NHNLNH+NL)].
(φrθ)λc,θ0=[πλc·NH·NLN0·(NHNL)]·θ(λc).
[φrθ]S=[φrθ]P,
nH=2.14;nL=1.47;n0=1,nH=3.45;nL=2.05;n0=1.47.
Glass(1.52)/(HL)92A(LH)9/Glass.
Glass(1.52)/(HL)8H2AH(LH)8/Glass.
Glass(1.52)/(HL)64S(LH)6S(HL)64S(LH)6S(HL)64S(LH)6/Glass.
Glass(1.52)/(LS)7L2SL(SL)7S(LS)7L2SL(SL)7S(LS)7L2SL(SL)7/Glass.
Glass(1.52)/(HS)3H2SH(SH)3S(HS)3H2SH(SH)3S(HS)3H2SH(SH)3/Glass.

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