Abstract

A generalized study has been carried out on the modeling of a Fabry–Perot microcavity for sensing applications. Different analytical models on transmission characteristics of a Fabry–Perot microcavity are established by using plane-wave-based techniques, such as the Macleod characteristic matrix technique, the transfer matrix technique, and Smith’s technique. A novel Gaussian-optics-based model for a Fabry–Perot microcavity illuminated by a laser beam is then developed and validated. The influence of laser beam waist on microcavity optical response is investigated, and the required minimal beam waist size is explored to ensure a useful optical response for sensing applications that can be accurately predicted by plane-wave optics. Also, the perturbations of microcavity performance induced by different types of microcavity mirror imperfections are discussed, based on the novel optical model. The prototype of the proposed Fabry–Perot microcavity for sensing applications has been successfully fabricated and characterized.

© 2005 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. W. N. Macpherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fiber Fabry–Perot sensors with inexact phase steps,” Appl. Opt. 39, 1382–1388 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2004 (1)

W. J. Wang, D. G. Guo, R. M. Lin, X. W. Wang, “A single-chip diaphragm-type miniature Fabry–Perot pressure sensor with improved cross-sensitivity to temperature,” Meas. Sci. Technol. 15, 905–910 (2004).
[CrossRef]

2003 (3)

W. J. Wang, R. M. Lin, T. T. Sun, D. G. Guo, Y. Ren, “Performance-enhanced Fabry–Perot microcavity structure with a novel non-planar diaphragm,” Microelectron. Eng. 70, 102–108 (2003).
[CrossRef]

Z. M. Wu, G. Q. Xia, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon: a nonresonant case,” Opt. Laser Technol. 35, 123–126 (2003).
[CrossRef]

Z. M. Wu, G. Q. Xia, H. Q. Zhou, J. W. Wu, M. L. Liu, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon,” Opt. Laser Technol. 35, 1–4 (2003).
[CrossRef]

2000 (3)

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

J. Han, J. Y. Kim, T. S. Kim, J. S. Kim, “Performance of Fabry–Perot microcavity structures with corrugated diaphragms,” Sens. Actuators, A 79, 162–172 (2000).
[CrossRef]

W. N. Macpherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fiber Fabry–Perot sensors with inexact phase steps,” Appl. Opt. 39, 1382–1388 (2000).
[CrossRef]

1999 (2)

J. Han, “Novel fabrication and characterization method of Fabry–Perot microcavity pressure sensors,” Sens. Actuators, A 75, 168–175 (1999).
[CrossRef]

J. Park, M. G. Kim, “High-performance fiber-optic Fabry–Perot pressure sensor with Si3N4∕SiO2∕Si3N4 diaphragm,” Jpn. J. Appl. Phys. Part 1 38, 1562–1564 (1999).
[CrossRef]

1998 (1)

E. T. Carlen, C. H. Mastrangelo, “Statistical model for spatial correlation in thin film deposition and reactive growth,” IEEE Trans. Semicond. Manuf. 11, 511–521 (1998).
[CrossRef]

1996 (1)

1995 (3)

1994 (1)

1992 (1)

1991 (1)

1981 (1)

P. D. Atherton, N. K. Reay, J. Ring, T. R. Hicks, “Tunable Fabry–Perot filters,” Opt. Eng. 20, 806–814 (1981).
[CrossRef]

1976 (1)

H. A. Macleod, “Thin film narrow band optical filters,” Thin Solid Films 34, 335–342 (1976).
[CrossRef]

1974 (1)

G. S. Bhatnagar, K. Singh, B. N. Gupta, “Transmission profile of a Fabry–Perot interferometer suffering from asymmetric surface defects,” Nouv. Rev. Opt. 5, 237–240 (1974).
[CrossRef]

1966 (1)

1958 (1)

Abeles, F.

F. Abeles, Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, 1967).

Abu-Aljarayesh, I.

Abu-Safia, H.

Allen, R. M.

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

Al-Tahtamouni, R.

Atherton, P. D.

P. D. Atherton, N. K. Reay, J. Ring, T. R. Hicks, “Tunable Fabry–Perot filters,” Opt. Eng. 20, 806–814 (1981).
[CrossRef]

Barton, J. S.

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

W. N. Macpherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fiber Fabry–Perot sensors with inexact phase steps,” Appl. Opt. 39, 1382–1388 (2000).
[CrossRef]

Belsley, K. L.

K. L. Belsley, D. R. Huber, J. Goodman, “All-passive interferometric fiber-optic pressure sensor,” presented at the Annual Meeting of the Instrument Society of America, 1986, Paper 86-2801.

Bhatnagar, G. S.

G. S. Bhatnagar, K. Singh, B. N. Gupta, “Transmission profile of a Fabry–Perot interferometer suffering from asymmetric surface defects,” Nouv. Rev. Opt. 5, 237–240 (1974).
[CrossRef]

Bretenaker, F.

Carlen, E. T.

E. T. Carlen, C. H. Mastrangelo, “Statistical model for spatial correlation in thin film deposition and reactive growth,” IEEE Trans. Semicond. Manuf. 11, 511–521 (1998).
[CrossRef]

Coker, J. E.

W. H. Quick, K. A. James, J. E. Coker, “Fiber optics sensing techniques,” presented at the First International Conference on Optical Fiber Sensors, London, April 1983.

Cotteverte, J. C.

Gander, M. J.

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

Gonzalez, F.

Goodman, J.

K. L. Belsley, D. R. Huber, J. Goodman, “All-passive interferometric fiber-optic pressure sensor,” presented at the Annual Meeting of the Instrument Society of America, 1986, Paper 86-2801.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1968).

Guo, D. G.

W. J. Wang, D. G. Guo, R. M. Lin, X. W. Wang, “A single-chip diaphragm-type miniature Fabry–Perot pressure sensor with improved cross-sensitivity to temperature,” Meas. Sci. Technol. 15, 905–910 (2004).
[CrossRef]

W. J. Wang, R. M. Lin, T. T. Sun, D. G. Guo, Y. Ren, “Performance-enhanced Fabry–Perot microcavity structure with a novel non-planar diaphragm,” Microelectron. Eng. 70, 102–108 (2003).
[CrossRef]

Gupta, B. N.

G. S. Bhatnagar, K. Singh, B. N. Gupta, “Transmission profile of a Fabry–Perot interferometer suffering from asymmetric surface defects,” Nouv. Rev. Opt. 5, 237–240 (1974).
[CrossRef]

Han, J.

J. Han, J. Y. Kim, T. S. Kim, J. S. Kim, “Performance of Fabry–Perot microcavity structures with corrugated diaphragms,” Sens. Actuators, A 79, 162–172 (2000).
[CrossRef]

J. Han, “Novel fabrication and characterization method of Fabry–Perot microcavity pressure sensors,” Sens. Actuators, A 75, 168–175 (1999).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1965).

Hicks, T. R.

P. D. Atherton, N. K. Reay, J. Ring, T. R. Hicks, “Tunable Fabry–Perot filters,” Opt. Eng. 20, 806–814 (1981).
[CrossRef]

Huber, D. R.

K. L. Belsley, D. R. Huber, J. Goodman, “All-passive interferometric fiber-optic pressure sensor,” presented at the Annual Meeting of the Instrument Society of America, 1986, Paper 86-2801.

James, K. A.

W. H. Quick, K. A. James, J. E. Coker, “Fiber optics sensing techniques,” presented at the First International Conference on Optical Fiber Sensors, London, April 1983.

Jones, J. D. C.

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

W. N. Macpherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fiber Fabry–Perot sensors with inexact phase steps,” Appl. Opt. 39, 1382–1388 (2000).
[CrossRef]

Kao, T. W.

Kidd, S. R.

Kim, J. S.

J. Han, J. Y. Kim, T. S. Kim, J. S. Kim, “Performance of Fabry–Perot microcavity structures with corrugated diaphragms,” Sens. Actuators, A 79, 162–172 (2000).
[CrossRef]

Kim, J. Y.

J. Han, J. Y. Kim, T. S. Kim, J. S. Kim, “Performance of Fabry–Perot microcavity structures with corrugated diaphragms,” Sens. Actuators, A 79, 162–172 (2000).
[CrossRef]

Kim, M. G.

J. Park, M. G. Kim, “High-performance fiber-optic Fabry–Perot pressure sensor with Si3N4∕SiO2∕Si3N4 diaphragm,” Jpn. J. Appl. Phys. Part 1 38, 1562–1564 (1999).
[CrossRef]

Kim, T. S.

J. Han, J. Y. Kim, T. S. Kim, J. S. Kim, “Performance of Fabry–Perot microcavity structures with corrugated diaphragms,” Sens. Actuators, A 79, 162–172 (2000).
[CrossRef]

Kim, Y. M.

Y. M. Kim, D. P. Neikirk, “Micromachined Fabry–Perot cavity pressure transducer,” IEEE Photonics Technol. Lett. 7, 1471–1473 (1995).
[CrossRef]

Kogelnik, H.

Lanternier, T.

Le Floch, A.

Li, T.

Lin, R. M.

W. J. Wang, D. G. Guo, R. M. Lin, X. W. Wang, “A single-chip diaphragm-type miniature Fabry–Perot pressure sensor with improved cross-sensitivity to temperature,” Meas. Sci. Technol. 15, 905–910 (2004).
[CrossRef]

W. J. Wang, R. M. Lin, T. T. Sun, D. G. Guo, Y. Ren, “Performance-enhanced Fabry–Perot microcavity structure with a novel non-planar diaphragm,” Microelectron. Eng. 70, 102–108 (2003).
[CrossRef]

Liu, M. L.

Z. M. Wu, G. Q. Xia, H. Q. Zhou, J. W. Wu, M. L. Liu, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon,” Opt. Laser Technol. 35, 1–4 (2003).
[CrossRef]

Macleod, H. A.

H. A. Macleod, “Thin film narrow band optical filters,” Thin Solid Films 34, 335–342 (1976).
[CrossRef]

Macpherson, W. N.

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

W. N. Macpherson, S. R. Kidd, J. S. Barton, J. D. C. Jones, “Phase demodulation in optical fiber Fabry–Perot sensors with inexact phase steps,” Appl. Opt. 39, 1382–1388 (2000).
[CrossRef]

Mastrangelo, C. H.

E. T. Carlen, C. H. Mastrangelo, “Statistical model for spatial correlation in thin film deposition and reactive growth,” IEEE Trans. Semicond. Manuf. 11, 511–521 (1998).
[CrossRef]

Moreno, F.

Neikirk, D. P.

Y. M. Kim, D. P. Neikirk, “Micromachined Fabry–Perot cavity pressure transducer,” IEEE Photonics Technol. Lett. 7, 1471–1473 (1995).
[CrossRef]

Nichelatti, E.

Nussbaum, A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice Hall, 1976).

Owen, C. L.

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

Park, J.

J. Park, M. G. Kim, “High-performance fiber-optic Fabry–Perot pressure sensor with Si3N4∕SiO2∕Si3N4 diaphragm,” Jpn. J. Appl. Phys. Part 1 38, 1562–1564 (1999).
[CrossRef]

Phillips, R. A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice Hall, 1976).

Poirson, J.

Quick, W. H.

W. H. Quick, K. A. James, J. E. Coker, “Fiber optics sensing techniques,” presented at the First International Conference on Optical Fiber Sensors, London, April 1983.

Reay, N. K.

P. D. Atherton, N. K. Reay, J. Ring, T. R. Hicks, “Tunable Fabry–Perot filters,” Opt. Eng. 20, 806–814 (1981).
[CrossRef]

Ren, Y.

W. J. Wang, R. M. Lin, T. T. Sun, D. G. Guo, Y. Ren, “Performance-enhanced Fabry–Perot microcavity structure with a novel non-planar diaphragm,” Microelectron. Eng. 70, 102–108 (2003).
[CrossRef]

Ring, J.

P. D. Atherton, N. K. Reay, J. Ring, T. R. Hicks, “Tunable Fabry–Perot filters,” Opt. Eng. 20, 806–814 (1981).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1968).

Salvetti, G.

Singh, K.

G. S. Bhatnagar, K. Singh, B. N. Gupta, “Transmission profile of a Fabry–Perot interferometer suffering from asymmetric surface defects,” Nouv. Rev. Opt. 5, 237–240 (1974).
[CrossRef]

Smith, S. D.

Sun, T. T.

W. J. Wang, R. M. Lin, T. T. Sun, D. G. Guo, Y. Ren, “Performance-enhanced Fabry–Perot microcavity structure with a novel non-planar diaphragm,” Microelectron. Eng. 70, 102–108 (2003).
[CrossRef]

Taylor, H. F.

Verdeyn, J. T.

J. T. Verdeyn, Laser Electronics, 2nd ed. (Prentice Hall, 1989).

Wang, W. J.

W. J. Wang, D. G. Guo, R. M. Lin, X. W. Wang, “A single-chip diaphragm-type miniature Fabry–Perot pressure sensor with improved cross-sensitivity to temperature,” Meas. Sci. Technol. 15, 905–910 (2004).
[CrossRef]

W. J. Wang, R. M. Lin, T. T. Sun, D. G. Guo, Y. Ren, “Performance-enhanced Fabry–Perot microcavity structure with a novel non-planar diaphragm,” Microelectron. Eng. 70, 102–108 (2003).
[CrossRef]

Wang, X. W.

W. J. Wang, D. G. Guo, R. M. Lin, X. W. Wang, “A single-chip diaphragm-type miniature Fabry–Perot pressure sensor with improved cross-sensitivity to temperature,” Meas. Sci. Technol. 15, 905–910 (2004).
[CrossRef]

Watson, A. J.

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

Wu, J. W.

Z. M. Wu, G. Q. Xia, H. Q. Zhou, J. W. Wu, M. L. Liu, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon,” Opt. Laser Technol. 35, 1–4 (2003).
[CrossRef]

Wu, Z. M.

Z. M. Wu, G. Q. Xia, H. Q. Zhou, J. W. Wu, M. L. Liu, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon,” Opt. Laser Technol. 35, 1–4 (2003).
[CrossRef]

Z. M. Wu, G. Q. Xia, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon: a nonresonant case,” Opt. Laser Technol. 35, 123–126 (2003).
[CrossRef]

Xia, G. Q.

Z. M. Wu, G. Q. Xia, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon: a nonresonant case,” Opt. Laser Technol. 35, 123–126 (2003).
[CrossRef]

Z. M. Wu, G. Q. Xia, H. Q. Zhou, J. W. Wu, M. L. Liu, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon,” Opt. Laser Technol. 35, 1–4 (2003).
[CrossRef]

Yusuf, N. A.

Zhou, H. Q.

Z. M. Wu, G. Q. Xia, H. Q. Zhou, J. W. Wu, M. L. Liu, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon,” Opt. Laser Technol. 35, 1–4 (2003).
[CrossRef]

Appl. Opt. (6)

IEEE Photonics Technol. Lett. (1)

Y. M. Kim, D. P. Neikirk, “Micromachined Fabry–Perot cavity pressure transducer,” IEEE Photonics Technol. Lett. 7, 1471–1473 (1995).
[CrossRef]

IEEE Trans. Semicond. Manuf. (1)

E. T. Carlen, C. H. Mastrangelo, “Statistical model for spatial correlation in thin film deposition and reactive growth,” IEEE Trans. Semicond. Manuf. 11, 511–521 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. Part 1 (1)

J. Park, M. G. Kim, “High-performance fiber-optic Fabry–Perot pressure sensor with Si3N4∕SiO2∕Si3N4 diaphragm,” Jpn. J. Appl. Phys. Part 1 38, 1562–1564 (1999).
[CrossRef]

Meas. Sci. Technol. (2)

W. N. Macpherson, M. J. Gander, J. S. Barton, J. D. C. Jones, C. L. Owen, A. J. Watson, R. M. Allen, “Blast-pressure measurement with a high-bandwidth fibre optic pressure sensor,” Meas. Sci. Technol. 11, 95–102 (2000).
[CrossRef]

W. J. Wang, D. G. Guo, R. M. Lin, X. W. Wang, “A single-chip diaphragm-type miniature Fabry–Perot pressure sensor with improved cross-sensitivity to temperature,” Meas. Sci. Technol. 15, 905–910 (2004).
[CrossRef]

Microelectron. Eng. (1)

W. J. Wang, R. M. Lin, T. T. Sun, D. G. Guo, Y. Ren, “Performance-enhanced Fabry–Perot microcavity structure with a novel non-planar diaphragm,” Microelectron. Eng. 70, 102–108 (2003).
[CrossRef]

Nouv. Rev. Opt. (1)

G. S. Bhatnagar, K. Singh, B. N. Gupta, “Transmission profile of a Fabry–Perot interferometer suffering from asymmetric surface defects,” Nouv. Rev. Opt. 5, 237–240 (1974).
[CrossRef]

Opt. Eng. (1)

P. D. Atherton, N. K. Reay, J. Ring, T. R. Hicks, “Tunable Fabry–Perot filters,” Opt. Eng. 20, 806–814 (1981).
[CrossRef]

Opt. Laser Technol. (2)

Z. M. Wu, G. Q. Xia, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon: a nonresonant case,” Opt. Laser Technol. 35, 123–126 (2003).
[CrossRef]

Z. M. Wu, G. Q. Xia, H. Q. Zhou, J. W. Wu, M. L. Liu, “Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon,” Opt. Laser Technol. 35, 1–4 (2003).
[CrossRef]

Opt. Lett. (1)

Sens. Actuators, A (2)

J. Han, “Novel fabrication and characterization method of Fabry–Perot microcavity pressure sensors,” Sens. Actuators, A 75, 168–175 (1999).
[CrossRef]

J. Han, J. Y. Kim, T. S. Kim, J. S. Kim, “Performance of Fabry–Perot microcavity structures with corrugated diaphragms,” Sens. Actuators, A 79, 162–172 (2000).
[CrossRef]

Thin Solid Films (1)

H. A. Macleod, “Thin film narrow band optical filters,” Thin Solid Films 34, 335–342 (1976).
[CrossRef]

Other (8)

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1965).

K. L. Belsley, D. R. Huber, J. Goodman, “All-passive interferometric fiber-optic pressure sensor,” presented at the Annual Meeting of the Instrument Society of America, 1986, Paper 86-2801.

W. H. Quick, K. A. James, J. E. Coker, “Fiber optics sensing techniques,” presented at the First International Conference on Optical Fiber Sensors, London, April 1983.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice Hall, 1976).

F. Abeles, Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, 1967).

J. T. Verdeyn, Laser Electronics, 2nd ed. (Prentice Hall, 1989).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1968).

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic diagram of a Fabry–Perot microcavity for sensing applications, (b) simplified mathematical model used in the Gaussian-optics-based modeling.

Fig. 2
Fig. 2

Proposed Fabry–Perot microcavity structure.

Fig. 3
Fig. 3

Transmittance comparisons of different plane-wave-based models and the novel Gaussian-optics-based model on a Fabry–Perot microcavity under different incidence angles (a) 0°, (b) 20°, (c) 40°, (d) 50°.

Fig. 4
Fig. 4

Transmittance fluctuations of an unpressurized Fabry–Perot microcavity illuminated by a laser beam with different ratios of Gaussian beam waist size to cavity length ranging from 0.1 to 1000 under different incidence angles.

Fig. 5
Fig. 5

Influence of laser beam waist on resultant Fabry–Perot transmittance under different incidence angles (a) 0°, (b) 20°, (c) 40°, (d) 50°.

Fig. 6
Fig. 6

Schematic view of mirror imperfections.

Fig. 7
Fig. 7

Transmittance perturbations of a Fabry–Perot microcavity induced by (a) initial residual spherical curvature up to 50 nm (b), nonparallelism up to 50 nm , (c) surface roughness of 10, 15, and 20 nm , (d) surface roughness up to 20 nm .

Fig. 8
Fig. 8

(a) Isometric cross-sectional view of a single Fabry–Perot microcavity using an SDCD and (b) top view of a 5 × 5 Fabry–Perot microcavity array using an SDCD.

Fig. 9
Fig. 9

Schematic view of the proposed optical measurement setup.

Fig. 10
Fig. 10

Simulated and measured transmittance as a function of operating wavelength.

Fig. 11
Fig. 11

Simulated and measured (a) transmittance and (b) reflectance as a function of applied pressure under a room temperature of 25 ° C ( λ = 1300 nm ) .

Equations (55)

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( B C ) = ( i = 1 q [ cos δ i j η i sin δ i j η j sin δ i cos δ i ] ) ( 1 η s ) ,
n 0 sin θ 0 = n i sin θ i = n s sin θ s .
R = ( η 0 Y η 0 + Y ) ( η 0 Y η 0 + Y ) * = ( η 0 B C η 0 B + C ) ( η 0 B C η 0 B + C ) * ,
T = 4 η 0 Re ( η m ) ( η 0 B + C ) ( η 0 B + C ) * ,
( E 0 + E 0 ) = { i = 0 q 1 1 τ ( i ) ( i + 1 ) [ 1 ρ ( i ) ( i + 1 ) ρ ( i ) ( i + 1 ) 1 ] [ exp ( i δ i ) 0 0 exp ( i δ i ) ] } 1 τ ( q ) ( s ) [ 1 0 ρ ( q ) ( s ) 0 ] ( E s + 0 ) ,
ρ ( i ) ( i + 1 ) = n i cos θ i n i + 1 cos θ i + 1 n i cos θ i + n i + 1 cos θ i 1 ,
τ ( i ) ( i + 1 ) = 2 n i cos θ i n i cos θ i + n i + 1 cos θ i + 1 ,
n 0 sin θ 0 = n i sin θ i = n i + 1 sin θ i + 1 = n s sin θ s .
t [ t 11 t 12 t 21 t 22 ] = { i = 0 q 1 1 τ ( i ) ( i + 1 ) [ 1 ρ ( i ) ( i + 1 ) ρ ( i ) ( i + 1 ) 1 ] [ exp ( i δ i ) 0 0 exp ( i δ i ) ] } 1 τ ( q ) ( s ) [ 1 0 ρ ( q ) ( s ) 0 ] .
E s + E 0 + = 1 t 11 ,
T = n s cos θ s n 0 cos θ 0 ( 1 t 11 ) * ( 1 t 11 ) .
ρ = ρ 1 + ρ 2 exp ( 2 i δ ) 1 + ρ 1 ρ 2 exp ( 2 i δ ) , τ = τ 1 τ 2 exp ( i δ ) 1 + ρ 1 ρ 2 exp ( 2 i δ ) ,
δ = ( 2 π n d cos θ ) λ ,
ρ m = ρ m exp ( i φ ρ m ) , τ m = ρ m exp ( i φ τ m ) ,
ρ f = ρ f exp ( i φ ρ f ) , τ f = ρ f exp ( i φ τ m ) .
T = n s cos θ s n 0 cos θ 0 τ m 2 τ f 2 ( 1 ρ m ρ f ) 2 + 4 ρ m ρ f sin 2 ( φ ρ m + φ ρ f 2 δ ) ,
n 0 sin θ 0 = n g sin θ g = n s sin θ s .
E i ( x , y ) = A exp ( x 2 + y 2 ω 2 ) exp [ i π λ R b ( x 2 + y 2 ) ] .
E t ( x , y ) = p = 0 B p exp [ i ( p + 1 2 ) ψ ] exp [ ( x p Δ x ) 2 + y 2 ω 2 ] exp { j π λ R b [ ( x p Δ x ) 2 + y 2 ] } ,
B p = τ m τ f ( ρ m ρ f ) p A
ψ = ( 4 π n g g cos θ g ) λ
Δ x = 2 g tan θ g cos θ 0
ρ m = ρ m exp ( i φ ρ m ) , τ m = ρ m exp ( i φ τ m ) ,
ρ f = ρ f exp ( i φ ρ f ) , τ f = ρ f exp ( i φ τ m ) .
E t ( x , y ) = p = 0 τ ρ p A exp [ i ( φ + p ψ ) ] exp [ ( x p Δ x ) 2 + y 2 ω 2 ] exp { i π λ R b [ ( x p Δ x ) 2 + y 2 ] } ,
φ = φ τ m + φ τ f ψ 2 , ψ = φ ρ m + φ ρ f ψ ,
τ = τ m τ f , ρ = ρ m ρ f .
T = n s cos θ s n 0 cos θ 0 d y d x E t ( x , y ) 2 d y d x E i ( x , y ) 2 .
T = n s cos θ s n 0 cos θ 0 τ 2 k = 0 ρ 2 k { 1 + 2 p = 1 k cos ( 2 p ψ ) exp [ ( 2 p ) 2 Δ x 2 2 ω 0 2 ] + 2 ρ p = 0 k cos [ ( 2 p + 1 ) ψ ] exp [ ( 2 p + 1 ) 2 Δ x 2 2 ω 0 2 ] } .
T ̂ = i = 1 n T ( ψ + ψ e i ) A m i A m ,
P ( h r ) = 1 σ 2 π exp ( h r 2 2 σ r 2 ) ,
T ( d ) = T ( d h r ) P ( h r ) d h r = n s ( cos θ s ) τ 2 n 0 ( cos θ 0 ) σ r 2 π k = 0 ρ 2 k { 1 + 2 p = 1 k exp [ ( 2 p ) 2 Δ x 2 2 ω 0 2 ] cos ( 2 p ψ 8 π p cos θ g λ h r ) + 2 p = 1 k exp [ ( 2 p + 1 ) 2 Δ x 2 2 ω 0 2 ] cos [ ( 2 p + 1 ) ψ 4 π ( 2 p + 1 ) cos θ g λ h r ] } exp ( h r 2 2 σ r 2 ) d h r .
exp ( p x 2 ) sin ( q x ) d x = l = 1 ( 1 ) l + 1 l ! ( 2 l ) ! q 2 l 1 p l ,
0 exp ( p x 2 ) cos ( q x ) d x = exp ( q 2 4 p ) 2 π p ,
0 exp ( p x 2 ) d x = 1 2 π p .
T ( d ) = n s cos θ s n 0 cos θ 0 τ 2 k = 0 ρ 2 k ( 1 + 4 σ r 2 π p = 1 k exp [ ( 2 p ) 2 Δ x 2 2 ω 0 2 ] [ A 1 cos ( 2 p ψ ) + B 1 sin ( 2 p ψ ) ] + 4 ρ σ r 2 π p = 1 k exp [ ( 2 p + 1 ) 2 Δ x 2 2 ω 0 2 ] { A 2 cos [ ( 2 p + 1 ) ψ ] + B 2 sin [ ( 2 p + 1 ) ψ ] } ) ,
A 1 = 1 2 exp { [ ( 8 π p σ r cos θ g ) λ ] 2 2 } ,
B 1 = l = 1 ( 1 ) l + 1 l ! ( 2 l ) ! [ ( 8 π p cos θ g ) λ ] 2 l 1 ( 2 σ r 2 ) l ,
A 2 = 1 2 exp ( { [ 4 π ( 2 p + 1 ) σ cos θ r ] λ } 2 2 ) ,
B 2 = l = 1 ( 1 ) l + 1 l ! ( 2 l ) ! { [ 4 π ( 2 p + 1 ) cos θ g ] λ } 2 l 1 ( 2 σ r 2 ) l .
N p p = 0 τ ρ p A exp [ i ( φ + p ψ ) ] exp [ ( x p Δ x ) 2 + y 2 ω 2 ] exp { i π λ R [ ( x p Δ x ) 2 + y 2 ] } ,
E t ( x , y ) 2 = p = 0 q = 0 N p N q * .
E t ( x , y ) 2 = k = 0 ( v = 0 2 k N 2 k v N v * + v = 0 2 k + 1 N 2 k + 1 v N v * ) .
E t ( x , y ) 2 = k = 0 [ N k N k * + p = 1 k ( N k p N k + p * + N k + p N k p * ) + p = 0 k ( N k + 1 + p N k p * + N k p N k + p + 1 * ) ] .
N k N k * = τ 2 ρ 2 p A 2 exp { 2 [ ( x p Δ x ) 2 + y 2 ] ω 2 } ,
N k p N k + p * + N k + p N k p * = 2 τ 2 ρ 2 k A 2 exp { [ x ( k p ) Δ x ] 2 + [ x ( k + p ) Δ x ] 2 + 2 y 2 ω 2 } × cos ( 2 p ψ + π { [ x ( k p ) Δ x ] 2 [ x ( k + p ) Δ x ] 2 } λ ρ ) ,
N k + 1 + p N k p * + N k p N k + 1 + p * = 2 τ 2 ρ 2 k + 1 A 2 exp { [ x ( k + 1 + p ) Δ x ] 2 + [ x ( k p ) Δ x ] 2 + 2 y 2 ω 2 } × cos ( ( 2 p + 1 ) ψ + π { [ x ( k p ) Δ x ] 2 [ x ( k + 1 + p ) Δ x ] 2 } λ ρ ) .
exp ( a x 2 ) cos ( 2 b x ) d x = π a exp ( b 2 a ) ,
exp ( a x 2 ) sin ( 2 b x ) d x = 0 ,
exp ( 2 x 2 + y 2 ω 2 ) d x d y = ω 2 π 2 ,
N k N k * d x d y = τ 2 ρ 2 k A 2 ω 2 π 2 ,
( N k p N k + p * + N k + p N k p * ) d x d y = 2 τ 2 ρ 2 k A 2 cos ( 2 p ψ ) exp { ( 2 p ) 2 Δ x 2 2 [ 1 ω 2 + ( τ ω ρ λ ) 2 ] } ,
( N k + 1 + p N k p * + N k + 1 + p N k p * ) d x d y = 2 τ 2 ρ 2 k + 1 A 2 cos [ ( 2 p + 1 ) ψ ] exp { ( 2 p + 1 ) 2 Δ x 2 2 [ 1 ω 2 + ( π ω ρ λ ) 2 ] } .
1 ω 0 2 = 1 ω 2 + ( π ω ρ λ ) 2 ,
T = n s cos θ s n 0 cos θ 0 τ 2 k = 0 ρ 2 k { 1 + 2 p = 1 k cos ( 2 p ψ ) exp [ ( 2 p ) 2 Δ x 2 2 ω 0 2 ] + 2 ρ p = 0 k cos [ ( 2 p + 1 ) ψ ] exp [ ( 2 p + 1 ) 2 Δ x 2 2 ω 0 2 ] } .

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