Abstract

I present a new derivation of the analytic form for the phase shift near resonance and the optical penetration length upon reflection from a distributed dielectric mirror consisting of a quarter-wave stack. The requirement of proper termination to achieve high reflectivity is suspended to investigate large optical penetration depths. Separate equations, derived for N and N + 1/2 layer pairs, are convenient for the design of tunable Fabry-Perot filters with a specified tuning range. The analysis is also applicable to distributed Bragg reflectors, vertical-cavity surface-emitting lasers, and resonant photodiodes. I show that the penetration length can sharply reduce the overly broad free spectral range of an ultrathin Fabry-Perot filter that might be useful in applications such as tunable wavelength filters for wavelength division multiplexing applications. The results also demonstrate regimes of zero dispersion and of superluminal reflection in the dielectric mirrors, which are of particular interest in photonic bandgap structures.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. R. Mallinson, J. H. Jerman, “Miniature micromachined Fabry-Perot interferometers in silicon,” Electron. Lett. 23, 1041–1043 (1987).
    [CrossRef]
  2. E. C. Vail, G. S. Li, W. Yuen, C. J. Chang-Hasnain, “GaAs micromachined widely tunable Fabry-Perot filters,” Electron. Lett. 31, 228–229 (1995).
    [CrossRef]
  3. A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8, 393–395 (1996).
    [CrossRef]
  4. P. Tayebati, P. Wang, D. Vakhshoori, R. N. Sacks, “Microelectromechanical tunable filters with 0.47 nm linewidth and 70 nm tuning range,” Electron. Lett. 34, 76–78 (1998).
    [CrossRef]
  5. D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
    [CrossRef]
  6. B. G. Kim, E. Garmire, “Comparison between the matrix method and the coupled-wave method in the analysis of Bragg reflector structures,” J. Opt. Soc. Am. A 9, 132–136 (1992).
    [CrossRef]
  7. B. G. Kim, E. Garmire, “Effect of front-facet reflections on the reflectivity of Bragg reflectors,” Opt. Lett. 16, 1065–1067 (1991).
    [CrossRef] [PubMed]
  8. L. R. Brovelli, U. Keller, “Simple analytical expressions for the reflectivity and penetration depth of a Bragg mirror between arbitrary media,” Opt. Commun. 116, 343–350 (1995).
    [CrossRef]
  9. S. Longhi, M. Marano, P. Laporta, M. Belmonte, P. Crespi, “Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap,” Phys. Rev. E 65, 045602 (2002).
    [CrossRef]
  10. E. Garmire, “Optical nonlinearities in semiconductors,” in Nonlinear Optics in Semiconductors I, E. Garmire, A. Kost, eds. (Academic, New York, 1999), p. 140.
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 52–70.
  12. Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
    [CrossRef]

2002

S. Longhi, M. Marano, P. Laporta, M. Belmonte, P. Crespi, “Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap,” Phys. Rev. E 65, 045602 (2002).
[CrossRef]

1998

P. Tayebati, P. Wang, D. Vakhshoori, R. N. Sacks, “Microelectromechanical tunable filters with 0.47 nm linewidth and 70 nm tuning range,” Electron. Lett. 34, 76–78 (1998).
[CrossRef]

1996

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8, 393–395 (1996).
[CrossRef]

1995

L. R. Brovelli, U. Keller, “Simple analytical expressions for the reflectivity and penetration depth of a Bragg mirror between arbitrary media,” Opt. Commun. 116, 343–350 (1995).
[CrossRef]

E. C. Vail, G. S. Li, W. Yuen, C. J. Chang-Hasnain, “GaAs micromachined widely tunable Fabry-Perot filters,” Electron. Lett. 31, 228–229 (1995).
[CrossRef]

1994

Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
[CrossRef]

1992

D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[CrossRef]

B. G. Kim, E. Garmire, “Comparison between the matrix method and the coupled-wave method in the analysis of Bragg reflector structures,” J. Opt. Soc. Am. A 9, 132–136 (1992).
[CrossRef]

1991

1987

S. R. Mallinson, J. H. Jerman, “Miniature micromachined Fabry-Perot interferometers in silicon,” Electron. Lett. 23, 1041–1043 (1987).
[CrossRef]

Babic, D. I.

D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[CrossRef]

Belmonte, M.

S. Longhi, M. Marano, P. Laporta, M. Belmonte, P. Crespi, “Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap,” Phys. Rev. E 65, 045602 (2002).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 52–70.

Brovelli, L. R.

L. R. Brovelli, U. Keller, “Simple analytical expressions for the reflectivity and penetration depth of a Bragg mirror between arbitrary media,” Opt. Commun. 116, 343–350 (1995).
[CrossRef]

Chang-Hasnain, C. J.

E. C. Vail, G. S. Li, W. Yuen, C. J. Chang-Hasnain, “GaAs micromachined widely tunable Fabry-Perot filters,” Electron. Lett. 31, 228–229 (1995).
[CrossRef]

Chen, L.

Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
[CrossRef]

Corzine, S. W.

D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[CrossRef]

Crespi, P.

S. Longhi, M. Marano, P. Laporta, M. Belmonte, P. Crespi, “Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap,” Phys. Rev. E 65, 045602 (2002).
[CrossRef]

Garmire, E.

Haronian, D.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8, 393–395 (1996).
[CrossRef]

Hu, K.

Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
[CrossRef]

Jerman, J. H.

S. R. Mallinson, J. H. Jerman, “Miniature micromachined Fabry-Perot interferometers in silicon,” Electron. Lett. 23, 1041–1043 (1987).
[CrossRef]

Karim, Z.

Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
[CrossRef]

Keller, U.

L. R. Brovelli, U. Keller, “Simple analytical expressions for the reflectivity and penetration depth of a Bragg mirror between arbitrary media,” Opt. Commun. 116, 343–350 (1995).
[CrossRef]

Kim, B. G.

Kyriakakis, C.

Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
[CrossRef]

Laporta, P.

S. Longhi, M. Marano, P. Laporta, M. Belmonte, P. Crespi, “Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap,” Phys. Rev. E 65, 045602 (2002).
[CrossRef]

Li, G. S.

E. C. Vail, G. S. Li, W. Yuen, C. J. Chang-Hasnain, “GaAs micromachined widely tunable Fabry-Perot filters,” Electron. Lett. 31, 228–229 (1995).
[CrossRef]

Lo, Y. H.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8, 393–395 (1996).
[CrossRef]

Longhi, S.

S. Longhi, M. Marano, P. Laporta, M. Belmonte, P. Crespi, “Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap,” Phys. Rev. E 65, 045602 (2002).
[CrossRef]

Madhukar, A.

Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
[CrossRef]

Mallinson, S. R.

S. R. Mallinson, J. H. Jerman, “Miniature micromachined Fabry-Perot interferometers in silicon,” Electron. Lett. 23, 1041–1043 (1987).
[CrossRef]

Marano, M.

S. Longhi, M. Marano, P. Laporta, M. Belmonte, P. Crespi, “Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap,” Phys. Rev. E 65, 045602 (2002).
[CrossRef]

Mozdy, E.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8, 393–395 (1996).
[CrossRef]

Sacks, R. N.

P. Tayebati, P. Wang, D. Vakhshoori, R. N. Sacks, “Microelectromechanical tunable filters with 0.47 nm linewidth and 70 nm tuning range,” Electron. Lett. 34, 76–78 (1998).
[CrossRef]

Tanguay, A. R.

Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
[CrossRef]

Tayebati, P.

P. Tayebati, P. Wang, D. Vakhshoori, R. N. Sacks, “Microelectromechanical tunable filters with 0.47 nm linewidth and 70 nm tuning range,” Electron. Lett. 34, 76–78 (1998).
[CrossRef]

Tran, A. T. T. D.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8, 393–395 (1996).
[CrossRef]

Vail, E. C.

E. C. Vail, G. S. Li, W. Yuen, C. J. Chang-Hasnain, “GaAs micromachined widely tunable Fabry-Perot filters,” Electron. Lett. 31, 228–229 (1995).
[CrossRef]

Vakhshoori, D.

P. Tayebati, P. Wang, D. Vakhshoori, R. N. Sacks, “Microelectromechanical tunable filters with 0.47 nm linewidth and 70 nm tuning range,” Electron. Lett. 34, 76–78 (1998).
[CrossRef]

Wang, P.

P. Tayebati, P. Wang, D. Vakhshoori, R. N. Sacks, “Microelectromechanical tunable filters with 0.47 nm linewidth and 70 nm tuning range,” Electron. Lett. 34, 76–78 (1998).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 52–70.

Yuen, W.

E. C. Vail, G. S. Li, W. Yuen, C. J. Chang-Hasnain, “GaAs micromachined widely tunable Fabry-Perot filters,” Electron. Lett. 31, 228–229 (1995).
[CrossRef]

Zhu, Z. H.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8, 393–395 (1996).
[CrossRef]

Appl. Phys. Lett.

Z. Karim, C. Kyriakakis, A. R. Tanguay, K. Hu, L. Chen, A. Madhukar, “Externally deposited phase-compensating dielectric mirrors for asymmetric Fabry-Perot cavity tuning,” Appl. Phys. Lett. 64, 2913–2915 (1994).
[CrossRef]

Electron. Lett.

S. R. Mallinson, J. H. Jerman, “Miniature micromachined Fabry-Perot interferometers in silicon,” Electron. Lett. 23, 1041–1043 (1987).
[CrossRef]

E. C. Vail, G. S. Li, W. Yuen, C. J. Chang-Hasnain, “GaAs micromachined widely tunable Fabry-Perot filters,” Electron. Lett. 31, 228–229 (1995).
[CrossRef]

P. Tayebati, P. Wang, D. Vakhshoori, R. N. Sacks, “Microelectromechanical tunable filters with 0.47 nm linewidth and 70 nm tuning range,” Electron. Lett. 34, 76–78 (1998).
[CrossRef]

IEEE J. Quantum Electron.

D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[CrossRef]

IEEE Photon. Technol. Lett.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8, 393–395 (1996).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

L. R. Brovelli, U. Keller, “Simple analytical expressions for the reflectivity and penetration depth of a Bragg mirror between arbitrary media,” Opt. Commun. 116, 343–350 (1995).
[CrossRef]

Opt. Lett.

Phys. Rev. E

S. Longhi, M. Marano, P. Laporta, M. Belmonte, P. Crespi, “Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap,” Phys. Rev. E 65, 045602 (2002).
[CrossRef]

Other

E. Garmire, “Optical nonlinearities in semiconductors,” in Nonlinear Optics in Semiconductors I, E. Garmire, A. Kost, eds. (Academic, New York, 1999), p. 140.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 52–70.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Geometry for symmetric FP cavity of physical length L c with an internal refractive index of n i and identical QWS mirrors each consisting of N quarter-wave layer pairs of alternating refractive indices n 1 and n 2. The final refractive index outside the QWS is n f . The internal electric field observes a reflection coefficient of r exp(iφ) at each QWS mirror. All the refractive indices can have any values, and the extension to an asymmetric cavity is straightforward.

Fig. 2
Fig. 2

Phase of the total reflection coefficient φ (solid curve) and reflectivity R (crosses) as a function of fractional wave-vector deviation δk/ k o (or fractional wavelength deviation δλ/λo). The peak value of the reflectivity shown here is R o = 0.988. The graph is drawn assuming the number of layer pairs N = 5, and the refractive indices are n 1 = 2.2, n 2 = 1.4, n i = 1 and n f = 3.45.

Fig. 3
Fig. 3

Normalized optical mirror penetration length L M (in units of λo/4) for several different numbers of layer pairs N and for two cases in which n f differs. Also shown are typical reflectivities for both cases RA and RB (for N = 6 layer pairs). Light is incident from the air: (a) HL configuration with n 1 = 1.38 and a varying ratio of n 2/n 1 and (b) LH configuration with n 2 = 1.38 and a varying ratio of n 1/n 2.

Fig. 4
Fig. 4

Normalized optical mirror length for n 1 = 1.38 and a varying ratio of n 2/n 1 in the LH configuration, with a substrate refractive index of n f n s = 3.45 and an initial refractive index n i = 1. Examples of N = 4, 6, and 11 layer pairs are shown. Also shown are the respective reflectivities R. This graph shows regions of zero and negative optical penetration length, as well as a region of large positive mirror length. The poles in the function are located at R = 0. The negative L M corresponds to superluminal reflection, which can occur for small reflectivities.

Fig. 5
Fig. 5

Normalized optical mirror penetration length L M ′ for N = 6 layer pairs, comparing N layer pairs (heavy line) and N + 1/2 layer pairs (ticked line). The scaled reflectivity is also shown as thin curves: (a) the LH configuration in which N + 1/2 layer pairs provide large reflectivity and remove the pole and (b) the HL configuration in which N layer pairs produce high reflectivity, and an additional layer reduces the reflectivity but introduces a pole.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

2kmLOPL+2φm=m2π,
ΔkFSR=πLOPL+φ/k.
ΔkFSR=π/Leff.
Leff=LOPL+LMa+LMb =LOPL+1/2φa/k+1/2φb/k.
λ-=2πkm-π/2Leff-1=1/λm+1/4Leff-1,
λ+=2πkm+π/2Leff-1=1/λm-1/4Leff-1.
λ+-λ-=λmλm/2Leff1-λm/4Leff2-1.
MP=M1M2=cos ϑ1sin ϑ1/n1-n1 sin ϑ1cos ϑ1cos ϑ2sin ϑ2/n2-n2 sin ϑ2cos ϑ2,
MN=M1M2N.
MN=M00M01M10M11,
r=iM00+nfM01ni-infM11-M10iM00+nfM01ni+infM11-M10,
t=-2ininfiM00+nfM01ni+infM11-M10.
R=|r|2=ninfM01+M102+M00ni-nfM112ninfM01-M102+M00ni+nfM112,
tan φ=-2niM00M10+nf2M11M01ninfM012-M102+M00ni2-nfM112.
M=-δj1-δj2/2/n-n1-δj2/2-δj.
MP=-n2/n1-δ1/n2-δ2/n1δ2n1+δ1n2-n1/n2.
ko1/n1n1/k=ko lnn1/k =λo lnn1/λ,
δ1=1+λ lnn1/λπ/2δk/ko.
MP=-1abδ-cδd,
M2=n2/n121/n1+1/n2n2/n1+n1/n2δ-n1+n2n2/n1+n1/n2δn1/n22.
tan φN=2niaNcN-nf2dNbNni2a2N-nf2d2N δ.
δφ=2niaNcN-nf2dNbNni2a2N-nf2d2Nδk/koπ/2.
bN=baN-1+aN-2d+aN-3d2adN-2+dN-1bFN,
cN=caN-1+aN-2d+aN-3d2adN-2+dN-1cFN,
FN=aN-1+aN-3+aN-5a-N+3+a-N+1,
aN-1FN=a2N-1+a2N-2+1=a2N-1/a2-1.
δφ/δϑ=2 n1n21-nf2/n1n2a-2N1-a-2Nnin2-n11-nf/ni2a-4N,
LMδφ/δϑλo/8,
LM=n1n21-nf2/n1n2n1/n22N1-n1/n22Nnin2-n11-nf/ni2n1/n24Nλo/4.
niLT=c/21/2νn1/nin2n2-n1 ×1-nf/n22n1/n2m-11-n1/n2m1-n1/ni2nf/n22n1/n22m-2,
LLH=n1n2nin2-n11-nf2/n1n2+1n1/n22Nλo/4,
LHL=nin1-n21-n1n2/nf2+1n2/n12Nλo/4,
RN=niaN-nfdN2niaN+nfdN2=1-nf/nin1/n22N21+nf/nin1/n22N2,
TN=1-RN=4ni/nfn2/n12N+nf/nin1/n22N+2.
THL=4ni/nfn2/n12N,
TLH=4nf/nin1/n22N,
δkko<λo16LM .
MN+1/2=-1N+1δaN+bNn1-aN/n1n1dNδdN+cN/n1,
LN+1/2=1-n1n2nf2n1n22N+1/21-n1n22N+1/21-n2n1nfni2n1n24N+1/2×n1n2nin2-n1λo4.
RN+1/2=1-n2n1/nfnin1/n22N+1/221+n2n1/nfnin1/n22N+1/22.
Leff=LOPL+LM1+LM2.

Metrics