Abstract

We present a method for tailoring the lasing spectrum of a Fabry–Perot laser through the introduction of a low density of weakly reflective features along the length of the optical cavity. Using a transfer-matrix approach, the positions of the features are obtained from a self-consistent solution of the corresponding inverse problem at first order in the effective index step introduced. Theoretical examples are given describing how a single-mode laser cavity and a two-color laser are designed. Experimental measurements show that the method enables the realization of single-mode semiconductor lasers with high spectral purity at a predetermined wavelength. We also demonstrate that cavities designed according to our first-order prescription are robust at larger values of the effective index step where effects at second order cannot be neglected.

© 2006 Optical Society of America

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References

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  1. H. Kogelnik and C. V. Shank, "Coupled wave theory of distributed feedback laser diodes," J. Appl. Phys. 43, 2327-2335 (1972).
    [Crossref]
  2. G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, 1986).
  3. A. Talneau, J. Charil, and A. Ougazzaden, "Multiple distributed feedback operation at 1.55μm with uniform output powers in a single laser diode," Appl. Phys. Lett. 75, 600-602 (1999).
    [Crossref]
  4. S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
    [Crossref]
  5. D. A. Kozlowski, J. S. Young, J. M. C. England, and R. G. S. Plumb, "Singlemode 1.3μm Fabry-Pérot lasers by mode suppression," Electron. Lett. 31, 648-650 (1995).
    [Crossref]
  6. B. Corbett and D. McDonald, "Ridge waveguide single longitudinal mode Fabry-Pérot lasers by modal perturbation," Electron. Lett. 31, 2181-2182 (1995).
    [Crossref]
  7. R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999).
    [Crossref]
  8. S. O'Brien and E. P. O'Reilly, "Theory of improved spectral purity in index patterned Fabry-Pérot lasers," Appl. Phys. Lett. 86, 201101 (2005).
    [Crossref]
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).
  10. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).
  11. D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
    [Crossref]
  12. S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
    [Crossref]
  13. M. Tani, O. Morikawa, S. Matsuura, and M. Hangyo, "Generation of terahertz radiation by photomixing with dual- and multiple-mode lasers," Semicond. Sci. Technol. 20, 151-163 (2005).
    [Crossref]

2005 (2)

S. O'Brien and E. P. O'Reilly, "Theory of improved spectral purity in index patterned Fabry-Pérot lasers," Appl. Phys. Lett. 86, 201101 (2005).
[Crossref]

M. Tani, O. Morikawa, S. Matsuura, and M. Hangyo, "Generation of terahertz radiation by photomixing with dual- and multiple-mode lasers," Semicond. Sci. Technol. 20, 151-163 (2005).
[Crossref]

2004 (1)

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

2001 (1)

D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
[Crossref]

2000 (1)

S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
[Crossref]

1999 (2)

A. Talneau, J. Charil, and A. Ougazzaden, "Multiple distributed feedback operation at 1.55μm with uniform output powers in a single laser diode," Appl. Phys. Lett. 75, 600-602 (1999).
[Crossref]

R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999).
[Crossref]

1995 (2)

D. A. Kozlowski, J. S. Young, J. M. C. England, and R. G. S. Plumb, "Singlemode 1.3μm Fabry-Pérot lasers by mode suppression," Electron. Lett. 31, 648-650 (1995).
[Crossref]

B. Corbett and D. McDonald, "Ridge waveguide single longitudinal mode Fabry-Pérot lasers by modal perturbation," Electron. Lett. 31, 2181-2182 (1995).
[Crossref]

1972 (1)

H. Kogelnik and C. V. Shank, "Coupled wave theory of distributed feedback laser diodes," J. Appl. Phys. 43, 2327-2335 (1972).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, 1986).

Avrutin, E. A.

D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).

Brundermann, E.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Charil, J.

A. Talneau, J. Charil, and A. Ougazzaden, "Multiple distributed feedback operation at 1.55μm with uniform output powers in a single laser diode," Appl. Phys. Lett. 75, 600-602 (1999).
[Crossref]

Coleman, J. J.

S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
[Crossref]

Corbett, B.

B. Corbett and D. McDonald, "Ridge waveguide single longitudinal mode Fabry-Pérot lasers by modal perturbation," Electron. Lett. 31, 2181-2182 (1995).
[Crossref]

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, 1986).

England, J. M. C.

D. A. Kozlowski, J. S. Young, J. M. C. England, and R. G. S. Plumb, "Singlemode 1.3μm Fabry-Pérot lasers by mode suppression," Electron. Lett. 31, 648-650 (1995).
[Crossref]

Feced, R.

R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999).
[Crossref]

Hangyo, M.

M. Tani, O. Morikawa, S. Matsuura, and M. Hangyo, "Generation of terahertz radiation by photomixing with dual- and multiple-mode lasers," Semicond. Sci. Technol. 20, 151-163 (2005).
[Crossref]

Havenith, M.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Hoffmann, M.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Hoffmann, S.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Huber, A. E.

S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
[Crossref]

Kira, M.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Koch, S. W.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Kogelnik, H.

H. Kogelnik and C. V. Shank, "Coupled wave theory of distributed feedback laser diodes," J. Appl. Phys. 43, 2327-2335 (1972).
[Crossref]

Kozlowski, D. A.

D. A. Kozlowski, J. S. Young, J. M. C. England, and R. G. S. Plumb, "Singlemode 1.3μm Fabry-Pérot lasers by mode suppression," Electron. Lett. 31, 648-650 (1995).
[Crossref]

Marsh, J. H.

D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
[Crossref]

Matsuura, S.

M. Tani, O. Morikawa, S. Matsuura, and M. Hangyo, "Generation of terahertz radiation by photomixing with dual- and multiple-mode lasers," Semicond. Sci. Technol. 20, 151-163 (2005).
[Crossref]

Matus, M.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

McDonald, D.

B. Corbett and D. McDonald, "Ridge waveguide single longitudinal mode Fabry-Pérot lasers by modal perturbation," Electron. Lett. 31, 2181-2182 (1995).
[Crossref]

McDougall, S. D.

D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
[Crossref]

Moloney, J. V.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Morikawa, O.

M. Tani, O. Morikawa, S. Matsuura, and M. Hangyo, "Generation of terahertz radiation by photomixing with dual- and multiple-mode lasers," Semicond. Sci. Technol. 20, 151-163 (2005).
[Crossref]

Moskalenko, A. S.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Muriel, M. A.

R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999).
[Crossref]

O'Brien, S.

S. O'Brien and E. P. O'Reilly, "Theory of improved spectral purity in index patterned Fabry-Pérot lasers," Appl. Phys. Lett. 86, 201101 (2005).
[Crossref]

O'Reilly, E. P.

S. O'Brien and E. P. O'Reilly, "Theory of improved spectral purity in index patterned Fabry-Pérot lasers," Appl. Phys. Lett. 86, 201101 (2005).
[Crossref]

Ougazzaden, A.

A. Talneau, J. Charil, and A. Ougazzaden, "Multiple distributed feedback operation at 1.55μm with uniform output powers in a single laser diode," Appl. Phys. Lett. 75, 600-602 (1999).
[Crossref]

Plumb, R. G. S.

D. A. Kozlowski, J. S. Young, J. M. C. England, and R. G. S. Plumb, "Singlemode 1.3μm Fabry-Pérot lasers by mode suppression," Electron. Lett. 31, 648-650 (1995).
[Crossref]

Roh, S. D.

S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
[Crossref]

Saito, S.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Sakai, K.

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Shank, C. V.

H. Kogelnik and C. V. Shank, "Coupled wave theory of distributed feedback laser diodes," J. Appl. Phys. 43, 2327-2335 (1972).
[Crossref]

Street, M. W.

D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
[Crossref]

Swint, R. B.

S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
[Crossref]

Talneau, A.

A. Talneau, J. Charil, and A. Ougazzaden, "Multiple distributed feedback operation at 1.55μm with uniform output powers in a single laser diode," Appl. Phys. Lett. 75, 600-602 (1999).
[Crossref]

Tani, M.

M. Tani, O. Morikawa, S. Matsuura, and M. Hangyo, "Generation of terahertz radiation by photomixing with dual- and multiple-mode lasers," Semicond. Sci. Technol. 20, 151-163 (2005).
[Crossref]

Thayne, I. G.

D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Woo, J. S.

S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
[Crossref]

Yanson, D. A.

D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
[Crossref]

Yeoh, T. S.

S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
[Crossref]

Young, J. S.

D. A. Kozlowski, J. S. Young, J. M. C. England, and R. G. S. Plumb, "Singlemode 1.3μm Fabry-Pérot lasers by mode suppression," Electron. Lett. 31, 648-650 (1995).
[Crossref]

Zervas, M. N.

R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999).
[Crossref]

Appl. Phys. Lett. (3)

A. Talneau, J. Charil, and A. Ougazzaden, "Multiple distributed feedback operation at 1.55μm with uniform output powers in a single laser diode," Appl. Phys. Lett. 75, 600-602 (1999).
[Crossref]

S. O'Brien and E. P. O'Reilly, "Theory of improved spectral purity in index patterned Fabry-Pérot lasers," Appl. Phys. Lett. 86, 201101 (2005).
[Crossref]

S. Hoffmann, M. Hoffmann, E. Brundermann, M. Havenith, M. Matus, J. V. Moloney, A. S. Moskalenko, M. Kira, S. W. Koch, S. Saito, and K. Sakai, "Four-wave mixing and direct terahertz emission with two-color semiconductor lasers," Appl. Phys. Lett. 84, 3585-3587 (2004).
[Crossref]

Electron. Lett. (2)

D. A. Kozlowski, J. S. Young, J. M. C. England, and R. G. S. Plumb, "Singlemode 1.3μm Fabry-Pérot lasers by mode suppression," Electron. Lett. 31, 648-650 (1995).
[Crossref]

B. Corbett and D. McDonald, "Ridge waveguide single longitudinal mode Fabry-Pérot lasers by modal perturbation," Electron. Lett. 31, 2181-2182 (1995).
[Crossref]

IEEE J. Quantum Electron. (2)

R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999).
[Crossref]

D. A. Yanson, M. W. Street, S. D. McDougall, I. G. Thayne, J. H. Marsh, and E. A. Avrutin, "Ultrafast harmonic mode-locking of monolithic compound-cavity laser diodes incorporating photonic bandgap reflectors," IEEE J. Quantum Electron. 38, 1-11 (2001).
[Crossref]

IEEE Photon. Technol. Lett. (1)

S. D. Roh, T. S. Yeoh, R. B. Swint, A. E. Huber, J. S. Woo, and J. J. Coleman, "Dual-wavelength InGaAs-GaAs ridge waveguide distributed Bragg reflector lasers with tunable mode speration," IEEE Photon. Technol. Lett. 12, 1307-1309 (2000).
[Crossref]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, "Coupled wave theory of distributed feedback laser diodes," J. Appl. Phys. 43, 2327-2335 (1972).
[Crossref]

Semicond. Sci. Technol. (1)

M. Tani, O. Morikawa, S. Matsuura, and M. Hangyo, "Generation of terahertz radiation by photomixing with dual- and multiple-mode lasers," Semicond. Sci. Technol. 20, 151-163 (2005).
[Crossref]

Other (3)

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, 1986).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).

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Figures (11)

Fig. 1
Fig. 1

One-dimensional model of a FP laser cavity of length L c and including N index steps. The cavity effective index is n 1 while the additional features providing the index step have effective index n 2 . All cavity sections are numbered 2 N + 1 i 1 beginning on the left. The additional features are also numbered with an index j. The matrix T relates the left- and right-moving fields inside the cavity at the cavity mirrors. The complex mirror reflectivities are r 1 and r 2 as shown.

Fig. 2
Fig. 2

Index step feature (region 2) of length L, centered at position z 0 with refractive index n 2 . The surrounding regions 1 and 3 have refractive index n 1 . The matrix F relates the right- and left-moving electric fields after the right interface and before the left interface as shown.

Fig. 3
Fig. 3

(a) Threshold gain of modes in wavenumber space in the ideal single-mode case. The parameters are a = 20 modes and τ = 0.036 . (b) Fourier transform (which determines the feature density function) corresponding to the threshold gain spectrum of (a).

Fig. 4
Fig. 4

Upper panel: Laser cavity schematic indicating the locations of the reflective features along one side of the device center in the single-mode case for r 1 = r 2 . (a) Feature density function that we approximate. (b) Calculated threshold gain of modes for the laser cavity schematically pictured in the upper panel. The numerical value of the index step Δ n = 0.005 . The horizontal line is at the value of the mirror losses of the plain cavity.

Fig. 5
Fig. 5

Schematic illustration of the subcavities considered in our model of a FP laser cavity with a spatially varying refractive index. The complex mirror reflectivities are such that φ 1 = φ 2 = 0 and the feature length is assumed to be quarter-wave (QW) such that Δ n sin θ 2 j < 0 . The upper graph shows the FP resonance in the cavity of length L c with quarter-wave features on either side of the device center. The lower graph shows the half-wave (HW) resonance in the long subcavity formed by the rightmost feature and the consequent quarter-wave resonance formed in the shorter subcavity. The subcavity between the two features is then half-wave.

Fig. 6
Fig. 6

Upper panel: Laser cavity schematic indicating the locations of the reflective features along one side of the device center in the two-color case with r 1 > r 2 . (a) Feature density function that we approximate. The horizontal line is at the zero of the function. (b) Calculated threshold gain of modes for the laser cavity schematically pictured in the upper panel. The numerical value of the index step Δ n = 0.005 . The horizontal line is at the value of the mirror losses of the plain cavity.

Fig. 7
Fig. 7

(a) Feature density function that we approximate. Inset: Laser cavity schematic indicating the locations of the reflective features. (b) Calculated threshold gain of modes for the laser cavity schematically pictured in the inset of (a). The horizontal line is at the value of the mirror losses of the plain cavity.

Fig. 8
Fig. 8

(a) Laser spectrum of the index-patterned device of Fig. 7 at twice the threshold. (b) Spectrum at twice the threshold of an equivalent FP laser.

Fig. 9
Fig. 9

(a) Below the threshold laser spectrum of the index-patterned device of Fig. 7. (b) Below the threshold spectrum of an equivalent FP laser.

Fig. 10
Fig. 10

Calculated threshold gain of modes for the laser cavity schematically pictured in the upper panel of Fig. 4. The numerical value of the index step is Δ n = 0.015 . The horizontal line is at the value of the mirror losses of the plain cavity. (b) Contributions to the threshold gain of modes at first order in the index step (filled circles) and at second order where the change in threshold is given by the square of the cavity resonance shift (open circles).

Fig. 11
Fig. 11

(a) Cavity resonance shift of modes for the laser cavity schematically pictured in the upper panel of Fig. 4. The numerical value of the index step is Δ n = 0.015 . (b) Contributions to the calculated threshold gain at second order, γ m ( 2 ) , proportional to the quantity γ m ( 1 ) (open circles) and given by the coupling between pairs of features ( j , k ) (filled circles).

Tables (1)

Tables Icon

Table 1 Device Parameters for the Theoretical Example of Figs. 4, 6, 10

Equations (53)

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[ E + ( z 0 L 2 ϵ ) E ( z 0 L 2 ϵ ) ] = F [ E + ( z 0 + L 2 + ϵ ) E ( z 0 + L 2 + ϵ ) ] ,
F = T 12 P ( θ 2 ) T 23 ,
T i j = ( t i j ) 1 [ 1 r i j r i j 1 ] ,
t i j = 2 n i n i + n j , r i j = n i n j n i + n j ,
P ( θ i ) = [ exp ( i θ i ) 0 0 exp ( i θ i ) ] .
F j = P ( θ 2 j ) q sin θ 2 j σ y + i [ 1 q + ] sin θ 2 j σ z ,
σ y = [ 0 i i 0 ] , σ z = [ 1 0 0 1 , ] ,
[ E + ( ϵ ) E ( ϵ ) ] = T [ E + ( L c ϵ ) E ( L c ϵ ) ] .
T = P ( θ 1 ) F 1 P ( θ 3 ) F N 1 P ( θ 2 N 1 ) F N P ( θ 2 N + 1 ) .
n 1 = n ,
n 2 = n + Δ n .
q + = 1 2 ( n n + Δ n + n + Δ n n ) = 1 + 1 2 [ ( Δ n n ) 2 ] ,
q = 1 2 ( n n + Δ n n + Δ n n ) = Δ n n + 1 2 [ ( Δ n n ) 2 ] .
F j = P ( θ 2 j ) + Δ n n sin θ 2 j σ y 1 2 ( Δ n n ) 2 sin θ 2 j ( σ y + i σ z ) .
σ y P ( θ ) = [ P ( θ ) ] 1 σ y
T = P ( Σ θ i ) + ( Δ n n ) 2 k > j = 1 N sin θ 2 j sin θ 2 k P 0 j [ P j k ] 1 P k ( N + 1 ) + [ Δ n n 1 2 ( Δ n n ) 2 ] j = 1 N sin θ 2 j P 0 j [ P j ( N + 1 ) ] 1 σ y 1 2 ( Δ n n ) 2 j = 1 N sin θ 2 j P 0 j P j ( N + 1 ) i σ z .
P j k = P ( i = 2 j + 1 2 k 1 θ i ) .
μ [ r 1 1 ] = T [ 1 r 2 ] ,
T 11 T 22 r 1 r 2 = T 21 r 1 T 12 r 2 .
1 = r 1 r 2 exp ( 2 i Σ θ i ) [ 1 + ( Δ n n ) 2 k > j = 1 N sin θ 2 j sin θ 2 k exp ( 2 i ϕ jk ) ] + i Δ n n j sin θ 2 j [ r 1 exp ( 2 i ϕ j ) + r 2 exp ( 2 i ϕ j + ) ] ( Δ n n ) 2 k > j = 1 N sin θ 2 j sin θ 2 k exp ( 2 i ϕ j k ) .
1 r 1 r 2 exp ( 2 i Σ θ i ) = 0 .
λ m = 2 n L c m ,
γ m ( 0 ) = 1 L c log 1 r 1 r 2 .
Σ θ i = ϕ j + ϕ j + = m π + δ m ( φ 1 + φ 2 ) 2 ,
γ t ( m ) = γ m ( 0 ) + Δ n n γ m ( 1 ) + ( Δ n n ) 2 γ m ( 2 ) ,
γ m ( 1 ) = 1 L c r 1 r 2 j = 1 N sin θ 2 j [ A j sin 2 α j + A j + sin 2 α j + ] ,
γ m ( 2 ) = γ m ( 1 ) 1 r 1 r 2 j = 1 N ϵ j sin θ 2 j [ A j sin 2 α j A j + sin 2 α j + ] + 2 L c { [ δ m ( 1 ) ] 2 + k > j = 1 N sin θ 2 j sin θ 2 k sinh [ ( ϵ k ϵ j ) L c γ m ( 0 ) ] cos 2 ( α j + α k + ) } .
δ m = Δ n n δ m ( 1 ) + ( Δ n n ) 2 δ m ( 2 ) ,
δ m ( 1 ) = 1 2 r 1 r 2 j = 1 N sin θ 2 j [ A j cos 2 α j + A j + cos 2 α j + ] ,
δ m ( 2 ) = L c 2 γ m ( 1 ) { δ m ( 1 ) + 1 r 1 r 2 j = 1 N ϵ j sin θ 2 j [ A j cos 2 α j A j + cos 2 α j + ] } k > j = 1 N sin θ 2 j sin θ 2 k cosh [ ( ϵ k ϵ j ) L c γ m ( 0 ) ] sin 2 ( α j + α k + ) .
sin ( 2 α j ± ) = ± cos m π × { cos [ ( φ 1 φ 2 ) 2 ] v ( ϵ j , m , φ 1 , φ 2 ) sin [ ( φ 1 φ 2 ) 2 ] w ( ϵ j , m , φ 1 , φ 2 ) } ,
v ( ϵ j , m , φ 1 , φ 2 ) = cos ( 2 ϵ j m π ) sin [ ϵ j ( φ 1 + φ 2 ) ] sin ( 2 ϵ j m π ) cos [ ϵ j ( φ 1 + φ 2 ) ] ,
w ( ϵ j , m , φ 1 , φ 2 ) = cos ( 2 ϵ j m π ) cos [ ϵ j ( φ 1 + φ 2 ) ] + sin ( 2 ϵ j m π ) sin [ ϵ j ( φ 1 + φ 2 ) ] .
sin ( 2 ϕ j ) = sin ( 2 ϕ j + ) cos ( m π ) sin ( 2 ϵ j m π ) .
γ m ( 1 ) = 1 L c r 1 r 2 j = 1 N sin θ 2 j ( A j A j + ) sin 2 ϕ j .
cos ( m π ) sin ( 2 ϵ j m π ) = cos m 0 π cos Δ m π × [ sin 2 π ϵ j m 0 cos 2 π ϵ j Δ m + cos 2 π ϵ j m 0 sin 2 π ϵ j Δ m ] .
f ( ϵ j ) = A j A j + = r 1 exp ( ϵ j L c γ m ( 0 ) ) r 2 exp ( ϵ j L c γ m ( 0 ) )
γ m ( 1 ) sinc ( Δ m ) = 1 2 1 2 cos [ 2 π ϵ Δ m ] d ϵ .
γ m ( 1 ) exp [ π τ 2 ( Δ m ) 2 ] n = sinc ( Δ m n a ) ,
A n ϵ min ϵ j [ f ( x ) ] 1 Γ n ( x ) d x = j 1 2 ,
ϕ j = η j + ( N j + 1 2 ) β Δ n n 1 + N β Δ n n θ i .
L 2 j = ( s + 1 2 ) λ m 0 2 n 2 ,
γ m ( 1 ) sinc ( Δ m + a 2 ) + sinc ( Δ m a 2 ) ,
F j q + P ( θ 2 j ) q σ y .
T = q + N P 0 ( N + 1 ) q q + N 1 j = 12 N P 0 j [ P j ( N + 1 ) ] 1 σ y + q 2 q + N 2 k > j = 1 N P 0 j [ P j k ] 1 P k ( N + 1 ) q 3 q + N 3 l > k > j = 1 N P 0 j [ P j k ] 1 P k l [ P l ( N + 1 ) ] 1 σ y + , ( 1 ) N q N P 01 [ P 12 ] 1 P 23 [ P N ( N + 1 ) ] ± 1 σ y N .
A k = N ! ( N k ) ! k ! ( Δ n n ) k .
R k , k + 1 A k A k + 1 = k + 1 N k n Δ n .
cos ( 2 ϕ j ) = cos ( 2 ϕ j + ) cos ( m π ) cos ( 2 ϵ j m π ) = cos m 0 π cos Δ m π × [ cos 2 ϵ j m 0 π cos 2 ϵ j Δ m π sin 2 ϵ j m 0 π sin 2 ϵ j Δ m π ] .
δ m ( 1 ) cos m 0 π cos Δ m π sgn ϵ j sin ( 2 ϵ j Δ m π ) .
cos 2 ( α j + α k + ) = cos [ 2 ( ϵ k ϵ j ) m π ] cos [ ( ϵ k ϵ j ) ( φ 1 + φ 2 ) ] + sin [ 2 ( ϵ k ϵ j ) m π ] sin [ ( ϵ k ϵ j ) ( φ 1 + φ 2 ) ] ,
cos [ 2 ( ϵ k ϵ j ) m π ] = cos [ 2 ( ϵ k ϵ j ) m 0 π ] cos [ 2 ( ϵ k ϵ j ) Δ m π ] sin [ 2 ( ϵ k ϵ j ) m 0 π ] sin [ 2 ( ϵ k ϵ j ) Δ m π ] ,
sin [ 2 ( ϵ k ϵ j ) m π ] = sin [ 2 ( ϵ k ϵ j ) m 0 π ] cos [ 2 ( ϵ k ϵ j ) Δ m π ] + cos [ 2 ( ϵ k ϵ j ) m 0 π ] sin [ 2 ( ϵ k ϵ j ) Δ m π ] .
cos 2 ( α j + α k + ) = sin ( 2 ϵ j m 0 π ) sin ( 2 ϵ k m 0 π ) cos [ 2 ( ϵ k ϵ j ) Δ m π ] .

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