Abstract

We derive macroscopic equations of motion for the slowly varying electric field amplitude in three-dimensional active nonlinear optical nanostructures. We show that the microscopic Maxwell equations and polarization dynamics can be simplified to a macroscopic one-dimensional problem in the direction of group velocity. For a three-level active material, we derive the steady-state equations for normal mode frequency, threshold pumping, nonlinear Bloch mode amplitude, and lasing in photonic crystals. Our analytical results accurately recapture the results of exact numerical methods.

© 2013 Optical Society of America

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  1. B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
    [CrossRef]
  2. S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
    [CrossRef]
  3. P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, Phys. Rev. B 73, 165125 (2006).
    [CrossRef]
  4. S. L. Chua, Y. Chong, A. D. Stone, M. Soljacic, and J. B. Abad, Opt. Express 19, 1539 (2011).
    [CrossRef]
  5. H. Takeda and S. John, Phys. Rev. A 83, 053811 (2011).
    [CrossRef]
  6. A. Kaso and S. John, Phys. Rev. E 74, 046611 (2006).
    [CrossRef]
  7. A. Kaso and S. John, Phys. Rev. A 76, 053838 (2007).
    [CrossRef]
  8. L. Florescu, K. Busch, and S. John, J. Opt. Soc. Am. B 19, 2215 (2002).
    [CrossRef]
  9. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Chap. 3.
  10. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass, 1977), Chap. 8.
  11. A. E. Siegman, Lasers (University Science Books, 1986), Chap. 24, Sect. 4.
  12. In the semiclassical laser theory ∂A/∂t and the loss (σ) are both assumed one order smaller than ωA and are referred to as first-order terms (Refs. [10,11]). Their product, A/∂t, is a second-order term that can be neglected. A more systematic ordering and elimination of second-order terms can be accomplished by a multiscale expansion method as discussed in [8].
  13. C. M. de Sterke and J. E. Sipe, Phys. Rev. A 38, 5149 (1988).
    [CrossRef]
  14. C. M. Bowden and G. P. Agrawal, Opt. Commun. 100, 147 (1993).
    [CrossRef]
  15. Y. C. Lan, Appl. Phys. Lett. 88, 071109 (2006).
    [CrossRef]
  16. P. W. Milonni and J. H. Eberly, Laser Physics, 2nd ed. (Wiley, 2010).
  17. K. Sakoda, Phys. Rev. B 52, 7982 (1995).
    [CrossRef]
  18. J. F. Cornwell, Group Theory in Physics (Academic, 1997).
  19. S. John and R. Wang, Phys. Rev. A 78, 043809 (2008).
    [CrossRef]
  20. S. Eyderman, S. John, and A. Deinega, J. Appl. Phys. 113, 154315 (2013).
    [CrossRef]

2013

S. Eyderman, S. John, and A. Deinega, J. Appl. Phys. 113, 154315 (2013).
[CrossRef]

2011

S. L. Chua, Y. Chong, A. D. Stone, M. Soljacic, and J. B. Abad, Opt. Express 19, 1539 (2011).
[CrossRef]

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

H. Takeda and S. John, Phys. Rev. A 83, 053811 (2011).
[CrossRef]

2008

S. John and R. Wang, Phys. Rev. A 78, 043809 (2008).
[CrossRef]

2007

A. Kaso and S. John, Phys. Rev. A 76, 053838 (2007).
[CrossRef]

2006

Y. C. Lan, Appl. Phys. Lett. 88, 071109 (2006).
[CrossRef]

A. Kaso and S. John, Phys. Rev. E 74, 046611 (2006).
[CrossRef]

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, Phys. Rev. B 73, 165125 (2006).
[CrossRef]

2002

1995

K. Sakoda, Phys. Rev. B 52, 7982 (1995).
[CrossRef]

1993

C. M. Bowden and G. P. Agrawal, Opt. Commun. 100, 147 (1993).
[CrossRef]

1988

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 38, 5149 (1988).
[CrossRef]

Abad, J. B.

Agrawal, G. P.

C. M. Bowden and G. P. Agrawal, Opt. Commun. 100, 147 (1993).
[CrossRef]

Andreani, L. C.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Badolato, A.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Bermel, P.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, Phys. Rev. B 73, 165125 (2006).
[CrossRef]

Bouwmeester, D.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Bowden, C. M.

C. M. Bowden and G. P. Agrawal, Opt. Commun. 100, 147 (1993).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Chap. 3.

Busch, K.

Choi, Y.-S.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Chong, Y.

Chua, S. L.

Cornwell, J. F.

J. F. Cornwell, Group Theory in Physics (Academic, 1997).

de Sterke, C. M.

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 38, 5149 (1988).
[CrossRef]

Deinega, A.

S. Eyderman, S. John, and A. Deinega, J. Appl. Phys. 113, 154315 (2013).
[CrossRef]

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Laser Physics, 2nd ed. (Wiley, 2010).

Ellis, B.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

Eyderman, S.

S. Eyderman, S. John, and A. Deinega, J. Appl. Phys. 113, 154315 (2013).
[CrossRef]

Fink, Y.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, Phys. Rev. B 73, 165125 (2006).
[CrossRef]

Florescu, L.

Hallerm, E. E.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

Harris, J.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

Hennessy, K.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Hu, E. L.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Joannopoulos, J. D.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, Phys. Rev. B 73, 165125 (2006).
[CrossRef]

John, S.

S. Eyderman, S. John, and A. Deinega, J. Appl. Phys. 113, 154315 (2013).
[CrossRef]

H. Takeda and S. John, Phys. Rev. A 83, 053811 (2011).
[CrossRef]

S. John and R. Wang, Phys. Rev. A 78, 043809 (2008).
[CrossRef]

A. Kaso and S. John, Phys. Rev. A 76, 053838 (2007).
[CrossRef]

A. Kaso and S. John, Phys. Rev. E 74, 046611 (2006).
[CrossRef]

L. Florescu, K. Busch, and S. John, J. Opt. Soc. Am. B 19, 2215 (2002).
[CrossRef]

Kaso, A.

A. Kaso and S. John, Phys. Rev. A 76, 053838 (2007).
[CrossRef]

A. Kaso and S. John, Phys. Rev. E 74, 046611 (2006).
[CrossRef]

Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass, 1977), Chap. 8.

Lan, Y. C.

Y. C. Lan, Appl. Phys. Lett. 88, 071109 (2006).
[CrossRef]

Lidorikis, E.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, Phys. Rev. B 73, 165125 (2006).
[CrossRef]

Mayer, M. A.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Laser Physics, 2nd ed. (Wiley, 2010).

Petroff, P. M.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Rakher, M. T.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Sakoda, K.

K. Sakoda, Phys. Rev. B 52, 7982 (1995).
[CrossRef]

Sargent, M.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass, 1977), Chap. 8.

Sarmiento, T.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass, 1977), Chap. 8.

Shambat, G.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 24, Sect. 4.

Sipe, J. E.

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 38, 5149 (1988).
[CrossRef]

Soljacic, M.

Stone, A. D.

Strauf, S.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Takeda, H.

H. Takeda and S. John, Phys. Rev. A 83, 053811 (2011).
[CrossRef]

Vuckovic, J.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

Wang, R.

S. John and R. Wang, Phys. Rev. A 78, 043809 (2008).
[CrossRef]

Appl. Phys. Lett.

Y. C. Lan, Appl. Phys. Lett. 88, 071109 (2006).
[CrossRef]

J. Appl. Phys.

S. Eyderman, S. John, and A. Deinega, J. Appl. Phys. 113, 154315 (2013).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photonics

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Hallerm, and J. Vučković, Nat. Photonics 5, 297 (2011).
[CrossRef]

Opt. Commun.

C. M. Bowden and G. P. Agrawal, Opt. Commun. 100, 147 (1993).
[CrossRef]

Opt. Express

Phys. Rev. A

S. John and R. Wang, Phys. Rev. A 78, 043809 (2008).
[CrossRef]

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 38, 5149 (1988).
[CrossRef]

H. Takeda and S. John, Phys. Rev. A 83, 053811 (2011).
[CrossRef]

A. Kaso and S. John, Phys. Rev. A 76, 053838 (2007).
[CrossRef]

Phys. Rev. B

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, Phys. Rev. B 73, 165125 (2006).
[CrossRef]

K. Sakoda, Phys. Rev. B 52, 7982 (1995).
[CrossRef]

Phys. Rev. E

A. Kaso and S. John, Phys. Rev. E 74, 046611 (2006).
[CrossRef]

Phys. Rev. Lett.

S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96, 127404 (2006).
[CrossRef]

Other

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Chap. 3.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass, 1977), Chap. 8.

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 24, Sect. 4.

In the semiclassical laser theory ∂A/∂t and the loss (σ) are both assumed one order smaller than ωA and are referred to as first-order terms (Refs. [10,11]). Their product, A/∂t, is a second-order term that can be neglected. A more systematic ordering and elimination of second-order terms can be accomplished by a multiscale expansion method as discussed in [8].

J. F. Cornwell, Group Theory in Physics (Academic, 1997).

P. W. Milonni and J. H. Eberly, Laser Physics, 2nd ed. (Wiley, 2010).

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Figures (1)

Fig. 1.
Fig. 1.

(a) Schematic of 2D PC and the active dopant. (b) The band structure of E polarization. The insets shows the magnitude of the magnetic field (whose components are in the 2D periodic plane) for the modes at the X point. The modes at X are labeled with their symmetry representations. (c) The gain line shape function of Er3+ (red curve), and for an active medium that has 1000 times larger line width than that of Er3+ (dark blue). (d) The intensity ratio as a function of pumping for the first Bloch mode (with symmetry representation A1). Blue line: exact self-consistent method [5]; open circles: perturbation theory [6]; triangles: presented method.

Equations (12)

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

××ψm(r)[ωm2/c2]ε(r)ψm(r)=0,
E(r,t)=A(r,t)ψ(r)eiωt,
P(r,t)=B(r,t)ψ(r)eiωt,
××E(r,t)+ε(r)c22E(r,t)t2+μ0σ(r)E(r,t)t+μ0η(r)2P(r,t)t2=0.
A××ψ+(A)·ψ(A·)ψ=2iωc2εψAt+ω2c2ψA(iγ+ηχ+δε).
ic22ωq·A+εψ2At=ω2(iγ+ηχ+δε)Aψ2.
vg·A+At=ω2{iγψ2+ηχψ2+δ}A.
γψ2+ηIm(χ)ψ2=0;ηRe(χ)ψ2+δ=0.
χ(r)=g0(p1p+1)(ωω0)T2i1+[(ωω0)T2]2+A2ψ2(r)/[Is(p+1)],
ωm2/ω2=1+(ωω0)γψ2T2.
Nth=γψ2g0ηψ2[1+(ωω0)2T22].
γψ2(p+1)g0(p1)=ηψ21+[(ωω0)T2]2+A2ψ2(r)/Is(p+1).

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