Abstract

Electromagnetic Bloch modes are used to describe the field distribution of light in periodic media that cannot be adequately approximated by effective macroscopic media. These modes explicitly take into account the spatial modulation of the medium and therefore contain the full physical information at any specific location in the medium. For instance, the propagation velocity of light can be determined locally, and it is not an invariant of space, as it is often implicitly assumed when definitions such as that of the group velocity vgr=dω/dk are used (where ω is the angular frequency and k is the Bloch index of a monochromatic mode). Spatially invariant light velocities can only be expected if the medium is assumed to show an effective behavior similar to a homogeneous material (where a plane-wave ansatz would be more appropriate). This inevitably leads to the question: what exactly is dω/dk of a Bloch mode, if it is not the group velocity? The answer is the average group velocity. This is not a trivial observation, and it has to be taken into account, for instance, when the enhancement of nonlinear effects induced by slow light is estimated. The example of a Kerr nonlinearity is studied, and we show formally that using the average group velocity can lead to an underestimation of the effect. Furthermore, this article critically reviews the concepts of energy and phase velocity. In particular, the different interpretations of phase velocity that exist in the literature are unified using a generic definition of the quantity.

© 2013 Optical Society of America

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References

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  1. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  2. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  3. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed., Springer Series in Optical Sciences (Springer, Berlin, 2005).
  4. J. D. Joannopoulos and S. G. Johnson, Photonic Crystals, 2nd ed. (Princeton University, 2008).
  5. A simple way of illustrating the difference in nature between wave vector and Bloch index is to consider the analogy to solid-state physics. The momentum of a given electronic Bloch state |ψ〉 is not simply given by ℏ multiplied by the Bloch index of the mode (as is the case for the plane-wave momentum of a free electron), but by 〈ψ|p^|ψ〉, which contains a weighted sum over all plane-wave components [6].
  6. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks/Cole, 1976).
  7. K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media, Springer Series in Optical Sciences (Springer, 2006).
  8. K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences (Springer, 2009).
  9. Sometimes an individual Bloch mode is mistakenly interpreted as a pulse train formed by a harmonic wave and a periodic envelope function uω(x). This interpretation cannot reflect the phenomenon of pulse propagation, since uω(x) is stationary in space (it is independent of time).
  10. L. Brillouin, Wave Propagation and Group Velocity, Pure and Applied Physics (Academic, 1960).
  11. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  12. M. J. Lighthill, “Group velocity,” IMA J. Appl. Math. 1, 1–28 (1965).
    [CrossRef]
  13. T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
    [CrossRef]
  14. J. Li, L. O’Faolain, I. H. Rey, and T. F. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express 19, 4458–4463 (2011).
    [CrossRef]
  15. It is sometimes argued that phase fronts cannot be unambiguously defined because a Bloch mode is the result of multiple plane waves (some of them counterpropagating). Equation (10) clearly shows that this implication is incorrect.
  16. G. B. Whitham, “Group velocity and energy propagation for three-dimensional waves,” Commun. Pure Appl. Math. 14, 675–691 (1961).
    [CrossRef]
  17. Note that, due to the periodicity of uω(x) in Eq. (2), replacing k with k+m(2π/Γ) will result in a new function uω(x)e−im(2π/Γ)x with a fully periodic phase function φ(ω,x).
  18. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
    [CrossRef]
  19. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics (Pergamon, 1984).
  20. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B 22, 1179–1190 (2005).
    [CrossRef]

2011 (1)

2007 (1)

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
[CrossRef]

2005 (1)

2000 (1)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

1979 (1)

1977 (1)

1965 (1)

M. J. Lighthill, “Group velocity,” IMA J. Appl. Math. 1, 1–28 (1965).
[CrossRef]

1961 (1)

G. B. Whitham, “Group velocity and energy propagation for three-dimensional waves,” Commun. Pure Appl. Math. 14, 675–691 (1961).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks/Cole, 1976).

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity, Pure and Applied Physics (Academic, 1960).

Dunbar, L. A.

Ferrini, R.

Houdré, R.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Joannopoulos, J. D.

J. D. Joannopoulos and S. G. Johnson, Photonic Crystals, 2nd ed. (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos and S. G. Johnson, Photonic Crystals, 2nd ed. (Princeton University, 2008).

Krauss, T. F.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics (Pergamon, 1984).

Li, J.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics (Pergamon, 1984).

Lighthill, M. J.

M. J. Lighthill, “Group velocity,” IMA J. Appl. Math. 1, 1–28 (1965).
[CrossRef]

Lombardet, B.

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks/Cole, 1976).

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

O’Faolain, L.

Oughstun, K. E.

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media, Springer Series in Optical Sciences (Springer, 2006).

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences (Springer, 2009).

Rey, I. H.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed., Springer Series in Optical Sciences (Springer, Berlin, 2005).

Whitham, G. B.

G. B. Whitham, “Group velocity and energy propagation for three-dimensional waves,” Commun. Pure Appl. Math. 14, 675–691 (1961).
[CrossRef]

Yariv, A.

Yeh, P.

Commun. Pure Appl. Math. (1)

G. B. Whitham, “Group velocity and energy propagation for three-dimensional waves,” Commun. Pure Appl. Math. 14, 675–691 (1961).
[CrossRef]

IMA J. Appl. Math. (1)

M. J. Lighthill, “Group velocity,” IMA J. Appl. Math. 1, 1–28 (1965).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Phys. D (1)

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (1)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

Other (12)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics (Pergamon, 1984).

Note that, due to the periodicity of uω(x) in Eq. (2), replacing k with k+m(2π/Γ) will result in a new function uω(x)e−im(2π/Γ)x with a fully periodic phase function φ(ω,x).

It is sometimes argued that phase fronts cannot be unambiguously defined because a Bloch mode is the result of multiple plane waves (some of them counterpropagating). Equation (10) clearly shows that this implication is incorrect.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed., Springer Series in Optical Sciences (Springer, Berlin, 2005).

J. D. Joannopoulos and S. G. Johnson, Photonic Crystals, 2nd ed. (Princeton University, 2008).

A simple way of illustrating the difference in nature between wave vector and Bloch index is to consider the analogy to solid-state physics. The momentum of a given electronic Bloch state |ψ〉 is not simply given by ℏ multiplied by the Bloch index of the mode (as is the case for the plane-wave momentum of a free electron), but by 〈ψ|p^|ψ〉, which contains a weighted sum over all plane-wave components [6].

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks/Cole, 1976).

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media, Springer Series in Optical Sciences (Springer, 2006).

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences (Springer, 2009).

Sometimes an individual Bloch mode is mistakenly interpreted as a pulse train formed by a harmonic wave and a periodic envelope function uω(x). This interpretation cannot reflect the phenomenon of pulse propagation, since uω(x) is stationary in space (it is independent of time).

L. Brillouin, Wave Propagation and Group Velocity, Pure and Applied Physics (Academic, 1960).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

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Figures (1)

Fig. 1.
Fig. 1.

Box of length L and cross-section area A. An electromagnetic wave propagates along x. The energy density in the box is U, and the energy flux density through the area A is S=S·x^, where x^ is the unit vector along the x coordinate.

Tables (1)

Tables Icon

Table 1. Definitions of the Group, Energy, and Phase Velocity of Bloch Modes, According to Sections 35a

Equations (24)

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Hω=hω(y,z)ei[k(ω)xωt],
Hω=hω(y,z)uω(x)ei[k(ω)xωt],
H(0,t)=G(ω)hωuω(0)eiωtdω.
H(x¯,t+t¯)=G(ω)hωuω(x¯)ei[k(ω)x¯ω(t+t¯)]dω.
H(x¯,t+t¯)=H(0,t)eiφ0
k(ω)x¯ωt¯=φ0
dnϕdωn=0for alln1.
dkdω(ω)x¯t¯=0.
vgr=x¯t¯=(dkdω(ω))1.
Hω=hω(y,z)|uω(x)|ei[k(ω)x+φ(ω,x)ωt]=hω(y,z)|uω(x)|eiΦω(x,t)
H(0,t)=G(ω)hωuωeiΦω(0,t)dω
H(x¯,t+t¯)=G(ω)hωuωeiΦω(x¯,t+t¯)dω.
dkdω(ω)x¯+dφdω(ω,x¯)t¯=0.
ddx¯(dkdω(ω)x¯+dφdω(ω,x¯))·vgr=1.
vgr(ω)1=1Γxx+Γvgr(ω,x¯)1dx¯=1Γxx+Γddx¯(dkdω(ω)x¯+dφdω(ω,x¯))dx¯=dkdω(ω)+1Γddω(φ(ω,x+Γ)φ(ω,x))=dkdω(ω).
Δφ=0ΓωcB(1/vgr(x))2dx=ΓωcB(1/vgr)2ΓωcB1/vgr2ΓωcB(1/vgr)2.
Δφ=0ΓωcB˜I(x)2dx,
v=ΔrΔt=1Δt1dr=1Δtdrdtdt=1Δtvdt.
1Δrvdr=1Δrdrdtdr=1Δrdrdtdrdtdt=ΔtΔr(1Δtv2dt)=v2vv.
1Δrv1dr=1Δrdtdrdr=1Δr1dt=ΔtΔr=v1,
vph(x)=tΦωxΦω=ωk(ω)+xφ(ω,x).
vph=ωk(ω)+m2πΓ,
ven=ASTdydzAULdydz.
ven=ASTdydzAUTΓdydz.

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