Fourier analysis of Bloch wave propagation in photonic crystals

Benoît Lombardet; L. Andrea Dunbar; Rolando Ferrini; Romuald Houdré

doi:10.1364/JOSAB.22.001179

Journal of the Optical Society of America B
Vol. 22,
Issue 6,
pp. 1179-1190
(2005)
•https://doi.org/10.1364/JOSAB.22.001179

Fourier analysis of Bloch wave propagation in photonic crystals

Benoît Lombardet, L. Andrea Dunbar, Rolando Ferrini, and Romuald Houdré

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Benoît Lombardet, L. Andrea Dunbar, Rolando Ferrini, and Romuald Houdré, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B 22, 1179-1190 (2005)

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Abstract

The standard representation of an optical field propagating in a photonic crystal (PhC) is an electromagnetic Bloch wave. We present a description of these waves based on their Fourier transform into a series of electromagnetic plane waves. The contribution of each plane wave to the global energy and group velocity is detailed, and the valid domain of this decomposition is discussed. This description brings new insight to the fundamental properties of light propagation in PhCs. Most notably, it permits a continuous description of light propagation from the homogenous medium to the strongly modulated PhC case. It also provides an original physical understanding of negative refraction in PhCs and resolves inconsistencies that result from band folding.

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Wave Properties (1D)	Electromagnetic Bloch Wave	$n th$ Component	Electromagnetic Plane Wave
Magnetic field	$H_{k} = Σ_{n} H_{n (k)}$	$H_{n (k)} (x) = h_{n (k)} H_{0} \exp (i k_{n} x) e_{z}$	$H_{k} (x) = \exp (i k x) e_{z}$
Electric field	$E_{k} = Σ_{n} E_{n (k)}$	$E_{n (k)} (x) = μ_{0} c h_{n (k)} H_{0} \frac{ω}{k_{n} c} \exp (i k_{n} x) e_{y}$	$E_{k} (x) = μ_{0} c H_{0} \frac{ω}{k c} \exp (i k x) e_{y}$
Time–space-averaged energy density	${⟨ E_{k} ⟩}_{t, s} = Σ_{n} E_{n (k)} = \frac{1}{2} μ_{0} H_{0}^{2}$	$E_{n (k)} = \frac{1}{2} μ_{0} {∣ h_{n (k)} ∣}^{2} H_{0}^{2}$	${⟨ E_{k} ⟩}_{t, s} = \frac{1}{2} μ_{0} H_{0}^{2}$
Time–space-averaged Poynting vector	${⟨ S_{k} ⟩}_{t, s} = Σ_{n} S_{n (k)}$	$S_{n (k)} = \frac{1}{2} μ_{0} {∣ h_{n (k)} ∣}^{2} H_{0}^{2} \frac{ω}{k_{n}} e_{x}$	${⟨ S_{k} ⟩}_{t, s} = \frac{1}{2} μ_{0} H_{0}^{2} \frac{ω}{k} e_{x}$
Group velocity	$v_{g} = Σ_{n} {∣ h_{n (k)} ∣}^{2} v_{n}$	$v_{n} = \frac{ω}{k_{n}} e_{x}$	$v_{g} = \frac{ω}{k} e_{x}$

Wave Properties (2D)	Electromagnetic Bloch Wave	$(n, m) th$ Component	Electromagnetic Plane Wave
Magnetic field	$H_{k} = Σ_{n, m} H_{n, m (k)}$	$H_{n, m (k)} (r) = h_{n, m (k)} H_{0} \exp (i k_{n, m} \cdot r)$	$H_{k} (r) = H_{0} \exp (i k \cdot r)$
Electric field	$E_{k} = Σ_{n, m} E_{n, m (k)}$	$E_{n, m (k)} (r) = μ_{0} ∣ h_{n, m (k)} ∣ H_{0} \frac{ω}{∣ k_{n, m} ∣} \exp (i k_{n, m} \cdot r) e_{z}$	$E_{k} (r) = μ_{0} H_{0} \frac{ω}{∣ k ∣} \exp (i k \cdot r) e_{z}$
Time-space-averaged energy density	${⟨ E_{k} ⟩}_{t, s} = Σ_{n, m} E_{n, m (k)} = \frac{1}{2} μ_{0} H_{0}^{2}$	$E_{n, m (k)} = \frac{1}{2} μ_{0} {∣ h_{n, m (k)} ∣}^{2} H_{0}^{2}$	${⟨ E_{k} ⟩}_{t, s} = \frac{1}{2} μ_{0} H_{0}^{2}$
Time-space-averaged Poynting vector	${⟨ S_{k} ⟩}_{t, s} = Σ_{n, m} S_{n, m (k)}$	$S_{n, m (k)} = \frac{1}{2} μ_{0} {∣ h_{n, m (k)} ∣}^{2} H_{0}^{2} \frac{ω}{k_{n, m}^{2}} k_{n, m}$	${⟨ S_{k} ⟩}_{t, s} = \frac{1}{2} μ_{0} H_{0}^{2} \frac{ω}{k^{2}} k$
Group velocity	$v_{g} = Σ_{n, m} {∣ h_{n, m (k)} ∣}^{2} v_{n, m}$	$v_{n, m} = \frac{ω}{k_{n, m}^{2}} k_{n, m}$	$v_{g} = \frac{ω}{k^{2}} k$

Wave Properties (1D)	Electromagnetic Bloch Wave	$n th$ Component	Electromagnetic Plane Wave
Magnetic field	$H_{k} = Σ_{n} H_{n (k)}$	$H_{n (k)} (x) = h_{n (k)} H_{0} \exp (i k_{n} x) e_{z}$	$H_{k} (x) = \exp (i k x) e_{z}$
Electric field	$E_{k} = Σ_{n} E_{n (k)}$	$E_{n (k)} (x) = μ_{0} c h_{n (k)} H_{0} \frac{ω}{k_{n} c} \exp (i k_{n} x) e_{y}$	$E_{k} (x) = μ_{0} c H_{0} \frac{ω}{k c} \exp (i k x) e_{y}$
Time–space-averaged energy density	${⟨ E_{k} ⟩}_{t, s} = Σ_{n} E_{n (k)} = \frac{1}{2} μ_{0} H_{0}^{2}$	$E_{n (k)} = \frac{1}{2} μ_{0} {∣ h_{n (k)} ∣}^{2} H_{0}^{2}$	${⟨ E_{k} ⟩}_{t, s} = \frac{1}{2} μ_{0} H_{0}^{2}$
Time–space-averaged Poynting vector	${⟨ S_{k} ⟩}_{t, s} = Σ_{n} S_{n (k)}$	$S_{n (k)} = \frac{1}{2} μ_{0} {∣ h_{n (k)} ∣}^{2} H_{0}^{2} \frac{ω}{k_{n}} e_{x}$	${⟨ S_{k} ⟩}_{t, s} = \frac{1}{2} μ_{0} H_{0}^{2} \frac{ω}{k} e_{x}$
Group velocity	$v_{g} = Σ_{n} {∣ h_{n (k)} ∣}^{2} v_{n}$	$v_{n} = \frac{ω}{k_{n}} e_{x}$	$v_{g} = \frac{ω}{k} e_{x}$

Wave Properties (2D)	Electromagnetic Bloch Wave	$(n, m) th$ Component	Electromagnetic Plane Wave
Magnetic field	$H_{k} = Σ_{n, m} H_{n, m (k)}$	$H_{n, m (k)} (r) = h_{n, m (k)} H_{0} \exp (i k_{n, m} \cdot r)$	$H_{k} (r) = H_{0} \exp (i k \cdot r)$
Electric field	$E_{k} = Σ_{n, m} E_{n, m (k)}$	$E_{n, m (k)} (r) = μ_{0} ∣ h_{n, m (k)} ∣ H_{0} \frac{ω}{∣ k_{n, m} ∣} \exp (i k_{n, m} \cdot r) e_{z}$	$E_{k} (r) = μ_{0} H_{0} \frac{ω}{∣ k ∣} \exp (i k \cdot r) e_{z}$
Time-space-averaged energy density	${⟨ E_{k} ⟩}_{t, s} = Σ_{n, m} E_{n, m (k)} = \frac{1}{2} μ_{0} H_{0}^{2}$	$E_{n, m (k)} = \frac{1}{2} μ_{0} {∣ h_{n, m (k)} ∣}^{2} H_{0}^{2}$	${⟨ E_{k} ⟩}_{t, s} = \frac{1}{2} μ_{0} H_{0}^{2}$
Time-space-averaged Poynting vector	${⟨ S_{k} ⟩}_{t, s} = Σ_{n, m} S_{n, m (k)}$	$S_{n, m (k)} = \frac{1}{2} μ_{0} {∣ h_{n, m (k)} ∣}^{2} H_{0}^{2} \frac{ω}{k_{n, m}^{2}} k_{n, m}$	${⟨ S_{k} ⟩}_{t, s} = \frac{1}{2} μ_{0} H_{0}^{2} \frac{ω}{k^{2}} k$
Group velocity	$v_{g} = Σ_{n, m} {∣ h_{n, m (k)} ∣}^{2} v_{n, m}$	$v_{n, m} = \frac{ω}{k_{n, m}^{2}} k_{n, m}$	$v_{g} = \frac{ω}{k^{2}} k$

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Journal of the Optical Society of America B