Dispersive models for the finite-difference time-domain method: design, analysis, and implementation

Charles Hulse; André Knoesen

doi:10.1364/JOSAA.11.001802

Journal of the Optical Society of America A
Vol. 11,
Issue 6,
pp. 1802-1811
(1994)
•https://doi.org/10.1364/JOSAA.11.001802

Dispersive models for the finite-difference time-domain method: design, analysis, and implementation

Charles Hulse and André Knoesen

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Charles Hulse and André Knoesen, "Dispersive models for the finite-difference time-domain method: design, analysis, and implementation," J. Opt. Soc. Am. A 11, 1802-1811 (1994)

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Abstract

Digital signal processing techniques are used to design, analyze, and implement discrete models of polarization dispersion in finite-difference time-domain (FDTD) simulations. These methods are warranted by the extreme importance of dispersion to the propagation behavior of femtosecond-duration optical pulses. Input-invariant and frequency-approximation methods of designing discrete analogs of continuous-time dispersive electromagnetic systems are presented. Our methods are shown to unify existing design techniques. The inherent accuracy of each dispersive design is quantified by truncation error and frequency response methods, and stability analysis of the total FDTD system is found through root locus techniques. Implementation of the design is accomplished by algebraic manipulation of the system function in the frequency domain, resulting in canonical, partial fraction, or cascade structures that minimize the number of stored variables and provide a trade-off between efficiency and sensitivity to finite precision.

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	Debye Relaxation	Lorentzian Resonance
General Form	$\frac{A z + B}{z - exp (- Δ t / τ)}$	$\frac{{C_{z}}^{2} + D z + E}{[exp (δ Δ t / 2)] z^{2} - {2 cos [Δ t {({ω_{0}}^{2} - δ^{2} / 4)}^{1 / 2}]} z + exp (- δ Δ t / 2)}$
Impulse Invariant	A = ∊_∞ + (∊_s−∊_∞)Δt/τ	C = ∊_∞ [exp(δ Δt /2)]
	B = − ∊_∞ exp(−Δt/ τ)	D = (∊_s − ∊_∞){(ω₀ Δt)² sinc[Δt(ω₀²-δ²/4) ^1/2] } − 2∊_∞{cos[Δt (ω₀² − δ²/4) ^1/2 }
		E = ∊_∞ [exp(−δ Δt /2)]
Step Invariant	A = ∊_∞	C =∊_∞ [exp(δ Δt /2)]
	B = (∊_s − ∊_∞) − ∊_s exp(−Δt/ τ)	D = (∊_s − ∊_∞){exp(δ Δt/2) − (δ Δt /2)sinc[Δt(ω₀² − δ²/4)^1/2]} − (∊_s + ∊_∞){cos[Δt (ω₀² − δ²/4) ^1/2]}
		E = (∊_s − ∊_∞){(δ Δt/2)sinc[Δt (ω₀² − δ²/4) ^1/2] − cos[Δt (ω₀² − δ²/4) ^1/2]}− (∊_s + ∊_∞)exp(−δ Δt/2)

	Debye Relaxation	Lorentzian Resonance
Forward Difference	$\frac{∊_{\infty} z - (∊_{\infty} - ∊_{s} Δ t / τ)}{z - (1 - Δ t / τ)}$	$\frac{∊_{\infty} z^{2} - ∊_{\infty} (2 - δ Δ t / τ) z + [∊_{\infty} (1 - δ Δ t / τ) + ∊_{s} {(ω_{0} Δ t)}^{2}]}{z^{2} - (2 - δ Δ t / τ) z + [1 - δ Δ t / τ + {(ω_{0} Δ t)}^{2}]}$
Backward Difference	$\frac{(∊_{\infty} - ∊_{s} Δ t / τ) z - ∊_{\infty}}{(1 - Δ t / τ) z - 1}$	$\frac{[∊_{\infty} (1 + δ Δ t) + ∊_{s} {(ω_{0} Δ t)}^{2}] z^{2} - ∊_{\infty} (2 + δ Δ t) z + ∊_{\infty}}{[1 + δ Δ t + {(ω_{0} Δ t)}^{2}] z^{2} - (2 + δ Δ t) z + 1}$
Centered Difference	$\frac{∊_{\infty} z^{2} + (2 ∊_{s} Δ t / τ) z - ∊_{\infty}}{z^{2} + (2 Δ t / τ) z - 1}$	$\frac{∊_{\infty} z^{4} - ∊_{\infty} (2 δ Δ t) z^{3} - [2 ∊_{\infty} - 4 ∊_{\infty} {(ω_{0} Δ t)}^{2}] z^{2} - ∊_{\infty} (2 δ Δ t) z + ∊_{\infty}}{z^{4} - (2 δ Δ t) z^{3} - [2 - 4 {(ω_{0} Δ t)}^{2}] z^{2} - (2 δ Δ t) z + 1}$
Bilinear	$\frac{(2 ∊_{\infty} + ∊_{s} Δ t / τ) z - (2 ∊_{\infty} - ∊_{s} Δ t / τ)}{(2 + Δ t / τ) z - (2 - Δ t / τ)}$	$\frac{[∊_{\infty} (4 + 2 δ Δ t) + ∊_{s} {(ω_{0} Δ t)}^{2}] z^{2} - [8 ∊_{\infty} - 2 ∊_{s} {(ω_{0} Δ t)}^{2}] z + [∊_{\infty} (4 - 2 δ Δ t) + ∊_{\infty} {(ω_{0} Δ t)}^{2}]}{[4 + 2 δ Δ t + ∊_{s} {(ω_{0} Δ t)}^{2}] z^{2} - [8 - 2 {(ω_{0} Δ t)}^{2}] z + [4 - 2 δ Δ t + ∊_{\infty} {(ω_{0} Δ t)}^{2}]}$

	Debye Relaxtion	Lorentzian Resonance
Forward Difference	DΔt	LΔt
Backward Difference	−DΔt	−LΔt
Centered Difference	$\frac{- j ω D}{3} Δ t^{2}$	$\frac{- j ω L}{3} Δ t^{2}$
Bilinear	$\frac{- j ω D}{6} Δ t^{2}$	$\frac{- j ω L}{6} Δ t^{2}$
Impulse Invariant	$\frac{(∊_{s} - ∊_{\infty})}{2 τ} Δ t$	$\frac{- (∊_{s} - ∊_{\infty}) {ω_{0}}^{2}}{12} Δ t^{2}$
Step Invariant	$\frac{- (∊_{s} - ∊_{\infty}) j ω}{2 (1 + j ω τ)} Δ t$	$\frac{- (∊_{s} - ∊_{\infty}) j ω {ω_{0}}^{2}}{2 ({ω_{0}}^{2} + j ω δ - ω^{2})} Δ t$

	Debye Relaxation	Lorentzian Resonance
General Form	$\frac{A z + B}{z - exp (- Δ t / τ)}$	$\frac{{C_{z}}^{2} + D z + E}{[exp (δ Δ t / 2)] z^{2} - {2 cos [Δ t {({ω_{0}}^{2} - δ^{2} / 4)}^{1 / 2}]} z + exp (- δ Δ t / 2)}$
Impulse Invariant	A = ∊_∞ + (∊_s−∊_∞)Δt/τ	C = ∊_∞ [exp(δ Δt /2)]
	B = − ∊_∞ exp(−Δt/ τ)	D = (∊_s − ∊_∞){(ω₀ Δt)² sinc[Δt(ω₀²-δ²/4) ^1/2] } − 2∊_∞{cos[Δt (ω₀² − δ²/4) ^1/2 }
		E = ∊_∞ [exp(−δ Δt /2)]
Step Invariant	A = ∊_∞	C =∊_∞ [exp(δ Δt /2)]
	B = (∊_s − ∊_∞) − ∊_s exp(−Δt/ τ)	D = (∊_s − ∊_∞){exp(δ Δt/2) − (δ Δt /2)sinc[Δt(ω₀² − δ²/4)^1/2]} − (∊_s + ∊_∞){cos[Δt (ω₀² − δ²/4) ^1/2]}
		E = (∊_s − ∊_∞){(δ Δt/2)sinc[Δt (ω₀² − δ²/4) ^1/2] − cos[Δt (ω₀² − δ²/4) ^1/2]}− (∊_s + ∊_∞)exp(−δ Δt/2)

	Debye Relaxation	Lorentzian Resonance
Forward Difference	$\frac{∊_{\infty} z - (∊_{\infty} - ∊_{s} Δ t / τ)}{z - (1 - Δ t / τ)}$	$\frac{∊_{\infty} z^{2} - ∊_{\infty} (2 - δ Δ t / τ) z + [∊_{\infty} (1 - δ Δ t / τ) + ∊_{s} {(ω_{0} Δ t)}^{2}]}{z^{2} - (2 - δ Δ t / τ) z + [1 - δ Δ t / τ + {(ω_{0} Δ t)}^{2}]}$
Backward Difference	$\frac{(∊_{\infty} - ∊_{s} Δ t / τ) z - ∊_{\infty}}{(1 - Δ t / τ) z - 1}$	$\frac{[∊_{\infty} (1 + δ Δ t) + ∊_{s} {(ω_{0} Δ t)}^{2}] z^{2} - ∊_{\infty} (2 + δ Δ t) z + ∊_{\infty}}{[1 + δ Δ t + {(ω_{0} Δ t)}^{2}] z^{2} - (2 + δ Δ t) z + 1}$
Centered Difference	$\frac{∊_{\infty} z^{2} + (2 ∊_{s} Δ t / τ) z - ∊_{\infty}}{z^{2} + (2 Δ t / τ) z - 1}$	$\frac{∊_{\infty} z^{4} - ∊_{\infty} (2 δ Δ t) z^{3} - [2 ∊_{\infty} - 4 ∊_{\infty} {(ω_{0} Δ t)}^{2}] z^{2} - ∊_{\infty} (2 δ Δ t) z + ∊_{\infty}}{z^{4} - (2 δ Δ t) z^{3} - [2 - 4 {(ω_{0} Δ t)}^{2}] z^{2} - (2 δ Δ t) z + 1}$
Bilinear	$\frac{(2 ∊_{\infty} + ∊_{s} Δ t / τ) z - (2 ∊_{\infty} - ∊_{s} Δ t / τ)}{(2 + Δ t / τ) z - (2 - Δ t / τ)}$	$\frac{[∊_{\infty} (4 + 2 δ Δ t) + ∊_{s} {(ω_{0} Δ t)}^{2}] z^{2} - [8 ∊_{\infty} - 2 ∊_{s} {(ω_{0} Δ t)}^{2}] z + [∊_{\infty} (4 - 2 δ Δ t) + ∊_{\infty} {(ω_{0} Δ t)}^{2}]}{[4 + 2 δ Δ t + ∊_{s} {(ω_{0} Δ t)}^{2}] z^{2} - [8 - 2 {(ω_{0} Δ t)}^{2}] z + [4 - 2 δ Δ t + ∊_{\infty} {(ω_{0} Δ t)}^{2}]}$

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Equations (58)

Journal of the Optical Society of America A