Surface Electromagnetic Waves on Metals and Polar Insulators: Some Comments

C. A. Ward; R. J. Bell; R. W. Alexander; G. S. Kovener; I. Tyler

doi:10.1364/AO.13.002378

Applied Optics
Vol. 13,
Issue 10,
pp. 2378-2381
(1974)
•https://doi.org/10.1364/AO.13.002378

Surface Electromagnetic Waves on Metals and Polar Insulators: Some Comments

C. A. Ward, R. J. Bell, R. W. Alexander, G. S. Kovener, and I. Tyler

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C. A. Ward, R. J. Bell, R. W. Alexander, G. S. Kovener, and I. Tyler, "Surface Electromagnetic Waves on Metals and Polar Insulators: Some Comments," Appl. Opt. 13, 2378-2381 (1974)

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Abstract

The important parameters governing the propagation of surface electromagnetic waves (SEW) on metals or polar insulators have been theoretically investigated. General approximations for the propagation distance, decay distance, and penetration depth are presented in summary tables. It is found that the propagation distance of SEW can be used to study overlayer materials on metals, that the penetration depth of SEW fields into a metal is governed by the classical skin depth at low frequencies and by the plasma frequency in the visible wavelength region, that surface phonons can propagate over a centimeter on PbS, PbTe, and ferroelectrics, and that the propagation distances of SEW increase with decreasing temperature.

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Equations (23)

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Parameter	Visible and Infrared (σ ≳ 300 cm⁻¹), σ_p² ≫ σ², ∊₁² ≫ ∊₂²			Far Infrared (σ ≲ 150 cm⁻¹), σ_p² ≫ σ², ∊₂² ≫ ∊₁²
Parameter	Vaccum overlayer η₁ = 1, η₂ = 0	Nonabsorbing overlayer η₁ ≥ 1, η₂ = 0	Absorbing overlayer, but \|∊₁\| ≫ η₁	Vaccum overlayer η₁ = 1, η₂ = 0	Nonabsorbing overlayer η₁ ≥ 1, η₂ = 0
Propagation distance along surface, L_x	$\begin{array}{l} \approx \frac{{∊_{1}}^{2}}{2 π σ ∊_{2}} {(1 + \frac{1}{∊_{1}})}^{\frac{3}{2}} \\ {[∊_{\infty} - {(σ_{p} / σ)}^{2}]}^{2} \\ \approx \frac{\times {(σ / σ_{p})}^{2}}{2 π σ_{τ}} \\ \approx \frac{{σ_{p}}^{2}}{2 π σ_{τ} σ^{2}} \approx \frac{{σ_{p}}^{2} A}{2 π σ^{2} T^{2}} \end{array}$ (1)	$\begin{array}{l} \approx \frac{{∊_{1}}^{2}}{2 π σ ∊_{2}} {(\frac{1}{η_{1}} + \frac{1}{∊_{1}})}^{\frac{3}{2}} \\ \approx \frac{{σ_{p}}^{2}}{2 π σ_{τ} σ^{2}} {(\frac{1}{η_{1}} - \frac{σ^{2}}{{σ_{p}}^{2}})}^{\frac{3}{2}} \end{array}$ (4)	$\begin{array}{l} \approx \frac{1}{2 π σ} (\frac{{\sqrt{η}}_{1} {∊_{1}}^{2}}{η_{2} {∊_{1}}^{2} + ∊_{2} {η_{1}}^{2}}) \\ \approx [\sqrt{η_{1}} / (α n)] (Ref . 13) \\ \approx 1 / α (Ref . 14) \end{array}$ (7)	$\begin{array}{l} \approx ∊_{2} / 2 π σ \\ \approx \frac{{(∊_{\infty} - \frac{{σ_{p}}^{2}}{σ^{2}})}^{2} \frac{σ^{2}}{{σ_{p}}^{2}}}{2 π σ_{τ}} \\ \approx {σ_{p}}^{2} / (2 π σ_{τ} σ^{2}) \end{array}$ (10)	$\begin{array}{l} \approx ∊_{2} / ({2 π σ η}_{1}^{\frac{3}{2}}) \\ \approx {σ_{p}}^{2} / (2 π σ_{τ} {η_{1}}^{\frac{3}{2}} σ^{2}) \end{array}$ (13)
Decay into overlayer, 1/k_a₂_z	$\begin{array}{l} \approx \frac{{\| ∊_{1} + 1 \|}^{\frac{1}{2}}}{2 π σ} \\ \approx σ_{p} / (2 π σ^{2}) \end{array}$ (2)	$\begin{array}{l} \approx \frac{{\| ∊_{1} + η_{1} \|}^{\frac{1}{2}}}{2 π σ η_{1}} \\ \approx \frac{{σ_{p}}^{2} {(1 - η_{1} σ^{2} / {σ_{p}}^{2})}^{\frac{1}{2}}}{2 π σ^{2} η_{1}} \\ \approx \frac{σ_{p}}{2 π σ^{2} η_{1}} (Ref . 12) \end{array}$ (5)	(8)	$\begin{array}{l} \approx {(2 ∊_{2})}^{\frac{1}{2}} / (2 π σ) \\ \approx \frac{σ_{p} \sqrt{∊_{\infty}}}{\sqrt{2 π} \sqrt{σ_{τ}} σ^{\frac{3}{2}}} \end{array}$ (11)	(14)
Penetration into SEW medium, 1/k_b₂_z	$\begin{array}{l} \approx \frac{{\| ∊_{1} + 1 \|}^{\frac{1}{2}}}{2 π σ \| ∊_{1} \|} \\ \approx 1 / (2 π σ_{p}) \end{array}$ (3)	$\begin{array}{l} \approx \frac{{\| ∊_{1} + η_{1} \|}^{\frac{1}{2}}}{2 π σ \| ∊_{1} \|} \\ \approx \frac{{(1 - η_{1} σ^{2} / {σ_{p}}^{2})}^{\frac{1}{2}}}{2 π σ_{p}} \end{array}$ (6)	(9)	$\begin{array}{l} \approx {(2 / ∊_{2})}^{\frac{1}{2}} / (2 π σ) \\ \approx {[\frac{σ_{τ}}{2 π^{2} ∊_{\infty} {σ_{p}}^{2} σ}]}^{\frac{1}{2}} = δ_{c} \end{array}$ (12)	(15)

Crystal	σ_TO (cm⁻¹)	δ∊	σ_Γ(cm⁻¹)	L_x(cm)	Ref.
LiF	306	7.35	16.2	0.072	5
NaCl	175	3.6	9.6	0.059	5
InSb	185	2.0	1.3	0.24	6
CdS	234	3.5	4.0	0.14	6
PbS	72	123.0	15.1	1.3	7
PbTe	31	268.0	17.0	2.5	8
CaF₂	260	4.7	10.4	0.072	5
BaTiO₃	42(200°C)	1250.0	168.0	1.2	16
	27(145°C)	3400.0	149.0	3.6	16
	19.5(127°C)	5500.0	120.0	7.3	16
SrTiO₃	88(300K)	310.0	35.2	1.4	16
	44(88K)	1250.0	61.7	3.2	16

σ	= wavenumber (cm⁻¹);
∊	= ∊₁ + i∊₂, SEW medium dielectric constant;
η	= η₁ + iη₂, overlayer material dielectric constant
	= (n² − K²) + i (2nK), where n = index of refraction of overlayer, K = extinction index of overlayer;
k_x	= k₁_x + ik₂_x = complex propagation constant along surface;
k_x² + k_z²	= ${\begin{cases} η \\ ∊ \end{cases}} \frac{ω^{2}}{c^{2}} {\begin{cases} overlayer, η, \\ SEW medium, ∊; \end{cases}$
L_x	= 1/(2k₂_x) = 1/e value for power drop along surface
	= propagation distance of SEW;
1/k_a₂_z	= decay distance of the field in the overlayer material;
1/k_b₂_z	= penetration depth of the field into the SEW supporting medium;
σ_p²	= n′e²/(πc²m*) (plasma frequency squared, cm⁻²);
σ_τ	= 1/(2πcτ) (free carrier damping, cm⁻¹)
	= T₁/A (for low temperatures for metals,15 where A is the proportionality constant);
σ_γ	= Γ/(2πc) (phonon damping, cm⁻¹);
α	= 4πσK = absorption coefficient of overlayer material;
∊_∞	= high frequency dielectric constant;
∊₀	= low frequency dielectric constant;
δ∊	= ∊₀ − ∊_∞ ≡ coupling strength;
δ_c	= [c²/(2πσ₀ω)]^1/2 = [(ρc)/(4π²σ)]^1/2
	= classical skin depth of a metal;
σ₀	= dc conductivity
ρ	= dc resistivity;
ω	= 2πcσ = angular frequency;
ω_To	= 2πcσ_TO = transverse optical frequency;
ω_Lo	= 2πcσ_LO = longitudinal optical phonon frequency;
Γ	= phonon damping (angular frequency);
τ	= scattering frequency for electrons or holes;
T	= temperature (K).

Parameter	Visible and Infrared (σ ≳ 300 cm⁻¹), σ_p² ≫ σ², ∊₁² ≫ ∊₂²			Far Infrared (σ ≲ 150 cm⁻¹), σ_p² ≫ σ², ∊₂² ≫ ∊₁²
Parameter	Vaccum overlayer η₁ = 1, η₂ = 0	Nonabsorbing overlayer η₁ ≥ 1, η₂ = 0	Absorbing overlayer, but \|∊₁\| ≫ η₁	Vaccum overlayer η₁ = 1, η₂ = 0	Nonabsorbing overlayer η₁ ≥ 1, η₂ = 0
Propagation distance along surface, L_x	$\begin{array}{l} \approx \frac{{∊_{1}}^{2}}{2 π σ ∊_{2}} {(1 + \frac{1}{∊_{1}})}^{\frac{3}{2}} \\ {[∊_{\infty} - {(σ_{p} / σ)}^{2}]}^{2} \\ \approx \frac{\times {(σ / σ_{p})}^{2}}{2 π σ_{τ}} \\ \approx \frac{{σ_{p}}^{2}}{2 π σ_{τ} σ^{2}} \approx \frac{{σ_{p}}^{2} A}{2 π σ^{2} T^{2}} \end{array}$ (1)	$\begin{array}{l} \approx \frac{{∊_{1}}^{2}}{2 π σ ∊_{2}} {(\frac{1}{η_{1}} + \frac{1}{∊_{1}})}^{\frac{3}{2}} \\ \approx \frac{{σ_{p}}^{2}}{2 π σ_{τ} σ^{2}} {(\frac{1}{η_{1}} - \frac{σ^{2}}{{σ_{p}}^{2}})}^{\frac{3}{2}} \end{array}$ (4)	$\begin{array}{l} \approx \frac{1}{2 π σ} (\frac{{\sqrt{η}}_{1} {∊_{1}}^{2}}{η_{2} {∊_{1}}^{2} + ∊_{2} {η_{1}}^{2}}) \\ \approx [\sqrt{η_{1}} / (α n)] (Ref . 13) \\ \approx 1 / α (Ref . 14) \end{array}$ (7)	$\begin{array}{l} \approx ∊_{2} / 2 π σ \\ \approx \frac{{(∊_{\infty} - \frac{{σ_{p}}^{2}}{σ^{2}})}^{2} \frac{σ^{2}}{{σ_{p}}^{2}}}{2 π σ_{τ}} \\ \approx {σ_{p}}^{2} / (2 π σ_{τ} σ^{2}) \end{array}$ (10)	$\begin{array}{l} \approx ∊_{2} / ({2 π σ η}_{1}^{\frac{3}{2}}) \\ \approx {σ_{p}}^{2} / (2 π σ_{τ} {η_{1}}^{\frac{3}{2}} σ^{2}) \end{array}$ (13)
Decay into overlayer, 1/k_a₂_z	$\begin{array}{l} \approx \frac{{\| ∊_{1} + 1 \|}^{\frac{1}{2}}}{2 π σ} \\ \approx σ_{p} / (2 π σ^{2}) \end{array}$ (2)	$\begin{array}{l} \approx \frac{{\| ∊_{1} + η_{1} \|}^{\frac{1}{2}}}{2 π σ η_{1}} \\ \approx \frac{{σ_{p}}^{2} {(1 - η_{1} σ^{2} / {σ_{p}}^{2})}^{\frac{1}{2}}}{2 π σ^{2} η_{1}} \\ \approx \frac{σ_{p}}{2 π σ^{2} η_{1}} (Ref . 12) \end{array}$ (5)	(8)	$\begin{array}{l} \approx {(2 ∊_{2})}^{\frac{1}{2}} / (2 π σ) \\ \approx \frac{σ_{p} \sqrt{∊_{\infty}}}{\sqrt{2 π} \sqrt{σ_{τ}} σ^{\frac{3}{2}}} \end{array}$ (11)	(14)
Penetration into SEW medium, 1/k_b₂_z	$\begin{array}{l} \approx \frac{{\| ∊_{1} + 1 \|}^{\frac{1}{2}}}{2 π σ \| ∊_{1} \|} \\ \approx 1 / (2 π σ_{p}) \end{array}$ (3)	$\begin{array}{l} \approx \frac{{\| ∊_{1} + η_{1} \|}^{\frac{1}{2}}}{2 π σ \| ∊_{1} \|} \\ \approx \frac{{(1 - η_{1} σ^{2} / {σ_{p}}^{2})}^{\frac{1}{2}}}{2 π σ_{p}} \end{array}$ (6)	(9)	$\begin{array}{l} \approx {(2 / ∊_{2})}^{\frac{1}{2}} / (2 π σ) \\ \approx {[\frac{σ_{τ}}{2 π^{2} ∊_{\infty} {σ_{p}}^{2} σ}]}^{\frac{1}{2}} = δ_{c} \end{array}$ (12)	(15)

Crystal	σ_TO (cm⁻¹)	δ∊	σ_Γ(cm⁻¹)	L_x(cm)	Ref.
LiF	306	7.35	16.2	0.072	5
NaCl	175	3.6	9.6	0.059	5
InSb	185	2.0	1.3	0.24	6
CdS	234	3.5	4.0	0.14	6
PbS	72	123.0	15.1	1.3	7
PbTe	31	268.0	17.0	2.5	8
CaF₂	260	4.7	10.4	0.072	5
BaTiO₃	42(200°C)	1250.0	168.0	1.2	16
	27(145°C)	3400.0	149.0	3.6	16
	19.5(127°C)	5500.0	120.0	7.3	16
SrTiO₃	88(300K)	310.0	35.2	1.4	16
	44(88K)	1250.0	61.7	3.2	16

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Tables (2)

Equations (23)

Applied Optics