2. PARAMETRIC COUPLING

Consider the multilayered structure of glass (x < −a), metal (x < 0), and free space (x > 0). A laser is obliquely incident from the glass side on the glass–metal interface having field, given by (Liu & Tripathi, Reference Liu and Tripathi2000)

(1)$${\vec E_{\rm l}} \cong {\vec A_{0{\rm L}}}{e^{[ - {{\{ (x + a)\sin {{\rm \theta} _i} - z\cos {{\rm \theta} _i}\}} ^2}/{b^{^{\prime}2}}]}}{e^{ - i({{\rm \omega} _0}t - {{\vec k}_0}.\vec r)}}$$

where ‘a’ is the width of the metal, θ_i is the angle of incidence and b′ = bcosθ_i and b is the spot size of laser. The incident laser excites a SPW on metal free space interface (x = 0) propagating along z-direction (as shown in Fig. 1). The electric field profile of the SPW $({{\rm \omega} _0},{\vec k_{0{\rm z}}};\,{\vec k_0} = {\vec k_{0{\rm z}}}\hat z - i{{\rm \alpha} _{\rm 0}}\hat x)$ is given by

(2)$${\vec E_0} = {\vec A_{0{\rm z}}}(x)e ^{- i({{\rm \omega} _0}t - {k_{0{\rm z}}}\hat z)}$$

where ${A_{{\rm 0z}}} = {A_{0{\rm s}}}\left( {\hat z - \displaystyle{{i{k_{0{\rm z}}}} \over {{{\rm \alpha} _0}}}\hat x} \right){e^{{{\rm \alpha} _0}x}}$ (x < 0, metal)

$A_{0z} = A_{0{\rm s}}\bigg( {\hat z} + \displaystyle{{i{k_{0z}}} \over {{\rm \alpha}_{0}^{\prime}{\hat x}} }\bigg)e^{ - {\rm \alpha}_{0}^{\prime}x}$ (x > 0, vacuum)

Fig. 1. Schematic of surface plasma wave Compton scattering at the vacuum–metal interface.

The amplitude of the wave for the free space at x = 0 and z = 0 is given by

(3)$${A_{0{\rm s}}} = \displaystyle{{{A_{0{\rm L}}}b} \over {2\sqrt {\rm \pi}}} {T_{00}}({{\rm \sigma} _1} + i{{\rm \sigma} _2})I$$

where T ₀₀ = (2iψ₁e ^α₀a)/(1+iψ₁)

$${{\rm \psi} _1} = \displaystyle{{{{\rm \eta} _{\rm g}}} \over {{{\left \vert {1 + {{{\rm \varepsilon ^{\prime}}}_0}(1 - 1/{\rm \eta} _{\rm g}^2 )} \right \vert} ^{1/2}}}}$$

$${{\rm \psi} _2} = \displaystyle{{2(1 + i{{\rm \psi} _1}){{\rm \omega} _0}{{\left \vert {{{{\rm \varepsilon ^{\prime}}}_0}} \right \vert} ^{3/2}}} \over {(1 - i{{\rm \psi} _1})c{{\left \vert {1 + {{{\rm \varepsilon ^{\prime}}}_0}} \right \vert} ^{1/2}}({{{\rm \varepsilon ^{\prime}}}_0}^2 - 1)}}$$

$$I = \int\limits_{ - \infty} ^\infty {\displaystyle{{{e^{ - {{\rm p}^2}}}(\,p - {{\rm \sigma} _1}b/2 + i{{\rm \sigma} _2}b/2)} \over {({{(\,p - {{\rm \sigma} _1}b/2)}^2} + {{({{\rm \sigma} _2}b/2)}^2})}}} {\rm dp}$$

In the above expression η_g = 1.5, ψ₂exp(−2α₀a) = σ₁ + iσ₂, α₀² = k _0z² − (ω₀²/c ²)ε₀′ and α₀^′2 = k _0z² − (ω₀²/c ²). The dielectric constant of the metal at frequency ω₀ is given by ε₀^′ = ε_l − ω_p²/ω₀². ε_l and ${{\rm \omega} _{\rm p}}\, = \,\sqrt {{{{n_0}{e^2}} / {m{\rm \varepsilon}}}} $ are the lattice permittivity of metal and plasma frequency, respectively, ε = 8.854 × 10⁻¹² f/m, n ₀ = 5.85 × 10²⁸/m³ for silver metal; −e and m is the charge and effective mass of electron, respectively. On applying conditions of continuity ε₀^′E _0x and E _0z at x = 0 along x and z directions, the dispersion relation of the SPW is given by

(4)$${k_{0z}}\, = \,\displaystyle{{{{\rm \omega} _0}} \over c}\sqrt {\displaystyle{{{{{\rm \varepsilon ^{\prime}}}_0}} \over {1 + {{{\rm \varepsilon ^{\prime}}}_0}}}} $$

The pump SPW imparts oscillatory velocity to electrons$({\vec v_0} = e{\vec E_0}/mi{{\rm \omega} _0})$. At resonance, the pump SPW decay into a quasi-electrostatic plasma wave $({\rm \omega}, {\vec k_z})$ of potential (ϕ),

(5)$${\rm \phi} \, = \,A(x){e^{ - i({\rm \omega} t - {k_{\rm z}}\hat z)}}$$

and a sideband SPW $({{\rm \omega} _1},{\vec k_{1z}};{\vec k_1} = {k_{1z}}\hat z - i{{\rm \alpha} _1}\hat x)$ of electric field $({\vec E_1})$$

(6)$${\vec E_1} = {\vec A_{1{\rm z}}}(x)e ^{- i({{\rm \omega }_1}t - {k_{1{\rm z}}}\hat z)} $$

where ${A_{1{\rm z}}} = {A_1}\left( {\hat z - \displaystyle{{i{k_{1{\rm z}}}} \over {{{\rm \alpha} _{\rm 1}}}}\hat x} \right){e^{{\rm \alpha} 1x}}$ (x < 0, metal)

${A_{1{\rm z}}} = {A_1}\left( {\hat z + \displaystyle{{i{k_{1{\rm z}}}} \over {{\rm \alpha} {{\rm ^{\prime}}_{\rm 1}}}}\hat x} \right){e^ -} ^{{\rm \alpha} 1x} $ (x > 0, vacuum)

where α₁² = k _1z² − (ω₁²/c ²)ε₁ and α₁^′2 = k _1z² − (ω₁²/c ²). The dielectric constant of sideband SPW at frequency ω₁ is ε₁ = ε_l − ω_p²/ω₁². The phase matching conditions for parametric decay are ${\vec k_{\rm z}} = {\vec k_{0{\rm z}}} + {\vec k_{1{\rm z}}}$ and ω = ω₀ + ω₁. The sideband SPW imparts oscillatory velocity to electrons $({\vec v_1} = e{\vec E_1}/mi{{\rm \omega} _1})$ and couples nonlinearly with the pump wave to exert a ponderomotive force on electrons at frequency ω, which is given by

(7)$${\vec F_{\rm p}} = - m[\vec v.\nabla \vec v\,] - e[\vec v \times \vec B]$$

Replace $\vec v$ by ${\vec v_0} + {\vec v_1}$ and ${\vec B}$ by ${\vec B_0} + {\vec B_1}$ in the above equation, where ${\vec B_0} = i{\vec k_0} \times {\vec E_0}/i{{\rm \omega} _0}$ and ${\vec B_1} = i{\vec k_1} \times {\vec E_1}/i{{\rm \omega} _1}$ for pump and sideband SPW, respectively. On substituting these values into Eq. (7), we obtained

(8)$${\vec F_{\rm p}} = e\nabla {\Phi _{\rm p}}$$

where ${\Phi _{\rm p}}\, = \,{A_{0s}}(x)\;{A_1}(x)\left( {\displaystyle{e \over {2m{{\rm \omega} _0}{{\rm \omega} _1}}}} \right)\left( {1 \!- \displaystyle{{{k_{0z}}{k_{1z}}} \over {{{\rm \alpha} _0}{{\rm \alpha} _1}}}} \right){e^{ - i({\rm \omega} t - {k_{\rm z}}z)}}$, termed as ponderomotive potential.

This ponderomotive force along with self-consistent low-frequency field on the metal free electrons drive plasma oscillations at frequency ω. Hence, the equation of motion of electron becomes

$$m\displaystyle{{\partial {{\vec v}_{\rm \omega}}} \over {\partial t}} = e\nabla ({\rm \phi} + {\Phi _{\rm p}}) - m{\rm \upsilon} \,{\vec v_{\rm \omega}} $$

where υ is the collision frequency. Here, the collisions are considered effective only for low-frequency plasma wave for which phase velocity is close to the Fermi velocity of free electrons. For high-frequency SPWs collisional effects are neglected. The oscillatory velocity $({\vec v_{\rm \omega}} )$ of electron at frequency (ω) is obtained by solving above equation of motion, given by

(9)$${\vec v_{\rm \omega}} = - \displaystyle{{e\left[ { - i\nabla {\Phi _{\rm p}} - i\displaystyle{{\partial {\rm \phi}} \over {\partial x}}\hat x + {k_z}{\rm \phi} \,\hat z} \right]} \over {m({\rm \omega} + i{\rm \upsilon} )}}$$

Substituting the value of oscillatory velocity from Eq. (9) into continuity equation $(\partial n/\partial t + \nabla \cdot ({n_0}{\vec v_{\rm \omega}} ) = 0)$, we obtain the density perturbation (n) due to oscillatory motion of metal electrons.where χ_e = −ω_p²/ω (ω +iυ).

n ₀ is the electron density inside the metal. Using n in Poisson's equation, we get

$${\nabla ^2}{\rm \phi} = \displaystyle{{ - {\chi _e}} \over {({{\rm \varepsilon} _{\rm l}} + {\chi _e})}}\left[ { - k_z^2 + {{({{\rm \alpha} _0} + {{\rm \alpha} _1})}^2}} \right]{\Phi _{\rm p}}$$

The density perturbations couples with oscillatory movement of metal electrons at frequency ω₀ (due to pump SPW) and excites a nonlinear current density $(\vec J_1^{{\,\,\rm NL}} )$ at the sideband frequency ω₁, which is given by

(11)$$\vec J_1^{{\,\,\rm NL}} = \displaystyle{1 \over 2}env_0^{\ast} $$

On substituting values of n and $v_0^{\ast} $ ($v_0^{\ast} = - e{\vec E_0}/mi{{\rm \omega} _0}$, oscillatory velocity of electrons at frequency ω₀), the nonlinear current density is given by

(12)$$\vec J_1^{{\,\,\rm NL}} = \displaystyle{{ie{{\rm \chi} _e}} \over {8{\rm \pi} m{{\rm \omega} _0}}}\left( {\displaystyle{{{{\rm \varepsilon} _{\rm l}}} \over {{{\rm \varepsilon} _{\rm l}} + {{\rm \chi} _e}}}} \right){\nabla ^2}{\Phi _{\rm p}}\vec E_0^{\ast} $$

This nonlinear current density $(\vec J_1^{{\,\,\rm NL}} )$ is responsible for the growth of sideband SPW whose characteristic equation can be derived by solving wave equation. The wave equation governing electric field of SPW at the sideband frequency can be written as

(13)$${\nabla ^2}{\vec E_1} - \nabla (\nabla \cdot {\vec E_1}) = \displaystyle{{4{\rm \pi}} \over {{c^2}}}\displaystyle{{\partial \vec J_{1}^{\,\,{\rm NL}}} \over {\partial t}} + \displaystyle{{{{\rm \varepsilon} _1}} \over {{c^2}}}\displaystyle{{{\partial ^2}{{\vec E}_1}} \over {\partial {t^2}}}$$

Taking the divergence of the Eq. (13), we obtain

(14)$$\nabla \cdot {\vec E_1} = \displaystyle{{4{\rm \pi} i} \over {{{\rm \omega} _1}{{\rm \varepsilon} _1}}}\nabla \cdot \vec J_1^{{\,\,\rm NL}} $$

Substituting Eq. (14) into Eq. (13), we obtain

(15)$${\nabla ^2}{\vec E_1} - \displaystyle{{{{\rm \varepsilon} _1}} \over {{c^2}}}\displaystyle{{{\partial ^2}{{\vec E}_1}} \over {\partial {t^2}}} = \displaystyle{{4{\rm \pi}} \over {{c^2}}}\displaystyle{{\partial \vec J_1^{{\,\,\rm NL}}} \over {\partial t}} + \displaystyle{{4{\rm \pi}} \over {i{{\rm \omega} _1}{{\rm \varepsilon} _1}}}\nabla (\nabla \cdot \vec J_1^{{\,\,\rm NL}} )$$

After the Fourier analysis of Eq. (15), we have

(16)$$\displaystyle{{{\rm \omega} _1^2 - {\rm \omega} _{1{\rm r}}^2} \over {{c^2}}}{{\rm \varepsilon} _1}{\vec E_1} = \displaystyle{{4{\rm \pi} i{{\rm \omega} _1}} \over {{c^2}}}\vec J_1^{{\,\,\rm NL}} \displaystyle{{4{\rm \pi} i{{\rm \omega} _1}} \over {{\rm \omega} _1^2 {{\rm \varepsilon} _1}}}\nabla (\nabla \cdot \vec J_1^{{\,\,\rm NL}} )$$

Taking the dot product of Eq. (16) with $\vec E_1^{\ast} $ and integrating from −∞ to ∞ with respect to x, we have

(17)$$\eqalign{& \displaystyle{{{\rm \omega }_1^2 - {\rm \omega }_{1{\rm r}}^2 } \over {{c^2}}}\left[ {\int_{ - \infty }^0 {{{\rm \varepsilon }_1}{{\vec E}_1} \cdot \vec E_1^{\ast} {\rm d}x} + \int_0^\infty {{{\vec E}_1} \cdot \vec E_1^{\ast} {\rm d}x} } \right] \cr & \quad = - \displaystyle{{4{\rm \pi }i{{\rm \omega }_1}} \over {{c^2}}}\int_{ - \infty }^0 {\vec J_{1}^{\,\,{\rm NL}} \cdot \vec E_1^{\ast} {\rm d}x} - \displaystyle{{4{\rm \pi }i{{\rm \omega }_1}} \over {{\rm \omega }_1^2 {{\rm \varepsilon }_1}}} \cr & \quad \quad \int_{ - \infty }^0 {\nabla (\nabla \cdot \vec J_{1}^{\,\,{\rm NL}} )} \cdot \vec E_1^{\ast} {\rm d}x} $$

Here, ε₁ = ε_l − ω_p²/ω₁² for metal (x < 0) and ε₁ = 1 for vacuum (x > 0). Further, substituting the value of nonlinear current from Eq. (12). Then Eq. (17)

(18)$$\eqalign{ {\rm \omega }_1^2 - {\rm \omega }_{1{\rm r}}^2 & = - \displaystyle{{{e^2}{k^2}{{\rm \chi }_e}} \over {4{m^2}{\rm \omega }_0^2 {\rm \omega }_1^2 {{\rm \varepsilon }_1}}}\left( {\displaystyle{{{{\rm \varepsilon }_{\rm l}}} \over {{{\rm \varepsilon }_{\rm l}} + {{\rm \chi }_e}}}} \right)\left( {1 - \displaystyle{{{k_{0z}}{k_{1z}}} \over {{{\rm \alpha }_0}{{\rm \alpha }_1}}}} \right)\left( {\displaystyle{{{{\rm \alpha }_1}{{{\rm \alpha '}}_1}} \over {{{\rm \alpha }_0} + {{\rm \alpha }_1}}}} \right) \cr & \quad \displaystyle{{\left[ {{\rm \omega }_1^2 {{\rm \varepsilon }_1} - {c^2}k_{1z}^2 + {c^2}{{(2{{\rm \alpha }_0} + {{\rm \alpha }_1})}^2}} \right]} \over {({{\rm \varepsilon }_1}{{{\rm \alpha '}}_1} - {{\rm \alpha }_1})}}\vert{E_0}\vert{^2}} $$

where χ_e = − ω_p²/ω (ω +iυ), ${k_{{\rm 0z}}} = {{\rm \omega} _0}/c\sqrt {{{{\rm \varepsilon '}}_0}/1 + {{{\rm \varepsilon '}}_0}} $, ${k_{1{\rm z}}} = $$ {{\rm \omega} _1}/c\sqrt {{{\rm \varepsilon} _1}/1 + {{\rm \varepsilon} _1}} $, α₀² = k _0z² − (ω₀²/c ²)ε₀^′, α₁² = k _1z² − (ω₁²/c ²)ε₁ and α₁^′2 = k _1z² − (ω₁²/c ²). Eq. (18) is the dispersion relation for Compton process. Growth rate (γ) for the process can be deduced by solving Eq. (18) that comes out as follows:

(19)$$\eqalign{ {\rm \gamma }\, & = \, - \displaystyle{{{e^2}{k^2}{\rm \varepsilon }_l^2 {\rm \omega} {\rm \upsilon }} \over {8{m^2}{\rm \omega }_0^2 {\rm \omega }_{\rm p}^{\rm 2} {{({\rm \omega } - {{\rm \omega }_0})}^3}}}\left( {1 - \displaystyle{{{k_{0z}}{k_{1z}}} \over {{{\rm \alpha }_0}{{\rm \alpha }_1}}}} \right)\left( {\displaystyle{{{{\rm \alpha }_1}{{{\rm \alpha '}}_1}} \over {{{\rm \alpha }_0} + {{\rm \alpha }_1}}}} \right) \cr & \quad \left( {\displaystyle{{{{({\rm \omega } - {{\rm \omega }_0})}^2}{{\rm \varepsilon }_1} - {c^2}k_{{\rm 1z}}^{\rm 2} + {c^2}{{(2{{\rm \alpha }_0} + {{\rm \alpha }_1})}^2}} \over {({{\rm \varepsilon }_1}{{{\rm \alpha '}}_1} - {{\rm \alpha }_1})}}} \right){\left\vert {{E_0}} \right\vert^2}} $$

where ω₁ = ω_1r+iγ  and k ² = (α₀ + α₁)² − (k _0z + k _1z)².

Eq. (19) is normalized and numerically solved using the parameters ε_l = 4, T _e = 63600 K (Fermi temperature for silver) and ω_p = 2.17 × 10¹⁵ rad/s. The growth rate of Compton process increases with the frequency of incident laser and turns out to be 2.5 × 10⁻⁵ ω_p at laser frequency ω₀ = 0.35ω_p for incident laser amplitude A _0L = 0.3 mω_pc/e, laser spot size b = 100c/ω_p. Here, width of the metal is (aω₀/c) = 0.5 and free electron density of silver metal is n ₀ = 5.85 × 10²⁸/m³. The frequency of quasi-static plasma mode can be calculated from the resonance condition ${\vec k_{0{\rm z}}}\, = \,{\vec k_{\rm z}} - {\vec k_{1{\rm z}}}$ where ${\vec k_{\rm z}} = {\rm \omega} /{v_{\rm F}}$. Here, v _F is the Fermi velocity of electrons for silver. Figure 2 depicts the variation of plasma mode frequency (ω/ω_p) with pump frequency (ω/ω_p). Here, plasma frequency increases linearly with frequency of pump wave. This is due to the fact that with increase in pump frequency, there is corresponding increase in its wave number. Hence, to satisfy the resonance conditions, there should be a related increase in the plasma mode frequency. Amplitude of SPW (eA _0S/mω_pc) is plotted as a function of incident laser spot size (bω_p/c) on varying width of the metal film (bω₀/c) and amplitude of laser (eA _0L/mω_pc) in Figures 3a and 3b, respectively. Figure 3a shows that for smaller values of spot size, the amplitude of SPW increases linearly and saturates at higher values of bω_p/c. From Eq. (3), the amplitude of SPW is larger than the laser amplitude for the condition bω_p/c > exp(aω₀/c), where b and a are the spot size of laser and the width of the metal, respectively (Liu & Tripathi, Reference Liu and Tripathi2000). On increasing b and keeping a fixed, the inequality grows, resulting in increased amplitude of the SPW as observed in Figure 3a. The saturation value of amplitude is larger for metal films of larger width and it occurs at larger spot size of the laser. For (aω₀/c = 0.5), the amplitude of SPW further increases with the laser amplitude as observed in Figure 3b. As the laser intensity increases, evanescent wave provides higher oscillatory velocity to the electrons and couple with the pump wave to exert ponderomotive on electron at (${\rm \omega}, \,{\vec k_{\rm z}}$) and excites the SPW of higher amplitude. Enhanced amplitude of SPW at metal free space interface is reported by Raether (Reference Raether1988) and Liu and Tripathi, (Reference Liu and Tripathi1998).

Fig. 2. Variation of normalized plasma wave frequency (ω/ω_p) with normalized pump wave frequency (ω₀/ω_p).

Fig. 3. Plot of surface plasma wave amplitude versus laser spot size (bω_p/c) on varying (a) width of the metal layer (bω₀/c) for (eA _0L/mω_pc) = 0.3 and (b) laser amplitude (eA _0L/mω_pc) for (aω₀/c) = 0.5.

Figure 4 shows the variation of SPW amplitude with pump frequency and width of the metal. Other parameters are (eA _0L/mω_pc) = 0.3 and (bω_p/c) = 100. As the metal thickness increases, the evanescent field has to penetrate longer distances to excite the SPW. Also, the condition [bω₀/c > exp(aω₀/c)] approaches towards equality on increasing a, resulting in decrease of SPW amplitude with the metal width as observed in Figure 4. In Figures 5a and 5b, growth rate (γ/ω_p) of the Compton process is plotted as a function of pump frequency for various combinations of laser spot size and its amplitude for (aω₀/c) = 0.5. The growth rate of the Compton process increases with the pump frequency because SPW is strongly localized near the interface and propagates along the interface with low attenuation. Increased amplitude of the SPW at the metal free space interface as observed in Figures 3a and 3b enhances the ponderomotive force [Eq. (8)] on electrons. As a result, the growth rate increases with incident laser spot size and its amplitude as shown in Figures 5a and 5b, respectively. Lee and Cho (Reference Lee and Cho1999) theoretically investigated the decay of high-frequency light wave into two daughter SWs and reported the increase in growth rate of the surface wave with the light wave frequency. Drake et al. (Reference Drake, Baldis, Kruer, Williams, Estabrook, Johnston and Young1990) have experimentally studied the stimulated Compton scattering on electrons from laser produced plasma and reported that stimulated Compton scattering increases with the laser intensity. Variation of growth rate of the Compton process with width of the metal is plotted in Figure 6. Other parameters are same as Figure 4. Increased width of the metal surface leads to lesser amplitude of SPW at the metal free space interface and lesser ponderomotive force on electrons to drive the quasi-static plasma mode, resulting in decrease of growth rate with the metal width.

Fig. 4. Plot of surface plasma wave amplitude versus pump frequency and width of metal. The other parameters are (eA _0L/mω_pc) = 0.3 and (bω_p/c) = 100.

Fig. 5. Plot of growth rate (γ/ω_p) of the Compton process versus pump frequency on varying (a) spot size of laser for (eA _0L/mω_pc) = 0.3 and (b) its amplitude for (bω_p/c) = 100 at (aω₀/c) = 0.5.

Fig. 6. Variation of growth rate of the Compton process versus pump frequency on varying width of the metal. The other parameters are (eA _0L/mω_pc) = 0.3 & (bω_p/c) = 100.