2. FORMULATION AND GOVERNING EQUATIONS

Consider a vacuum–plasma interface at z = 0; z < 0 being vacuum and z > 0 a uniform underdense plasma of density n ₀ that is set in a ${\bf B}( = B_0\hat y)$ transverse magnetic field. A p-polarized laser beam (propagating in the x–z plane) incidents on the interface at an angle of incidence θ as shown in Figure 1.

(1) $$E_0 = (\hat x\cos {\rm \theta} - \hat z\sin {\rm \theta} )\displaystyle{1 \over 2}E_0\exp [i(k_{0x}x + k_{0z}z - {\rm \omega} _0t)] + c.c,$$

where E ₀ is the amplitude of the incident electric field, k _0x  = (ω₀/c)sin θ and k _0z  = (ω₀/c)cos θ are the wavenumber components along the x and z directions, respectively. The electric and magnetic fields of the transmitted laser can be written as

(2) $$\eqalign{E_1 = (\hat x({\rm \epsilon} _1 - \mathop {\sin} \nolimits^2 {\rm \theta} )^{1/2} - \hat z\sin {\rm \theta} )\displaystyle{1 \over 2}E_1\exp [i(k_{1x}x + k_{1z}z - {\rm \omega} _0t)] + c.c,}$$

(3) $$B_1 = ck_1 \times E_1/{\rm \omega} _0,$$

where k _1x  = k _0x , $k_{1z} = ({\rm \omega} _0^2 /c^2{\rm \epsilon} _1 - k_{1x}^2 )^{1/2}$ , and ${\rm \epsilon} _1 = k_1^2 c^2/{\rm \omega} _0^2 $ is the dielectric constant at frequency ω₀. $E_1 = 2\sqrt {{\rm \epsilon} _1} E_0/({\rm \epsilon} _1+ \sqrt {{\rm \epsilon} _1 - \mathop {\sin} \nolimits^2 {\rm \theta}} /\cos {\rm \theta} )$ is the amplitude of the fundamental electric field transmitted into the plasma. The SH electric field can be written as

(4) $$\eqalign{E_2 = (\hat x({\rm \epsilon} _2 - \mathop {\sin} \nolimits^2 {\rm \theta} )^{1/2} - \hat z\sin {\rm \theta} )\displaystyle{1 \over 2}E_2\exp [i(k_{2x}x + k_{2z}z - 2{\rm \omega} _0t)] + c.c,}$$

where k _2x  = (2ω₀sin θ)/c, $k_{2z} = (4{\rm \omega} _0^2 /c^2{\rm \epsilon} _2 - k_{2x}^2 )^{1/2}$ . Furthermore ${\rm \epsilon} _2 = k_2^2 c^2/4{\rm \omega} _0^2 $ and E ₂ are the dielectric constants at frequency 2ω₀ and the SH wave amplitude, respectively. The wave equation governing the propagation of electromagnetic waves through the plasma is given by

(5) $$\nabla ^2{\bf E} - \nabla (\nabla. {\bf E}) - \displaystyle{1 \over {c^2}}\displaystyle{{\partial ^2{\bf E}} \over {\partial t^2}} = \displaystyle{{4{\rm \pi}} \over {c^2}}\displaystyle{{\partial {\bf J}} \over {\partial t}},$$

where J is the plasma electron current density. The governed equations on electron motion are given by the set of relativistic cold electron fluid equation

(6) $$\displaystyle{{d({\rm \gamma} {\bi v})} \over {dt}} = - \displaystyle{{e{\bf E}} \over m} - \displaystyle{e \over {mc}}{\bi v} \times ({\bf B} + {\bf B}_{\bf 0}),$$

and the continuity equation

(7) $$\displaystyle{{\partial n} \over {\partial t}} + \nabla. (n{\bi v}) = 0,$$

where γ is the relativistic factor, v and n are the velocity and density of plasma electrons, B and B ₀ are the magnetic fields corresponding to the radiation laser pulse and a self-generated magnetic field. We would like to start with perturbative theory and expand all quantities in orders of the radiation field amplitude. By making use of Eq. (6), the velocity components in the x and z directions are, respectively, given by

(8) $$\displaystyle{{\partial v_{1x}} \over {\partial t}} = - \displaystyle{{eE_x} \over m} + {\rm \omega} _{\rm c}v_{1z},$$

and

(9) $$\displaystyle{{\partial v_{1z}} \over {\partial t}} = \displaystyle{{eE_z} \over m} - {\rm \omega} _{\rm c}v_{1x},$$

where ω_c(= eB ₀/mc) represents the cyclotron frequency and E _x,z  = E _1(x,z) + E _2(x,z) in which E _1(x,z) and E _2(x,z) are the amplitudes of electric field in fundamental and SH. By making use of Eqs (2)–(4), we can solve Eqs (8) and (9) to derive the first order of velocity components as

(10) $$\eqalign{v_{1x} &= \left( {\displaystyle{{i{\rm \omega} _0^2 a_{1x}c} \over {{\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2}} + \displaystyle{{{\rm \omega} _0{\rm \omega} _{\rm c}a_{1z}c} \over {{\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2}}} \right)e^{i(k_{1x}x + k_{1z}z - {\rm \omega} _0t)} \cr & \quad + \left( {\displaystyle{{2i{\rm \omega} _0^2 a_{2x}c} \over {{\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2}} + \displaystyle{{{\rm \omega} _0{\rm \omega} _{\rm c}a_{2z}c} \over {{\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2}}} \right)e^{i(k_{2x}x + k_{2z}z - 2{\rm \omega} _0t)} + {\rm c}{\rm. c}{\rm.},} $$

(11) $$\eqalign{v_{1z} &= \left( {\displaystyle{{{\rm \omega} _0{\rm \omega} _{\rm c}a_{1x}c} \over {{\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2}} - \displaystyle{{i{\rm \omega} _0^2 a_{1z}c} \over {{\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2}}} \right)e^{i(k_{1x}x + k_{1z}z - {\rm \omega} _0t)} \cr & \quad + \left( {\displaystyle{{{\rm \omega} _0{\rm \omega} _{\rm c}a_{2x}c} \over {{\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2}} - \displaystyle{{2i\omega _0^2 a_{2z}c} \over {{\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2}}} \right)e^{i(k_{2x}x + k_{2z}z - 2{\rm \omega} _0t)} + {\rm c}{\rm. c}{\rm.}} $$

Where a _j(x,z) = eE _j(x,z)/mω₀ c (j = 1,2) are the components of normalized amplitude of the transmitted fundamental and SH electric field.

Fig. 1. Schematic representation of second-harmonic generation process.

By following the same steps, the expression for velocity components in second order is given by

(12) $$\displaystyle{{\partial v_{2x}} \over {\partial t}} = \displaystyle{e \over {mc}}v_{1z}B_{1y} - v_{1x}\displaystyle{{\partial v_{1x}} \over {\partial x}} - v_{1z}\displaystyle{{\partial v_{1x}} \over {\partial z}} + {\rm \omega} _{\rm c}v_{2z},$$

and

(13) $$\displaystyle{{\partial v_{2z}} \over {\partial t}} = - \displaystyle{e \over {mc}}v_{1x}B_{1y} - v_{1z}\displaystyle{{\partial v_{1z}} \over {\partial z}} - v_{1x}\displaystyle{{\partial v_{1z}} \over {\partial x}} - {\rm \omega} _{\rm c}v_{2x}.$$

The second-order velocity components are obtained by using of Eqs (10) and (11) and simultaneous solutions of Eqs (12) and (13) as bellow

(14) $$\eqalign{ v_{2x}{\rm } &= \displaystyle{{{\rm \epsilon }_1a_1^2 \omega _0^2 c^2} \over {{(\omega _{\rm c}^2 - \omega _0^2 )}^2(\omega _{\rm c}^2 - 4\omega _0^2 )}}\left[ {\displaystyle{1 \over {\omega _0}}(2\omega _0^4 - \omega _{\rm c}^4 - 4\omega _0^2 \omega _{\rm c}^2 )k_{1x}} \right. \cr & \quad \left. - i\omega _{\rm c}(4\omega _{\rm c}^2 - \omega _0^2 )k_{1z} \right] e^{2i(k_{1x}x + k_{1z}z - \omega _0t)} + {\rm c}.{\rm c}.,}$$

(15) $$\eqalign{ v_{2z} &= \displaystyle{{{\rm \epsilon }_1a_1^2 \omega _0^2 c^2} \over {{(\omega _{\rm c}^2 - \omega _0^2 )}^2(\omega _{\rm c}^2 - 4\omega _0^2 )}} \left[ i\omega _{\rm c}(4\omega _{\rm c}^2 - \omega _0^2 )k_{1x}. \right.\quad \cr & \quad \left. { + \displaystyle{1 \over {\omega _0}}(2\omega _0^4 - \omega _{\rm c}^4 - 4\omega _0^2 \omega _{\rm c}^2 )k_{1z}} \right]e^{2i(k_{1x}x + k_{1z}z - \omega _0t)} + {\rm c}.{\rm c}.}$$

Using Eq. (7), the first-order density perturbation is given as follows

(16) $$\eqalign{ n_1 &= \displaystyle{{n_0a_1{\rm \omega} _{\rm c}k_1^2 c^2} \over {{\rm \omega} _0({\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2 )}}e^{i(k_{1x}x + k_{1z}z - {\rm \omega} _0t)} \cr & \quad + \displaystyle{{n_0a_2{\rm \omega} _{\rm c}k_2^2 c^2} \over {4{\rm \omega} _0({\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2 )}}e^{i(k_{2x}x + k_{2z}z - 2{\rm \omega} _0t)} + {\rm c}{\rm. c}.} $$

We find from Eq. (16) that the density perturbation disappears in the absence of external magnetic field. The plasma current density ${\bf J} = - en{\bi v}$ components can be derived as

(17) $$\eqalign{J_x &= - \displaystyle{{n_0e{\rm \omega} _0c} \over {{\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2}} (i{\rm \omega} _0a_{1x} + {\rm \omega} _{\rm c}a_{1z})e^{i(k_{1x}x + k_{1z}z - {\rm \omega} _0t)} \cr & \quad - \displaystyle{{n_0e{\rm \omega} _0c} \over {{\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2}} (2i{\rm \omega} _0a_{2x} + {\rm \omega} _{\rm c}a_{2z})e^{i(k_{2x}x + k_{2z}z - 2{\rm \omega} _0t)} \cr & \quad - \displaystyle{{n_0ec{\rm \epsilon} _1a_1^2 {\rm \omega} _0^2} \over {{({\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2 )}^2({\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2 )}}\left[ { - 3i{\rm \omega} _{\rm c}({\rm \omega} _0^2 + {\rm \omega} _{\rm c}^2 )k_{1z}c} \right. \cr & \quad \left. {\, +\, 2{\rm \omega} _0({\rm \omega} _0^2 - 4{\rm \omega} _{\rm c}^2 )k_{1x}c} \right]e^{2i(k_{1x}x + k_{1z}z - {\rm \omega} _0t)} + {\rm c}{\rm. c}.,} $$

and

(18) $$\eqalign{J_z &= - \displaystyle{{n_0e{\rm \omega} _0c} \over {{\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2}} ({\rm \omega} _{\rm c}a_{1x} - i{\rm \omega} _0a_{1z})e^{i(k_{1x}x + k_{1z}z - {\rm \omega} _0t)} \cr & \quad - \displaystyle{{n_0e{\rm \omega} _0c} \over {{\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2}} ({\rm \omega} _{\rm c}a_{2x} - 2i{\rm \omega} _0a_{2z})e^{i(k_{2x}x + k_{2z}z - 2{\rm \omega} _0t)} \cr & \quad - \displaystyle{{n_0ec{\rm \epsilon} _1a_1^2 {\rm \omega} _0^2} \over {{({\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2 )}^2({\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2 )}}\left[ {2{\rm \omega} _0({\rm \omega} _0^2 - 4{\rm \omega} _{\rm c}^2 )k_{1z}c} \right. \cr & \quad + 3i{\rm \omega} _{\rm c}({\rm \omega} _0^2 + {\rm \omega} _{\rm c}^2 )k_{1x}c]e^{2i(k_{1x}x + k_{1z}z - {\rm \omega} _0t)} + {\rm c}{\rm. c}{\rm.}} $$

Substituting the linear parts of current density into the wave equation, we get to the linear dispersion relations for fundamental and SH.

(19) $$k_1^2 c^2 = {\rm \omega} _0^2 - \displaystyle{{{\rm \omega} _{\rm p}^2 ({\rm \omega} _0^2 - {\rm \omega} _{\rm p}^2 )} \over {{\rm \omega} _0^2 - {\rm \omega} _{\rm p}^2 - {\rm \omega} _{\rm c}^2}}, $$

and

(20) $$k_2^2 c^2 = 4{\rm \omega} _0^2 - \displaystyle{{{\rm \omega} _{\rm p}^2 (4{\rm \omega} _0^2 - {\rm \omega} _{\rm p}^2 )} \over {4{\rm \omega} _0^2 - {\rm \omega} _{\rm p}^2 - {\rm \omega} _{\rm c}^2}}. $$

Equations (19) and (20) introduce the dispersion relation for extraordinary waves in fundamental and SH propagating perpendicular to the magnetic field.

3. SH GENERATION

In order to evaluate the amplitude of the SH field, the source currents, Eqs (17) and (18), can be used in the wave Eq. (5) and equate the SH terms on both sides,

(21) $$\displaystyle{{\partial ^2a_{2x}(z)} \over {\partial z^2}} + {\rm \chi} _1\displaystyle{{\partial a_{2x}(z)} \over {\partial z}} + {\rm \chi} _2a_{2x}(z) + {\rm \chi} _3a_{2z}(z) = {\rm \chi} _4e^{i\Delta k.z},$$

and

(22) $$\displaystyle{{\partial ^2a_{2z}(z)} \over {\partial z^2}} + {\rm \chi} _1\displaystyle{{\partial a_{2z}(z)} \over {\partial z}} + {\rm \chi} _2a_{2z}(z) - {\rm \chi} _3a_{2x}(z) = {\rm \chi} _5e^{i\Delta k.z},$$

where

$$\eqalign{{\rm \chi} _1 &= 2ik_{2z}, \cr {\rm \chi} _2 &= \displaystyle{{{\rm \omega} _{\rm c}^2 {\rm \omega} _{\rm p}^4} \over {c^2(4{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 )(4{\rm \omega} _0^2 - {\rm \omega} _{\rm p}^2 - {\rm \omega} _{\rm c}^2 )}}, \cr {\rm \chi} _3 &= \displaystyle{{i{\rm \omega} _{\rm c}{\rm \omega} _{\rm p}^4 (4{\rm \omega} _0^2 - {\rm \omega} _{\rm p}^2 )} \over {2{\rm \omega} _0c^2(4{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 )(4{\rm \omega} _0^2 - {\rm \omega} _{\rm p}^2 - {\rm \omega} _{\rm c}^2 )}}, \cr {\rm \chi} _4 &= \displaystyle{{2{\rm \varepsilon} _1a_1^2 {\rm \omega} _0^2 {\rm \omega} _{\rm p}^2} \over {c^2{({\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2 )}^2({\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2 )}} \cr &\quad\times\left[ {2i{\rm \omega} _0({\rm \omega} _0^2 - 4{\rm \omega} _{\rm c}^2 )k_{1x}c + 3{\rm \omega} _{\rm c}({\rm \omega} _0^2 + {\rm \omega} _{\rm c}^2 )k_{1z}c} \right], \cr {\rm \chi} _5 &= \displaystyle{{2{\rm \varepsilon} _1a_1^2 {\rm \omega} _0^2 {\rm \omega} _{\rm p}^2} \over {c^2{({\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2 )}^2({\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2 )}} \cr &\quad \times\left[ {3{\rm \omega} _{\rm c}({\rm \omega} _0^2 + {\rm \omega} _{\rm c}^2 )k_{1x}c - 2i{\rm \omega} _0({\rm \omega} _0^2 - 4{\rm \omega} _{\rm c}^2 )k_{1z}c} \right].} $$

In this case, the wavenumber of the SH is more than twice as large as that for the fundamental wave at which Δk = 2k _1z  − k _2z is called the wavevector mismatch for SH radiation. We assume that ∂a _2(x,z)(z)/∂z considerably changes larger than the laser wavelength, means that ∂² a _2(x,z)(z)/∂z ² ≪ k _2z ∂a _2(x,z)(z)/∂z, which is known as the slowly varying envelope approximation (SVEA) and is almost valid in non-linear processes. Furthermore, assuming a _1(x,z) depletes very slightly with z so that the quantity $a_{1(x,z)}^2 $ can be considered to be independent of z. Therefore, solution or Eqs (21) and (22) lead us to the normalized amplitudes for the phase-mismatch SH inside the plasma and vacuum as

(23) $$ a_{2x}(z) = \left\{ \matrix{ \left[ a_{2x}(0)\cos \left( {{\chi _3} \over {\chi _1}}z \right) + a_{2z}(0)\sin \left( {{\chi _3} \over {\chi _1}}z \right) + \alpha \left( q_1\cos \left( {{\chi _3} \over {\chi _1}}z \right) \right.\right. \hfill \cr \left.\left. - q_2\sin \left( {{\chi _3} \over {\chi _1}}z \right) + q_2e^{(\chi _2 + i\chi _1\Delta k/(\chi _1)z)} \right)\right]e^{( - (\chi _2/\chi _1)z)}, \; \; \quad z \gt 0, \hfill \cr {A_{2r}} , \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad z \lt 0, \hfill} \right.$$

and

(24) $$ a_{2z}(z) = \left\{ \matrix{ \left[ a_{2z}(0)\cos \left( {{\chi _3} \over {\chi _1}}z \right) + a_{2x}(0)\sin \left( {{\chi _3} \over {\chi _1}}z \right) + \alpha \left( q_1\sin \left( {{\chi _3} \over {\chi _1}}z \right) \right.\right. \hfill \cr \left.\left. - q_2\cos \left( {{\chi _3} \over {\chi _1}}z \right) + q_2e^{(\chi _2 + i\chi _1\Delta k/(\chi _1)z)} \right)\right]e^{( - (\chi _2/\chi _1)z)}, \; \; \quad z \gt 0, \hfill \cr {A_{2r}^{\prime}} , \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad z \lt 0, \hfill} \right.$$

where

$$q_1 = {\rm \chi} _3{\rm \chi} _5 - {\rm \chi} _2{\rm \chi} _4 - i{\rm \chi} _1{\rm \chi} _4\Delta k,$$

$$q_2 = {\rm \chi} _3{\rm \chi} _4 + {\rm \chi} _2{\rm \chi} _5 + i{\rm \chi} _1{\rm \chi} _5\Delta k,$$

$${\rm \alpha} = \displaystyle{1 \over {{\rm \chi} _1\Delta k(2i{\rm \chi} _2 - {\rm \chi} _1\Delta k)}},$$

Here A _2r and $A_{2{\rm r}}^{\prime} $ are reflected SH amplitudes in x and z directions and a _2x (0) and a _2z (0) refer to the amplitudes of SH wave components at z = 0. Using $\nabla. a_2 = 0$ in the vacuum, we find $A_{2{\rm r}}^{\prime} = (k_{0x}/k_{0z})A_{2{\rm r}}$ . The magnetic field amplitude of the SH radiation derived as

(25) $$B_{2y}(z) = \left\{ {\matrix{ {\displaystyle{{mc^2} \over {2ie}}\left[ {ik_{2z}a_{2x}(z) + 2ik_{1x}a_{2z}(z) + e^{ - ({\rm \chi} _2/{\rm \chi} _1)z}( - {\rm \varphi} (z)a_{2x}(0) } \right.} \hfill & {} \hfill \cr {\left. {+ {\rm \phi} (z)a_{2z}(0) - {\rm \alpha} [q_1({\rm \varphi} (z) + i\Delta ke^{({\rm \chi} _2 + i{\rm \chi} _1\Delta k/({\rm \chi} _1)z)})} \right.}\hfill & {} \hfill \cr {\left. {+ q_2{\rm \phi} (z)])} \right],} \hfill \quad { z \gt 0,} & {} \hfill \cr{ - \displaystyle{{mc{\rm \omega} _0} \over {e\cos {\rm \theta}}} A_{2{\rm r}},} \hfill\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {z \lt 0,} \hfill \cr}} \right.$$

where

$$\eqalign{{\rm \varphi} (z) &= \displaystyle{{{\rm \chi} _2} \over {{\rm \chi} _1}}\cos \left( {\displaystyle{{{\rm \chi} _3} \over {{\rm \chi} _1}}z} \right) + \displaystyle{{{\rm \chi} _3} \over {{\rm \chi} _1}}\sin \left( {\displaystyle{{{\rm \chi} _3} \over {{\rm \chi} _1}}z} \right), \cr {\rm \phi} (z) &= \displaystyle{{{\rm \chi} _2} \over {{\rm \chi} _1}}\sin \left( {\displaystyle{{{\rm \chi} _3} \over {{\rm \chi} _1}}z} \right) - \displaystyle{{{\rm \chi} _3} \over {{\rm \chi} _1}}\cos \left( {\displaystyle{{{\rm \chi} _3} \over {{\rm \chi} _1}}z} \right).} $$

In order to obtain the amplitudes a _2x (0), a _2z (0), and A _2r, we apply the proper boundary conditions for electric and magnetic fields E _2x and B _2y and displacement vector εE _2z at z = 0. We get

(26) $$A_{2{\rm r}} = - a_{2x}(0),$$

(27) $$\eqalign{ {{2i\omega _0} \over {c\cos (\theta )}}A_{2{\rm r}} &+ \left[ {\displaystyle{{\chi _2} \over {\chi _1}} - ik_{2z}} \right]a_{2x}(0) + \left[ {\displaystyle{{\chi _3} \over {\chi _1}} - 2ik_{1x}} \right]a_{2z}(0) \cr &= \alpha \left[ {q_2\displaystyle{{\chi _3} \over {\chi _1}} - q_1\left( {\displaystyle{{\chi _2 + i\chi _1\Delta k} \over {\chi _1}}} \right)} \right].}$$

Integrating the Poisson equation $\nabla. {\rm \epsilon} {\bf E} = - en_1$ , we will get the following relation

(28) $$\displaystyle{{k_{0x}} \over {k_{0z}}}A_{2{\rm r}} - {\rm \epsilon} _2a_{2z}(0) = - \displaystyle{{i{\rm \omega} _{\rm c}{\rm \omega} _{\rm p}^2 k_2^2 c} \over {32{\rm \pi} k_{1x}{\rm \omega} _0^2 (4{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 )}}a_2(0).$$

The right-hand side of Eq. (28) is very small for underdense plasma $({\rm \omega} _{\rm p}^2 /{\rm \omega} _0^2 {\rm \ll} 1)$ , and we can neglect from this term. Therefore, we obtain from the set of Eqs (26–28) the SH waves amplitudes at plasma surface as

(29) $$\eqalign{ a_{2x}(0) &= - A_{2{\rm r}} \cr &= \displaystyle{{{\rm \alpha} [q_2({\rm \chi} _3/{\rm \chi} _1) - q_1({\rm \chi} _2 + i{\rm \chi} _1\Delta k/{\rm \chi} _1)]} \over {{\rm \chi} _2/{\rm \chi} _1 - ik_{2z} - 2i{\rm \omega} _0/c\cos ({\rm \theta} ) - \tan ({\rm \theta} )/{\rm \varepsilon} _2({\rm \chi} _3/{\rm \chi} _1 - 2ik_{1x})}},}$$

and

(30) $$a_{2z}(0) = \displaystyle{{{\rm \alpha} \tan ({\rm \theta} )/{\rm \epsilon} _2[q_1({\rm \chi} _2 + i{\rm \chi} _1\Delta k/{\rm \chi} _1) - q_2({\rm \chi} _3/{\rm \chi} _1)]} \over {{\rm \chi} _2/{\rm \chi} _1 - ik_{2z} - 2i{\rm \omega} _0/c\cos ({\rm \theta} ) - \tan ({\rm \theta} )/{\rm \epsilon} _2({\rm \chi} _3/{\rm \chi} _1 - 2ik_{1x})}}.$$

The well-known SH conversion efficiency (η) is given by

(31) $${\rm \eta} = \displaystyle{{{\rm \mu} _2} \over {{\rm \mu} _1}}\displaystyle{{ \vert a_2 \vert ^2} \over { \vert a_1 \vert ^2}},$$

where ${\rm \mu} _1 = \sqrt {{\rm \epsilon} _1} $ and ${\rm \mu} _2 = \sqrt {{\rm \epsilon} _2} $ are the refractive indexes corresponding to the fundamental and SH radiations, respectively.

3.1 Normal incidence

It is straightforward to show that for normal incidence of laser pulse, the coefficients χ₂, χ₃, and χ₅ do not appear in differential Eqs (21) and (22). Furthermore, just the x component of SH wave and y component of wavevector exist. Therefore, coupled Eqs (21) and (22) reduce to an differential equation for laser field amplitude in x direction as bellow

(32) $${\rm \chi} _1\displaystyle{{\partial a_2(z)} \over {\partial z}} = {\rm \chi} _4^{\prime} e^{i\Delta k.z},$$

where the coefficient χ₄ is changed to

$${\rm \chi} _4^{\prime} = \displaystyle{{6k_1{\rm \omega} _{\rm c}{\rm \omega} _0^2 {\rm \omega} _{\rm p}^2 a_1^2 ({\rm \omega} _0^2 + {\rm \omega} _{\rm c}^2 )} \over {c{({\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2 )}^2({\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2 )}}.$$

The detailed derivation of Eq. (32) is shown in Appendix. We obtain the transmitted SH radiation amplitude by solution of Eq. (32) at normal incidence as bellow

(33) $$a_2(z) = a_2(0) - \displaystyle{{i\chi _4^{\prime}} \over {k_2\Delta k}}e^{i\Delta kz/2}\sin \left( {\displaystyle{{\Delta k.z} \over 2}} \right).$$

Following Eqs (23), (25)–(27), for normal incidence, a ₂(0) can be derived as bellow

(34) $$a_2(0) = \displaystyle{{c{\rm \chi} _4^{\prime}} \over {2k_2(2{\rm \omega} _0 + k_2c)}}.$$

Above equations show that the amplitude of SH radiation is different from previous report by Jha et al. (Reference Jha, Mishra, Raj and Upadhyay2007). This difference is due to the reflection of SH radiation from vacuum–plasma interface, and changes by plasma density, strength of magnetic field, and laser intensity. Furthermore, the reflected SH radiation amplitude at normal incidence is given by

(35) $$a_2^{\prime} = - \displaystyle{{c{\rm \chi} _4^{\prime}} \over {2k_2(2{\rm \omega} _0 + k_2c)}}.$$

Therefore, we obtain conversion efficiencies of the SH in both sides of plasma boundary as

(36) $$\eqalign{ {\rm \eta} (z \gt 0) &= \displaystyle{{c{\rm \chi} _4^{^{\prime}2}} \over {2{\rm \omega} k_2a_0^2}} \displaystyle{{\mathop {\sin} \nolimits^2 (\Delta k.z/2)} \over {\Delta k^2}} + \displaystyle{{c{\rm \chi} _4^{\prime} a_2(0)} \over {{\rm \omega} \Delta ka_0^2}} \mathop {\sin} \nolimits^2 \left( {\displaystyle{{\Delta k.z} \over 2}} \right) \cr &+ \displaystyle{{\sqrt {{\rm \epsilon} _2} a_2^2 (0)} \over {a_0^2}}, }$$

(37) $${\rm \eta} (z \lt 0) = \displaystyle{{9k_1^2 {\rm \omega} _{\rm c}^2 {\rm \omega} _0^4 {\rm \omega} _{\rm p}^4 a_1^4 {({\rm \omega} _0^2 + {\rm \omega} _{\rm c}^2 )}^2} \over {k_2^2 a_0^2 {({\rm \omega} _{\rm c}^2 - 4{\rm \omega} _0^2 )}^2{(2{\rm \omega} _0 + k_2c)}^2{({\rm \omega} _{\rm c}^2 - {\rm \omega} _0^2 )}^4}}.$$

It is obvious from Eq. (36), the first term is in complete agreement with Jha et al. (Reference Jha, Mishra, Raj and Upadhyay2007), and the second and third terms are obtained by taking into account the reflected SH components contribution.

3.2 Non-magnetized plasma

It is clear that in the absence of the magnetic field the coefficients of χ₂ and χ₃ are zero and the other coefficients change to

$$\eqalign{{\rm \chi} _1 &= 2ik_{2z}, \cr {{{\rm \chi} ^{\prime \prime}}}_{\!\!\!4} &= - \displaystyle{{i{\rm \epsilon} _1k_{1x}a_1^2 {\rm \omega} _{\rm p}^2} \over {{\rm \omega} _0c}}, \cr {{{\rm \chi} ^{\prime \prime}}}_{\!\!\!5} &= \displaystyle{{i{\rm \epsilon} _1k_{1z}a_1^2 {\rm \omega} _{\rm p}^2} \over {{\rm \omega} _0c}}.} $$

In this case, the dispersion relations of fundamental and SH waves are expressed by Eqs (19) and (20) with ω_c = 0. After applying these changes in Eqs (23–30), the transmitted and reflected SH radiation amplitude are written as

(38) $$a_2(z) = \left[ {Ae^{i(\Delta k.z)}\mathop {\sin} \nolimits^2 \left( {\displaystyle{{\Delta k.z} \over 2}} \right) + iBe^{i\Delta k.z/2}\sin \left( {\displaystyle{{\Delta k.z} \over 2}} \right) - C} \right]^{1/2},$$

and

(39) $${a}^{\prime}_2 = \displaystyle{{i{\rm \omega} _{\rm p}^2 a_1^2 {\rm \epsilon} _1{\rm \epsilon} _2k_{1x}} \over {4{\rm \omega} _0^2 k_{2z}D}},$$

where

$$\eqalign{A &= \displaystyle{{{\rm \omega} _{\rm p}^4 a_1^4 {\rm \epsilon} _1^3} \over {k_{2z}^2 \Delta k^2c^4}}, \cr B &= - \displaystyle{{{\rm \omega} _{\rm p}^4 a_1^4 {\rm \epsilon} _1^2 k_{1x}^2 ({\rm \epsilon} _2k_{0z} + k_{1z})} \over {2{\rm \omega} _0^4 k_{2z}^2 \Delta kD}}, \cr C &= \displaystyle{{{\rm \omega} _{\rm p}^4 a_1^4 {\rm \epsilon} _1^2 k_{1x}^2 c^2({\rm \epsilon} _2^2 k_{0z}^2 + k_{1x}^2 )} \over {16{\rm \omega} _0^6 k_{2z}^2 D^2}}, \cr D &= \displaystyle{{k_{1x}^2 c^2} \over {{\rm \omega} _0^2}} - {\rm \epsilon} _2\left( {1 + \displaystyle{{k_{0z}k_{2z}c^2} \over {2{\rm \omega} _0^2}}} \right).} $$

We get the conversion efficiencies of the SH inside the plasma as well as vacuum as bellow:

(40) $$\eqalign{{\rm \eta} (z \gt 0) &= \left[ {\displaystyle{{{\rm \epsilon} _2} \over {a_0^4}} (A(A + 2B)\mathop {\sin} \nolimits^4 \left( {\displaystyle{{\Delta k.z} \over 2}} \right) + (B(B + 2C) } \right. \cr & \quad \left. {- 2AC\cos (\Delta k.z))\mathop {\sin} \nolimits^2 \left( {\displaystyle{{\Delta k.z} \over 2}} \right) + C^2)} \right]^{1/2},} $$

(41) $$\eqalign{&{\rm \eta} (z \lt 0) \cr&= \displaystyle{{{\rm \omega} _{\rm p}^4 a_0^2 {\rm \epsilon} _1^4 {\rm \epsilon} _2^2 \mathop {\cos} \nolimits^4 {\rm \theta} \mathop {\sin} \nolimits^2 {\rm \theta} {(\mathop {\sin} \nolimits^2 \theta - {\rm \epsilon} _2(1 + \cos \theta \sqrt {{\rm \epsilon} _2 - \mathop {\sin} \nolimits^2 {\rm \theta}} ))}^{ - 2}} \over {4{\rm \omega} ^4({\rm \epsilon} _2 - \mathop {\sin} \nolimits^2 {\rm \theta} )({\rm \epsilon} _1\cos {\rm \theta} + \sqrt {{\rm \epsilon} _1 - \mathop {\sin} \nolimits^2 {\rm \theta}} )^4}}.}$$

It is obvious from Eq. (38), for B = C = 0, the SH efficiency is in complete agreement with Jha and Agrawal (Reference Jha and Agrawal2014).

It should be noted that, Eq. (40) is obtained by taking into account the reflected SH components contribution. Consequently, the introduced SH conversion efficiency by Eq. (40) is different with previous report in which the reflected SH contribution was missed (Jha & Agrawal, Reference Jha and Agrawal2014). The order of deviation will be discussed in the next section.

4. NUMERICAL RESULTS

In this section, we would like to determine the conversion of a fraction of a laser beam to its phase-mismatch SH. To do this, we consider the pump laser a Nd:Yag with intensity I ₀ ≤ 10¹⁷ W/cm² (a ₀ ≤ 0.3) and frequency ω₀ = 1.88 × 10¹⁵ rad/s and or wavelength λ = 1 μm.

Figure 2 depicted variation of conversion efficiency $({\rm \eta} \% )$ with respect to the normalized propagation distance z/λ. Figure 2 shows that the conversion efficiency is periodic in z for a given value of Δk, so we have the points with maximum and minimum electric field amplitudes for the SH radiation. The maximum power efficiency (η_max) takes place in the coherence length z _c = π/Δk. In Figure 2a, 2b, the red dashed and black dotted lines are plotted based on the previous report (Jha & Agrawal, Reference Jha and Agrawal2014) and Eq. (40), in the absence of magnetic field, respectively. It seems from the figure that for incidence angle near the critical, the deviation increases; however, for angle far from the critical (Fig. 2a), our result is in complete agreement with previous report.

Fig. 2. Variation of transmitted second-harmonic conversion efficiency $({\rm \eta} \% )$ as a function of normalized propagation distance z/λ for different values of incidence angle (a) θ = 20 and (b) θ = 80 (θ ≈ θ_c). The previous report (Jha & Agrawal, Reference Jha and Agrawal2014) (dotted line) is compared with our result in the absence of the magnetic field (dashed line), respectively. The other parameters are a ₀ = 0.1, ω_p/ω₀ = 0.1.

In Figure 3, we plotted the relative difference of the maximum efficiency ( $\Delta {\rm \eta} _{{\rm max}} = \vert{{\rm \eta} ^{\prime}}_{{\rm max}} - {\rm \eta} _{{\rm max}}\vert /\bar {\rm \eta} _{{\rm max}}$ ), where ${{\rm \eta} ^{\prime}}_{{\rm max}}$ and η_max refer to the maximum efficiencies reported based on Jha and Agrawal (Reference Jha and Agrawal2014) and Eq. (40) also $\bar {\rm \eta} _{{\rm max}} = ({{\rm \eta} ^{\prime}}_{{\rm max}} + {\rm \eta}_{{\rm max}})/2$ , as a function of incidence angle. We found from the figure that the deviation reaches to the maximum around 10%, near to the critical angle. The results show that deviation depends strongly to the laser field intensity and plasma density, too. For example, for a ₀ = 0.3 and plasma density near ω_p/ω₀ = 0.5, deviation can be increased up to the 36%, which is considerably large.

Fig. 3. Variation of Δη_max as a function of incident angle θ. The parameters are chosen as a ₀ = 0.1, ω_p/ω₀ = 0.1, and ω_c/ω₀ = 0.

Figure 4 reveals the influence of self-generated transverse magnetic field on the SH power. The figure indicates SH variation in terms of the normalized propagation distance z/λ. The dashed and solid lines are plotted for a non-magnetized and a magnetized plasma, respectively. It is understood that the maximum conversion efficiency is decreased by magnetic field. Taking into account this fact that such strong magnetic fields can be generated in laser plasma interaction, therefore, we expect that in a real physical system, the SH power becomes less than the power reported for a non-magnetized plasma (Jha & Agrawal, Reference Jha and Agrawal2014).

Fig. 4. Variation of transmitted second-harmonic conversion efficiency $({\rm \eta} \% )$ as a function of normalized propagation distance z/λ. The dashed and solid lines are plotted for non-magnetized and magnetized ω_c/ω₀ = 0.2 (B ₀ = 22MG) plasmas, respectively. The other parameters are ω_p/ω₀ = 0.1 and θ = 20.

We compare SH conversion efficiency for normally and obliquely incident of a laser pulse on vacuum–plasma interface in Figure 5, in terms of the self-generated magnetic field strength ω_c/ω₀. Figure 5a is in complete agreement with previous report (Jha et al., Reference Jha, Mishra, Raj and Upadhyay2007). It seems that the magnetic field intensifies the SH power for normal incidence of a laser pulse, while SH power is weakened by magnetic field for obliquely incident. Moreover, Figure 5 clearly shows that the order of SH power is ten times greater for obliquely incident. Figure 5a demonstrates that the magnetic field is essentially required for SH generation in normal incidence of a laser pulse on plasma surface (Jha et al., Reference Jha, Mishra, Raj and Upadhyay2007).

Fig. 5. Variation of maximum conversion efficiency $({\rm \eta} _{{\rm max}}\% )$ as a function of magnetic field strength ω_c/ω₀ for a ₀ = 0.3, ω_p/ω₀ = 0.1, (a) for normal incidence, and (b) for oblique incidence θ = 20.

Figure 6 shows variations of transmitted maximum conversion efficiency $({\rm \eta} _{{\rm max}}\% )$ and reflected conversion efficiency $({\rm \eta} \% )$ in terms of incident angle varying from θ ≈ 0^° up to critical angle for different values of ω_c/ω₀. The solid and dashed lines are plotted for a magnetized and non-magnetized plasmas. Figure 6a shows that the SH power is smaller in a magnetized plasma. As the figures show that the efficiency of SH radiation increases very smoothly with incidence angle, however, it drastically increases near the critical angle. Figure 6b indicates that the influence of magnetic field, on the reflected SH power, is very poor and just near the critical angle a minor decrement ( $ \approx\!\! 1\% $ ) is observable.

Fig. 6. Variations of (a) transmitted maximum conversion efficiency $({\rm \eta} _{{\rm max}}\% )$ and (b) reflected conversion efficiency $({\rm \eta} \% )$ , as a function of incidence angle for a ₀ = 0.1, ω_p/ω₀ = 0.1 and different values of magnetic field strength. The dashed and solid lines are plotted for B ₀ = 0 and 22 MG, respectively.

In Figure 7, we plotted variations of transmitted maximum conversion efficiency $({\rm \eta} _{{\rm max}}\% )$ and reflected conversion efficiency $({\rm \eta} \% )$ as a function of plasma density ω_p/ω₀ for different values of incident angle. This figure predicts that for a given angle of incidence θ and magnetic field strength, the SH power sharply increases for both transmitted and reflected components when the plasma density satisfies the required condition θ ≈ θ_c. For example, for given parameters in Figure 7, the condition θ ≈ θ_c is satisfied for angles θ = 83^° and θ = 77^° in plasma densities ω_p/ω₀ ≈ 0.1 and ω_p/ω₀ ≈ 0.2, respectively. We find from the Figures 6 and 7, the transmitted SH power is more than twice with respect to the reflected component; however, reflected component is more appropriate for technical and experimental applications.

Fig. 7. Variations of (a) transmitted maximum conversion efficiency $({\rm \eta} _{{\rm max}}\% )$ and (b) reflected conversion efficiency $({\rm \eta} \% )$ , as a function of ω_p/ω₀ for a ₀ = 0.1 and B ₀ = 22 MG for different values of incident angle. The dashed and solid lines are plotted for θ = 83 and 77, respectively.

5. SUMMERY AND CONCLUSION

In conclusion, we investigated the possibility of SH generation in the interaction of an obliquely p-polarized laser pulse on a vacuum-magnetized plasma interface. In view of this fact that strong megagauss magnetic fields can be generated in a laser plasma interaction, we modeled the self-generated magnetic field as a transverse magnetic field. We focused our attention to transmitted as well as reflected SH components. We shown that if the transmitted and reflected SH are considered as a simultaneous process, the results deviated from previous reports, especially near the critical angle (Jha et al., Reference Jha, Mishra, Raj and Upadhyay2007; Jha & Agrawal, Reference Jha and Agrawal2014). The deviation can be increased by laser intensity and plasma density. Our analysis revealed a sharp increase for both transmitted and reflected SH components near the critical angle. We have demonstrated that transverse magnetic field had destructive role and the SH power decreased slightly with magnetic field strength enhancement. The analytical investigations show that the transmitted and reflected SH amplitudes are proportional to laser intensity ( $I \propto a_1^2 $ ), which as expected. If we kept the angel of incident and magnetic field strength constant and changed the plasma density, we found that the SH power for both transmitted and reflected components sharply intensified when the angle of incidence satisfied the required condition θ ≈ θ_c. In view of this fact that the critical angle reduced by increasing plasma density, for a large incident angle the maximum power is generated at smaller plasma density. A comparison between transmitted and reflected SH revealed that the conversion efficiency is more than twice for transmitted component; however, the reflected component is more proper for technical and experimental applications.