3.1. The σ_L Polarization

The Cartesian axes in the S(t, x, y, z) and S′(t′, x′, y′, z′) systems are parallel. In virtue of Eq. (2), the S′ system moves with the velocity – $\vert {\bar{V}_0 } \vert$ along the oz axis (see Fig. 1).

From Eqs. (1) and (3) it follows that the four-dimensional wave vector in the S system is $\lpar {\rm \omega} _L /c\comma \; 0\comma\; \vert {\bar{k}_L } \vert \sin{\rm \theta} _L\comma $$\vert {\bar{k}_L } \vert \cos{\rm \theta} _L \rpar $. In virtue of the Lorentz transformation, given by Eq. (11.22) in Jackson (Reference Jackson1999), it follows that the components of the four-dimensional vector in the S′ system are as follows:

(8)$${\rm \omega}^{\prime }_L={\rm \gamma} _0 {\rm \omega} _L \lpar {1+\vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta} _L } \rpar \ {\rm and}\ \vert {\bar{k}^{\prime }_L } \vert ={\rm \omega}^{\prime }_L /c\comma \;$$

(9)$$k^{\prime }_{Lx^{\prime }}=0\comma \; k^{\prime }_{Ly^{\prime }}=\vert \bar{k}_L\vert \sin {\rm \theta} _L\ {\rm and} \ k^{\prime }_{Lz^{\prime }}={\rm \gamma}_0 \vert \bar{k}_L \vert \lpar\! \cos {\rm \theta} _L+\vert {\bar{\rm \beta}_0} \vert \rpar \comma \;$$

where

(10)$$\bar{\rm \beta}_0=- \displaystyle{{\vert \bar{V}_0 \vert } \over c}\bar{k}\quad {\rm and} \quad {\rm \gamma} _0=\lpar 1 - {\rm \beta} _0^2 \rpar ^{ - {1 \over 2}}.$$

Since the phase of the electromagnetic wave is a relativistic invariant (Jackson, Reference Jackson1999), we have

(11)$${\rm \eta}={\rm \omega} _L t - \bar{k}_L \cdot \bar{r}+{\rm \eta} _i={\rm \omega}^{\prime }_L t^{\prime} - \bar{k}^{\prime }_L \cdot \bar{r}^{\prime }+{\rm \eta} _i={\rm \eta}^{\prime }\comma \;$$

where $\bar{r}$ and $\bar{r}^{\prime }$ are the position vectors of the electron in the two systems. Since the relations between the spacetime four-dimensional vectors in the systems S′ and S are $ct^{\prime}={\rm \gamma} _0 \lpar ct\,+ $$\vert {\bar{\rm \beta}_0} \vert z \rpar $, $z^{\prime}={\rm \gamma} _0 \lpar {z+\vert {\bar{\rm \beta}_0 } \vert ct} \rpar $, x′ = x and y′ = y, it is easy to see that these relations and Eqs. (8) and (9) verify (11).

We write equations (11.149) from Jackson (Reference Jackson1999), which give the Lorentz transformation of the fields, in the International System, and using Eq. (1), we obtain the the following expressions for the components of the electromagnetic field in the S′ system:

(12)$$\eqalign {\bar{E}^{\prime}_L&={\rm \gamma} _0 \left( 1+\vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta} _L\right) E_M \cos {\rm \eta}^{\prime} \bar{i} \; \cr \quad \bar{B}^{\prime}_L&=\left[ {\rm \gamma} _0 \left(\! \cos {\rm \theta} _L+\vert {\bar{\rm \beta}_0 } \vert\right) \bar{j} - \sin {\rm \theta} _L \bar{k}\,\right] \displaystyle{{E_M } \over c}\cos {\rm \eta}^{\prime }\; \cr \quad &=\lpar {k^{\prime}_{Lz^{\prime}} \bar{j} - k^{\prime}_{Ly^{\prime}} \bar{k}}\rpar \displaystyle{{E_M } \over {c\vert {\bar{k}_L } \vert }}\cos {\rm \eta}^{\prime }.}$$

The equations of motion written for the electron are

(13)$$\eqalign {m {d \over dt^{\prime}} \lpar {\rm \gamma} ^{\prime} v^{\prime}_{x^{\prime}}\rpar &=-{\rm \gamma} _0 \lpar 1+\vert \bar{\rm \beta}_0\vert \cos {\rm \theta} _L\rpar eE_M \cos {\rm \eta}^{\prime}\cr & \quad +\lpar k^{\prime}_{Ly^{\prime}}v^{\prime}_{y^{\prime }}+ k^{\prime}_{Lz^{\prime}}v^{\prime}_{z^{\prime }}\rpar \displaystyle {{eE_M}\over {c \vert \bar{k}_L\vert}}\cos {\rm \eta}^{\prime}\comma \; }$$

(14)$$m\displaystyle{d \over {dt^{\prime}}}\lpar {{\rm \gamma} ^{\prime}v^{\prime}_{y^{\prime}} }\rpar =- k^{\prime}_{Ly^{\prime}} v^{\prime}_{x^{\prime}} \displaystyle{{eE_M } \over {c\vert {\bar{k}_L } \vert }}\cos{\rm \eta}^{\prime }\comma \;$$

(15)$$m\displaystyle{d \over {dt^{\prime}}}\lpar {{\rm \gamma} ^{\prime}v^{\prime}_{z^{\prime}} }\rpar =- k^{\prime}_{Lz^{\prime}} v^{\prime}_{x^{\prime}} \displaystyle{{eE_M } \over {c\vert {\bar{k}_L } \vert }}\cos{\rm \eta}^{\prime}.$$

Using (3), (6), and (8), the equations of motion can be written

(16)$$\displaystyle {d \over dt^{\prime}}\lpar {\rm \gamma} ^{\prime}{\rm \beta}^{\prime}_{x^{\prime}}\rpar =-a{\rm \omega}^{\prime}_L \cos {\rm \eta} ^{\prime}+a{\rm \omega}^{\prime}_L \left(\displaystyle {k^{\prime}_{Ly^{\prime}}\over \vert \bar{k}^{\prime}_L\vert}{\rm \beta} ^{\prime }_{y^{\prime}}+\displaystyle {k^{\prime}_{Lz^{\prime}}\over \vert \bar{k}^{\prime}_L\vert}{\rm \beta} ^{\prime }_{z^{\prime}}\right)\cos {\rm \eta}^{\prime}\comma \;$$

(17)$$\displaystyle{d \over {dt^{\prime}}}\lpar {{\rm \gamma}^{\prime}{\rm \beta}^{\prime}_{y^{\prime}} }\rpar =- a{\rm \omega}^{\prime}_L \displaystyle{{k^{\prime}_{Ly^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}{\rm \beta}^{\prime}_{x^{\prime}} \cos{\rm \eta}^{\prime}\comma \;$$

(18)$$\displaystyle{d \over {dt^{\prime}}}\lpar {{\rm \gamma}^{\prime}{\rm \beta}^{\prime}_{z^{\prime}} }\rpar =- a{\rm \omega}^{\prime}_L \displaystyle{{k^{\prime}_{Lz^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}{\rm \beta}^{\prime}_{x^{\prime}} \cos{\rm \eta}^{\prime}.$$

We multiply (16), (17), and (18), respectively, by β′_x′, by β′_y′, and by β′_z′. Taking into account that β′_x′² + β′_y′² + β′_z′² = 1 − 1/γ′², their sum leads to

(19)$$\displaystyle{{d{\rm \gamma}^{\prime}} \over {dt^{\prime}}}=- a{\rm \omega}^{\prime}_L {\rm \beta}^{\prime}_{x^{\prime}} \cos{\rm \eta}^{\prime}.$$

From (17) and (19) we obtain

(20)$$\displaystyle{d \over {dt^{\prime}}}\lpar {{\rm \gamma}^{\prime}{\rm \beta}^{\prime}_{y^{\prime}} }\rpar =\displaystyle{{k^{\prime}_{Ly^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}\displaystyle{{d{\rm \gamma}^{\prime}} \over {dt^{\prime}}}.$$

Integrating this relation with respect to time between 0 and t′, and taking into account the initial conditions (7), we have:

(21)$${\rm \beta}^{\prime}_{y^{\prime}}=\displaystyle{{k^{\prime}_{Ly^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}\left( {1 - \displaystyle{1 \over {{\rm \gamma}^{\prime}}}}\right) .$$

Similarly, from (18) and (19) we obtain

(22)$${\rm \beta}^{\prime}_{z^{\prime}}=\displaystyle{{k^{\prime}_{Lz^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}\left({1 - \displaystyle{1 \over {{\rm \gamma}^{\prime}}}}\right).$$

On the other hand, the differentiation of the phase η′, given by (11), leads to

(23)$$\eqalign{\displaystyle {d{\rm \eta}^{\prime}\over dt^{\prime}} &= {\rm \omega}^{\prime}_L-k^{\prime}_{Ly^{\prime}}c{\rm \beta}^{\prime}_{y^{\prime}}-k^{\prime}_{Lz^{\prime}}c{\rm \beta}^{\prime}_{z^{\prime}} \cr &= {\rm \omega}^{\prime}_L\left(1-\displaystyle{{k^{\prime}_{Ly^{\prime}}}\over {\vert \bar{k}^{\prime}_L\vert}}{\rm \beta}_{y^{\prime}}-\displaystyle{{k^{\prime}_{Lz^{\prime}}}\over {\vert \bar{k}^{\prime}_L\vert}}{\rm \beta}_{z^{\prime}}\right)\comma\;}$$

and from (16) and (23) we obtain

(24)$$\displaystyle{d \over {dt^{\prime}}}\lpar {{\rm \gamma}^{\prime}{\rm \beta}^{\prime}_{x^{\prime}} }\rpar =- a \cos{\rm \eta}^{\prime}\displaystyle{{d{\rm \eta}^{\prime}} \over {dt^{\prime}}}.$$

We integrate (24) with respect to time between 0 and t′, and taking into the account the initial conditions (7), we obtain

(25)$${\rm \beta}^{\prime}_{x^{\prime}}=\displaystyle{{\,f^{\prime}_1 } \over {{\rm \gamma}^{\prime}}}\ {\rm where}\ f^{\prime}_1=- a\lpar {\sin{\rm \eta}^{\prime}-\sin{\rm \eta} _i }\rpar .$$

We substitute the expressions of β′_x′, β′_y′, and β′_z′, respectively, from Eqs. (25), (21), and (22) in β′_x′² + β′_y′² + β′_z′² = 1 − 1/γ′² and obtain the expression of γ′:

(26)$${\rm \gamma} ^{\prime }=1+\displaystyle{{\,f^{\prime 2}_1 } \over 2}.$$

From (21) and (26) respectively, from (22) and (26), we obtain the expressions of β′_y′ and β′_z′. With the aid of the Eqs. (8) and (9), we have:

(27)$${\rm \beta}^{\prime}_{y^{\prime}}=\displaystyle{{\,f^{\prime}_2 } \over {{\rm \gamma}^{\prime}}}\ {\rm where}\ f^{\prime}_2=\displaystyle{{\sin{\rm \theta} _L } \over {{\rm \gamma} _0 \lpar {1+\vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta} _L }\rpar }} \cdot \displaystyle{{\,f^{\prime 2}_1 } \over 2}\comma \;$$

and

(28)$${\rm \beta}^{\prime}_{z^{\prime}}=\displaystyle{{\,f^{\prime}_3 } \over {{\rm \gamma}^{\prime}}}\ {\rm where}\ f^{\prime}_3=\displaystyle{{\cos{\rm \theta} _L+\vert {\bar{\rm \beta}_0 } \vert } \over {1+\vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta} _L }} \cdot \displaystyle{{\,f^{\prime 2}_1} \over 2}.$$

From Eq. (16), we obtain ${\dot {\rm \beta}}^{\prime}_{x^{\prime}}$, where the dot signifies the derivation with respect to time:

(29)$$\eqalign {{\dot {\rm \beta}}_{x^{\prime}}&=\displaystyle {1\over {\rm \gamma} ^{\prime}}\left[-{\rm \beta}^{\prime}_{x^{\prime}}\displaystyle {d{\rm \gamma}^{\prime}\over dt^{\prime}}-a{\rm \omega}^{\prime}_L \cos{\rm \eta}^{\prime} +a{\rm \omega}^{\prime}_L \left(\displaystyle {k^{\prime}_{Ly^{\prime}}\over \vert \bar{k}^{\prime}_L\vert}{\rm \beta} ^{\prime }_{y^{\prime}}+\displaystyle {k^{\prime}_{Lz^{\prime}}\over \vert \bar{k}^{\prime}_L\vert}{\rm \beta} ^{\prime }_{z^{\prime}}\right)\cos {\rm \eta}^{\prime}\right].}$$

From (21) and (22) we obtain $k^{\prime}_{Ly^{\prime}} {\rm \beta}^{\prime}_{y^{\prime}}+k^{\prime}_{Lz^{\prime}} {\rm \beta}^{\prime}_{z^{\prime}}= $$\vert {\bar{k}^{\prime}_L } \vert \lpar {1 - 1/{\rm \gamma}^{\prime}}\rpar $. Introducing this expression in (29), together with the expressions of dγ′/dt′ and β′_x′, given respectively, by (19) and (25), we obtain

(30)$${\dot {\rm \beta}}^{\prime}_{x^{\prime}}={\rm \omega}^{\prime}_L g^{\prime}_1\, {\rm where}\, g^{\prime}_1=\displaystyle{a\over {\rm \gamma} ^{\prime 3}}\left(\displaystyle {\,f^{\prime 2}_1\over 2}-1\right)\cos {\rm \eta}^{\prime}.$$

Similarly, from (17) and (18) we obtain ${\dot {\rm \beta}}^{\prime}_{y^{\prime}}$ and ${\dot {\rm \beta}}^{\prime}_{z^{\prime}}$:

(31)$${\dot {\rm \beta}}^{\prime}_{y^{\prime}}={\rm \omega}^{\prime}_L g^{\prime}_2 \, {\rm where} \, g^{\prime}_2=-\displaystyle{\sin {\rm \theta} _L\over {\rm \gamma}_0 \lpar 1+\vert \bar{\rm \beta}_0\vert \cos {\rm \theta}_L\rpar } \cdot \displaystyle{\,f^{\prime}_1\over {\rm \gamma}^{\prime 3}}a \cos{\rm \eta}^{\prime}\comma \;$$

and

(32)$$\dot {\rm \beta}^{\prime}_{z^{\prime}}={\rm \omega}^{\prime}_Lg^{\prime}_3 \, {\rm where}\, g^{\prime}_3=-\displaystyle{\cos {\rm \theta}_L+\vert \bar{\rm \beta}_0\vert \over 1+\vert {\rm \beta}_0\vert \cos {\rm \theta} _L} \cdot \displaystyle {\,f^{\prime}_1\over {\rm \gamma} ^{\prime 3}}a \cos {\rm \eta}^{\prime}.$$

The analysis of the expressions of f′₁, γ′, f′₂, f′₃, g′₁, g′₂, and g′₃, given, respectively, by Eqs. (25)–(28) and (30)–(32), reveals that the quantities β′_x′, β′_y′, β′_z′, $\dot {\rm \beta}^{\prime}_{x^{\prime}}$, $\dot {\rm \beta}^{\prime}_{y^{\prime}}$, and $\dot {\rm \beta}^{\prime}_{z^{\prime}}$ are periodical functions of only one variable, that is the phase η′.

3.2. The π_L Polarization

From Eqs. (3) and (5) it follows that the four-dimensional wave vector in the S system is $\lpar {\rm \omega} _L /c\comma \; \vert {\bar{k}_L } \vert \sin{\rm \theta} _L\comma \; 0\comma $$ \vert {\bar{k}_L } \vert \cos{\rm \theta} _L \rpar $. With the aid of the Lorentz transformations, we obtain the components of the four-dimensional vector in the S′ system, denoted by ω′_L/c, k′_Lx′, k′_Ly′, k′_Lz′), as follows (see Fig. 2):

(33)$${\rm \omega}^{\prime}_L={\rm \gamma} _0 {\rm \omega} _L \lpar {1+\vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta} _L }\rpar \ {\rm and}\ \vert {\bar{k}^{\prime}_L } \vert ={\rm \omega}^{\prime}_L/c\comma \;$$

(34)$$k^{\prime}_{Lx^{\prime}}=\vert\bar{k}_L\vert \sin {\rm \theta}_L\comma \; k^{\prime}_{Ly^{\prime}}=0 \ {\rm and}\ k^{\prime}_{Lz^{\prime}}={\rm \gamma}_0 \vert\bar{k}_L\vert \lpar \!\cos {\rm \theta} _L+\vert\bar{\rm \beta}_0\vert\rpar \comma \;$$

where $\bar{\rm \beta}_0$ and γ₀ are given by (10).

The phase of the electromagnetic wave is a relativistic invariant (Jackson, Reference Jackson1999), and Eq. (11) remains valid. Since the relations between the spacetime four-dimensional vectors in the systems S’ and S are the same as in the previous subsection, it is easy to see that these relations and Eqs. (33) and (34) verify (11).

With the aid of the Lorentz transformation of the fields, and using Eq. (5), we obtain the the following expressions for the components of the electromagnetic field in the S′ system:

(35)$$\eqalign {\bar{E}^{\prime}_L&=\left[ {\rm \gamma} \left( \!\cos {\rm \theta}_L+\vert \bar{\rm \beta}_0\vert\right) \bar{i}-\sin {\rm \theta}_L\bar{k}\right] \cr \quad E_M \cos {\rm \eta}^{\prime}&=\lpar k^{\prime}_{Lz^{\prime}}\bar{i}-k^{\prime}_{Lx^{\prime}}\bar{k}\rpar \displaystyle {E_M\over \vert \bar{k}_L\vert} \cos {\rm \eta}^{\prime}\comma \; \cr \quad \bar{B}^{\prime}_L&={\rm \gamma}_0\displaystyle{E_M\over c}\left( 1+\vert \bar{\rm \beta}_0\vert\! \cos {\rm \theta}_L\right) \cos {\rm \eta}^{\prime}\bar{j}.}$$

With the aid of (3), (33), and (35), the equations of the motion of the electron can be written:

(36)$$m\displaystyle{d \over {dt^{\prime}}}\left( {{\rm \gamma}^{\prime}v^{\prime}_{x^{\prime}} }\right) =- eE_M \displaystyle{{{\rm \omega}^{\prime}_L } \over {{\rm \omega} _L }} \cdot \displaystyle{{k^{\prime}_{Lz^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}\cos{\rm \eta}^{\prime}+\displaystyle{{eE_M } \over c} \cdot \displaystyle{{{\rm \omega}^{\prime}_L } \over {{\rm \omega} _L }}v^{\prime}_{z^{\prime}} \cos{\rm \eta}^{\prime}\comma \;$$

(37)$$m\displaystyle{d \over {dt^{\prime}}}\left( {{\rm \gamma}^{\prime}v^{\prime}_{z^{\prime}} }\right) =eE_M \displaystyle{{{\rm \omega}^{\prime}_L } \over {{\rm \omega} _L }} \cdot \displaystyle{{k^{\prime}_{Lx^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}\cos{\rm \eta}^{\prime} - \displaystyle{{eE_M } \over c} \cdot \displaystyle{{{\rm \omega}^{\prime}_L } \over {{\rm \omega} _L }}v^{\prime}_{z^{\prime}} \cos{\rm \eta}^{\prime}\comma \;$$

where v′_y′ = 0 and

(38)$${\rm \gamma}^{\prime}=\left( 1 - {\rm \beta}^{\prime 2}_{x^{\prime}} - {\rm \beta}^{\prime 2}_{z^{\prime}}\right) ^{ - {1 \over 2}}\comma \;$$

with β′_x′ = v′_x′/c and β′_z′ = v′_z′/c.

With the aid of (6), the equations of motion become

(39)$$\displaystyle{d \over {dt^{\prime}}}\lpar {{\rm \gamma}^{\prime}{\rm \beta}^{\prime}_{x^{\prime}} }\rpar =- a{\rm \omega}^{\prime}_L \displaystyle{{k^{\prime}_{Lz^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}\cos{\rm \eta}^{\prime}+a{\rm \omega}^{\prime}_L {\rm \beta}^{\prime}_{z^{\prime}} \cos{\rm \eta}^{\prime}\comma \;$$

(40)$$\displaystyle{d \over {dt^{\prime}}}\lpar {{\rm \gamma}^{\prime}{\rm \beta}^{\prime}_{z^{\prime}} }\rpar =a{\rm \omega}^{\prime}_L \displaystyle{{k^{\prime}_{Lx^{\prime}} } \over {\vert {\bar{k}^{\prime}_L } \vert }}\cos{\rm \eta}^{\prime} - a{\rm \omega}^{\prime}_L {\rm \beta}^{\prime}_{x^{\prime}} \cos{\rm \eta}^{\prime}.$$

We solve this system with the aid of the substitution

(41)$${\rm \beta}^{\prime}_{x^{\prime}}={\rm \beta} _v \sin{\rm \xi}+{\rm \beta} _u \cos{\rm \xi} \, {\rm and}\, {\rm \beta}^{\prime}_{z^{\prime}}={\rm \beta} _v \cos{\rm \xi} - {\rm \beta} _u \sin{\rm \xi}\comma \;$$

where

(42)$$\eqalign {\sin {\rm \xi}&=\displaystyle {k^{\prime}_{Lx^{\prime}}\over \vert \bar{k}^{\prime}_L\vert }=\displaystyle {\sin {\rm \theta} _L\over {\rm \gamma}_0 \left( 1+\vert \bar{\rm \beta}_0\vert \cos {\rm \theta} _L\right) } \, {\rm and}\, \cr \quad \cos {\rm \xi}&=\displaystyle {k^{\prime}_{Lz^{\prime}}\over \vert \bar{k}^{\prime}_L\vert }=\displaystyle {\cos {\rm \theta} _L+\vert \bar{\rm \beta}_0\vert \over \left( 1+\vert\bar{\rm \beta}_0\vert \cos {\rm \theta} _L\right) }.}$$

After substitution of (41) in (39) and (40), we multiply (39) and (40), respectively, by cos ξ and − sin ξ. The sum of these equations leads to (43) shown below. Similarly, by multiplication of the same equations by sin ξ and cos ξ, we obtain (44) shown below.

(43)$$\displaystyle{d \over {dt^{\prime}}}\left( {{\rm \gamma}^{\prime}{\rm \beta} _u }\right) =- a{\rm \omega}^{\prime}_L \cos{\rm \eta}^{\prime}+a{\rm \omega}^{\prime}_L {\rm \beta} _v \cos{\rm \eta}^{\prime}\comma \;$$

(44)$$\displaystyle{d \over {dt^{\prime}}}\left( {{\rm \gamma}^{\prime}{\rm \beta} _v }\right) =- a{\rm \omega}^{\prime}_L {\rm \beta} _u \cos{\rm \eta}^{\prime}.$$

We multiply (43) and (44), respectively, by β_u and β_v. Taking into account that β_u² + β_v² = β′_x′² + β′_z′² = 1 − 1/γ′², their sum leads to

(45)$$\displaystyle{{d{\rm \gamma}^{\prime}} \over {dt^{\prime}}}=- a{\rm \omega}^{\prime}_L {\rm \beta} _u \cos{\rm \eta}^{\prime}.$$

On the other hand, the differentiation of the phase η′, given by (11), taking into account (34), (41), and (42), leads to

(46)$$\displaystyle{{d{\rm \eta}^{\prime}} \over {dt^{\prime}}}={\rm \omega}^{\prime}_L - k^{\prime}_{Lx^{\prime}} c{\rm \beta}^{\prime}_{x^{\prime}} - k^{\prime}_{Lz^{\prime}} c{\rm \beta}^{\prime}_{z^{\prime}}={\rm \omega}^{\prime}_L \lpar {1 - {\rm \beta} _v }\rpar .$$

From (44) and (45) we have d(γ′β_v)/dt′ = dγ′/dt′. Integrating this equation, and taking into account the initial conditions (7) and Eq. (46), we have

(47)$$\displaystyle{1 \over {{\rm \gamma}^{\prime}}}=1 - {\rm \beta} _v=\displaystyle{1 \over {{\rm \omega}^{\prime}_L }}\displaystyle{{d{\rm \eta}^{\prime}} \over {dt^{\prime}}}.$$

Introducing 1 − β_v from (47) in (43), we obtain d(γ′β_u) /dt′ = −a cos η′ dη′/dt′. We integrate this equation, taking into the account the initial conditions (7), and obtain

(48)$${\rm \beta} _u=\displaystyle{\,f \over {{\rm \gamma}^{\prime}}}\comma \;$$

where

(49)$$\,f=- a\lpar {\sin{\rm \eta}^{\prime} - \sin{\rm \eta} _i }\rpar \comma \;$$

From (45) and (48) we obtain

(50)$$\displaystyle{{d{\rm \gamma}^{\prime}} \over {dt^{\prime}}}=- a{\rm \omega}^{\prime}_L \displaystyle{\,f \over {{\rm \gamma}^{\prime}}}\cos{\rm \eta}^{\prime}.$$

From (47) and (48), taking into account that β_u² + β_v² = β′_x′² + β′_z′² = 1 − 1/γ′², we have

(51)$${\rm \gamma}^{\prime}=1+\displaystyle{{\,f^2 } \over 2}\comma \;$$

and from (47) and (51) we obtain

(52)$${\rm \beta} _v=\displaystyle{{\,f^2 } \over {2{\rm \gamma}^{\prime}}}.$$

Introducing (48) and (52) in (41), we obtain the expressions of the normalized velocities, as follows:

(53)$$\eqalign{{\rm \beta}^{\prime}_{x^{\prime}}&=\displaystyle{\,f^{\prime}_1 \over {\rm \gamma}^{\prime}}\ {\rm where}\ f^{\prime}_1=- \displaystyle{\cos{\rm \theta} _L+\vert\bar{\rm \beta}_0\vert \over 1+\vert \bar{\rm \beta}_0 \vert \cos{\rm \theta} _L} \cr & \quad f+\displaystyle{\sin{\rm \theta} _L \over {\rm \gamma} _0 \lpar 1+\vert \bar{\rm \beta}_0 \vert \cos {\rm \theta} _L\rpar } \displaystyle{\,f^2 \over 2}\comma \; }$$

(54)$$\eqalign{{\rm \beta}^{\prime}_{z^{\prime}}&=\displaystyle{\,f^{\prime}_3 \over {\rm \gamma}^{\prime}}\ {\rm where}\ f^{\prime}_3=- \displaystyle{\sin {\rm \theta} _L \over {\rm \gamma} _0 \lpar 1+\vert \bar{\rm \beta}_0\vert \cos{\rm \theta} _L \rpar }\cr & \quad f+\displaystyle{\cos {\rm \theta} _L+\vert \bar{\rm \beta}_0 \vert \over 1+\vert \bar{\rm \beta}_0 \vert \cos {\rm \theta} _L} \displaystyle{\,f^2 \over 2}.}$$

From Eq. (39), we obtain ${\dot {\rm \beta}}^{\prime}_{x^{\prime}}$:

(55)$${\dot {\rm \beta}}^{\prime}_{x^{\prime}}=\displaystyle{1\over {\rm \gamma}^{\prime}}\left(\!\!{-}{\rm \beta}^{\prime}_{x^{\prime}}\displaystyle {d{\rm \gamma}^{\prime}\over dt^{\prime}}-a{\rm \omega}^{\prime}_L\displaystyle {k^{\prime}_{Lz^{\prime}}\over \vert \bar{k}_L\vert}\cos {\rm \eta}^{\prime}+a{\rm \omega}^{\prime}_L{\rm \beta}^{\prime}_{z^{\prime}}\cos {\rm \eta}^{\prime}\right).$$

Introducing the expressions of dγ′/dt′, β′_x′ and β′_z′, given respectively by (50), (53), and (54), in (55), we obtain

(56)$$\dot{\rm \beta}^{\prime}_{x^{\prime}}={\rm \omega}^{\prime}_L g^{\prime}_1\, {\rm where}\, g^{\prime}_1=\displaystyle{a\over {\rm \gamma}^{\prime}}\left(\displaystyle{\,f f^{\prime}_1\over {\rm \gamma}^{\prime}}+\displaystyle{\,f^{\prime}_3\over {\rm \gamma}^{\prime}}-\displaystyle{\cos {\rm \theta}_L+\vert \bar{\rm \beta}_0\vert \over 1+\vert \bar{\rm \beta}_0\vert \cos {\rm \theta} _L}\right)\cos {\rm \eta}^{\prime}.$$

Similarly, from (40) we obtain $\dot {\rm \beta}^{\prime}_{z^{\prime}}$:

(57)$$\eqalign{\dot {\rm \beta}^{\prime}_{z^{\prime}}&={\rm \omega}^{\prime}_L g^{\prime}_3 \ {\rm where}\ g^{\prime}_3=\displaystyle{a\over {\rm \gamma}^{\prime}}\left[\displaystyle{\,f f^{\prime}_3\over {\rm \gamma}^{\prime 2}}-\displaystyle{\,f^{\prime}_1\over {\rm \gamma}^{\prime}}\right. \cr & \quad \left. +\displaystyle{\sin {\rm \theta}_L\over {\rm \gamma}_0\left( 1+\vert \bar{\rm \beta}_0\vert \cos {\rm \theta}_L\right) }\right]\cos {\rm \eta}^{\prime}.}$$

We see that, in virtue of the expressions of f, γ′, f′₁, f′₃, g′₁, and g′₃, given, respectively, by Eqs. (49), (51), (53), (54), (56), and (57), the quantities β′_x′, β′_z′, ${\dot {\rm \beta}}^{\prime}_{x^{\prime}}$, and ${\dot {\rm \beta}}^{\prime}_{z^{\prime}}$ are periodical functions of only one variable, that is the phase η′.

The term appearing in the Compton relation written in the S′ system that reflects the quantum behavior is ${\rm \omega}^{\prime}_L \hbar\! /\lpar mc^2\rpar ={\rm \gamma} _0 \lpar {1+\vert \bar{\rm \beta}_0 \vert \cos{\rm \theta} _L }\rpar \lsqb {\rm \omega} _L \hbar\! /\lpar mc^2\rpar \rsqb $ for both the σ_L and π_L polarizations of the electromagnetic field. We recall that this paper considers initial energies of the electron smaller than 100 MeV, which correspond to γ₀ < 200. In this case, the above quantum term is negligible compared to unity. Consequently, the Compton model is at the classical limit and, therefore, using the Thomson model, on which our analysis is based, is justified.

4.1. Spectral Components of the Electromagnetic Field in the S′ System

In order to obtain the intensity of the electrical field generated by the motion of the electron, we introduce the expressions of $\bar{\rm \beta}^{\prime}$ and $\dot {\bar{\rm \beta}}^{\prime}$, calculated in the previous section, in the Liènard-Wiechert relation:

(58)$$\bar{E}^{\prime}=\displaystyle{-e\over 4{\rm \pi} {\rm \varepsilon}_0cR^{\prime}}\cdot \displaystyle {\bar{n}^{\prime}\times \left[\left(\bar{n}^{\prime}-\bar{\rm \beta}^{\prime}\right)\times \dot {\bar{\rm \beta}}^{\prime}\right]\over \left(1-\bar{n}^{\prime} \cdot \bar{\rm \beta}^{\prime}\right)^3}\comma \;$$

where R′ is the distance from the electron to the observation point (the detector) and $\bar{n}^{\prime}$ is the versor of the direction electron-detector. We calculate the right-hand side of Eq. (58) at time t′. In virtue of the significance of the quantities entering in the Liènard-Wiechert equation, it results that the field $\bar{E}^{\prime}$ corresponds to the time t′ + R′/c and we have $\bar{E}^{\prime}=\bar{E}^{\prime}\lpar \bar{r}^{\prime}+\bar{r}^{\prime}\comma \; t^{\prime}+R^{\prime}/c\rpar $, where $\bar{r}^{\prime}=R^{\prime}\bar{n}^{\prime}$. The inequality R′ > >r′ is strongly fulfilled (Jackson, Reference Jackson1999). We write the vector $\bar{n}^{\prime}$ in a spherical coordinate system, so its components are written as follows:

(59)$$n^{\prime}_{x^{\prime}}=\sin{\rm \theta}^{\prime} \cos{\rm \phi}^{\prime}\comma \; n^{\prime}_{y^{\prime}}=\sin{\rm \theta}^{\prime}\sin{\rm \phi}^{\prime} \; \quad {\rm and}\quad n^{\prime}_{z^{\prime}}=\cos{\rm \theta}^{\prime}\comma \;$$

where θ′ is the azimuthal angle between the $\bar{n}^{\prime}$ and $\bar{k}$ versors and ϕ′ is the polar angle in the plane x′y′.

Introducing the components of $\bar{\rm \beta}^{\prime}$ and $\dot {\bar{\rm \beta}}^{\prime}$, given, respectively, by f′₁/γ′, f′₂/γ′, f′₃/γ′ and ω′_Lg′₁, ω′_Lg′₂, ω′_Lg′₃, in (58), and taking into account that f′₂ = 0 and g′₂ = 0 for π_L polarization, we obtain the following expression of the intensity of the scattered electric field in the system S′ (Popa, Reference Popa2011):

(60)$$\bar{E}^{\prime}=\displaystyle{K^{\prime} \over F^{\prime 3}_1}\left( {h^{\prime}_1 \bar{i}+h^{\prime}_2 \bar{j}+h^{\prime}_3 \bar{k}}\right) \quad {\rm with}\quad K^{\prime}=\displaystyle{{ - e{\rm \omega}^{\prime}_L } \over {4{\rm \pi} {\rm \varepsilon} _0 cR^{\prime}}}\comma \;$$

where

(61)$$h^{\prime}_1=F^{\prime}_2 \left( {n^{\prime}_{x^{\prime}} - \displaystyle{{\,f^{\prime}_1 } \over {{\rm \gamma}^{\prime}}}}\right) - F^{\prime}_1\, g^{\prime}_1\comma \;$$

(62)$$h^{\prime}_2=F^{\prime}_2 \left( {n^{\prime}_{y^{\prime}} - \displaystyle{{\,f^{\prime}_2 } \over {{\rm \gamma}^{\prime}}}}\right) - F^{\prime}_1\, g^{\prime}_2\comma \;$$

(63)$$h^{\prime}_3=F^{\prime}_2 \left( {n^{\prime}_{z^{\prime}} - \displaystyle{{\,f^{\prime}_3 } \over {{\rm \gamma}^{\prime}}}}\right) - F^{\prime}_1\, g^{\prime}_3\comma \;$$

with

(64)$$F^{\prime}_1=1 - n^{\prime}_{x^{\prime}} \displaystyle{{\,f^{\prime}_1 } \over {{\rm \gamma}^{\prime}}} - n^{\prime}_{y^{\prime}} \displaystyle{{\,f^{\prime}_2 } \over {{\rm \gamma}^{\prime}}} - n^{\prime}_{z^{\prime}} \displaystyle{{\,f^{\prime}_3 } \over {{\rm \gamma}^{\prime}}}\comma \;$$

(65)$$F^{\prime}_2=n^{\prime}_{x^{\prime}} g^{\prime}_1+n^{\prime}_{y^{\prime}} g^{\prime}_2+n^{\prime}_{z^{\prime}} g^{\prime}_3.$$

From Eqs. (60)–(65) we see that the components of the field given by Eq. (58) are periodic composite functions of only one variable, which is η′, and they can be developed in Fourier series.

We expand the components of the electric field $\bar{E}^{\prime}$ in Fourier series and find the expression of the intensity of harmonics j of the electric field, which is denoted by $\bar{E}^{\prime}_j$:

(66)$$\bar{E}^{\prime}_j=\bar{E}^{\prime}_{\,js}+\bar{E}^{\prime}_{\,jc}\comma \;$$

where

(67)$$\eqalign {\bar{E}^{\prime}_{\,js}& =K^{\prime}\left( f^{\prime}_{1sj}\bar{i}+f^{\prime}_{2sj}\bar{j}+f^{\prime}_{3sj}\bar{k}\right)\! \sin j{\rm \eta}^{\prime}\comma \; \cr & \quad \bar{E}^{\prime}_{\,jc}=K^{\prime}\left( f^{\prime}_{1cj}\bar{i}+f^{\prime}_{2cj}\bar{j}+f^{\prime}_{3cj}\bar{k}\right)\! \cos j{\rm \eta}^{\prime}\comma \; }$$

(68)$$\,f^{\prime}_{{\rm \alpha} sj}=\displaystyle{1 \over {\rm \pi} }\int_0^{2{\rm \pi} } \displaystyle{{h^{\prime}_{\rm \alpha} } \over {F^{\prime 3}_1}}\sin j{\rm \eta}^{\prime}d{\rm \eta}^{\prime} \ {\rm and}\ f^{\prime}_{{\rm \alpha} cj}=\displaystyle{1 \over {\rm \pi} }\int _0^{2{\rm \pi} } \displaystyle{{h^{\prime}_{\rm \alpha} } \over {F^{\prime 3}_1 }}\cos j{\rm \eta}^{\prime}d{\rm \eta}^{\prime}\comma \;$$

with α = 1,2,3 and j = 1,2,3,….

It is easy to verify that, in virtue of Eqs. (58)–(65), we have $\bar{E}^{\prime}\bar{n}^{\prime}=0$. We multiply both equations (67) by $\bar{n}^{\prime}$, taking into account the previous equation and (68), and obtain:

(69)$$\bar{E}^{\prime}_{\,js} \bar{n}^{\prime}=0\ {\rm and}\ \bar{E}^{\prime}_{\,jc} \bar{n}^{\prime}=0.$$

Eqs. (66)–(68) can be simplified, due to the following symmetry properties. It is easy to see that, in virtue of the relations from the previous section, taking into account (61)–(65), for σ_L and π_L polarizations, the functions h′α/F′₁³ are of the form F(η′) = f(sin η′ − sin η_i) cos η′. Since the function F has the symmetry F(η′) = −F(π − η′), it is easy to see that

(70)$$f^{\prime}_{{\rm \alpha} sj} = 0 \ {\rm when}\ j = 1\comma \; 3\comma \; 5\comma \; \ldots \ {\rm and}\ f^{\prime}_{{\rm \alpha} cj} = 0\ {\rm when}\ j = 2\comma \; 4\comma \; 6\comma \; \ldots$$

Since the phase of the j harmonics of the electromagnetic field is jη′, it follows that, in virtue of Eq. (11), the angular frequency and the wave vector of the radiation scattered in the $\bar{n}^{\prime}$ direction, corresponding to the j harmonics, are

(71)$${\rm \omega}^{\prime}_j=j{\rm \omega}^{\prime}_L\comma \; \bar{k}^{\prime}_j=j\vert {\bar{k}^{\prime}_L } \vert \bar{n}^{\prime}\ {\rm and}\ {\rm \omega}^{\prime}_j=c\vert {\bar{k}^{\prime}_j } \vert.$$

The corresponding magnetic field is

(72)$$\bar{B}^{\prime}_j=\bar{n}^{\prime} \times \displaystyle{{\bar{E}^{\prime}_j } \over c}\comma \; \bar{B}^{\prime}_{\,js}=\bar{n}^{\prime} \times \displaystyle{{\bar{E}^{\prime}_{\,js} } \over c}\ {\rm and}\ \bar{B}^{\prime}_{\,jc}=\bar{n}^{\prime} \times \displaystyle{{\bar{E}^{\prime}_{\,jc} } \over c}.$$

4.2. Spectral Components of the Electromagnetic Field in the S System

We calculate the components of the electromagnetic field corresponding to an arbitrary value of j following a procedure similar to that presented in (Popa, Reference Popa2011) and used there to calculate the fundamental component of the electromagnetic field in the laboratory system when j = 1. We use again the Lorentz transformations of the fields given by Eq. (11.149) from (Jackson, Reference Jackson1999) to calculate the component $\bar{E}_{\,js}$ in the laboratory system S. From (72), we obtain for an arbitrary value of j:

(73)$$\eqalign{\bar{E}_{\,js}& ={\rm \gamma} _0 \left( {\bar{E}^{\prime}_{\,js} - \bar{\rm \beta}_0 \times c\bar{B}^{\prime}_{\,js} }\right) - \displaystyle{{{\rm \gamma} _0^2 } \over {{\rm \gamma} _0+1}}\bar{\rm \beta}_0 \left( {\bar{\rm \beta}_0 \cdot \bar{E}^{\prime}_{\,js}}\right) =\cr &\quad {\rm \gamma}_0 \left( 1-\vert {\rm \beta}_0\vert \cos {\rm \theta}^{\prime}\right) \bar{E}^{\prime}_{\,js}+K^{\prime}_{{\rm \gamma}_0}\vert {\rm \beta} _0\vert f^{\prime}_{3sj}\cr & \quad \sin j{\rm \eta}^{\prime}\left(\bar{n}^{\prime}-\displaystyle{{\rm \gamma}_0\vert {\rm \beta} _0\vert \over {\rm \gamma} _0+1}\bar{k}\right).}$$

Similarly, we have:

(74)$$\eqalign {\bar{E}_{\,jc}&={\rm \gamma}_0 \left( 1-\vert {\rm \beta}_0\vert \cos {\rm \theta}^{\prime}\right) \bar{E}^{\prime}_{\,jc} +K^{\prime}{{\rm \gamma}_0}\vert {\rm \beta} _0\vert f^{\prime}_{3cj}\cos j{\rm \eta}^{\prime}\left(\bar{n}^{\prime}-\displaystyle{{\rm \gamma}_0\vert {\rm \beta} _0\vert \over {\rm \gamma} _0+1}\bar{k}\right).}$$

From (59), (66)–(68), (73), and (74) and taking into account that the phase of the electromagnetic field is a relativistic invariant, namely η = η′, we obtain the expression for the intensity of the fundamental electrical field in the system S:

(75)$$\bar{E}_j=\bar{E}_{\,js}+\bar{E}_{\,jc}\comma \;$$

where

(76)$$\eqalign {\bar{E}_{\,js}& =K^{\prime}\left( {I_{1sj} \bar{i}+I_{2sj} \bar{j}+I_{3sj} \bar{k}}\right) \sin j{\rm \eta} \ {\rm and}\, \cr & \quad \bar{E}_{\,jc}=K^{\prime}\left({I_{1cj} \bar{i}+I_{2cj} \bar{j}+I_{3cj} }\right) \bar{k}\cos j{\rm \eta} \comma \; }$$

with

(77)$$I_{1sj}={\rm \gamma} _0 \left( {1 - \vert {{\rm \beta} _0 } \vert \cos{\rm \theta}^{\prime}}\right) f^{\prime}_{1sj}+{\rm \gamma} _0 \vert {{\rm \beta} _0 } \vert \sin{\rm \theta}^{\prime}\cos {\rm \phi}^{\prime}f^{\prime}_{3sj}\comma \;$$

(78)$$I_{1cj}={\rm \gamma} _0 \left( {1 - \vert {{\rm \beta} _0 } \vert \cos{\rm \theta}^{\prime}}\right) f^{\prime}_{1cj}+{\rm \gamma} _0 \vert {{\rm \beta} _0 } \vert \sin{\rm \theta}^{\prime}\cos {\rm \phi}^{\prime}f^{\prime}_{3cj}\comma \;$$

(79)$$I_{2sj}={\rm \gamma} _0 \left( {1 - \vert {{\rm \beta} _0 } \vert \cos {\rm \theta}^{\prime}}\right) f^{\prime}_{2sj}+{\rm \gamma} _0 \vert {{\rm \beta} _0 } \vert \sin {\rm \theta}^{\prime}\sin {\rm \phi}^{\prime}f^{\prime}_{3sj}\comma \;$$

(80)$$I_{2cj}={\rm \gamma} _0 \left( {1 - \vert {{\rm \beta} _0 } \vert \cos {\rm \theta}^{\prime}}\right) f^{\prime}_{2cj}+{\rm \gamma} _0 \vert {{\rm \beta} _0 } \vert \sin {\rm \theta}^{\prime}\sin {\rm \phi}^{\prime}f^{\prime}_{3cj}\comma \;$$

(81)$$I_{3sj}=f^{\prime}_{3sj}\comma \;$$

(82)$$I_{3cj}=f^{\prime}_{3cj}.$$

From (70) and (75)–(82), we obtain

(83)$$I_{1sj}=I_{2sj}=I_{3sj}=0\ {\rm and}\ \bar{E}_j=\bar{E}_{\,jc} \ {\rm when}\ j=1\comma \; 3\comma \; 5\comma \; \ldots\comma \;$$

and

(84)$$I_{1cj}=I_{2cj}=I_{3cj}=0\ {\rm and}\ \bar{E}_j=\bar{E}_{\,js} \ {\rm when}\ j=2\comma \; 4\comma \; 6\comma \; \ldots$$

5.1. Energetic and Angular Relations

With the aid of Eqs. (59) and (71), we find that the four-dimensional wave vector of the j-th harmonics, in the S′ system, is $\lpar {{\rm \omega}^{\prime}_j /c\comma \; \vert {\bar{k}^{\prime}_j } \vert \sin {\rm \theta}^{\prime}\cos {\rm \phi}^{\prime}\comma \; \vert {\bar{k}^{\prime}_j } \vert \sin {\rm \theta}^{\prime}\sin {\rm \phi}^{\prime}\comma \; \vert {\bar{k}^{\prime}_j } \vert \cos {\rm \theta}^{\prime}}\rpar $. With the aid of the Lorentz equations, we calculate the components of this vector in the S system, denoted by (ω_j/c, k _jx, k _jy, k _jz). We obtain

(85)$$\displaystyle{{{\rm \omega} _j } \over c}={\rm \gamma} _0 \left({\displaystyle{{{\rm \omega}^{\prime}_j } \over c}+\bar{\rm \beta}_0 \cdot \bar{k}^{\prime}_j }\right)={\rm \gamma} _0 \displaystyle{{{\rm \omega}^{\prime}_j } \over c}\left( {1 - \vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta}^{\prime}}\right) \comma \;$$

(86)$$k_{\,jz}={\rm \gamma} _0 \left({k^{\prime}_{\,jz^{\prime}} - \vert {{\rm \beta} _0 } \vert \displaystyle{{{\rm \omega}^{\prime}_j } \over c}}\right)={\rm \gamma} _0 \vert {\bar{k}^{\prime}_j } \vert \left( {\cos {\rm \theta}^{\prime} - \vert {\bar{\rm \beta}_0 } \vert }\right) \comma \;$$

(87)$$k_{\,jx}=k^{\prime}_{\,jx^{\prime}}=\vert {\bar{k}^{\prime}_j } \vert \sin{\rm \theta}^{\prime}\cos{\rm \phi}^{\prime}\, {\rm and}\, k_{\,jy}=k^{\prime}_{\,jy^{\prime}}=\vert {\bar{k}^{\prime}_j } \vert \sin{\rm \theta}^{\prime}\sin{\rm \phi}^{\prime}.$$

From (85)–(87) we obtain

(88)$$\displaystyle{{{\rm \omega} _j } \over c}=\vert {\bar{k}_j } \vert.$$

We denote by $\bar{n}$ the versor of the direction in which the radiation is emitted in the S system. Its components are:

(89)$$n_x=\cos{\rm \theta} \sin{\rm \phi}\comma \; \quad n_y=\sin{\rm \theta}\sin{\rm \phi}\; \quad {\rm and}\quad n_z=\cos{\rm \theta}.$$

From (86) and (89) we have $k_{\,jz}=\vert {\bar{k}_j } \vert \cos{\rm \theta}={\rm \gamma} _0 \vert {\bar{k}^{\prime}_j } \vert (\cos{\rm \theta}^{\prime} -$$ \vert {\bar{\rm \beta}_0} \vert ) $. From (71), (85), and (88) we obtain $\vert {\bar{k}_j } \vert /\vert {\bar{k}^{\prime}_j } \vert = $${\rm \gamma} _0 \left( {1 - \vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta}^{\prime}}\right) $. From these relations, we deduce the relation between the angles θ and θ′, as follows:

(90)$$\cos {\rm \theta}=\displaystyle{\cos{\rm \theta}^{\prime} - \vert \bar{\rm \beta}_0 \vert \over 1 - \vert \bar{\rm \beta}_0 \vert \cos {\rm \theta}^{\prime}}\, {\rm and}\, \sin {\rm \theta}=\displaystyle{\sin {\rm \theta}^{\prime} \over {\rm \gamma} _0 \left( 1 - \vert \bar{\rm \beta}_0 \vert \cos {\rm \theta}^{\prime}\right) }.$$

We note that these equations are identical to (5.6) from Landau and Lifschit (Reference Landau and Lifschits1959), in spite of the fact that they are deduced in a completely different way.

In order to find the relation between ϕ and ϕ′, we observe that, in virtue of Eqs. and (8), we have $k_{\,jx}=\vert {\bar{k}_j } \vert \sin{\rm \theta} \cos{\rm \phi}=\vert {\bar{k}^{\prime}_j } \vert \sin{\rm \theta}^{\prime}\cos{\rm \phi}^{\prime}$ and $k_{\,jy}={\bar{k}_j } \vert \sin{\rm \theta} \sin{\rm \phi}=$$\vert {\bar{k}^{\prime}_j } \vert \sin{\rm \theta}^{\prime}\sin{\rm \phi}^{\prime}$. The ratio of these relations leads to

(91)$${\rm \phi} = {\rm \phi}^{\prime}.$$

It is easy to prove, that in virtue of equations (73), (74), (89), (90), and (91), the following relations are valid:

(92)$$\bar{E}_{\,js} \bar{n}=0\hbox{ and }\bar{E}_{\,jc} \bar{n}=0 .$$

With the aid of Eqs. (8) (which is the same for both the σ_L and π_L polarizations), (71), and (85) we obtain the following expressions of the angular frequency and of the wavelength of the j-th component of the scattered radiation, in an arbitrary direction:

(93)$$\eqalign {{\rm \omega} _j & =j{\rm \omega} _L {\rm \gamma} _0^2 \left( {1+\vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta} _L }\right) \left( {1 - \vert {{\rm \beta} _0 } \vert \cos{\rm \theta}^{\prime}}\right)\ {\rm and}\cr & \quad {\rm \lambda} _j = \displaystyle{{{\rm \lambda} _L } \over {\,j{\rm \gamma} _0^2 \left({1 +\vert \bar{\rm \beta}_0\vert \cos {\rm \theta} _L } \right)\left({1 - \vert \bar{\rm \beta}_0\vert \cos {\rm \theta} ^\prime } \right)}}}.$$

The energy of the quanta of the scattered radiation, in an arbitrary direction, is:

(94)$$W_j={\rm \omega} _j \hbar=j{\rm \omega} _L {\rm \gamma} _0^2 \left( {1+\vert {\bar{\rm \beta}_0 } \vert \cos{\rm \theta} _L }\right)\left( {1 - \vert {{\rm \beta} _0 } \vert \cos{\rm \theta}^{\prime}}\right) \hbar.$$

5.2. Polarization of the Electromagnetic Field of the Scattered Beam

The scattering plane is defined by the directions of the incident and scattered beams. The versors corresponding to these directions are $\bar{n} _L=\bar{k}_L /\vert {\bar{k}_L } \vert $ and $\bar{n} $. The versor normal to the scattering plane, which is the same as the versor of the electric field intensity, in the case of the σ polarization, denoted by $\bar{n} _{\rm \sigma} $, is $\bar{n} \times \bar{n} _L$. Its expression is different for the σ_L and π_L polarizations of the incident beam, so we will analyze separately these cases.

5.2.1. The Case of the σ_L Polarization of the Incident Field

In virtue of relations Eqs. (1) and (89), it follows that the versor normal to the scattering plane is

(95)$$\bar{n} _{\rm \sigma}=\left( {\cos{\rm \theta} _L n_y - \sin{\rm \theta} _L n_z }\right) \bar{i} - \cos{\rm \theta} _L n_x \bar{j}+\sin{\rm \theta} _L n_x \bar{k}.$$

In virtue of (76), (83), and (95), it follows that the angle ξ_o between the vector $\bar{E}_j$ and the direction of the σ polarization in the case of odd harmonics (when $\bar{E}_j=\bar{E}_{\,jc}$) is given by the following relation

(96)$$\eqalign {& \cos {\rm \xi} _e = \displaystyle{{\bar{E}}_{\,jc} \over {\bar{E}_{\,jc} }} \cdot \bar{n}_{\rm \sigma}\cr & \quad = \displaystyle{{\left({\cos {\rm \theta} _L n_y - \sin {\rm \theta} _L n_z } \right)I_{1cj} - \cos {\rm \theta} _L n_x I_{2cj} + \sin {\rm \theta} _L n_x I_{3cj} } \over {\sqrt {I_{1cj}^2 + I_{2cj}^2 + I_{3cj}^2 } }}.}$$

Similarly, the angle ξ_e between the vector $\bar{E}_j$ and the direction of the σ polarization, in the case of even harmonics (when $\bar{E}_j=\bar{E}_{\,js}$) is given by the relation

(97)$$\eqalign {& \cos {\rm \xi} _e = \displaystyle{{\bar{E}}_{\,js} \over \vert {\bar{E}_{\,js} }\vert} \cdot \bar{n}_{\rm \sigma} \cr & \quad = \displaystyle{\left(\! \cos {\rm \theta} _L n_y - \sin {\rm \theta} _L n_z\right) I_{1sj} - \cos {\rm \theta} _L n_x I_{2sj} + \sin {\rm \theta} _L n_x I_{3sj} \over \sqrt {I_{1sj}^2 + I_{2sj}^2 + I_{3sj}^2 }}.}$$

5.2.2. The Case of the π_L Polarization of the Incident Beam

From Eqs. (5) and (89), we obtain the following expression of the versor normal to the scattering plane

(98)$$\bar{n} _{\rm \sigma}=\cos{\rm \theta} _L n_y \bar{i}+\left( {\sin{\rm \theta} _L n_z - \cos{\rm \theta} _L n_x }\right) \bar{j} - \sin{\rm \theta} _L n_y \bar{k}.$$

From (76), (83), and (98), it follows that the angle between the vector $\bar{E}_j$ and the direction of the σ polarization, in the case of odd harmonics, is given by the relation

(99)$$\eqalign {& \cos {\rm \xi} _0 = \displaystyle{{\bar{E}}_{\,jc} \over \vert {\bar{E}_{\,jc} }\vert} \cdot \bar{n}_{\rm \sigma} \cr & \quad = \displaystyle{\cos {\rm \theta} _L n_y I_{1cj} + \left( \sin {\rm \theta} _L n_z - \cos {\rm \theta} _L n_x\right) I_{2cj} - \sin {\rm \theta} _L n_y I_{3cj} \over \sqrt {I_{1cj}^2 + I_{2cj}^2 + I_{3cj}^2 }}.}$$

Similarly, the angle between the vector $\bar{E}_j$ and the direction of the σ polarization, in the case of even harmonics, is given by the relation

(100)$$\eqalign {& \cos {\rm \xi} _e = \displaystyle{{\bar{E}}_{\,js} \over \vert {\bar{E}_{\,js} }\vert} \cdot \bar{n}_{\rm \sigma} \cr & \quad = \displaystyle{\cos {\rm \theta} _L n_y I_{1sj} + \left( \sin {\rm \theta} _L n_z - \cos {\rm \theta} _L n_x\right) I_{2sj} - \sin {\rm \theta} _L n_y I_{3sj} \over \sqrt {I_{1sj}^2 + I_{2sj}^2 + I_{3sj}^2 }}.}$$

We observe that Eqs. (92) and (95)–(100) define completely the polarization of the scattered electromagnetic field $\bar{E}_j$. This happens because the vector $\bar{E}_j$ is situated in the plane perpendicular on $\bar{n}$, and makes an angle ξ_o or ξ_e, respectively, for odd and even harmonics, with the direction of the σ polarization. If ξ_o = 0 or ξ_e = 0 the scattered electromagnetic field is σ polarized, while when ξ_o = π/2 or ξ_e = π/2 the scattered field is π polarized.

5.3. Intensities of the Harmonic Radiation Beams in the S′and S Systems

The average value of the intensity of the j-th component of the scattered radiation, normalized to ɛ₀cK′², in the system S′, is given by

(101)$$\underline I^{\prime}_j=\displaystyle{I_j \over {\rm \varepsilon} _0 cK^{\prime 2}}=\displaystyle{1 \over 2{\rm \pi} }\int _0^{2{\rm \pi} } \displaystyle{\bar{E}^{\prime 2}_j \over K^{\prime 2 }} d\lpar j{\rm \eta}^{\prime}\rpar .$$

Taking into account (66), we obtain the following relation:

(102)$$\underline I^{\prime}_j=\displaystyle{1 \over 2}\left(\,f^{\prime 2}_{1sj}+f^{\prime 2}_{1cj}+f^{\prime 2}_{2sj}+f^{\prime 2}_{2cj}+f^{\prime 2}_{3sj}+f^{\prime 2}_{3cj}\right).$$

In order to calculate the average value of the intensity of the j-th harmonics of the scattered radiation in the laboratory system, we calculate first the expression of $\bar{E}_j^2$, where $\bar{E}_j$ is the intensity of the electric field in the S system. The Lorentz transformation of the electric field intensity from the S′ to the S system is given by Eq. (11.149) of Jackson (Reference Jackson1999), and, written in the International System, leads to the following relation:

(103)$$\bar{E}_j^2=\left[{{\rm \gamma} _0 \left(\bar{E}^{\prime}_j - \bar{\rm \beta}_0 \times c\bar{B}^{\prime}_j\right)- \displaystyle{{\rm \gamma} _0^2 \over {\rm \gamma} _0+1}\bar{\rm \beta}_0 \left({\bar{\rm \beta}_0 \cdot \bar{E}^{\prime}_j}\right)} \right]^2.$$

Since $c\bar{B}_j^{\prime} = \bar{n}^{\prime} \times \bar{E}_j^{\prime} \comma \; \bar{n}^{\prime} \cdot \bar{E}_j^{\prime} = 0$ and $\bar{E}^{\prime} _j \cdot \left[{\bar{\rm \beta}_0 \times \left({\bar{n}^{\prime} \times \bar{E}_j^{\prime} } \right)} \right]= $$- \bar{E}_j^{\prime 2} \left({\bar{\rm \beta}_0 \cdot \bar{n}{^\prime} } \right)$, Eq. (103) becomes

(104)$$\eqalign {\bar{E}_j^2 &= {\rm \gamma}_0 ^2 \bar{E}_j^{\prime 2} \left[1+\bar{{\rm \beta}}_0^2 \sin^2 {\rm \alpha}_3 + {\displaystyle{{\rm \gamma}_0^2}\bar{{\rm \beta}}_0^4 \over \left( {\rm \gamma}_0+1\right) ^2} \cos^2 {\rm \alpha}_2\right. \cr & \quad \left. +2 \vert \bar{{\rm \beta}}_0\vert \cos {\rm \alpha}_1 - {\displaystyle {2{\rm \gamma} _{0}\bar{{\rm \beta}}_0^2 \over {\rm \gamma}_0+1}} \cos^2 {\rm \alpha}_2\right]\comma \; }$$

where α₁ is the angle between $\bar{{\rm \beta}}_0$ and $\bar{n}^{\prime}$, α₂ is the angle between $\bar{\rm \beta}_0 $ and $\bar{E}^{\prime}_j $, and α₃ is the angle between $\bar{\rm \beta}_0 $ and $\bar{n}^{\prime} \times \bar{E}^{\prime}_j $. It is easy to see that γ₀²β₀²/(γ₀ + 1)² − 2γ₀/(γ₀ + 1) = −1, and Eq. (104) can be written:

(105)$$\bar{E}_j^2={\rm \gamma} _0^2 \bar{E}^{\prime 2}_j \left(1+\bar{\rm \beta}_0^2 \sin^2 {\rm \alpha} _3 - \bar{\rm \beta}_0^2 \cos^2 {\rm \alpha} _2+2\vert \bar{\rm \beta}_0 \vert \cos {\rm \alpha} _1 \right).$$

The three vectors, $\bar{n}^{\prime}$, $\bar{E}^{\prime}_j $, and $\bar{n}^{\prime} \times \bar{E}^{\prime}_j $ form a right trihedral angle, therefore, cos² α₁ + cos² α₂ + cos² α₃ = 1 and we obtain:

(106)$$\bar{E}_j^2={\rm \gamma} _0^2 \bar{E}^{\prime 2}_j \left( {1+\vert {\bar{\rm \beta}_0 } \vert \cos{\rm \alpha} _1 }\right) ^2.$$

Since α₁ = π− θ′, from (106) we obtain

(107)$$\bar{E} _j^2=\bar{E}^{\prime 2}_j {\rm \gamma} _0^2 \left( 1 - \vert \bar{\rm \beta}_0 \vert \cos{\rm \theta}^{\prime}\right) ^2.$$

The intensity of the j-th harmonics of the scattered radiation is ${\rm \varepsilon} _0 c\bar{E}_j ^2 $ in the S system. Its averaged value, normalized to ɛ₀cK′², results from Eqs. (101), (102), and (107), as follows

(108)$$\eqalign {\underline I_j&=\displaystyle{I_j \over {\rm \varepsilon} _0 cK^{\prime 2 }}={\rm \gamma} _0^2 \lpar 1 - \vert {{\rm \beta} _0 } \vert \cos{\rm \theta}^{\prime}\rpar ^2 \displaystyle{1 \over {2{\rm \pi} }}\int_0^{2{\rm \pi} } \displaystyle{\bar{E}^{\prime 2}_j \over K^{\prime 2}}d\lpar j{\rm \eta}^{\prime}\rpar \cr & \quad =\displaystyle{1\over 2}{\rm \gamma}_0^2 \left( 1-\vert{\rm \beta}_0\vert \cos {\rm \theta}^{\prime}\right) ^2\cr & \quad \times \left(\,f^{\prime 2}_{1sj}+f^{\prime 2}_{1cj}+f^{\prime 2}_{2sj}+f^{\prime 2}_{2cj}+f^{\prime 2}_{3sj}+f^{\prime 2}_{3cj}\right)\comma \; }$$

or

$$\underline I_j={\rm \gamma} _0^2 \lpar {1 - \vert {{\rm \beta} _0 } \vert \cos{\rm \theta}^{\prime}}\rpar ^2 \underline I^{\prime}_j.$$

The analysis Eqs. (60)–(65) shows that the intensity components of the electric field in the S′ system, namely h′₁/F′₁³, h′₂/F′₁³, and h′₃/F′₁³, are composite functions of f′₁, f′₂, f′₃, g′₁, g′₂, g′₃, and γ′, which, in turn, are periodic functions of η′. Our method is based on this property, because the calculation of the composite functions and their integrals can be easily performed by simple computer programs.

When the incident electromagnetic field is σ_L polarized, our calculations are made using Mathematica 7 with the aid of the following algorithm. In the first step, we introduce the initial data of the system, which are a, γ₀, θ_L, η_i, ω_L θ′, ϕ′ and j. In the second step, we calculate the functions f′₁, γ′, f′₂, f′₃, g′₁, g′₂, g′₃, F′₁, F′₂, h′₁, h′₂, and h′₃. In the third step, we calculate the integrals (68), from which we obtain the f′_1sj, f′_1cj, f′_2sj, f′_2cj, f′_3sj, and f′_3cj. Eqs. (102) and (109) lead, finally, to the values of $\underline I^{\prime}_j $ and $\underline I_j$. The relative errors of our calculations are of the order 10⁻¹².

When the incident electromagnetic field is π_L polarized, the calculations are very similar to those performed for the σ_L polarization. There is one difference: in the second step of the algorithm, the order of the calculation of the composite functions is f, γ′, f′₁, f′₃, g′₁, and g′₃ (in this case f′₂ = g′₂ = 0).

6.1. The 180° and 90° Configurations, as Particular Cases of σ_L and π_L Polarizations

As the angle θ_L increases, the system will pass periodically through the two configurations of interaction having 180° or 90° geometries. We recall that these two geometries correspond to the cases in which the laser and electron beams collide, respectively, head on and perpendicularly. This holds for both the σ_L and π_L polarizations.

The value θ_L = 0 corresponds to the 180° geometry. In this case, the σ_L and π_L polarizations are identical, because they correspond to the same incident electromagnetic field, when $\bar{E}_L=E_M \cos{\rm \eta} \bar{i}\comma \; \bar{B}_L=B_M \cos{\rm \eta} \bar{j}$, and $\bar{k}_L=\vert {\bar{k}_L } \vert \bar{k}$.

The values θ_L = ±90° correspond to the 90° geometry. For these values of θ_L the σ_L and π_L polarizations behave differently. For the σ_L polarization, we have $\bar{E}_L=$$E_M \cos{\rm \eta} \bar{i}\comma \; \bar{B}_L=\mp B_M \cos{\rm \eta} \bar{k}$, and $\bar{k}_L=\pm \vert {\bar{k}_L } \vert \sin{\rm \theta} _L \bar{j}$, while for the π_L polarization, $\bar{E}_L=\mp E_M \sin{\rm \theta} _L \cos{\rm \eta} \bar{k}$, $\bar{B}_L=$$B_M \cos{\rm \eta} \bar{j}$, and $\bar{k}_L=\pm \vert {\bar{k}_L } \vert \sin{\rm \theta} _L \bar{i}$. Figure 3, obtained with the aid of Eq. (94), shows the variation of the quanta energy of the radiation scattered in the direction in which the electron beam moves (which corresponds to θ′ = θ= π) as function of θ_L. The energy is normalized to $j{\rm \omega} _L {\rm \gamma} _0^2 \lpar {1+\vert {\bar{\rm \beta}_0 } \vert }\rpar ^2 \hbar$. The 180° geometry corresponds to θ_L = 0°; 180°, while the 90° geometry corresponds to θ_L = 90°; 270°; these points have been marked accordingly on the curve illustrated in Figure 3. This figure shows that the energy corresponding to the 90° geometry is half the energy corresponding to the 180° geometry, for same values of ω_L and γ₀.

Fig. 3.

Variation of the quanta energy of the scattered radiation normalized to $j{\rm \omega}_L {\rm \gamma}_0^2 \lpar 1 + \vert \bar{{\rm \beta}}_0 \vert\rpar ^2 \hbar$ as function of θ_L, for θ′ = θ= 180° and γ₀ = 100. The point θ_L = 0° corresponds to the 180° geometry, while the points θ_L = 90° and θ_L = 270° correspond to the 90° geometry.

From the properties of the the Liènard-Wiechert equation, it results that the components of the scattered electromagnetic field emitted in a certain direction are rigorously determined. It follows that for a point on the curve shown in Figure 3, which corresponds a certain value of θ_L and to a certain geometry of the collision between the laser and electron beams, we can evaluate exactly the polarization of the scattered beam. The polarization, which corresponds to a given interaction geometry, results from Figure 4, in which we represent the variations of the angle ξ with θ_L, in the cases of the two σ_L and π_L polarizations, for odd and even harmonics. The angle ξ corresponds to the polarization of the electric field of the scattered radiation.

Fig. 4.

Variations of the angle ξ, which corresponds to the polarization of the electric field of the scattered radiation, as function of θ_L, for a = 4, γ₀ = 100, θ′ = θ= 180°, η_i = 20° and ω_L = 2.355 × 10¹⁵ rad/s. Curve 1 correspons to the σ_L polarization of odd harmonics, curve 2 correspons to the σ_L polarization of even harmonics, while curve 3 corresponds for the π polarization, for both the even and odd harmonics. The σ polarization of the scattered field corresponds to ξ = 0°, while the π polarization corresponds to ξ = ±90°.

We can find the polarization of the scattered field in a given geometry from Figures 3 and 4. We consider, for example, the 180° geometry, which corresponds to θ_L = 0 in Figure 3. We can see from Figure 4 that the scattered field is π polarized, because it corresponds to ξ = 90° for all cases, when the incident field is either σ_L or π_L polarized, for odd and even harmonics. For the 90° geometry, which corresponds to θ_L = 90° in Figure 3, we find from Figure 4 that the scattered field is σ polarized for even harmonics, because it corresponds to ξ = 0 on curve 2.

6.2. Comparison between Theory and Experiment

The initial data entering our model, which correspond to certain experiments reported in the literature, consists of the initial energy of the electron, the energy of the laser pulse, the radius of the laser beam, the duration of the laser pulse, and the wavelength of the laser pulse. These quantities are denoted, respectively, by E _i, W _L, r _L, τ_L, and λ_L. We use the relations ${\rm \gamma} _0=E_i /mc^2\comma \; \vert {\bar{\rm \beta}_0 } \vert =\sqrt {1 - 1/{\rm \gamma} _0^2 }\comma \; I_L=W_L /\lpar {\rm \pi} r_L^2 {\rm \tau} _L \rpar \comma $$ E_M=\sqrt {2I_L /\lpar {\rm \epsilon} _0 c\rpar }\comma \; {\rm \omega} _L=\lpar 2{\rm \pi} c\rpar /{\rm \lambda} _L$, and a = eE _M/(mcω) to calculate, respectively, ${\rm \gamma} _0\comma \; \vert {{\rm \beta} _0 } \vert $, I _L, E _M, ω_L, and a, where I _L is the intensity of the laser beam.

Figure 5 presents the angular distributions of the scattered radiations, which correspond to four experiments. The curves 1, 2, 3, and 4 from Figure 5 correspond, respectively, to the experiments (Babzien et al., Reference Babzien, Ben-Zvi, Kusche, Pavlishin, Pogorelski, Siddons and Yakimenko2006; Pogorelsky et al., Reference Pogorelsky, Ben-Zvi, Hirose, Kashiwagi, Yakimenko, Kusche, Siddons, Skaritka, Kumita, Tsunemi, Omori, Urakawa, Washio, Yokoya, Okugi, Liu, He and Cline2000; Anderson et al., Reference Anderson, Barty, Betts, Brown, Crane, Cross, Fittinghoff, Gibson, Hartemann, Kuba, Lesage, Rosenzweig, Slaughter, Springer and Tremaine2004; Kim et al., Reference Kim, Chattopadhay and Shank1994). The calculations are made for θ′ = π and for the values of θ_L, a, γ₀, λ_L, and ω_L, listen in caption of the figure. These calculations are made as follows. With the aid of Mathematica 7 and the procedure described in previous section, we calculate first the variation of the scattered beam intensity $\underline I_1$ as function of θ′, when θ′ takes values between 90° and 180°. With the aid of Eq. (90) we calculate the values of θ as function of θ′ and the values of the angle α given by the relation

(109)$${\rm \alpha}={\rm \pi}-{\rm \theta} .$$

Knowing the variations $\underline I_1 \lpar {\rm \theta}^{\prime}\rpar $ and ${\rm \alpha} \lpar {\rm \theta}^{\prime}\rpar $ we obtain the variation of $\underline I_1$ as function of α.

Fig. 5.

Variations of I ₁, normalized to their maximum values, with α. Curve 1 is for the 180° geometry, θ_L = 0°, γ₀ = 117.4, j = 1, a = 0.35, η_i = 0°, θ′ = 180°, ϕ′ = 0° and ω_L = 1.777 × 10¹⁴ rad/s; curve 2 is for the 180° geometry, θ_L = 0°, γ₀ = 117.4, j = 1, a = 0.053, η_i = 0°, θ′ = 180°, ϕ′ = 0° and ω_L = 1.777 × 10¹⁴ rad/s; curve 3 is for the 180° geometry, θ_L = 0°, γ₀ = 111.5, j = 1, a = 0.2012, η_i = 0°, θ′ = 180°, ϕ′ = 0° and ω_L = 2.355 × 10¹⁵ rad/s; curve 4 is for the 90° geometry, σ_L polarization, θ_L = 90°, γ₀ = 62.62, j = 1, a = 0.08369, η_i = 0°, θ′ = 180°, ϕ′ = 0° and ω_L = 2.355 × 10¹⁵ rad/s; curve 5 is for the same parameters as for curve 4, but the π_L polarization is used instead.

The values of the divergence angles of the scattered beams, which result from Figure 5, are almost identical to the experimental values. For example, in the case of curve 1, the divergence angle is 8 mrad. This value is in agreement with the experimental value which results from Figure 3a in Babzien (2006). Similarly, curves 2 and 3 show divergences of 8 mrad, which are in good agreement with the experimental data presented, respectively, at page 090702-4 in Pogorelsky et al. (Reference Pogorelsky, Ben-Zvi, Hirose, Kashiwagi, Yakimenko, Kusche, Siddons, Skaritka, Kumita, Tsunemi, Omori, Urakawa, Washio, Yokoya, Okugi, Liu, He and Cline2000) and in Figure 3 in Anderson et al. (Reference Anderson, Barty, Betts, Brown, Crane, Cross, Fittinghoff, Gibson, Hartemann, Kuba, Lesage, Rosenzweig, Slaughter, Springer and Tremaine2004).

In the same time, there is a good agreement between theoretical and experimental values of energies or wavelengths of the scattered radiations, which correspond to the data shown in Figure 5. For example, using the data for cases 1 and 3, we obtain, with the aid of Eq. (94), respectively, the values W ₁ = 6.45 keV and W ₁ = 77.1 keV. These values have to be compared, respectively, to the experimental values 6.5 keV (Babzien et al., Reference Babzien, Ben-Zvi, Kusche, Pavlishin, Pogorelski, Siddons and Yakimenko2006) and 78.5 keV (Anderson et al., Reference Anderson, Barty, Betts, Brown, Crane, Cross, Fittinghoff, Gibson, Hartemann, Kuba, Lesage, Rosenzweig, Slaughter, Springer and Tremaine2004). In the same mode, using the data for cases 2 and 4, we obtain, with the aid of Eq. (93), respectively, the values λ₁ = 1.9A and W ₁ = 1.02A. These values have to be compared, respectively, to the experimental values 1.8 A (Pogorelsky et al., Reference Pogorelsky, Ben-Zvi, Hirose, Kashiwagi, Yakimenko, Kusche, Siddons, Skaritka, Kumita, Tsunemi, Omori, Urakawa, Washio, Yokoya, Okugi, Liu, He and Cline2000) and 1 A (Kim et al., Reference Kim, Chattopadhay and Shank1994).

We evaluate now the properties of the second harmonic, resulted from the experiment corresponding to curve 1 from Figure 5. Using Eq. (94), in which we set j = 2, θ_L = 0, and θ′ = π, we obtain W ₂ = 12.9 keV. This value is in good agreement with the corresponding experimental value of W _2exp = 13 keV (Babzien et al., Reference Babzien, Ben-Zvi, Kusche, Pavlishin, Pogorelski, Siddons and Yakimenko2006).

For the aforementioned experiment, we calculate the angular distribution of the second harmonic, $\underline I_2$. We record in Figure 6 the angular variation of $\underline I_2$ as function of α. A comparison between our theoretical results shown in Figure 6 and the experimental results shown in Figure 3 in Babzien et al., (Reference Babzien, Ben-Zvi, Kusche, Pavlishin, Pogorelski, Siddons and Yakimenko2006) reveals that the angular distributions of $\underline I_2$ are the same in both cases. The distance between the two-peak pattern of the second harmonics radiation, shown in Figure 3b in Babzien et al., (Reference Babzien, Ben-Zvi, Kusche, Pavlishin, Pogorelski, Siddons and Yakimenko2006), corresponds to 6 mrad. This value is in good agreement with our calculated value, which results from Figure 6.

Fig. 6.

Variations with α of I ₂, normalized to its maximum value, for the 180° geometry, θ_L = 0°, γ₀ = 117.4, j = 2, a = 0.35, η_i = 0°, θ′ = 180°, ϕ′ = 0° and ω_L = 1.777 × 10¹⁴ rad/s.

For the same initial data, corresponding to the first case presented in Figure 5, we have calculated the polar distribution of the X-ray intensity, namely the variation of $\underline I_2$ with ϕ, for constant θ′. This variation, when the electromagnetic field is linearly polarized in the directions ox and oy, is shown in Figure 7. If we compare Figure 7 of our paper and Figure 4 in (Babzien et al., Reference Babzien, Ben-Zvi, Kusche, Pavlishin, Pogorelski, Siddons and Yakimenko2006), we will see a good agreement between the theoretical and experimental polar distributions of I ₂. We notice in both figures that these distributions are represented by two lobes, whose axes are parallel to the two directions of the field polarization.

Fig. 7.

The intensities of the second harmonics, I ₂, normalized to their maximum values, as functions of ϕ in the case of the head-on interaction between the electromagnetic field and relativistic electrons. The calculations are made for θ_L = 0, γ₀ = 117.4, ω_L = 1.777 × 10¹⁴ rad/s, θ′ = 130° and η_i = 0°. The field is characterized by a = 0.35 and is linearly polarized in the ox direction for curve 1, and in oy direction for curve 2.

We use Eqs. (102) and (109) to calculate the spectra of the beam scattered in the direction of the moving of the electron, in the inertial systems S′ and S, when $\underline I^{\prime}_j$ and $\underline I_j$ are normalized to their maximum values. This direction corresponds to the maximum intensity in the narrow emission cone of the scattered radiation. The spectra, which are calculated for three values of the a parameter, are represented in Figure 8. As it results from this figure, and from Eqs. (102) and (109), the spectra corresponding to $\underline I^{\prime}_j$ and $\underline I_j$ are identical.

Fig. 8.

Typical spectral distributions ot the scattered Thomson radiation, namely variations of I _j′ and I _j in the inertial systems S′ and S, normalized for their maximum values, for odd harmonics, E _i = 100 MeV, γ₀ = 195.8, θ_L = 0°, θ′ = 180°, ϕ′ = 0°, η_i = 0° and ω_L = 2.355 × 10¹⁵ rad/s. The spectra from curves 1, 2 and 3 correspond, respectively, to a = 3, a = 5 and a = 7. The spectra of I _j′ and I _j are identical. The continuous curves interpolate the discrete values of I _j′ and I _j, which are denoted by points.

This analysis reveals two properties. First, the spectrum has a specific shape that comprises a slowly increasing region for low order harmonics, followed by a maximum and a fast decreasing region for higher harmonics. Second, the maximum of the spectrum is shifting to shorter wavelengths, when a increases.

We recall that the spectrum given by $\underline I^{\prime}_j=\underline I^{\prime}_j \lpar j\rpar $ is calculated in the system S′ when a very intense laser field, characterized by relativistic parameters having values bigger than unity, interacts with an electron whose initial velocity is zero. We compare our theoretical prediction with the spectral curve obtained in an experiment presented in (Ta Phuok et al., Reference Ta Phuok, Rousse, Pittman, Rousseau, Malka, Fritzler, Umstadter and Hulin2003) that involved the interaction of a very intense laser beam and an electron having negligible initial velocity. The comparison of the curves illustrated in Figure 8 and Figure 2a in Ta Phuok et al., (Reference Ta Phuok, Rousse, Pittman, Rousseau, Malka, Fritzler, Umstadter and Hulin2003) shows that these curves have similar shapes. The second property stated above, namely that the maximum of the spectrum is shifting to shorter wavelengths when a increases, is in good agreement with the experimental data presented in Ta Phuok et al. (Reference Ta Phuok, Rousse, Pittman, Rousseau, Malka, Fritzler, Umstadter and Hulin2003).

We note that the energies of the radiations calculated in this example for low orders of harmonics with the aid of Eq. (94) are on the order of 1 MeV. For example, for j = 5, we obtain W ₅ = 1.189 MeV (see Fig. 8).