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Free electron transitions and light amplification
Volume
8, number
OPTICS COMMUNICATIONS
4
FREE ELECTRON
TRANSITIONS
V.M. BUIMISTROV
on free-free coefficient
1973
AND LIGHT AMPLIFICATION
and L.I. TRAHTENBERG
Received
A new mechanism of light amplification shown to be considerable. The amplification
August
6 June 1973
electron transitions is calculated.
This article is concerned with the inelastic collision of an electron with an atom in an electromagnetic field; this inelastic collision in the electromagnetic field may be accompanied by the induced radiation or absorption of photons. The amplitude of the transition is calculated in the second approximation of perturbation theory and is proportional to UF, where F is the interaction of the electrons with the electromagnetic field and U is the interaction of the free electron with the atom. The hamiltonian of the system is of the form
H=H&J+F,
is considered,
tion or absorption
and light amplification
is
of the photon
The “i” and “f’ subscripts correspond to the initial and final states respectively. The law of energy conservation is expressed by the equation: p;/2rn
tE‘ nf =p,2/2m +E ?1i Thw.
(3)
The variable (I,, which is included into the cross section, is expressed as an infinite sum over the intermediate atomic states
Ho=Hc+Ha, N
F = 2coswt[aVo+L,“],
La=z aVi, a = &E&) i= 1
Where Ha and He are hamiltonians of the atomic and the free electron, respectively, r0 = the free electron coordinate, r1, . ., rN = the atom electron coordinates, E, = the electric ve’ctor amplitude of the electromagnetic field, w = the light frequency. Perturbation theory is based on wave functions & and $,, which are eigenfunctions ofHa and H,,p is the momentum of the free electron; n = the se11of quantum numbers of the atomic states. The free electron energy is supposed to be considerably larger than the energy of the atom ionisation, therefore it is possible to consider the interaction U as a small perturbation and to neglect the exchange. After performing the calculation one obtains the following expression for the cross section of collision with radia-
(4) Where E, = the atomic state energy. Here the “t” sign corresponds to the radiation and the “-” to the absorption of the photon. It has been assumed and subsequently proved by detailed calculations, that when the frequency is close to one of the resonance frequencies (Eni, nf - En + fro" 0)in expression (2) the value of U, may be approximated by one resonance value. Assuming the width of the atomic level is to be equal to zero, formula (4) is true only for such values of frequencies which differ from the resonance values by more than the widths of the corresponding atomic levels. Suppose the gas is irradiated by a monochromatic electron beam. To simplify the analysis we shall consider only short time intervals after turning on the electron beam, when one may neglect the populations
289
OP’I‘I(‘S C‘OMMUNICATIONS
Volume 8, number 4
pilEo.
zYI=z
1--p-
hu - SE,! 1
Eni l,if. I
of all the levels except the ground one. Therefore the light absorption corresponding to the transition tlf + II and the usual mechanism of light amplification corresponding to the transition II + /Q may be neglected. Assume that I:‘,, N I:‘,,f + hw. Then the collision cross section with the photon radiation is resonant, while the absorption cross section of the photon is not in resonance since the atomic levels are not equidistant and consequently the atomic levels near the dotted positions I. 2 arc absent (fig. 1). The collision cross section of induced photon radiation is
x dcr,@+..Pi)cos20.
(5)
da,,,,i is the cross section of the inelastic collision without the photon which leads to the atomic transition /zi + ~;f‘i~~ is the oscillator strength for the transition tI + tq, 6’ = L(q, IT), hq =pi ~~p,. One can nedect the photon absorption during the transition of the atom tzi + tzf as a nonresonant process. Then the amplification coefficient is of the form
290
y = &2tnc,;!.
August 1973
(0)
otltli is the integral excitation cross section of the atom by the electron. wa the atomic unit of frequency, 1 the current density of the monochromatic electrons, IV, the atomic concentration. Iffi,, - 1O-i, ~7,~~~ - IO-l7 cm2. - 106, then IV, - 10’” mr3. ef - l@A/~n~,fiw,/n K- 1o-3 cm- ’ We will discuss an absorption mechanism, which seems to be essential because it is resonant. Consider the real transition of the atom to the level n, by simultaneous collision of the atom with the electron and the photon. The former intermediate level IEis now the final state tt;.. and the former final state tq is now the intermediate level tt’. Froni (4) we see that this process is resonant. The absorption cross section is expressed by (5), where the substitution tz + tz;, tq + tt’. is done. If the levels are classified as shown in fig. 1. the transition pzi+ ~1’ is strongly forbidden and the excitation cross sectloll is considerably less than the corresponding cross section in (5) for photon radiation. So, this absorption mechanism may be neglected in spite of its being resonant. The possibility of light amplification on the freefree transitions was considered approximately for the given field in [ 1. 21. The problern of light amplification by free- -free transitions in semiconductors was recently considered in [3]
References
111P. Mlrrcusc, Bell. Syst. Techn. J. 4 I (1962) 1557. 121F.V. Bunkin, A.E. Kazakov and M.V. Fyodorov, UPhN. 107 (1972) 559. 131 V.L. Bon&-Brucvich and M.L. Al-Sharnubi, Vestnik MGU, (1972) 616.
8, number
OPTICS COMMUNICATIONS
4
FREE ELECTRON
TRANSITIONS
V.M. BUIMISTROV
on free-free coefficient
1973
AND LIGHT AMPLIFICATION
and L.I. TRAHTENBERG
Received
A new mechanism of light amplification shown to be considerable. The amplification
August
6 June 1973
electron transitions is calculated.
This article is concerned with the inelastic collision of an electron with an atom in an electromagnetic field; this inelastic collision in the electromagnetic field may be accompanied by the induced radiation or absorption of photons. The amplitude of the transition is calculated in the second approximation of perturbation theory and is proportional to UF, where F is the interaction of the electrons with the electromagnetic field and U is the interaction of the free electron with the atom. The hamiltonian of the system is of the form
H=H&J+F,
is considered,
tion or absorption
and light amplification
is
of the photon
The “i” and “f’ subscripts correspond to the initial and final states respectively. The law of energy conservation is expressed by the equation: p;/2rn
tE‘ nf =p,2/2m +E ?1i Thw.
(3)
The variable (I,, which is included into the cross section, is expressed as an infinite sum over the intermediate atomic states
Ho=Hc+Ha, N
F = 2coswt[aVo+L,“],
La=z aVi, a = &E&) i= 1
Where Ha and He are hamiltonians of the atomic and the free electron, respectively, r0 = the free electron coordinate, r1, . ., rN = the atom electron coordinates, E, = the electric ve’ctor amplitude of the electromagnetic field, w = the light frequency. Perturbation theory is based on wave functions & and $,, which are eigenfunctions ofHa and H,,p is the momentum of the free electron; n = the se11of quantum numbers of the atomic states. The free electron energy is supposed to be considerably larger than the energy of the atom ionisation, therefore it is possible to consider the interaction U as a small perturbation and to neglect the exchange. After performing the calculation one obtains the following expression for the cross section of collision with radia-
(4) Where E, = the atomic state energy. Here the “t” sign corresponds to the radiation and the “-” to the absorption of the photon. It has been assumed and subsequently proved by detailed calculations, that when the frequency is close to one of the resonance frequencies (Eni, nf - En + fro" 0)in expression (2) the value of U, may be approximated by one resonance value. Assuming the width of the atomic level is to be equal to zero, formula (4) is true only for such values of frequencies which differ from the resonance values by more than the widths of the corresponding atomic levels. Suppose the gas is irradiated by a monochromatic electron beam. To simplify the analysis we shall consider only short time intervals after turning on the electron beam, when one may neglect the populations
289
OP’I‘I(‘S C‘OMMUNICATIONS
Volume 8, number 4
pilEo.
zYI=z
1--p-
hu - SE,! 1
Eni l,if. I
of all the levels except the ground one. Therefore the light absorption corresponding to the transition tlf + II and the usual mechanism of light amplification corresponding to the transition II + /Q may be neglected. Assume that I:‘,, N I:‘,,f + hw. Then the collision cross section with the photon radiation is resonant, while the absorption cross section of the photon is not in resonance since the atomic levels are not equidistant and consequently the atomic levels near the dotted positions I. 2 arc absent (fig. 1). The collision cross section of induced photon radiation is
x dcr,@+..Pi)cos20.
(5)
da,,,,i is the cross section of the inelastic collision without the photon which leads to the atomic transition /zi + ~;f‘i~~ is the oscillator strength for the transition tI + tq, 6’ = L(q, IT), hq =pi ~~p,. One can nedect the photon absorption during the transition of the atom tzi + tzf as a nonresonant process. Then the amplification coefficient is of the form
290
y = &2tnc,;!.
August 1973
(0)
otltli is the integral excitation cross section of the atom by the electron. wa the atomic unit of frequency, 1 the current density of the monochromatic electrons, IV, the atomic concentration. Iffi,, - 1O-i, ~7,~~~ - IO-l7 cm2. - 106, then IV, - 10’” mr3. ef - l@A/~n~,fiw,/n K- 1o-3 cm- ’ We will discuss an absorption mechanism, which seems to be essential because it is resonant. Consider the real transition of the atom to the level n, by simultaneous collision of the atom with the electron and the photon. The former intermediate level IEis now the final state tt;.. and the former final state tq is now the intermediate level tt’. Froni (4) we see that this process is resonant. The absorption cross section is expressed by (5), where the substitution tz + tz;, tq + tt’. is done. If the levels are classified as shown in fig. 1. the transition pzi+ ~1’ is strongly forbidden and the excitation cross sectloll is considerably less than the corresponding cross section in (5) for photon radiation. So, this absorption mechanism may be neglected in spite of its being resonant. The possibility of light amplification on the freefree transitions was considered approximately for the given field in [ 1. 21. The problern of light amplification by free- -free transitions in semiconductors was recently considered in [3]
References
111P. Mlrrcusc, Bell. Syst. Techn. J. 4 I (1962) 1557. 121F.V. Bunkin, A.E. Kazakov and M.V. Fyodorov, UPhN. 107 (1972) 559. 131 V.L. Bon&-Brucvich and M.L. Al-Sharnubi, Vestnik MGU, (1972) 616.