Journal of Experimental and Theoretical Physics, Vol. 95, No. 2, 2002, pp. 206–209. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 122, No. 2, 2002, pp. 241–244. Original Russian Text Copyright © 2002 by Antonov, Kondrat’ev.

NUCLEI, PARTICLES, AND THEIR INTERACTION

On the Stability of Electromagnetic Field in Inverted Media V. A. Antonova and B. P. Kondrat’evb, * aPulkovo

Observatory, Russian Academy of Sciences, Pulkovskoe shosse 65, St. Petersburg, 196140 Russia bUdmurt State University, Krasnoarmeœskaya ul. 71, Izhevsk, 426034 Russia *e-mail: [email protected] Received February 21, 2002

Abstract—The dispersion equation in the problem of electromagnetic field stability in an infinite inverted homogeneous medium is analyzed rigorously. The instability region for small quantum numbers k is established. It is found that, in contrast to the prevailing opinion, a continuous transition is also possible between polariton waves and electromagnetic waves upon a gradual variation of k; in this case, the two types of waves cannot be separated. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The instability of waves in inverted media has become the object of intense investigations. We are speaking of the situation described by V.V. Zheleznyakov, V.V. Kocharovskiœ, and Vl.V, Kocharovskiœ, [1, 2]. In spite of a detailed analysis of the equations for an electromagnetic field in a homogeneous infinite medium with inversion [1, 2], some aspects still remain unclear and will be investigated more thoroughly here. It should be recalled that, in the semiclassical approximation, the rf electromagnetic field in a substance is described by the following equations: 1 ∂B 1 ∂ ( E + 4πp ) 4πσ curl E = – --- ------- , curl B = --- ---------------------------- + ----------E, (1) c ∂t c ∂t c 2 ω 1 ∂ p 2 ∂p 2 -------2- + ----- ------ +  ω 0 + -----2 p = -----c-E. T 2 ∂t  4π ∂t T 2 2

(2)

Here, we use the notation adopted in [1, 2]: E and B are the electric and magnetic field vectors, p is the polarization vector of the substance, ω0 is the frequency of transition between two levels, 1/T2 is the broadening of the corresponding spectral line, and ωc is the plasma (cooperative) frequency of the substance in the given state. Above all, we are interested in the case of inverted 2 state, when ω c < 0. Irrespective of the direction of polarization vector, we obtain from Eqs. (1) and (2) 2 2 ωc k c 4πiσ 1 + ------------ – -------------------------------------– --------- = 0, 2 2 2 ω ( ω + i/T 2 ) – ω 0 ω

First, the condition for the existence of the instability region on the scale of wave numbers, 8πσ 2 ω c < – ---------- , T2

was derived in [1] not quite rigorously, but to a certain approximation. In fact, it permits a quite exact derivation on the basis of Eq. (3). Second, the boundedness of the instability region for large as well as small values of k was noted in [1, 2], but the second of these statements may be incorrect in the nonresonance case. Third, two types of waves (polariton and electromagnetic waves) are indicated in [1, 2] as two different entities on the basis of their physical differences. In actual practice, a continuous transition between them is sometimes possible upon a gradual change in k; consequently, a wave cannot be unambiguously attributed to a certain type of waves. In this paper, we fill these gaps. 2. CRITERION FOR THE EXISTENCE OF THE INSTABILITY ZONE For k ∞ and invariability of the remaining parameters, the difference between the two types of waves is manifested with maximum clarity. We can easily obtain the corresponding asymptotic form for all four roots of Eq. (3): 1 ω 1, 2 = ± kc – 2πiσ + O  --- ,  k

2

(3)

where ω is the frequency of the electromagnetic field and k is the wave number. Equation (3) was derived in [1], but the analysis of this equation made in [1, 2] has some drawbacks.

(4)

ω 3, 4

1 i = ± ω 0 – ----- + O  ----2 . k  T2

(5)

The solutions (5) obviously satisfy the stability criterion Imω < 0. We will gradually decrease the value of k. The instant of transition from stability to instability and back is characterized by a purely real-valued quan-

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ON THE STABILITY OF ELECTROMAGNETIC FIELD IN INVERTED MEDIA

tity ω. Then the separation of the real and imaginary components in Eq. (3) leads to two equations 2 2

k c -– 1 – --------2 ω

2 2 ωc ( ω

2 2 – 1/T 2 – ω 0 ) ----------------------------------------------------------------2 2 2 2 2 2 ( ω – 1/T 2 – ω 0 ) + 4ω /T 2

(6)

= 0,

2

2ωω c /T 2 4πσ ---------- + ----------------------------------------------------------------- = 0. 2 2 2 2 2 2 ω ( ω – 1/T – ω ) + 4ω /T 2

0

(7)

2

It is convenient to eliminate, first, the cumbersome denominator of fractions in Eqs. (6) and (7), which gives the values of the roots: 2 2πσT 2 ( ω 0

2 1/T 2 )

ω 0 + 1/T 2 ωc 4 --------------- + ----- > – -----------------------------------------------. 2πσT 2 ( 1 + 2πσT 2 ) 2πσT 2 T 22 2

2

(8)

The form of Eq. (3) rules out the possibility of ω = 0 and ω = ∞ at the very outset. The substitution of Eq. (8) into any of equations (6) or (7) allows us to obtain the required relation between the parameters at the critical point in the form of the quadratic equations for the auxiliary quantity K = k2c2:  ω 2c 4 1 2 2 - + -----2 K K + – 2  ω 0 + -----2 + ( 1 + 2πσT 2 )  -------------- T  2πσT 2 T  2

h = kc – ω 0 is introduced, and the quantities σ, |ωc |, and T 2 are also regarded as small. In this case, the roots ω from one of the pairs are also close to kc, and we can set –1

ω = kc + δ. The relation between the small correction δ and the renormalized wave number h is given (after disregarding quantities with a higher order of smallness) by the following simplified equation: ωc 2δ + 4πiσ – ----------------------------------- = 0, 2 ( δ + i/T 2 + h ) 2

(9)

2

4 1  2 ×  ω 0 + -----2  ---------------- + -----2 = 0.   T 2  2πσT 2 T 2 2 ωc

In some cases, the instability region does not exist at all. This is the case for a negative value of discriminant ∆ of Eq. (9) or for the presence of two real-valued but negative roots for K. To be more precise, ∆ < 0 for ωc 4 2 - < 4ω 0 , – -----2 < --------------T 2 2πσT 2

(12)

These relations can be conveniently illustrated for the particular case of resonance considered in detail in [1, 2]. In this case, a small correction

2

1 2 2 +  ω 0 + -----2 + 2πσT 2 ( 1 + 2πσT 2 )  T

2

If this condition is satisfied, the instability region has a finite left boundary on the K scale; otherwise, the instability region extends to infinitely small values of K since one of the roots becomes negative for k2c2.

2 2

+ +k c 2 ω = -------------------------------------------------------------. 1 + 2πσT 2

207

(13)

which is given in [1, 2] in a somewhat different form. In fact, it is a quadratic equation (the other two roots of Eq. (3) for ω are symmetric to those obtained by us and are close to –kc). The problem of the critical point of the stability–instability condition is solved as before: assuming that δ is purely real, we separate the real and imaginary components in Eq. (13). Under the same condition (4), both critical values for h are always realvalued:

2

and both roots for K are negative only if the coefficient of the first power of K in Eq. (9) has positive sign, i.e., ωc 4 2 ( ω 0 + 1/T 2 ) --------------- > – -----2 + -------------------------------. 1 + 2πσT 2 2πσT 2 T2 2

2

1 8πσ 2 h cr = ± ( 1 + 2πσT 2 ) ---------------- ω c + ---------- . 2πσT 2 T2

(10)

2

(11)

(14)

A simple analysis of this expression as a function of σ with respect to its extrema for the same values of the remaining parameters shows that the maximum and the minimum are attained at the levels 2 + µ ± ( 2 + µ ) ( µ – 14 ) σ = ------------------------------------------------------------- , 16πT 2

After the deduction of regions (10) and (11), we are 2 left with the range of ω c , namely, the region specified by inequality (4), in which a transition from instability to stability for any real K takes place. Thus, criterion (4) is confirmed by rigorous calculations.

where

3. BOUNDEDNESS OF THE INSTABILITY REGION FOR SMALL WAVE NUMBERS Under condition (4), the free term in Eq. (9) is positive if

Consequently, we can make the following refininement to Fig. 3 in [1]: the curve describing the dependence of σ on h has the form of an inverted suspended drop only for µ > 14; otherwise, we obtain a simple “hat” (in [1], only the condition µ > 4 is used explicitly).

(15)

µ = –ωc T 2 . 2

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4. MAXIMUM INCREMENT

2

If we separate the real and imaginary components in the small correction δ, δ = δ 1 + iδ 2 , the quantity δ2 will characterize the increment (if we choose solutions with δ2 > 0). The maximum of δ2 with respect to h can be obtained from the following considerations. We consider, along with Eq. (13), the complex conjugate relation

5. SEPARATION OF TWO TYPES OF WAVES As before, we confine our analysis to the region close to resonance. The solution of Eq. (13) is given by R i h σ = – πiσ – --------- – --- ± -------- , 2T 2 2 2

(21)

where

ωc 2δ∗ – 4πiσ – -------------------------------------- = 0, 2 ( δ∗ – i/T 2 + h ) 2

(16)

δ∗ = δ 1 – iδ 2 . We denote the left-hand sides of Eqs. (13) and (16) by Q(δ1, δ2, h) and Q*(δ1, δ2, h) and vary h as well as δ1 and δ2. Then we can write ∂Q ∂Q ∂Q -------- dδ 1 + -------- dδ 2 + -------dh ∂δ 2 ∂h ∂δ 1

(17)

∂Q∗ δQ∗ ∂Q∗ = ----------dδ 1 + ----------dδ 2 + ----------dh = 0. ∂δ 1 ∂δ 2 ∂h

For a value of h corresponding to the maximum increment, we must have dδ2 = 0, so that the solvability condition for system (17) has the form

2ih 1 2 2 2 R = h + -------- ( 1 – 2πσT 2 ) + ω c –  2πσ – ----- . (22)  T2 T 2 It is worth noting that ImR generally differs from zero and passes through zero only for h = 0 except in the special case mentioned in [1, 2], when 2πσT2 = 1. This means that R as a complex-valued quantity remains within a quadrant upon a change in h both for h > 0 (otherwise R would be real-valued) and for h < 0 (but the quadrant will be different). At the point h = 0, we have R2 < 0 and, hence, R passes through the imaginary axis at this moment. Let us now consider large values of h > 0, assuming, for the time being, that 2πσT2 > 1. From Eq. (22), we asymptotically obtain (see [3]) i 1 R = ± h + ----- ( 1 – 2πσT 2 ) + O  --- .  h T2

∂Q ∂Q -------- ------∂δ 1 ∂h ∂Q∗ ∂Q∗ ---------- ---------∂δ 1 ∂h

(18)

= 0.

According to the rule for differentiation of a complex function, we have ∂Q ∂Q -------- = -------, ∂δ ∂δ 1

for σ = 0 to zero for the known critical value ω c = −8πσ/T2.

∂Q∗ ∂Q∗ ---------- = ----------. ∂δ 1 ∂δ∗

Evaluating determinant (18) and carrying out elementary cancellation, we obtain h = 0. In this case, the roots of Eq. (13) are purely imaginary, δ = iy, and for a growing wave we have – ( 2πσ + 1/T 2 ) + ( 2πσ – 1/T 2 ) – ω c -. y = -------------------------------------------------------------------------------------------2 2

2

(19)

Thus, the only maximum of Imδ with respect to h is attained at the middle of the instability interval. At the same time, function (19) is monotonic with respect to σ. It decreases from the maximum value 1 1 1 2 y m = ---  – ----- + -----2 – ω c 2 T2 T2 

(20)

(23)

For the “upper” solution (corresponding to the larger value of Reδ), we chose the plus sign. Thus, Im R < 0 and R are in the fourth quadrant. At point h = 0, a transition to the third quadrant takes place, while, for h < 0 and large |h|, the position of R in the third quadrant indicates, according to the same formula (23), that we have also chosen the plus sign. Thus, the choice of the sign for R is matched on the left and right of scale h. The same is also true for 2πσT2 < 1 with the only difference that the values of R in the intermediate region pass through the upper (first and second) quadrants. The quantity X from formula (2.5) in [1] in the case of resonance has the following form in our notation: ωc δ + 2πiσ X = – --------------------------------------------- = – ---------------------. 2kc 8πkc ( δ + i/T 2 + h ) 2

(24)

This quantity tends to a finite value if we take the upper sign in formula (23) and to infinity if we take the lower sign. According to the terminology used in [1, 2], the latter case corresponds to a polariton wave and, accordingly, the former case corresponds to an electromagnetic wave. However, taking the above arguments into consideration, we can verify that the solution with the

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ON THE STABILITY OF ELECTROMAGNETIC FIELD IN INVERTED MEDIA

larger value of Imδ in the intermediate region corresponds to a polariton wave for 2πσT2 > 1 and to an electromagnetic wave for 2πσT2 < 1. This conclusion can be found in [1], but it is drawn on the basis of energy considerations, while our arguments are purely algebraic. The physical interpretation also changes in our approach. In the intermediate region (|h| < hcr from Eq. (14)), unstable solutions form a continuous set. The interpretation of a wave as an electromagnetic or a polariton wave must be treated as subjective. This difference becomes objective only for h ∞ far away from the instability region. 6. CONCLUSIONS A number of results obtained semi-intuitively in [1, 2] are confirmed by rigorous calculations. Above all, this concerns the criterion for the emergence of the instability region (4) on the scale of k. A qualitative separation of the cases when 2πσT2 > 1 and 2πσT2 < 1 is also confirmed, albeit not for the corresponding waves directly, but only for their continuous extension to the nonresonance region. In the instability region, however,

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the waist shown graphically in [1] and separating the two types of waves is not observed necessarily. In actual practice, a continuous transition may sometimes occur between polariton and electromagnetic waves upon a gradual change in k, and hence a wave cannot be assigned unambiguously to a certain type. The instability may also be manifested away from the resonance, but its region may or may not be bounded for small values of k. REFERENCES 1. V. V. Zheleznyakov, V. V. Kocharovskiœ, and Vl. V. Kocharovskiœ, Zh. Éksp. Teor. Fiz. 87, 1565 (1984) [Sov. Phys. JETP 60, 897 (1984)]. 2. V. V. Zheleznyakov, V. V. Kocharovskiœ, and Vl. V. Kocharovskiœ, Usp. Fiz. Nauk 159, 193 (1989) [Sov. Phys. Usp. 32, 835 (1989)]. 3. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968; Nauka, Moscow, 1987), Chap. I.4.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Translated by N. Wadhwa

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2002