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On the possibility of obtaining stimulated emission without inversion
VoIume 45A, number 2
PHYSICS LETTERS
10 September 1973
ON THE POSSIBILITY OF OBTAINING STIMULATED EMISSION WITHOUT INVERSIOP R.D. SHARMA* Physics Department, Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA Received 19 April 1973 It is shown that in a two photon emission at frequencies w and w’, amplification at frequency w can be achieved without population inversion if (i) w < w’ and N’, the number of black body photons at frequency w’, is much smaller than unity.
3
In this letter we would like to point out that ampli-
fication of radiation in a two photon process can occur without population inversion. The process pointed out here is very different from stimulated Raman scattering [I] because (i) the latter requires that, ignoring degeneracies, the population of the upper (lower) level be greater than that of the lower (upper) level for stimulated emission at anti-Stokes (Stokes) shifted frequency and (ii) the gain does not go to zero with the idler signal. To understand the physics involved, consider a three level system, (fig. 1). Level 3 is radiatively connected to the other two levels, 1 and 2. Single photon transitions between 1 and 2 are forbidden. Two photon transitions between these levels [2] are allowed, the number of transitions per unit time or the transition probability dw is given by: dw= (d13 4 @23 ~‘1 + Cdl3 - ~‘1 (d23 ~1 2 l
032
l
+ 0’
w3z p
-1 Fig. 1. Three level system to illustrate the idea of stimulated emission without population inversion. Level 3 is radiatively coupled to levels 1 and 2. Single photon transitions between levels 1 and 2 are prohibited. Two photon transitions w2r = w + w’ are considered in this paper.
d3k &(2n)=
Nw2 8n3,2 dwdo
where N is the number of incident photons of frequency w and polarization Y per unit volume: da=_-.-
00’3 Ii%4
l
(43*4 (d23- u’)+ (d13u’)(d23 u) 2
x
l
O32+O
032
(1) where w and w’ are the frequencies of the two photons emitted E2 -El = A(w + w’). u and U’ are unit vectors along the direction the the electric vector of photon w and w’ which are emitted in the solid angles do and do’; d13 = 1,2 are transition dipole moments involved. The cross section do for stimulated emission of photon of frequency w is obtained by dividing dw by:
+ 0’
l
do’. (2)
w32 to
To keep the algebra to basic minimum, let us assume that d13,d2,, and u are coincident. Expression (2) upon integration over do’ gives:
(T=$
wo’3
I(w32
t w’)-’
+ (~32
+ w)-‘i2
df3 dg3*(3)
Ii%?
The increase in the number of photons of frequency w per unit length, or gain, is given by: g= f
‘$=
tN’)
n2a(1
* The work was supported jointly by the Air Force Cambridge Research Laboratory and the Air Force Office of Scientific Research under contract No. AFOSR-73-2475. * Visiting Professor of Physics.
2
a3 1
-
nlaM--n2ar
2
( &>
- nlur
1t
1
(4)
153
Volume 45A, number 2
PHYSICS LETTERS
where N’ is the number of photons of frequency o’, nl and n2 are the number densities of atoms in level 1 and 2 and ur are Rayleigh scattering cross sections from levels 1 ahd 2. a
u
= $n
r1
,4
[(W32 + &I)-l
W112 df3 (5)
+
(a31 -
+
(~32 - ~)-l]
Ii%2
cd4 =%
12
-
W32
+ w)-’
2 d:3. (6)
?A4
If the temperature of the gas is such that N’, the number of blackbody photons at frequency w’, is much smaller than unity, eq. (4) becomes: g= n2a - n2u,,
- nlu,,
(7)
where we have taken cognizance of the fact that N is large by approximating (1 + l/N) by unity. A more quantitative criterion for the validity of eq. (7) is that dz3, stimulated emission is possiN’ 0) from eqs. (3), (5), (6) and (7), if n2wf3 [(W32 t
W)-l + (032
- n2w3 [(a32 + - nla3
[(a31
+ a’)-l]
W)-l+(032
2
-W)-l12
+ W>-’ t (W32 --W)-l12
> 0
(8)
If w Q 031 and w 4 ~32, eq. (8) becomes n2013 (c&
t c.o~~)2-n2w3/4u~2-nl~3/4a~l>0. (9)
The interesting thing about eqs. (8) and (9) is that the positive terms vary as u’~, the frequency of the discarded photon, while the two negative terms vary as a3, the frequency of the laser photon. Since w t 0’ = Wan, we can, for nonzero values of n2, make the left-hand side of eq. (9) positive by adjusting the relative values of o and a’. In the event that (n2/n 1) 3 (w 32 /o 31 )2, the condition for gain simplifies to (of/w)3 > a. Before computing the magnitude of the gain it ap pears desirable to pause and reflect on the physics of the situation. In the single photon situation the gain g is given by g = nUup - npu, where np and n, are the number densities of upper and lower levels and ua and up are the absorption and emission cross sections. Recalling that detailed balancing requires up E gpu and
154
10 September 1973
ua E gUu where gU and gQ are the degeneracies of the upper and the lower leves, we get g= o (n,gQ - n,g,).
(10)
Physically eq. (10) means that we obtain gain when the number of atoms in the upper state times the number of states available after emission of photons is greater than the number of atoms in the lower state times the number of states available after absorption of photon. The case under consideration is a very straightforward extension of eq. (10) if we also include in the number of states those available to discarded photon of frequency 0’. By agreeing to.work at temperatures and intensities at U’ low enough so that the number density of photons of frequency w’ in the cavity is much smaller than unity, we have made the loss term which corresponds to the second term of the right-hand side of eq. (10) small. And the loss terms corresponding to Rayleigh scattering are made small by making the ratio (o’/w) large. Since the density of states in unit frequency interval at frequency w is proportional to m2 the device of making (o’/w) large results in increasing the number of states available at frequency U’ larger so that the first term in eq. (7) dominates. Gain per unit length divided by n2, the number density of excited atoms is approximately equal to $r (ww’/R2c4) df2 ds3. Assuming d 12 = d,, = 1 debye and w * 1015 and o’ = 1016, the gain is g= 10d28 n2 cm-l. Although the gain is small, we should note that scaling of the gain as ww’ gives this idea potential for short wavelength region. The author is grateful to Professor Feld, Javan and Kumit for encouragement and hospitality at M.I.T. The constructive criticism of Professor Javan led to clarification of the ideas discussed here.
References [l] N. Bloembergen, Am. J. Phys. 35 (1967) 989. [Z] M. Goffer-Mayer, Annalen der Phys. 9 (1931) 401; A. Dalgarno, in Physics of the one- and twoelectron atoms, ed., F. Bopp and H. Kleinpoppen, (North-Holland Publishing Co., Amsterdam 1967) pp. 261-267; U.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Relativistic quantum theory (Pergamon Press, New York 1967) pp. 190-195.
PHYSICS LETTERS
10 September 1973
ON THE POSSIBILITY OF OBTAINING STIMULATED EMISSION WITHOUT INVERSIOP R.D. SHARMA* Physics Department, Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA Received 19 April 1973 It is shown that in a two photon emission at frequencies w and w’, amplification at frequency w can be achieved without population inversion if (i) w < w’ and N’, the number of black body photons at frequency w’, is much smaller than unity.
3
In this letter we would like to point out that ampli-
fication of radiation in a two photon process can occur without population inversion. The process pointed out here is very different from stimulated Raman scattering [I] because (i) the latter requires that, ignoring degeneracies, the population of the upper (lower) level be greater than that of the lower (upper) level for stimulated emission at anti-Stokes (Stokes) shifted frequency and (ii) the gain does not go to zero with the idler signal. To understand the physics involved, consider a three level system, (fig. 1). Level 3 is radiatively connected to the other two levels, 1 and 2. Single photon transitions between 1 and 2 are forbidden. Two photon transitions between these levels [2] are allowed, the number of transitions per unit time or the transition probability dw is given by: dw= (d13 4 @23 ~‘1 + Cdl3 - ~‘1 (d23 ~1 2 l
032
l
+ 0’
w3z p
-1 Fig. 1. Three level system to illustrate the idea of stimulated emission without population inversion. Level 3 is radiatively coupled to levels 1 and 2. Single photon transitions between levels 1 and 2 are prohibited. Two photon transitions w2r = w + w’ are considered in this paper.
d3k &(2n)=
Nw2 8n3,2 dwdo
where N is the number of incident photons of frequency w and polarization Y per unit volume: da=_-.-
00’3 Ii%4
l
(43*4 (d23- u’)+ (d13u’)(d23 u) 2
x
l
O32+O
032
(1) where w and w’ are the frequencies of the two photons emitted E2 -El = A(w + w’). u and U’ are unit vectors along the direction the the electric vector of photon w and w’ which are emitted in the solid angles do and do’; d13 = 1,2 are transition dipole moments involved. The cross section do for stimulated emission of photon of frequency w is obtained by dividing dw by:
+ 0’
l
do’. (2)
w32 to
To keep the algebra to basic minimum, let us assume that d13,d2,, and u are coincident. Expression (2) upon integration over do’ gives:
(T=$
wo’3
I(w32
t w’)-’
+ (~32
+ w)-‘i2
df3 dg3*(3)
Ii%?
The increase in the number of photons of frequency w per unit length, or gain, is given by: g= f
‘$=
tN’)
n2a(1
* The work was supported jointly by the Air Force Cambridge Research Laboratory and the Air Force Office of Scientific Research under contract No. AFOSR-73-2475. * Visiting Professor of Physics.
2
a3 1
-
nlaM--n2ar
2
( &>
- nlur
1t
1
(4)
153
Volume 45A, number 2
PHYSICS LETTERS
where N’ is the number of photons of frequency o’, nl and n2 are the number densities of atoms in level 1 and 2 and ur are Rayleigh scattering cross sections from levels 1 ahd 2. a
u
= $n
r1
,4
[(W32 + &I)-l
W112 df3 (5)
+
(a31 -
+
(~32 - ~)-l]
Ii%2
cd4 =%
12
-
W32
+ w)-’
2 d:3. (6)
?A4
If the temperature of the gas is such that N’, the number of blackbody photons at frequency w’, is much smaller than unity, eq. (4) becomes: g= n2a - n2u,,
- nlu,,
(7)
where we have taken cognizance of the fact that N is large by approximating (1 + l/N) by unity. A more quantitative criterion for the validity of eq. (7) is that dz3, stimulated emission is possiN’
W)-l + (032
- n2w3 [(a32 + - nla3
[(a31
+ a’)-l]
W)-l+(032
2
-W)-l12
+ W>-’ t (W32 --W)-l12
> 0
(8)
If w Q 031 and w 4 ~32, eq. (8) becomes n2013 (c&
t c.o~~)2-n2w3/4u~2-nl~3/4a~l>0. (9)
The interesting thing about eqs. (8) and (9) is that the positive terms vary as u’~, the frequency of the discarded photon, while the two negative terms vary as a3, the frequency of the laser photon. Since w t 0’ = Wan, we can, for nonzero values of n2, make the left-hand side of eq. (9) positive by adjusting the relative values of o and a’. In the event that (n2/n 1) 3 (w 32 /o 31 )2, the condition for gain simplifies to (of/w)3 > a. Before computing the magnitude of the gain it ap pears desirable to pause and reflect on the physics of the situation. In the single photon situation the gain g is given by g = nUup - npu, where np and n, are the number densities of upper and lower levels and ua and up are the absorption and emission cross sections. Recalling that detailed balancing requires up E gpu and
154
10 September 1973
ua E gUu where gU and gQ are the degeneracies of the upper and the lower leves, we get g= o (n,gQ - n,g,).
(10)
Physically eq. (10) means that we obtain gain when the number of atoms in the upper state times the number of states available after emission of photons is greater than the number of atoms in the lower state times the number of states available after absorption of photon. The case under consideration is a very straightforward extension of eq. (10) if we also include in the number of states those available to discarded photon of frequency 0’. By agreeing to.work at temperatures and intensities at U’ low enough so that the number density of photons of frequency w’ in the cavity is much smaller than unity, we have made the loss term which corresponds to the second term of the right-hand side of eq. (10) small. And the loss terms corresponding to Rayleigh scattering are made small by making the ratio (o’/w) large. Since the density of states in unit frequency interval at frequency w is proportional to m2 the device of making (o’/w) large results in increasing the number of states available at frequency U’ larger so that the first term in eq. (7) dominates. Gain per unit length divided by n2, the number density of excited atoms is approximately equal to $r (ww’/R2c4) df2 ds3. Assuming d 12 = d,, = 1 debye and w * 1015 and o’ = 1016, the gain is g= 10d28 n2 cm-l. Although the gain is small, we should note that scaling of the gain as ww’ gives this idea potential for short wavelength region. The author is grateful to Professor Feld, Javan and Kumit for encouragement and hospitality at M.I.T. The constructive criticism of Professor Javan led to clarification of the ideas discussed here.
References [l] N. Bloembergen, Am. J. Phys. 35 (1967) 989. [Z] M. Goffer-Mayer, Annalen der Phys. 9 (1931) 401; A. Dalgarno, in Physics of the one- and twoelectron atoms, ed., F. Bopp and H. Kleinpoppen, (North-Holland Publishing Co., Amsterdam 1967) pp. 261-267; U.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Relativistic quantum theory (Pergamon Press, New York 1967) pp. 190-195.