Semiclassical Theory of the Laser
6.2 Cavity Modes
Chapter 6
MIRROR MIRROR
ACTIVE MEDUIM
Z = L Z = 0
Fig. 6.1 A plane parallel resonator bounded by a pair of plane mirrors facing each other. The active medium is placed inside the resonator
∇ ×E= −∂B
∂t (6.1)
∇ ×H=Jf +∂D
∂t (6.2)
∇ ·D=ρf (6.3)
∇ ·B=0 (6.4)
where ρf represents the free charge density and Jf the free current density;
E, D, B, and H represent the electric field, electric displacement, magnetic induc- tion, and magnetic field, respectively. Inside the cavity we may assume
ρf =0 (6.5)
B=μ0H (6.6)
D=ε0E+P (6.7)
Jf =σE (6.8)
where P is the polarization, σ the conductivity, and ε0 and μ0 are the dielectric permittivity and magnetic permeability of free space. It will be seen that the conduc- tivity term leads to the medium being lossy which implies attenuation of the field;
we will assume that other losses like those due to diffraction and finite transmission at the mirrors are taken into account inσ. Now,
∇ ×(∇ ×E)= −μ0∂
∂t(∇ ×H)= −μ0∂J
∂t −μ0∂2D
∂t2 (6.9)
or
∇ ×(∇ ×E)+μ0σ∂E
∂t +ε0μ0∂2E
∂t2 = −μ0∂2P
∂t2 (6.10)
If we assume the losses to be small and the medium to be dilute, we may neglect the second term on the left-hand side and the term on the right-hand side of the above equation to approximately obtain
∇ ×(∇ ×E)+ε0μ0∂2E
∂t2 =0 (6.11)
6.2 Cavity Modes 123 Further since P is small, Eq. (6.3) gives
0= ∇ ·D≈ε0∇ ·E (6.12) Thus
∇ × ∇ ×E= −∇2E+ ∇(∇ ·E)≈ −∇2E (6.13) or
∇ × ∇ ×E≈ −∇2E= −∂2E
∂z2 (6.14)
where, in writing the last equation, we have neglected the x and y derivatives; this is justified when intensity variations in the directions transverse to the laser axis is small in distances ~λ, which is indeed the case (seeChapter 7). Thus Eq. (6.11) becomes
−∂2E
∂z2 + 1 c2
∂2E
∂t2 =0 (6.15)
where c = (ε0μ0)−1/2 represents the speed of light in free space. If we further assume a specific polarization of the beam, Eq. (6.15) becomes a scalar equation:
∂2E
∂z2 = 1 c2
∂2E
∂t2 (6.16)
which we solve by the method of separation of variables:
E(z, t)=Z(z)T(t) (6.17) to obtain
1 Z
d2Z dz2 = 1
c2 1 T
d2T
dt2 = −K2(say) (6.18)
Thus
Z(z)=A sin(Kz+θ) (6.19)
where the quantity K corresponds to the wave number. At the cavity ends (i.e., at z=0 and z=L), the field [and hence Z(z)] will vanish, giving
θ=0 and
K= nπ
L , n=1, 2, 3,. . . (6.20)
We designate different values of K by Kn,(n=1, 2, 3,. . .). The corresponding time dependence will be of the form
cos nt where
n=Knc= nπc
L (6.21)
Thus the complete solution of Eq. (6.16) would be given by E(z, t)=
n
Ancos( nt)sin(Knz) (6.22) If we next include the term describing the loses, we would have (instead of Eq.6.16)
∂2E
∂z2 −μ0σ∂E
∂t = 1 c2
∂2E
∂t2 (6.23)
We assume the same spatial dependence(∼sin Knz)and the time dependence to be of the form eintto obtain
2n−μ0σc2in− 2n=0 (6.24) or
n= 1 2
iμ0σc2±
−μ20σ2c4+4 2n
1/2
≈ ± n+iσ 2ε0
(6.25)
Thus the time dependence is of the form exp
− σ 2ε0
t e±i nt (6.26)
the first term describing the attenuation of the beam. In the expression derived above, the attenuation coefficient does not depend on the mode number n; however, in general, there is a dependence on the mode number which we explicitly indicate by writing the time-dependent factor as1
exp
− n 2Qn
t e±i nt (6.27)
where
1Because of the losses, the field in the cavity decays with time as exp
− nt 2Qn
and hence the energy decays as exp
− nt Qn
.Thus, the energy decays to 1
e of the value at t=0 in a time tc=Qn
nwhich is referred to as the cavity lifetime (see also Section 7.4).
6.2 Cavity Modes 125 Qn=ε0
σ n (6.28)
represents the quality factor (see Section 7.4). Thus the solution of Eq. (6.23) would be
E(z, t)=
n
Anexp
− n 2Qn
t cos ( nt)sin (Knz) (6.29) Finally, we try to solve the equation which includes the term involving the polarization:
∂2E
∂z2 −μ0σ ∂E
∂t − 1 c2
∂2E
∂t2 =μ0∂2P
∂t2 (6.30)
[cf. Eqs. (6.16) and (6.23)]. We assume E to be given by E= 1
2
n
{En(t)exp [−i(ωnt+φn(t))]+c.c.}sin Knz (6.31)
where c.c. stands for the complex conjugate (so that E is necessarily real), En(t) andφn(t) are real slowly varying amplitude and phase coefficients, andωn is the frequency of oscillation of the mode which may, in general, be slightly different from n. We assume P to be of the form
P=1 2
n
{Pn(t, z)exp [−i(ωnt+φn(t))]+c.c.} (6.32)
where Pn(t, z)may be complex but is a slowly varying component of the polar- ization. On substitution of E and P in Eq. (6.30), we get2 (after multiplying by c2)
2 nEn−i
σ
ε0 ωnEn−2iωnE˙n− ωn+ ˙φn
2
En=ω2n
ε0
pn(t) (6.33) pn(t)= 2
L L
0
Pn(t, z)sin Knz dz
where we have neglected small terms involvingE¨n,φ¨n,P¨n,E˙nφ˙n,σE˙n,σφ˙n,φ˙np˙n, andp˙n which are all of second order. Now, sinceωn will be very close to n, we may write
2 n−
ωn+ ˙φn
2
≈2ωn
n−ωn− ˙φn
(6.34)
2Actually we have equated each Fourier component; this follows immediately by multiplying Eq. (6.32) by sin Kmz and integrating from 0 to L.
Thus, equating real and imaginary parts of both sides of Eq. (6.33), we get ωn+ ˙φn− n
En(t)= −1 2
ωn
ε0
Re(pn(t)) (6.35) E˙n(t)+1
2 ωn
Qn En(t)= −ωn
2ε0
Im(pn(t)) (6.36)
where
Qn=ε0ωn
σ (6.37)
When pn=0,ωn= nand En(t)will decrease exponentially with time – consis- tent with our earlier findings. In general, if we define the susceptibilityχ through the equation
pn(t)=ε0χnEn(t)=ε0
χn +iχn
En(t) (6.38)
whereχn andχnrepresent, respectively, the real and imaginary parts ofχn, then ωn+ ˙φn= n−1
2ωnχn (6.39)
and
E˙n= −1 2
ωn
QnEn−1
2ωnχnEn(t) (6.40) The first term on the right-hand side of Eq. (6.40) represents cavity losses and the second term represents the effect of the medium filling the cavity. It can be easily seen that ifχnis positive, then the cavity medium adds to the losses. On the other hand ifχnis negative, the second term leads to gain.3If
−χn= 1
Qn (6.41)
the losses are just compensated by the gain and Eq. (6.41) is referred to as the threshold condition. If−χn>1
Qn, there would be a buildup of oscillation.
From Eq. (6.39), one may note that if we neglect the term φ˙n the oscillation frequency differs from the passive cavity frequency by−12ωnχn, which is known as the pulling term. In order to physically understand the gain and pulling effects due to the cavity medium, we consider a plane wave propagating through the cavity medium. Ifχnrepresents the electric susceptibility of the medium for the wave, then the permittivityεof the medium would be
3We will show in Section6.3thatχnis negative for a medium with a population inversion.
6.2 Cavity Modes 127 ε=ε0+ε0χn=ε0(1+χn) (6.42) This implies that the complex refractive index of the cavity medium is
ε ε0
1/2
=(1+χn)1/2≈
1+1
2χn =1+1 2χn + i
2χn (6.43) The propagation constant of the plane wave in such a medium would be
β= ω c
ε ε0
1/2
= ω c
1+1
2χn +1 2iω
cχn
=α+iδ (6.44)
where
α=ω c
1+1
2χn ; δ =1 2
ω
cχn (6.45)
Thus, a plane wave propagating along the z-direction would have a z dependence of the form
eiβz =eiαze−δz 6.46)
In the absence of the component due to the laser transitionχn =χn = 0 and the plane wave propagating through the medium undergoes a phase shift per unit length ofω/c. The presence of the laser transition contributes both to the phase change and to the loss or amplification of the beam. Thus ifχn is positive, thenδ is positive and the beam is attenuated as it propagates along the z-direction. On the other hand ifχnis negative, then the beam is amplified as it propagates through the medium.
As the response of the medium is stimulated by the field, the applied field and the stimulated response are phase coherent.
In addition to the losses or amplification caused by the cavity medium, there is also a phase shift caused by the real part of the susceptibilityχn. We will show in the next section thatχn is zero exactly at resonance, i.e., if the frequency of the oscillating mode is at the center of the atomic line and it has opposite signs on either side of the line center. This additional phase shift causes the frequencies of oscillation of the optical cavity filled with the laser medium to be different from the frequencies of oscillation of the passive cavity (i.e., the cavity in the absence of the laser medium). The actual oscillation frequencies are slightly pulled toward the center of the atomic line and hence the phenomenon is referred to as mode pulling.