chapter 13: quantum optics of lasers - cornell university · quantum optics for photonics and...

Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)

1

Chapter 13: Quantum Optics of Lasers

13.1 Introduction Consider a N two-level system interacting with a single mode of radiation in a cavity, as shown below.

In this Chapter, we will discuss the operation of a laser composed of the above two-level systems interacting with a single mode of radiation in a cavity. We assume that the two-level systems are “pumped” into a population inverted state using a third level, as shown below.

All electrons pumped form the lowest level into the third level relax down very quickly into the second level. Including the effects of cavity photon loss, population relaxation, decoherence, and pumping one can write the following set of equations,

tFtFttaktatki

T

tN

T

tN

dt

tNd

tFtFttaktatki

T

tN

T

tN

dt

tNd

PNp

PNp

ˆˆˆˆˆˆˆˆˆ

ˆˆˆˆˆˆˆˆˆ

1

211

1

212

Cavity Waveguide

z = 0

Non-radiative relaxation: T1

Spontaneous emission

Stimulated emission

Pump: TP

Level 1

Level 2

Level 3


2

ti

ti

o

o

etFtatNtNki

tT

idt

td

etFtNtNtaki

tT

idt

td

ˆˆˆˆˆ1ˆ

ˆˆˆˆˆ1ˆ

122

122

tiin

ppo

tiin

ppo

o

o

etStki

taidt

tad

etStki

taidt

tad

ˆ1ˆˆ

2

1ˆ

ˆ1ˆˆ

2

1ˆ

Here, the input vacuum fluctuation operators are,

tiLg

tiin

tiLg

tiin

oo

oo

etzbvetS

etzbvetS

,0ˆˆ

,0ˆˆ

The photons coming out of the cavity are described by the equations,

tiin

p

tiLg

p

tiRg

tiout

tiin

p

tiLg

p

tiRg

tiout

oo

oo

oo

oo

etStaetzbvta

etzbvetS

etStaetzbvta

etzbvetS

ˆˆ1,0ˆˆ1

,0ˆˆ

ˆˆ1,0ˆˆ1

,0ˆˆ

As discussed in an earlier Chapter, in the limit of strong decoherence one can integrate out the equations for the operators t̂ and t̂ and obtain the following operator rate equations,

)(ˆ)(ˆ)(ˆ)(ˆ

ˆˆˆˆ)(ˆ)(ˆ2ˆˆˆ

)(ˆ)(ˆ)(ˆ)(ˆ


121

211

121

212

tatFetFtae

tFtFtatatNtNgT

tN

T

tN

dt

tNd

tatFetFtae

tFtFtatatNtNgT

tN

T

tN

dt

tNd

spti

spti

PNdp

spti

spti

PNdp

oo

oo

tiin

psp

tid

po

tiin

psp

tid

po

oo

oo

etStFetNtNtagtaidt

tad

etStFetatNtNgtaidt

tad

ˆ1)(ˆ)(ˆ)(ˆ)(ˆˆ

2

1ˆ

ˆ1)(ˆˆ)(ˆ)(ˆˆ

2

1)(ˆ

12

12


3

)(ˆ)(ˆ)(ˆ)(ˆ1

)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ2ˆ)(ˆ

12

tatSeetSta

tatFetFtaetntNtNgtn

dt

tnd

intiti

inp

spti

spti

dp

oo

oo

where,

22

22

2

2

1

1gainaldifferenti

T

Tkg

od

Suppose the average populations, photon number, and output photon flux are,

p

outoutRtn

tStStFtF

tntn

tNtN

tNtN

ˆˆ,0ˆ,0

ˆ

ˆ

ˆ

11

22

The average populations and photon number obey the equations,

3 21

2

2 22

1 22

212

2121

211

2121

212

tdNtntNtNgdt

tdn

tNgtntNtNgT

tN

T

tN

dt

tdN

tNgtntNtNgT

tN

T

tN

dt

tdN

pd

ddp

gdp

In (1) and (2), it is convenient to assume that the spontaneous emission term is absorbed in the definition of relaxation (i.e. 22 Ngd is included in the term 12 TN ). Define, tNtNtNd 12 , and since

NtNtN 12 , one can write,

422 12

tNNtN

tNNtN dd

Subtract (2) from (1), to get,

54

1111

11tntNg

TTtN

TTN

dt

tdNdd

pd

p

d

so,

6

12 tNNgtntNg

dt

tdndd

pdd

Equations (5) and (6) describe the operation of a laser.


4

In the steady state (5) and (6) give,

bnNgTT

NTT

N ddp

dp

541111

011

bNNgnNg ddp

dd 6 1

20

Of course, one can use a computer to solve (5b) and (6b), which form a system of coupled non-linear equations for the population difference and the photon number. To gain insight, we obtain approximate solutions in different regimes.

13.2 Regimes of Operation for a Laser 13.2.1 No Pumping (1/Tp = 0) When the pumping rate is zero (i.e. 1/Tp = 0), one obtains the steady state solution,

0

021

n

NNN

NNd

When the pump is turned off, all electrons are sitting in the lower energy level and there are no photons inside the cavity. 13.2.2 Medium Offers Net Optical Loss (0 < 1/Tp < 1/T1 ) Suppose pumping rate is increased from zero. dN will increase from N but will remain negative if

1110 TTp . The photons in the cavity experience no net gain, but net loss (loss from stimulated

absorption and from photons escaping from the cavity). The photon number in the steady state will be very small. Therefore, in this regime 1110 TTp we can solve (5b) and (6b) by first solving (5b)

for dN assuming 0n , and then using the calculated value of dN in (6b) to get n (and then verify that n is indeed small). In other words, the population behaves as if no photons are present. Since the photon number is small in this regime, stimulated emission and absorption rates are negligible compared to the

Non-radiative relaxation: T1

Spontaneous emission

Stimulated emission

Pump: TP

Level 1

Level 2

Level 3


5

pump rate and the relaxation rate and that is why it is justified to assume 0n in (5b). Equation (5b) gives,

7

411

11

1

1

1

1

p

p

dp

pd TT

TTN

ngTT

TTNN

Using the above result in (6b) gives,

8

21 1

1

p

pd

ddp

dd

TT

TTNN

Ng

NNgn

dN vs pT1 as obtained from Equation (7) is plotted below.

For all values of pumping rate pT1 less than the relxation rate 11 T , dN is negative. This means that the

stimulated emission rate nNgd 22 is less than the stimulated absorption rate nNgd 12 . The material offers net loss to the photons, and photons are lost from the cavity not just by escaping into the waveguide but also by stimulated absorption. 13.2.3 Medium is Transparent (1/Tp = 1/T1 ) When the pumping rate PT1 equals the relaxation rate 11 T we have,

0 1

1

p

pd TT

TTNN

which implies, 12 NN

Transparency is the condition when the stimulated emission rate nNgd 22 is equal to the stimulated

absorption rate nNgd 12 . A transparent medium offers no net optical gain and no net optical loss. However, the photons in the cavity will still see a net loss because photons can escape from the cavity. But at least in transparency the material is not contributing to the optical loss. The photon number n is still small, and so the approximation 0n in (7) remains valid. 13.2.4 Medium Offers Net Optical Gain (1/Tp > 1/T1 )

N

-N

1/Tp 1/T1

Nd

Medium offers net optical loss

Medium offers net optical gain


6

When the pumping rate pT1 exceeds the relaxation rate 11 T , dN becomes positive (population

inversion). This means that the stimulated emission rate nNgd 22 is greater than the stimulated

absorption rate nNgd 12 , the material now provides net optical gain. Equation (6) shows that the net optical gain experienced by the photons in steady state equals,

pdd Ng

1

2

As long as pdd Ng 12 the photons in the cavity experience net loss since the loss due to photons

escaping from the cavity exceeds the gain from the population inverted medium. Consequently, the average number of photons n in the cavity is still small and therefore the approximation 0n in (7) remains valid. As the pumping rate is increased further, dN increases (as given by Equation (7)) and at some point dN becomes so large that the optical gain from the medium approaches the optical loss due to photons escaping from the cavity, i.e.,

p

dd Ng1

2

Consequently, the number of photons in the cavity, as given earlier by Equation (8),

8

21

ddp

d

Ng

NNdn

increases dramatically since the denominator approaches zero. In physical terms, the net photon generation rate through stimulated processes (given by nNg dd2 ) is becoming equal to the photon loss

rate due to photons escaping from the cavity (given by pn ). Photons are multiplying at a rate nearly

equal to the rate at which they are leaving the cavity. Consequently, a large population of photons builds up in the cavity in steady state. The relation,

ddp

d

Ng

NNdn

21

suggests that the photon number will becomes if dd Ng2 becomes equal to p1 , and even negative

if dd Ng2 were to exceed p1 . So it must be that dd Ng2 should never exactly equal p1 and must

always be less than p1 . How is this guaranteed? We go back to Equation (7),

7

411

11

1

1

1

1

p

p

dp

pd TT

TTN

ngTT

TTNN

When the photon number in the cavity is large the second approximate equality in the above expression no longer remains valid. A large photon number will ensure in steady state the value of dN , for a given pumping rate, is much smaller than the value predicted by the approximate relation,

p

pd TT

TTNN

1

1


7

The value of the pumping rate for which the above approximate relation (falsely) predicts that the material gain dd Ng2 will equal cavity photon loss p1 is called the threshold pumping rate,

12

1211

122

1

1

1

pd

pd

pth

ppth

pthddd

Ng

Ng

TT

TT

TTNgNg

The population difference value needed to make the material gain equal to the cavity loss is called the threshold population difference,

p

dthdNg1

2

The question then is what happens when the pumping rate pT1 approaches pthT1 and when pT1

exceeds pthT1 . This we address next.

13.2.5 Threshold and Lasing (1/Tp > 1/Tpth ) We know that when pT1 approaches pthT1 , dd Ng2 approaches p1 , and consequently n becomes

very large. A large photon number n means a large stimulated emission rate (given by nNg dd2 ). A large stimulated emission rate means that the population in the upper level is getting depleted at a large rate. Consequently, we cannot ignore the term that has n in the Equation,

ngTT

TTNN

dp

pd

411

11

1

1

Suppose pT1 is equal to or larger than pthT1 , and the material gain nNg dd2 is close to p1 . If

pT1 is now increased then the photon number increases so much that the accompanying increased

stimulated emission rate keeps the population difference dN from changing much at all. The net result is

that when the pumping rate pT1 is increased beyond pthT1 , the population difference does not change

at all and only the photon number increases, and the increase in the photon number is such as to keep the

N

-N

1/Tp 1/T1

Nd



1/Tpth

Ndth


8

value of population difference dN from changing much. This pinning of the population difference dN ,

and the material gain nNg dd2 , when the pumping rate pT1 rate is equal to or larger than the threshold

pumping rate pthT1 , is called gain saturation. The rapid increase in the photon number (when the

pumping rate pT1 is larger than the threshold pumping rate pthT1 ) is called lasing.

Approximate Model for Lasing: When the pumping rate pT1 exceeds the threshold pumping rate

pthT1 , the population difference does not change much and remains pinned at values close to the

threshold value thdN ,

p

dthddthd NgNN1

2

Of course, the above relation is not exactly true since the material gain nNg dd2 must always be less

than the cavity loss p1 . We need to find the cavity photon number when pT1 exceeds the threshold

pumping rate pthT1 . We should not use Equation (8) (or Equation (6b)),

ddp

d

Ng

NNdn

21

since our assumption of dthd NN will give an infinite value for n . Instead, we use Equation (5b) to determine n above threshold,

bnNgTT

NTT

N ddp

dp

541111

011

We can replace dN in the above equation by its approximate value dthN (where pthdd Ng 12 ) to

get,

pppdp

n

TTgTTN

2

11

2

1110

11

pthpd

pd

pppdp

TTg

Ngn

n

TTgTTN

11

4

12

211

2

1110

11

When pT1 exceeds the threshold pumping rate pthT1 , n increases linearly with the pumping rate, and

the population difference dN remains fixed at a value close to dthN . 13.2.6 Approximate model for 1/Tp < 1/Tpth (laser below threshold) Below threshold, the population difference increases with the pumping rate. The photon number also increases with the pumping rate but remains relatively small.

p

pd TT

TTNN

1

1


9

02

1

ddp

d

Ng

NNdn

13.2.7 Approximate model for 1/Tp > 1/Tpth (laser above threshold) Above threshold the population difference is pinned at the threshold population difference value. The photon number increases linearly with the pumping rate.

p

dthddthd NgNN1

2

pthpd

pd

TTg

Ngn

11

4

12

Since,

12

12

NNN

NNNd

above threshold the populations in both the upper and the lower level are also constant,

pd

dth

pd

dth

g

NNNN

g

NNNN

4

1

22

4

1

22

1

2

N

-N

1/Tp 1/T1

Nd



1/Tpth

Ndth


10

13.2.8 Output Photon Flux and Output Power Below threshold, the output power is small and is due to amplified spontaneous emission and one can approximate the output power as being zero. Above threshold, the flux of photons leaving the cavity is,

pthppd

pd

p TTg

Ngn 11

4

12

Therefore, the output power above threshold is.

pthppd

pdo

po TTg

NgnP

11

4

12

13.2.9 Exact Graphical Solution The steady state Equations (5b) and (6b),

bnNgTT

NTT

N ddp

dp

541111

011

bNNdnNg dp

dd 6 1

20

can be solved graphically to obtain solutions for both below and above threshold operation. The Figure below shows the graphical solution of Equations (5b) and (6b). Equation (5b) is plotted (dashed) for different values of the pump rate pT1 . Equation (6b) is plotted (solid) as a single line. It can be seen that

for pumping rates below the threshold pumping rate pthT1 , the population difference dN increases with

the pumping rate and the photon number remains very small. When the pumping rate exceeds the threshold pumping rate, the population difference dN remains fixed at values close to dthN and the photon number increases rapidly with the pumping rate.

1/Tp 1/T1

n

1/Tpth

Above threshold

Below threshold


11

13.2.10 Laser Stability above Threshold and Relaxation Oscillations The steady state solution of the laser rate equations above threshold is,

p

dthdthd dNNN1

2

pthpd

pd

TTg

Ngn

11

4

12

The questions is how stable are the above steady state values? In other words, if the populations or the photon number are perturbed from their steady state values do they return to their steady state values? To study the stability of the laser above threshold expand the population difference and photon number around their steady state values,

)()(

)()(

tnntn

tNNtN ddd

Plug the above expansions in the laser rate equations,

tntNg

TTtN

TTN

dt

tdNdd

pd

p

d 41111

11

12 tNNgtntNg

dt

tdndd

pdd

Since )(tn and )(tNd are small perturbations around the steady state, one can ignore terms that are of second order in the perturbations (i.e linearize the equations) to get,

Nd

n

Ndth -N

1/Tp=0

1/Tp=1/T1

1/Tp=1/Tpth

0

Increasing 1/Tp

1/Tp > 1/Tpth


12

)(

)(

02

24

11

)(

)(

)(1

22

12)(

)(4411

)(

1

02

thresholdabove2

1

tn

tN

ng

ngTT

tn

tN

dt

d

tnNgngtNdt

tnd

tnNgngTT

tNdt

tNd

d

d

pd

pd

pdd

dn

dd

dddp

dd

p

Let,

r1

differential relaxation time ngTT d

p4

11

1

)(

)(

02

21

)(

)(

tn

tN

ngtn

tN

dt

d d

d

prd

Solution: The above coupled differential equations give,

0)(1

0)(1

22

2

22

2

tndt

d

dt

d

tNdt

d

dt

d

rr

drr

Where,

p

dr

ng

42 relaxation oscillation frequency

The second order differential equations show that the perturbations )(tNd and/or )(tn are damped and the laser is stable above threshold against small perturbations. The general solution can be written as,

22

4

1

)cos()sin()(

)cos()sin()(

rrr

rt

rt

rt

rt

d

teDteCtn

teBteAtNrr

rr

where

The constants A,B,C and D are set by the initial conditions. Therefore if the population difference or the photon number are disturbed from their steady state values they return to their steady state values executing damped oscillations which are called relaxation oscillations. Since,

)(2 tNngdt

tnddd

during relaxation oscillations photon number oscillates 90 out of phase with the population difference oscillations. The Figure below depicts the relaxation oscillations that occur when the population


13

difference is suddenly increased from its steady state value (by momentarily increasing the pumping rate, for example).

13.3 Field Amplitude Equation for a Laser and Analogy with Second Order Phase Transitions In this Section, we look at the field amplitude and phase dynamics in a laser. We suppose that,

ti oetta ˆ

and therefore expand the destruction operator as follows,

0ˆˆˆ tataetta ti o where

We use the above expansion in the equation,

tiin

psp

tid

po

oo etStFetatNtNgtaidt

tad

ˆ1

)(ˆˆ)(ˆ)(ˆˆ2

1)(ˆ12

and take the average of the resulting equation to obtain,

tNtNtNttNgdt

tdd

pdd 12

ˆˆ where)(2

1)(

)(

Assuming 2ˆˆ ttata , the average population difference satisfies,

2

114

1111ttNg

TTtN

TTN

dt

tdNdd

pd

p

d

Suppose we are interested in field dynamics over time scales much longer than the population relaxation

times. One can then obtain a simple relation between the average population difference tNd and 2t

by setting 0dttdNd in the above equation. The result is,

2

1

1

)(411

11

)(

tgTT

TTNtN

dp

pd

The above equation can be used to eliminate tNd in the equation for t to get,

∆n(t)

∆Nd(t)


14

)(2

1

)(411

11

)(

2

1

1t

tgTT

TTNg

dt

td

pd

p

pd

)(t is a complex field amplitude, with an amplitude and a phase, or equivalently, with real and

imaginary parts. Let ),()()( 21 txitxt where )(1 tx and )(2 tx are real. We can also define a vector

)(tr

in the two dimensional 21 xx plane as,

ytxxtxtr ˆ)(ˆ)()( 21

and,

222

21

2 )()()( ttxtxtrtrtr

Then the equation for )(t can be written as,

)( trV

dt

trd

where,

1

1

1

1

2

112tan

11112

1

4)(

TT

dr

TTN

TT

drrV

p

p

p

p

The dynamical equation, )(rVdtrd

, shows that the vector )(tr

moves in the direction of the

steepest descent with respect to the function rV

and in steady state )(tr

will be at the point where the

function rV

has a minimum. Below, we examine the minima of rV

as a function of the pumping rate

pT1 .

13.3.1 Laser below Threshold (1/Tp < 1/Tpth ) Recall that the threshold pumping rate is,

)12(

)12(11

1

pd

pd

pth Ng

Ng

TT

For pthp TT 11 , the function rV

, plotted below, has only a single minimum at 0r

. Consequently,

below threshold the steady state solution is, 0ˆ)(ˆ)()( 21 ytxxtxtr

0ˆ ti oetta

Note that although photons due to amplified spontaneous emission are present in the cavity, the average field amplitude is zero since the field has no well-defined phase.


15

13.3.2 Laser above Threshold (1/Tp > 1/Tpth ) For pumping rate above the threshold pumping rate, the function rV

has a minima for all points in the

two dimensional 21 xx plane with values of r equal to minr where,

pthpd

pd

TTg

Ngr

11

4

)12(2min

Therefore, in steady state,

nTTg

Ngrt

pthpd

pd

numberphotonstatesteady

11

4

)12()( 2

min2

Note that the function rV

is radially symmetric and only depends on the distance from the center (i.e. on

)()( trtr and not on the direction of )(tr

). The contour of minimum values of rV

form a circle in the

two dimensional 21 xx of radius minr . Recall that,

ti oetta ˆ

Above threshold, the field operator has a non-zero average amplitude and a phase,

)(

tan22

2121

)(

)()()()()( 121

ti

txtxi

etr

etxtxtxitxt

The laser equations uniquely determine the average amplitude of the field in steady state but not the average phase of the field. In fact, there is no preference for any particular value of the phase )(t . Above

r

1/Tp < 1/Tpth

1/Tp > 1/Tpth

rrmin rmin

V(r)

(t)| = rmin 1/Tp > 1/Tpth

x1

x2

1/Tp < 1/T

pth 1/T

p > 1/T

pth


16

threshold, the field acquires an average value of the phase that is quite arbitrary. In other words, there is spontaneous breakdown of phase symmetry above threshold. This is quite similar to the breakdown of spin up-down symmetry in ferromagnets below the magnetic phase transition temperature. Although the spin-spin interactions in a ferromagnet can have complete up-down symmetry, the ferromagnetic state below the transition temperature has all spins either pointing in the upward direction or in the downward direction and the actual up or down direction selected is quite arbitrary. Thus, the lasing transition has many features in common with second order phase transitions. Since there is no restriction on the average phase t of the field above threshold, the smallest possible noise or perturbations can send it wandering on the circle shown in the Figure above. But the magnitude

of t is more or less fixed and equals n . We have already seen that any perturbation in the steady state value of n decays after executing relaxation oscillations.

13.4 Laser Phase Diffusion and the Schawlow-Townes Expression for the Laser Linewidth 13.4.1 Laser below Threshold (1/Tp < 1/Tpth ) Consider first a laser operating below threshold. The Heisenberg equation for the field operator is,

tiin

psp

ti

po

oo etStFetatNtNdtaidt

tad

ˆ1

)(ˆˆ)(ˆ)(ˆˆ2

1)(ˆ12

Here, the Langevin operator )(ˆ tFsp models the spontaneous emission noise and the Langevin

operator )(ˆ tSin models the vacuum fluctuations entering the cavity. If one is interested in phase fluctuations over long time scales (long compared to the population relaxation times) then to a very good approximation one can replace the operator )(ˆ)(ˆ

12 tNtN by its average time independent value,

dNtNtN 12ˆˆ

to get,

tiin

psp

ti

pddo

oo etStFetaNgidt

tad

ˆ1

)(ˆˆ2

1)(ˆ

Recall from an earlier Chapter that the spectrum of a quantum state of light is related to the Fourier transform of the first order coherence function,

i

i

etatatata

tatad

ettgdS

ˆˆˆˆ

ˆˆ

,1

The Heisenberg-Langevin equation above can be solved to get,

22

2

o

S


17

where the FWHM linewidth of the spectrum is,

ddp

Ng21

In steady state, the difference between the photon loss rate and the stimulated emission rate is the spontaneous emission rate,

2221

NgnNg dddp

Therefore,

n

NgNg d

ddp

222

1

spontaneous emission rate

As the laser approaches threshold, and the gain begins to approach the loss, the linewidth narrows. Right before threshold, when,

p

dd Ng1

2

using the definition of the spontaneous emission factor given earlier,

12

2

NN

Nnsp

we can write,

n

n

n

NgNg sp

p

ddd

p 12

21 2

13.4.2 Laser above Threshold (1/Tp > 1/Tpth ) Below threshold, both the in-phase and the out-of-phase quadratures of the noise operators affect the laser linewidth. Above threshold, the amplitude fluctuations are stabilized as a result of gain saturation and the amplitude and phase dynamics are decoupled. Consequently, only the out-of-phase quadratures of the noise operators affect the linewidth. Out-of-phase noise perturbations cause the laser phase to diffuse in time and this phase diffusion determines the laser linewidth. Consider a laser operating above threshold. Suppose at time t the field operator has an average value given by,

tiiti oo eeta ˆ

where is the average field amplitude and is the average phase of the field. We would like to study

the phase dynamics. The field operators obey the equations,

10ˆ1)(ˆ)(ˆ)(ˆ)(ˆˆ

2

1ˆ

9ˆ1)(ˆˆ)(ˆ)(ˆˆ

2

1)(ˆ

12

12

tiin

psp

tid

po

tiin

psp

tid

po

oo

oo

etStFetNtNtagtaidt

tad

etStFetatNtNgtaidt

tad

Recall that one can expand the field operator in terms of quadrature operators that are along and perpendicular to the direction specified by the average phase value,


18

tiitii

tii

oo

o

etxitxe

etxitxta

2

2

ˆˆ

ˆˆˆ

Also recall that the phase fluctuation operator is,

tx

t2ˆ

ˆ

In order to obtain an equation for the phase fluctuation operator we substitute the expansion of the field operator in terms of the quadrature fluctuation operators in Equations (9) and (10) and subtract the resulting equations. The result is,

i

etSetS

i

etFetFttNtNg

dt

td

iin

iin

p

isp

isp

pd

2

ˆ)(ˆ11

2

ˆ)(ˆ1ˆ2

1)(ˆ)(ˆ

ˆ12

Above threshold the average value of the population difference is fixed and equal to dthN . Any deviations from this average value are quickly restored. Therefore, if one is interested in phase fluctuations over long time scales (long compared to the population relaxation times) then to a very good

approximation one can replace the operator )(ˆ)(ˆ12 tNtN by its average value,

pd

dthd gNtNtNtN

2

1ˆˆ12

to get a much simpler equation for the phase fluctuation operator,

i

etSetS

i

etFetF

dt

tdi

ini

in

p

isp

isp

2

ˆ)(ˆ11

2

ˆ)(ˆ1ˆ

The above equation shows that the phase is kicked around by noise just like a particle undergoing diffusive motion. There is no mechanism that restores the phase to a specific value. The noise driving the phase is due to spontaneous emission and also due to vacuum fluctuations entering the cavity from the waveguide. The role played by spontaneous emission can be understood in a semi-classical spirit as follows. Every spontaneously emitted photon has a random phase. Therefore, after every spontaneous emission event the average field phasor gets a random kick whose component tangent to the field phasor

(t)| = rmin 1/Tp > 1/Tpth

x1

x2


19

contributes to the phase noise. Under the action of the noise, the field phasor wanders randomly on the circle of radius minr where,

nr 2min

2

The diffusion equation for the phase can be directly integrated to give,

'2

1

threshold above1

2'4

1

4

121

threshold above '4

1

4

122

'4

1

4

122'ˆˆ

22

22

22

2

ttn

n

Ngttn

NNttnNg

ttnNg

tt

sp

p

pdthd

pspp

dthdpspdthd

pspdd

Note

that the laser phase diffusion has almost equal contributions from spontaneous emission and vacuum fluctuations. We are now in a apposition to look at the laser linewidth,

i

i

etatatata

tatad

ettgdS

ˆˆˆˆ

ˆˆ

,1

For a laser above threshold we have,

2ˆˆˆˆ tatatata

and,

n

n

i

tt

i

titii

sp

po

o

o

ee

ee

eeetata

42

2

ˆˆ

2

ˆˆ2

ˆˆ

2

Therefore,


20

2

2

4

1

2

,

o

iin

n

i

eeed

ettgdS

o

sp

p

The frequency spectrum of the lasing mode is Lorentzian and the full-width at half-maximum (FWHM) laser linewidth equals,

n

nsp

p

2

1

The above expression for the laser linewidth is called the Schawlow-Townes expression (named after the inventors of the laser). Note that the laser linewidth, just after threshold is one-half the laser linewidth just before threshold. All lasers have linewidths that are equal to or greater than the value given by the Schawlow-Townes expression. A large number of average photons in the cavity implies a smaller linewidth. When the number of photons in the cavity is large the phase kick resulting from the addition of each spontaneously emitted photon is small and, therefore, the laser linewidth is narrow. The Schawlow-Townes expression is more often written in terms of the output power P of the laser,

P

nsp

p

o22

The above relation shows that high power lasers with long cavity photons lifetimes have narrower linewidths. 13.5 Photon Number and Photon Flux Noise of a Laser 13.5.1 Photon Number Noise inside the Laser Cavity We use the following three equations to find the photon number and photon flux noise of a laser operating above threshold,

)(ˆ)(ˆ)(ˆ)(ˆ


)(ˆ)(ˆ)(ˆ)(ˆ


121

211

121

212

tatFetFtae

tFtFtatatNtNgT

tN

T

tN

dt

tNd

tatFetFtae

tFtFtatatNtNgT

tN

T

tN

dt

tNd

spti

spti

PNdp

spti

spti

PNdp

oo

oo


21

)(ˆ)(ˆ)(ˆ)(ˆ1

)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ2ˆ)(ˆ

12

tatSeetSta

tatFetFtaetntNtNgtn

dt

tnd

intiti

inp

spti

spti

dp

oo

oo

Recall that one can write,

tFtNgtatFetFtae ndspti

spti oo ˆˆ2)(ˆ)(ˆ)(ˆ)(ˆ 2

where tFn̂ is a zero-mean noise source that describes the shot noise associated with the optical transitions,

'212'ˆˆ12 ttnNgnNgtFtF ddnn

We also write,

tFtatSeetSta vintiti

inp

oo ˆ)(ˆ)(ˆ)(ˆ)(ˆ1

where,

''ˆˆ ttn

tFtFp

vv

The vacuum fluctuations thus describe the shot noise associated with photon loss from the cavity. Consider a laser operating above threshold. We assume that,

tnntn

tNNtNtNNtN

tNNtNddd

ˆˆ

ˆˆˆˆ

ˆˆ

111

222

We linearize the non-linear equations for the photon number and the population difference operators. The linearized equations for the fluctuations become,

tvn

nPNd

d

prd

FtF

tFtFtF

tn

tN

ngtn

tNdt

dˆˆ

ˆ2ˆ2ˆ2

)(ˆ

)(ˆ

02

21

)(ˆ

)(ˆ

These equations are best solved in the Frequency domain. Upon Fourier transforming we get,

)(ˆ)(ˆ)(ˆ2)(ˆ2)(ˆ2

2

)(

)(ˆ

)(ˆ

1

2

2

vn

nPN

dr

d

FF

FFF

ing

iH

n

N

r

p

Here,

rr

r

iH

)(

)(22

2

p

dr

ng

42 ngTT d

pr4

111

1

The function )(H is the modulation response function of the laser. It describes the response of a laser to

perturbations of different frequency. The shape of the function )(H is plotted in the Figure below. The

Figure shows that a laser does not respond to perturbations of frequencies much larger than the laser relaxation oscillation frequency r .


22

The photon number noise spectral density can be found using,

)(ˆ'ˆ

2

' * nn

dS nn

p

ddd

pd

rnn

n

nNgnNgng

T

N

T

Nng

HS

r

r

212

12

212

1

1

224

2

2)1(24

44)2()(

)(

The mean square photon number fluctuations can be found by integrating the noise spectral density,

)(

2)0()(ˆ2

nnnn S

dRtn

Using,

12

2

NN

Nnsp

12

11NN

Nnsp

the normalized variance in the photon number is,

noise loss Photon

noise absorption and emission Stimulated

noise pump and ns transitioradiative-Non

pd

p

rr

p

sp

d

p

pr

p

spspr

ng

n

ngTT

T

n

T

n

n

tn

1

8

1

2

1

12

8

11

2

1

1)(ˆ

2

2

1

1

2

Above threshold, the value of spn is a constant (does not change with pumping) and its value is fixed by

the lasing condition,

10-2

10-1

100

10110

-2

10-1

100

101

/r

|H(

)|2 /

r4


23

p

ddNg1

2

At transparency, spn is infinite. If the medium is completely inverted then spn equals unity. In actual

lasers, the value of spn is somewhere between infinity and unity. Here, we will assume that 1spn

above threshold. Photon Number Noise Much Above Threshold: We now look at the photon number noise much above threshold, in the limit of strong pumping when 11,11 TTT pthp . The steady state photon number is,

pd

pd

pthpd

pd

Tg

Ng

TTg

Ngn

4

1211

4

12

Since, by assumption, 1spn , we have,

11222

sp

ddpddpd n

N

N

N

NNgNg

and therefore,

p

p

pd

pd

T

N

Tg

Ngn

24

12

This also means that,

p

d Tng

14

and,

ngngTT dd

pr44

111

1

Thus, in the limit 1spn and 11,11 TTT pthp , one can approximate the expression for the

variance in the photon number as,

12

1

02

1)(ˆ2

noise loss Photon


noise Pump n

tn

In the assumed limits, the dominant contributions to the laser noise come from the pumping process and from the shot noise associated with photon loss from the cavity. Interestingly, the contribution from the shot noise associated with the stimulated emission and absorption processes has disappeared. The reason for this is as follows. Suppose that in a certain time interval there are more stimulated emission events than given by the average stimulated emission rate. The extra stimulated emission events will end up depleting the gain in the two-level system medium. Consequently, the emission events in the successive time interval will be reduced in number because of the reduced gain in the medium. The result is that the total number of emission events in these two successive time intervals will be more or less close to the value dictated by the average stimulated emission rate. This instantaneous negative feedback from the gain medium helps reduce the noise from stimulated emission and absorption events. Noise from non-radiative transitions is negligible much above threshold compared to the noise from the pumping process. The above expression shows that much above threshold the variance in the photon number approaches the


24

mean photon number. This, as we know, is a characteristic of a coherent state. However, to infer from this result that the quantum state of radiation inside a laser cavity is a coherent state would be a mistake. As we have seen in earlier chapters that a statistical mixture of number states with a Poisson probability distribution will also exhibit the same photon number characteristics. 13.5.2 Photon Flux Noise outside the Laser Cavity The laser photon noise outside the cavity is different from the noise inside the cavity. Consider the noise form photon loss events. As seen in the previous Section, these events contribute half of the photon number noise inside the cavity much above threshold. The question arises if the photon loss events would also contribute noise to the photon flux noise outside the laser cavity. Suppose that in a certain time interval there are more photon loss events than dictated by the average photon loss rate from the cavity. The extra loss events will end up reducing the number of photons inside the cavity. Consequently, the photon loss events in the successive time interval will be reduced in number because of the reduced number of photons inside the cavity. The result is that the total number of photon loss events in these two successive time intervals will be more or less close to the value dictated by the average photon loss rate. This instantaneous negative feedback from the cavity helps reduce the photon flux noise outside the laser cavity. We will see how this comes out from the math in the discussion that follows.

The field in the waveguide is described by the operators,

tiin

p

tiLg

p

tiRg

tiout

oo

oo

etStaetzbvta

etzbvetS

ˆˆ1,0ˆˆ1

,0ˆˆ

The average photon flux escaping from the laser cavity is,

tStStaetSetStatn

tzF ininti

inti

inpp

Roo ˆˆˆˆˆˆ1ˆ

,0ˆ

p

outoutRtn

tStStzF

ˆˆˆ,0ˆ

The flux noise operator is,

tStStFtn

tzF ininvp

Rˆˆˆˆ

,0ˆ

The last term will never contribute upon averaging so we can neglect it and write,

tFtn

tzF vp

Rˆˆ

,0ˆ

The above equation is best solved in the frequency domain,

v

pR F

nzF ˆˆ

,0ˆ

Inside the cavity we know that,

Cavity Waveguide

z = 0


25

)(ˆ)(ˆ)(ˆ2)(ˆ2)(ˆ2

2

)(

)(ˆ

)(ˆ

1

2

2

vn

nPN

dr

d

FF

FFF

ing

iH

n

N

r

p

which implies,

)(ˆ)(ˆ4)(ˆ2)(ˆ22

)()(ˆ 11

2

vndPNdr

FiFingFFngH

nrr

And therefore,

rri

HFFingFFng

H

Fn

zF

prvndPNd

pr

vp

R

12

12

)(1)(ˆ)(ˆ4)(ˆ2)(ˆ22

)(

ˆˆ,0ˆ

The above expression is very interesting. If we consider the photon flux noise at high frequencies, such that r and 0H , we get,

)(ˆ,0ˆ vR FzF Recall that,

tFtatSeetSta vintiti

inp

oo ˆ)(ˆ)(ˆ)(ˆ)(ˆ1

which implies,

tzFn

zFzFd

S Rp

RRrFF RR,0ˆ),0(ˆ',0ˆ

2

' *

Thus, at frequencies much higher than the laser relaxation oscillation frequency, the noise in the output photon flux (as seen in the flux noise spectral density) is entirely due to the beating between the field emerging from the cavity and the reflected portion of the vacuum field incident on the cavity, and the resulting noise is exactly at the shot noise level. Next, we look at the photon flux noise spectral density at frequencies smaller than the laser relaxation oscillation frequency, when r and 1H . We

assume, as before, 1spn and strong pumping such that, 11,11 TTT pthp . First note that the

coefficient of the noise source )(ˆ vF goes to zero in this limit,

0)(

1 12

ri

H

pr

This shows that photon loss processes indeed do not contribute to photon flux noise outside the laser cavity at low frequencies. The result for the photon flux noise spectral density at low frequencies is,

1

0

0

1

noise loss Photon


noise Pump p

rFF

n

SRR

Interestingly, at low frequencies the photon flux noise is again at the shot noise level but the noise at low frequencies is entirely due to the pumping process. The noise accompanying pumping is not fundamental. In semiconductor lasers, for example, pumping is performed by an electrical current which takes electrons from the valence band into the conduction band. Electrical current pumping is essentially noiseless (if one


26

ignores the small amount of current noise from the circuit resistances). Therefore, one would expect that the low-frequency photon flux noise of electrically pumped semiconductor lasers to be much smaller than the shot noise level. This turns out to be the case. Photon flux noise from semiconductor lasers has been demonstrated to have values as much as ~13 dB below the shot noise level. Photon flux noise below the shot noise level is a characteristic of amplitude squeezed states. However, to infer from this result that the quantum state of radiation emerging from a semiconductor laser cavity is an amplitude squeezed state would be a mistake. A statistical mixture of number states with a sub-Poisson probability distribution will also exhibit the same photon number characteristics.

Increasing pumping Increasing pumping

Photon flux noise in lasers when the pump has shot noise (solid state and gas lasers)

Photon flux noise in lasers when the pump is noiseless (semiconductor lasers)

chapter 13: quantum optics of lasers - cornell university · quantum optics for photonics and...

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