Spectral Coherence and the Laser Linewidth

Markus Pollnau

Department of Materials and Nano Physics KTH  Royal Institute of Technology Stockholm, Sweden

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Resonator Mode

We assume a laterally fundamental, single longitudinal mode in one polarization (“the mode”), centered at frequency nL. 2 Mode volume: Vwmode   0 with length l and beam waist w0. All population densities and rates per unit time are averaged over the mode volume.

Speed of light: c c0 nr in the medium of refractive index nr that homogeneously fills the mode volume.

12 Resonator round-trip time: tRT  Dn FSR c with free spectral range DnFSR

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Resonator Losses, Photon Decay Time

T 0 1 ln 1 T  out T Outcoupling of light: out out (with mirror transmission Tout)  outtt RT RT

L 0 1 ln 1 L  RT L Intrinsic round-trip loss: RT RT  tt (with loss constant LRT) loss RT RT 1 1 1 Photon decay time:  c  out  loss

1 Photon decay rate: Rdecay  j  c Photon decay per round trip: (with number j of photons in the mode)

tRT c  t RT  out  t RT  loss jt0 tRT  j t 0 e  j t 0 e e  j  t 0 11  T out  L RT 

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Decay Time vs. Coherence Time

1.5 Exponential decay time   c 1 c defined from zero to infinity! 0.5

0

coh -0.5 Coherence time cc 2 defined from -1 -1.5 minus infinity to infinity! units] [arb. Field Electric 0 50 100 150 200 Time [arb. units]

1.5 coh cc 2 1

0.5

0

-0.5

-1

Electric Field [arb. units] [arb. Field Electric -1.5 -200 -100 0 100 200 Time [arb. units]

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Lorentzian-shaped Spectral Line

Time domain: t c Exponential decay with decay time c : jjt   t0  e

Fourier transform of its amplitude: 2  1 1it2 n n L  1 1 c  n  j t e dt 22   2 c j t0  t0 c 42n nLc   

Frequency domain: 2 Dn Lorentzian line shape with linewidth Dn : n   c c c  2 2 4n nLc  D n

Relationship between linewidth, photon-decay time, coherence time: 11 Dn c   coh 2 c  c

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Decay Time vs. Linewidth

11 Relationship decay time / linewidth: Dn c   coh 2 c  c

1.0 1.0 tau1 = 0.32 ms f1 0.8 0.8 f2 tau2 = 0.08 ms 0.6 0.6

0.4 0.4

0.2 0.2

Intensity (arb. units) Lorentzian Lorentzian Function 0.0 0.0 0 0.5 1 1.5 -20 -10 0 10 20 Time (ms) Angular Frequency (kHz)

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Q-Factor

The quality factor (Q-factor) of any resonator (be it mechanical, electrical, optical, …) is defined as: the energy stored in the resonator divided by the energy lost per oscillation cycle

E t j  t Q : 2stored  2   2 n   n  coh cEt 1 d L c L c lost    j t n L dt

Energy: E t  hnj  t

11 n Fourier transform: L Dn cc  coh  Q  2cc c D n

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Q-Factor

Two equivalent definitions:

Et  E n Q: 2stored  2 n   Q  2  stored  L cE t L c c E Dn lost  injectedcw c

Measurement by:

inserting a short pulse inserting broadband cw and observing its decay light and measuring the (“cavity ring-down measurement”) (Fabry-Pérot) linewidth

 c  Dnc

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Spectral Coherence in a Passive Mode

Mode passive

Q-factor Qc  ×n -1 Linewidth (Dnc)  ×2

Photon decay time c  ×1/2 coh Coherence time c  ×1/c coh Coherence length lc

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Stimulated Emission (and Absorption)

1,1

1,0

We assume that the transition cross sections 0,9

0,8 do not vary significantly over the spectral 0,7 linewidth of the lasing mode at frequency n . 0,6

L [normalized] 0,5

q 

, , 0,4 e

 0,3

0,2 Emission cross section: se 0,1 0,0 -0,003 -0,002 -0,001 0,000 0,001 0,002 0,003 (n-n )/n Absorption cross section: sa e e

Population density of upper laser level: N2

Population density of lower laser level: N1

Optical net gain per unit length: gssea N21 N

Stimulated-emission rate per unit time: Rst  cgj

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Photon Rates and Rate Equation

Number of photons in lasing mode: j

Stimulated-emission rate per unit time: Rst  cgj

1 Photon-decay rate per unit time: Rdecay  j  c

d 1 cw Photon rate equation: jRst  R decay  cg j  j  0 dt  c

1 Consequence: “The gain equals the losses!” cg   c

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group CW Laser: Gain Equals Losses ?

The assumption that in a cw laser the gain equals the losses is only an approximation which is often (but not always) fine to obtain with good accuracy laser threshold and slope efficiency.

Its consequences are that 1) the laser would emit a pure sine wave with an infinite coherence length, 2) its linewidth would become a delta function, 3) its Q-factor would assume an infinite value, 4) the photon number would build up and coherence would manifest itself only above threshold, and 5) the threshold inversion would depend only on the total resonator losses, but be independent of other decay channels.

None of these implications holds true for any laser that mankind has ever created!

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Atomic Behavior in an External Field

A. Einstein, Phys. Z. 18, 121 (1917)

Absorption Stimulated Emission Spontaneous Emission

E2, N2, 2 E2, N2, 2 E2, N2, 2

E1, N1 E1, N1 E1, N1

Stimulated processes Spontaneous process (driven by an external field) (driven by vacuum fluctuations)

Obviously we have neglected spontaneous emission!

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Spontaneous vs. Stimulated Emission

With one vacuum photon per mode, j 0  1 , the spontaneous-emission rate becomes

Rsp cs e N2 j 0 c s e N 2

In a 4-level laser: 4level Stimulated-emission rate: Rst cgj c s e N2 j

The rate constants cN s e 2 are identical, because spontaneous emission is stimulated emission driven by vacuum fluctuations !

4level RRcgcN j  s  cN s j  cNcN s  s j 1 st sp e2 e 2 e 2 e 2   You find this “extra” photon, representing spontaneous emission, in the textbooks on lasers by Siegman or Svelto (and others).

(In a 3-level laser, there is absorption of photons from the lower laser level, which has no equivalent in spontaneous absorption.)

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group CW Laser: Gain is Smaller than Losses!

Photon rate equation without / with spontaneous emission:

d 1 cw j R  R  cg j  j  0 dt st decay  c d 1 cw jRsp  R st  R decay  c s e N2  cg j  j  0 dt c The spontaneous-emission and stimulated-emission rates together compensate the loss rate.

Consequence: “The gain is smaller than the losses!”

1 Rdecay R st  R sp or   cgjs  c e N2  c

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Decay of Photons in Lasing Mode

Since in a cw laser the gain is smaller than the losses, cw1 cw Rdecay R st  R sp or   cgjs  c e N2  c photons present in the lasing resonator mode at any given time t1 decay according to: d 11cw t1 t 1 t 1 jt   Rdecay  R st     cg j t   j  t  0 dt cL

The gain elongates the photon-decay time from c in the passive resonator to L in the lasing resonator : 1 Lc 1 cg c

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group Laser Coherence Time and Linewidth

Because of gain, the photon-decay time is elongated

from c in the passive mode to L in the lasing mode:

Rdecay 1 cL   or   Rdecay R st1 cg c

Consequently, the laser coherence time is elongated:

cohRdecay 1 coh c2  c   or    L  2  L Rdecay R st1 cg c

Fourier transform  the laser linewidth is narrowed:

11RRdecay st Dnc    or  1 cg  c  D n L  22cR decay  L

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group The Finite Q-factor of a CW Laser

Reminder: The Q-factor of a resonator is defined as the energy stored in the resonator divided by the energy lost per oscillation cycle.

Equivalently, we define the Q-factor of a lasing resonator as the number

of photons stored in the resonator at any given time t1 divided by the number of these photons lost per oscillation cycle and not replaced coherently by stimulated emission:

t1 jcoh t coh n L QLLLLL: 2  2 n   n   1 d t Dn  j 1 t L n dt coh L

The net decay of photons with decay time L is compensated by incoupling of photons via spontaneous emission  like in a passive resonator that is excited by cw light, the photon number in the resonator is constant.

KTH  Royal Institute of Technology Markus Pollnau Rare-Earth Lasers and Dept. of Materials and Nano Physics [email protected] Integrated Devices Group