Lecture 9: Laser oscillators

Lecture 9: Laser oscillators



Theory of laser oscillation



Laser output characteristics



Pulsed lasers

References: This lecture follows the materials from Fundamentals of Photonics, 2nd _ed.,

2

Intro

 _{There are a wide variety of lasers, covering a spectral range}

from the soft X-ray (few nm) to the far infrared (hundreds of



m), delivering output powers from microwatts (or lower) to terawatts, operating from continuous wave (CW) to

femtosecond (even attosecond) pulses, and having spectral linewidths from just a few hertz to many terahertz.

 The gain media utilized include plasma, free electrons, ions, atoms, molecules, gases, liquids, solids, etc.

 _{The sizes range from microscopic, of the order of 10} _m3

(recently down to the order of sub



m3 for so-called

nanolasers), to gigantic, of an entire building, to stellar, of astronomical dimensions.

 An optical gain medium can amplify an optical field through

Intro

 _{The laser is an optical oscillator.}

 It comprises a resonant optical amplifier whose output is fed back to the input with matching phase.

 The oscillation process can be initiated by the presence at the amplifier input of even a small amount of noise that contains frequency components lying within the bandwidth of the

amplifier.

 This input is amplified and the output is fed back to the input, where it undergoes further amplification.

 The process continues until a large output is produced.

 The increase of the signal is ultimately limited by saturation of the amplifier gain, and the system reaches a steady state in which an output signal is created at the frequency of the

4

4

Laser oscillators

 In a practical laser device, it is generally necessary to have certain positive optical feedback in addition to optical

amplification provided by a gain medium.

 This requirement can be met by placing the gain medium in an optical resonator. The optical resonator provides selective feedback to the amplified optical field.

 _{In many lasers the optical feedback is provided by placing the}

gain medium inside a “Fabry-Perot” cavity, formed by using

two mirrors or highly reflecting surfaces

reflectivity (R₁ ~ 100 %) R₂ < 100 %

Intro

 _{Two conditions must be satisfied for oscillation to occur:}

 The amplifier gain must be greater than the loss in the feedback system s.t. net gain is incurred in a round trip through the feedback loop.

 The total phase shift in a single round trip must be a

multiple of 2 s.t. the feedback input phase matches the phase of the original input.

If these conditions are satisfied, the system becomes unstable and oscillation begins.

6

Intro

 _{As the power in the oscillator grows, the amplifier gain}

saturates and decreases below its initial value.

 A stable condition is reached when the reduced gain is equal to the loss.

 The gain then just compensates the loss s.t. the cycle of amplification and feedback is repeated without change and steady-state oscillation prevails.

6 gain loss Oscillator power Steady-state power

Intro



Because the

gain

and

phase shift

are functions of

frequency, the two oscillation conditions are satisfied

only at one or several frequencies, which are the

resonance frequencies of the oscillator.



The useful output is extracted by coupling a portion of

the power out of the oscillator.



An oscillator comprises:

 An amplifier with a gain-saturation mechanism

 A feedback system

 A frequency-selection mechanism

8

Intro



The laser is an

oscillator

in which the

amplifier

is the

pumped active medium.



Gain saturation

is a basic property of laser amplifiers.



Feedback

is enabled by placing the active medium in

an

optical resonator

, which in its simplest form

reflects the light back and forth between its mirrors.



Frequency selection

is jointly attained by the resonant

amplifier and the resonator, which admits only certain

modes

.



Output coupling

is attained by making one of the

resonator mirrors partially transmitting.

10

Laser amplification

 _{The laser amplifier is a narrowband coherent amplifier of}

light.

 Amplification is attained by stimulated emission from an atomic or molecular system with a transition whose

population is inverted (i.e. the upper energy level is more populated than the lower).

 The amplifier bandwidth is determined by the linewidth of the atomic transition, or by an inhomogeneous broadening mechanism (e.g. defects and strains and impurities in host solids)

 _{The laser amplifier is a distributed-gain device characterized}

by its gain coefficient (gain per unit length) (), which governs the rate at which the photon-flux density  (or the

Small-signal gain coefficient

 When the photon-flux density  is small, the gain coefficient

is

where N₀ = equilibrium population density difference (density of atoms in the upper energy state minus that in the lower state). Assumes degeneracy of the upper laser level equals that of the lower laser level (i.e. g₁=g₂). N₀ increases with

increasing pumping rate.

 () = transition cross section (_e() = _a() = ())

 _{sp = spontaneous lifetime}  g() = transition lineshape



₀(



)  N₀



(



)  N₀ c 2

12

Saturation photon-flux density

 As the photon-flux density increases, the amplifier enters a region of nonlinear operation. It saturates and its gain

decreases.

 The amplification process then depletes the initial population difference N₀, reducing it to

for a homogeneously broadened medium, where

12

N



N

0

1





/



_s

(



)



_s

(



)



1



_s



(



)

Saturation photon-flux density

_s saturation time constant, which depends on the decay times of the energy levels involved. _s ≈ _sp for four-level pumping,

Saturated gain coefficient

 _{The gain coefficient of the saturated amplifier is therefore}

reduced to (for homogeneous broadening)

 The laser amplification process also introduces a phase shift. When the lineshape is Lorentzian with linewidth ,

 The amplifier phase shift per unit length is



(



)



N



(



)





0

(



)

1





/



_s

(



)

ˆ

g

(



)







/ 2



(







₀

)

2



(





/ 2)

2



(



)









0







(



)

14 14

Gain coefficient and phase-shift coefficient for a laser amplifier with a Lorentzian lineshape function

Phase-shift coefficient () gain coefficient ()   _  _

Optical resonators

 _{Optical feedback is attained by placing the active medium in}

an optical resonator.

 A Fabry-Perot resonator, comprising two mirrors separated by a distance d, contains the active medium (refractive index n).

 Travel through the medium introduces a phase shift per unit length equal to the wavenumber k = 2n/c

 The resonator sustains only frequencies that correspond to a round-trip phase shift that is a multiple of 2.

k



2

d



2



n



c



2

d



q

2



16 16

Fabry-Perot resonators

• Only standing waves at discrete wavelengths exist in the cavity.

=> the laser wavelengths must match the cavity resonance wavelengths.

The resonance condition: 2nd = q 

where q is an integer (=1, 2, …), known as the longitudinal mode order, k = 2n/2n/c

d refractive index n

R₁ R₂ P_out R₁ R₂ P_out P_out P_{out Pout P} out P_out P_out

Resonant optical cavities

Fiber/waveguide ring resonator

18

18

Resonant optical cavities

 A linear cavity with two end mirrors is known as a Fabry-Perot cavity because it takes the form of a Fabry-Perot interferometer. In the case of semiconductor diodes, the diode end facets form the two end mirrors.

 A folded cavity can simply be a folded Fabry-Perot cavity

with a standing oscillating field.

 _{A folded cavity can also be a non-Fabry-Perot ring cavity}

that supports two independent oscillating fields traveling in opposite directions (clockwise, counterclockwise). Ring

cavity can be made of multiple mirrors in free space, or in the form of fiber/waveguide-based devices.

 The optical cavity can also comprise a distributed Bragg grating with distributed feedback. Distributed Feedback (DFB) diode lasers are the most common single-mode laser diodes for optical communications.

Resonant optical cavities

 _{In a ring cavity, an intracavity field completes one round trip}

by circulating inside the cavity in only one direction. The two contrapropagating fields that circulate in opposite

directions in a ring cavity are independent of each other even when they have the same frequency.

 _{In a Fabry-Perot cavity, an intracavity field has to travel the}

length of the cavity twice in opposite directions to complete a round trip.

 The time it takes for an intracavity field to complete one round trip in the cavity is called the round-trip time, T_F:

where l_RT is the round-trip optical path length (=2nd for

T

_F



l

RT

20

20

• The modes along the cavity axis is referred to as longitudinal modes. • Many ’s may satisfy the resonance condition => multimode cavity

The longitudinal mode spacing (free-spectral range):

 = 2 _{/ 2nd}  intensity q q-2 q+2  q+1 q-1 ….. …..

e.g. A semiconductor laser diode has a cavity length 400 m

with a refractive index of 3.5. The peak emission wavelength from the device is 0.8 m. Determine the longitudinal mode order

and the frequency spacing of the neighboring modes. • The longitudinal mode order q = 2nd/ ~ 3500

• The longitudinal mode frequencies:

 = _q = qc/2nd

• The mode spacing (free-spectral range) in frequency unit:

_q = c/2nd

22 22

Resonator losses

 The resonator also contributes to losses. Absorption and

scattering of light in the medium introduces a power loss per unit length (attenuation coefficient _s)

 In traveling a round trip through a resonator of length d, the photon-flux density is reduced by the factor

R₁R₂ exp(-2_sd)

where R₁ and R₂ are the reflectances of the two mirrors

 The overall power loss in one round trip can be described by a total effective distributed loss coefficient _r

Loss coefficients

_r = _s + _m1 + _m2

_m1 = (1/2d) ln(1/R₁)

_m2 = (1/2d) ln(1/R₂)

where _m1 and _m2 represent the contributions of mirrors 1 and 2.

 _{The contribution from both mirrors}

24 24

Photon lifetime and resonator linewidth

 Define photon lifetime (cavity lifetime) _p as the 1/e-power lifetime for photons inside the cavity of refractive index n:

exp(-_r _pc/n) = exp(-1)

 _p = n/_rc

 The resonator linewidth (FWHM)  is inversely proportional to the cavity lifetime

 = 1/2_p

 The cavity quality factor Q at resonance frequency _q is

Q = _q × (energy stored in the resonator/average power dissipation) = _q _p = _q/

 _{The finesse of the resonator}

F ≈ _q/

where _q = c/2nd

 When the resonator losses are small and the finesse is large



_q _q _q  _q c/2nd

26 26

Conditions for laser oscillation

 _{Two conditions must be satisfied for the laser to oscillate}

(lase):

 The gain condition determines the minimum population difference, and thus the pumping threshold required for lasing

 The phase condition determines the frequency (or frequencies) at which oscillation takes place

Gain Condition: Laser threshold

 _{The initiation of laser oscillation requires that the}

small-signal gain coefficient be greater than the loss coefficient

Or, the gain be greater than the loss.

 _{Translates this to the population difference}

where N_t is the threshold population difference. N_t, which is proportional to _r, determines the minimum pumping rate



₀

(



)





_r

N₀ 



0(



)



(



) 



_r

28

Gain condition: Laser threshold

 _r may be written in terms of the photon lifetime,

 Thus, N_t is given as

 The threshold population density difference is therefore

directly proportional to _r and inversely proportional to _p.

Higher loss (shorter photon lifetime) requires more vigorous pumping to attain lasing.

28



_r



n

c



_p

N

_t



n

Threshold population difference

 _{By using the transition cross section}

we find another expression for the threshold population difference,

 The threshold is lowest, and thus lasing is most readily attained, at the frequency where the lineshape function is largest, i.e., at its central frequency  = ₀.

 _{For a Lorentzian lineshape function, g(} _{) = 2/}



(



)  c 2 8



n2



2



_sp gˆ(



) N_t  8



n 3



2 c3



_sp



_p 1 ˆ g(



)

30

Threshold population difference

 _{The minimum population difference for oscillation at the central} frequency ₀ turns out to be

 N_t is directly proportional to the linewidth .

 If the transition is limited by lifetime broadening with a decay time _sp, and  = 1/2_sp

 This shows that the minimum threshold population difference required to attain oscillation is a simple function of the frequency  and the photon lifetime _p. Laser oscillation becomes more difficult to attain as the frequency increases. 30 N_t  2



n 3



2 c3 2







_sp



_p N_t  2



n 3



2 c3



_p  2



n2



2



_r c2

Phase condition: laser frequencies

 The phase condition of oscillation requires that the phase

shift of the laser light completing a cavity round-trip must be a multiple of 2

2kd + 2()d = 2q, q = 1,2,…

 If the contribution arising from the active laser atoms 2()d

is small, then the laser modes are given by the “cold” (or passive) cavity modes.

 In general, 2()d gives rise to a set of oscillation frequencies

_q’ that are slightly displaced from the cold-resonator frequencies _q.

 _{The cold-resonator modal frequencies are all pulled slightly}

32 32

Frequency pulling

 _{The laser oscillation frequencies fall near the cold-resonator}

modes – they are pulled slightly toward the atomic resonance central frequency ₀. Amplifier gain coefficient  Laser oscillation modes  Cold-resonator modes  _q  _   _

Laser output

characteristics

34

Laser power

 A laser pumped above the threshold exhibits a small-signal

gain coefficient ₀() that is greater than the loss coefficient _r.



₀(



) >



_r

 Laser oscillation may then begin, provided that the phase condition is satisfied.

2kd + 2()d = 2q, q = 1,2,…

 As the photon-flux density  inside the resonator increases, the

gain coefficient () begins to uniformly drop (for

homogeneously broadened media)

() = ₀() / (1 + /_s())

 _{As long as the gain coefficient remains larger than the loss}

coefficient, the photon flux continues to grow.

Laser oscillation: the unsaturated gain must exceed the loss

• sub-threshold

(incoherent emission)

• Threshold

(oscillation begins, start to emit coherent light) • above-threshold (increase in coherent output loss (assume constant) gain 

gain < loss gain = loss

loss loss

gain > loss

36

Laser oscillation

36 Loss _r  Allowed _modes  Resonator _modes   _q _

• Laser oscillation can occur only at frequencies for which the

small-signal gain coefficient exceeds the loss coefficient. Only a finite number of oscillation frequencies (₁, ₂, …, _m) are possible.

_ _ _

 _{At the moment the laser lases,}  _{= 0 so that} ₍_{) =} ₀₍_).

 As the oscillation builds up in time, the increase in  causes () to drop through gain saturation.

 When  reaches _r, the photon-flux density ceases its growth and steady-state conditions are attained.

     _s _  Laser turn-on _r loss coefficient  steady-state

Gain saturation

38 38  Gain clamping at the value of the loss.

 _{The steady-state laser internal photon-flux density} _is

therefore determined by equating the saturated gain coefficient to the loss coefficient

() = ₀() / (1 + /_s()) = _r

  = _s() (₀()/_r – 1), ₀() > _r

= 0, ₀() ≤ _r

 This is the mean number of laser photons per second crossing a unit area in both directions – laser photons traveling in both directions contribute to the saturation process. The photon-flux density for laser photons traveling in a single direction is thus /2. Spontaneous emission noise is neglected.

Steady-state internal photon-flux density

 As ₀() = N₀() and _r = N_t(), the steady-state internal photon-flux density  can be written as

 Below threshold, the laser photon-flux density is zero. Any increase in the pumping rate is manifested as an increase in the spontaneous-emission photon flux, but there is no

sustained oscillation.

 Above threshold, the steady-state internal laser photon-flux density is directly proportional to the initial population

difference N₀, and therefore increases with the pumping rate







_s

(



)

N

0

N

_t



1

















0

N₀ > N_t N₀ ≤ N_t

40

Steady-state internal photon-flux density

 If N₀ is twice the threshold value N_t, the photon-flux density is precisely equal to the saturation value _s(), which is the photon-flux density at which the gain coefficient decreases to half its maximum value.

 Laser oscillation occurs when N₀ exceeds N_t. The

steady-state value of N then saturates, clamping at the value N_t [just as ₀() is clamped at _r]. Above threshold,  is proportion to N₀ – N_t. 40 N N₀ N_t N_t Pumping rate Population difference  N₀ N_t _s Pumping rate Photon-flux density 2N_t

Output photon-flux density

 _{Only a portion of the steady-state internal photon-flux density leaves the} resonator in the form of useful light.

 _{The output photon-flux density} _{0 is that part of the internal photon-flux} density that propagates toward mirror 1 (/2) and is transmitted by it.  If the transmittance of mirror 1 is T, the output photon-flux density is

 The corresponding optical intensity of the laser output I₀ is

 _{The laser output power is}



_o



T



2

I

_o



h



T



2

42

Internal photon-number density

 The steady-state number of photons per unit volume inside the resonator N_p is related to the steady-state internal photon-flux density  (for photons traveling in both directions) by the simple relation

N_p = n/c

 The photon-number density corresponding to the steady-state internal photon-flux density in

where N_ps = _s()n/c is the photon-number density saturation value. 42         0 1 t s p p N N N N N₀ > N_t

Internal photon-number density

 Using the relations

_s() = 1/_s(), _r = (), _r = n/c_p and () = N() = N_t(), we can write steady-state photon number density as

 Interpretation: (N₀ – N_t) is the population difference (per unit volume) in excess of threshold, and (N₀ – N_t)/_s represents the rate at which photons are generated which, upon steady-state operation, is equal to the rate at which photons are lost, N_p/_p.

 The fraction _p/_s is the ratio of the rate at which photons are emitted (1/ ) to the rate at which they are lost (1/ ) .





s p t p

N

N

N









₀ N₀ > N_t

44

Internal photon-number density

 _{Upon ideal pumping conditions in a four-level laser system,}  _s ≈ _sp and N₀ ≈ R_sp, where R is the rate (s-1 _cm-3_{) at which}

atoms are pumped.

 We can rewrite the steady-state photon-number density as

where R_t = N_t/_sp is the threshold value of the pumping rate.

=> Upon steady-state conditions, the overall photon-density loss rate N_p/_p is equal to the excess pumping rate R – R_t.

t p p R R N    R > Rt

Output photon flux and efficiency

 If transmission through the laser output mirror is the only source of resonator loss (which is accounted for in _p), and V is the

volume of the active medium, the total output photon flux (photons per second) is

 If there are loss mechanisms other than through the output laser mirror, the output photon flux can be written as

where the extraction efficiency _e is the ratio of the loss arising from the extracted useful light to all of the total losses in the



_o



(

R



R

_t

)

V



_o





_e

(

R



R

_t

)

V

46

Output photon flux and efficiency

 _{If the useful light exits only through mirror 1,}

 If, T = 1 – R₁ << 1, the extraction efficiency

where we have defined 1/T_F = c/2nd, indicating that the

extraction efficiency _e can be understood in terms of the

photon lifetime to its round-trip travel time, multiplied by the mirror transmittance.

 The output laser power is

46



_e





m1



_r



c

2

nd



p

ln

1

R

₁



_e





p

T

_F

T

P

₀



h





₀





_e

h



(

R



R

_t

)

V

Output photon flux and efficiency

 _{Losses result from other sources as well, such as inefficiency}

in the pumping process.

 The power conversion efficiency _c (also called the overall efficiency or wall-plug efficiency) is defined at the ratio of the output optical power P_o to the supplied pump power P_p

 Because the laser output power increases linearly with pump power above threshold, the differential power-conversion

efficiency (also called the slope efficiency) is another measure of performance



_c



P

o

P

_p

48 48

Laser optical output vs. pumping

Light output (power)

Pumping Incoherent emission Coherent emission (Lasing) Threshold pumping _s

Spectral distribution

 The spectral distribution of the generated laser light is determined both by the spectral lineshape of the active

medium (homogeneous or inhomogeneous broadened) and by the resonator modes.

 The gain condition – ₀() > _r is satisfied for all

oscillation frequencies lying within a continuous spectral band of width B centered about the resonance frequency

₀. The bandwidth B increases with the spectral linewidth

 and the ratio ₀(₀)/_r. The precise relation depends on the shape of the function ₀().

 The phase condition – the oscillation frequency be one of the resonator modal frequencies _q (assuming mode

50

Spectral distribution

50 Loss _r  Allowed _modes  Resonator _modes   _q _

• Laser oscillation can occur only at frequencies for which the

small-signal gain coefficient exceeds the loss coefficient. Only a finite number of oscillation frequencies (₁, ₂, …, _m) are possible.

_ _ _

Spectral distribution

 The number of possible laser modes

M ≈ B/_q

 However, of these M possible modes, the number of modes that actually carry optical power depends on the nature of the

lineshape broadening mechanism.

 For an inhomogeneously broadened medium (e.g. HeNe, Nd:glass) all M modes oscillate (albeit at different powers).

 For a homogeneously broadened medium (e.g.

52

Laser linewidth

 _{The approximate FWHM linewidth of each laser mode might be}

expected to be the cavity resonance linewidth , but it turns out to be far smaller than this.

 The oscillating mode width can be orders of magnitude narrower

than the cavity mode linewidth.

 It is limited by the so-called Schawlow-Townes linewidth, which

decreases inversely as the optical power.

 _{This linewidth-narrowing effect is caused by the coherent nature}

of the stimulated emission and is a fundamental feature of lasers.

 _{Almost all lasers have linewidths far wider than the}

Schawlow-Townes limit as a result of extraneous effects such as acoustic and thermal fluctuations of the resonator mirrors, but the limit can be approached in carefully controlled experiments.

Schawlow-Townes relation

 _{A detailed analysis taking into account spontaneous emission yields the}

Schawlow-Townes relation for the linewidth of a laser mode in terms of the laser parameters:

where P_out is the output power of the laser mode being

considered and N_sp (≥ 1) is the spontaneous emission factor.

 _{The effect of spontaneous emission on the linewidth of an oscillating laser} mode enters the above relation through the population densities of the

upper and the lower laser levels in the form of the spontaneous emission factor.

 _{Because Nsp} ≥ _{1, the ultimate lower limit of the laser linewidth, which is} known as the Schawlow-Townes limit, is that given above with N_sp = 1.





_ST  2



h



(



)2

P_out Nsp 

h



54

Homogeneously broadened medium



Immediately after being turned on, all laser modes for

which the initial gain is greater than the loss begin to

grow.



Photon-flux densities



₁

,



₂

, …,



_M

are created in the

M modes.



Modes whose frequencies lie closest to the transition

central frequency



₀

grow most quickly and acquire

the highest photon-flux densities.



These photons interact with the medium and reduce

the gain by depleting the population difference. The

saturated gain is

54



(



)





0

(



)

1





j



_s

(



_j

)

j1 M



Growth of oscillation in an ideal homogeneously broadened medium _r _   _ _ _

• Immediately following laser turn-on, all modal frequencies for which the small-signal gain coefficient exceeds the loss coefficient begin to grow, with the

central modes growing at the highest rate. After a transient the gain

saturates so that the central modes continue to grow while the peripheral modes, for which the loss has become greater than the gain, are



56

Homogeneously broadened medium

 _{Because the gain coefficient is reduced uniformly, for}

modes sufficiently distant from the line center the loss becomes greater than the gain. These modes lose power while the more central modes continue to grow, albeit at a slower rate.

 _{Ultimately, only a single surviving mode maintains a gain}

equal to the loss, with the loss exceeding the gain for all other modes.

 _{Under ideal steady-state conditions, the power in this}

preferred mode remains stable, while laser oscillation at all other modes vanishes.

 The surviving mode has the frequency lying closest to ₀ (but not necessarily equal to



₀).

Spatial hole burning



In practice, however, homogeneously broadened

lasers do indeed oscillate on multiple modes

because

the different modes occupy different spatial portions of

the active medium.



When oscillation on the most central mode is

established, the gain coefficient can still exceed the

loss coefficient at those locations where the

standing-wave electric field of the most central mode vanishes.



This phenomenon is called

spatial hole burning

.



It allows another mode, whose peak fields are located

near the energy nulls of the central mode, the

58

Inhomogeneously broadened medium

 In an inhomogeneously broadened medium, the gain represents the

composite envelope of gains of different species of atoms.

 The situation immediately after laser turn-on is the same as in the

homogeneously broadened medium.

 _{Modes for which the gain is larger than the loss begin to grow and}

the gain decreases.

 If the spacing between the modes is larger than the width  of the

constituent atomic lineshape functions, different modes interact with different atoms.

 _{Atoms whose lineshapes fail to coincide with any of the modes are}

ignorant of the presence of photons in the resonator.

 Their population difference is therefore not affected and the gain

they provide remains the small-signal (unsaturated) gain.

Spectral hole burning

 Atoms whose frequencies coincide with modes deplete their

inverted population and their gain saturates, creating “holes” in the gain spectral profile.

 This process is known as spectral hole burning.

 _{This process of saturation by hole burning progresses}

independently for the different modes until the gain is equal to the loss for each mode in steady state.

 _{Modes do not compete because they draw power from different,}

rather than shared, atoms.

 _{Many modes oscillate independently, with the central modes}

burning deeper holes and growing larger.

 The number of modes is typically larger than that in

homogeneously broadened media as spatial hole burning generally sustains fewer modes than spectral hole burning.

60

Spatial distribution

 _{The spatial distribution of the emitted laser depends on the}

geometry of the resonator and on the shape of the active medium.

 _{So far we have ignored transverse spatial effects by assuming}

that the resonator is constructed of two parallel planar mirrors of infinite extent and that the space between them is filled with the active medium.

 _{In this idealized geometry the laser output is a plane wave}

propagating along the axis of the resonator. But this planar-mirror resonator is highly sensitive to misalignment.

Spatial distribution

 Laser resonators usually have spherical mirrors.

 _{The spherical-mirror resonator supports a Gaussian beam.}

 A laser using a spherical-mirror resonator may therefore give rise

to an output that takes the form of a Gaussian beam.

 _{The spherical-mirror resonator supports a set of transverse electric}

and magnetic modes denoted TEM_l,m,q.

 _{Each pair of indexes (l, m) defines a transverse mode with an}

associated spatial distribution.

 _{The (0, 0) transverse mode is the Gaussian beam.}

 _{Modes of a higher l and m form Hermite-Gaussian beams.}

 For a given (l, m), the index q defines a number of longitudinal

modes of the same spatial distribution but of different frequencies

_q, which are separated by the longitudinal-mode spacing _q = c/2nd, regardless of l and m.

 _{The resonance frequencies of two sets of longitudinal modes}

62

Spatial distribution

 The gains and losses for two transverse modes, e.g., (0,0) and (1,1),

usually differ because of their different spatial distributions. A mode can contribute to the output if it lies in the spectral band within which the small-signal gain coefficient exceeds the loss coefficient. There can be

multiple longitudinal modes for each transverse mode. 62

_0,0 _ _ _1,1 _ (0,0) modes (1,1) modes TEM_0,0 TEM_1,1 B_1,1 B_0,0

Spatial distribution

 _{Because of their different spatial distributions, different}

transverse modes undergo different gains and losses.

 The (0, 0) Gaussian mode is the most confined about the optical axis and therefore suffers the least diffraction loss at the boundaries of the mirrors.

 The (1, 1) mode vanishes at points on the optical axis. Thus, if the laser mirror were blocked by a small central

obstruction, the (1,1) mode would be completely unaffected, whereas the (0,0) mode would suffer significant loss.

 Higher-order modes occupy a larger volume and therefore can have larger gain.

 _{This difference between the losses and/or gains of different}

transverse modes in different geometries determine their

64

Spatial distribution

 In a homogeneous broadened laser, the strongest mode tends to suppress the gain for the other modes, but spatial hole

burning can permit a few longitudinal modes to oscillate.

 _{Transverse modes can have substantially different spatial}

distributions so that they can readily oscillate simultaneously.

 A mode whose energy is concentrated in a given transverse spatial region saturates the atomic gain in that region, thereby burning a spatial hole there.

 Two transverse modes that do not spatially overlap can

coexist without competition because they draw their energy from different atoms. Partial spatial overlap between

different transverse modes and atomic migrations (as in gases) allow for mode competition.

 _{Lasers are often designed to operate on a single transverse}

mode. This is usually the (0, 0) Gaussian mode because it has the smallest beam diameter and can be focused to the smallest spot size. Oscillation on higher-order modes can be desirable for purposes such as generating large optical power.

Polarization

 _{Each (l, m, q) mode has two degrees of freedom,}

corresponding to two independent orthogonal polarizations.

 These two polarizations are regarded as two independent modes.

 Because of the circular symmetry of the spherical-mirror resonator, the two polarization modes of the same l and m have the same spatial distributions.

 If the resonator and the active medium provide equal gains and losses for both polarizations, the laser will oscillate on the two modes simultaneously, independently, and with the same intensity. The laser output is then unpolarized.

66

Pulsed lasers

Pulsed lasers

 _{It is sometimes desirable to operate lasers in a pulsed mode as}

the optical power can be greatly increased when the output pulse has a limited duration.

 _{Lasers can be made to emit optical pulses with durations as}

short as femtoseconds; the durations can be further

compressed to the attosecond regime by making use of nonlinear-optical techniques.

 Maximum pulse-repetition rates reach more than 100 GHz.

 Maximum pulse energies reach from fJ to MJ, while peak powers extend to more than 10 MW and peak intensities reach 10 TW/cm2.

 _{Some lasers can only be operated in a pulsed mode as CW}

68

Methods of pulsing lasers



The most direct method of obtaining pulsed

light from a laser is to use a CW laser in

conjunction with an external modulator that

transmits the light only during selected short

time intervals.



This method has two drawbacks:



the scheme is inefficient as it blocks energy

during the off-time of the pulse train.



the peak power of the pulse cannot exceed the

steady power of the CW source.

Methods of pulsing lasers

 _{More efficient pulsing schemes are based on turning the laser}

itself on and off by means of an internal modulation process, designed so that energy is stored during the off-time and

released during the on-time.

 Energy may be stored either in the resonator, in the form of light that is periodically permitted to escape, or in the atomic system, in the form of a population inversion that is released periodically by allowing the system to oscillate.

 These schemes permit short laser pulses to be generated with peak powers far in excess of the constant power delivered by CW lasers.

 Four common methods used for the internal modulation of laser light are: gain switching, Q-switching, cavity dumping and mode locking.

70

Example of mode-locked lasers



Ti:sapphire is a popular mode-locked laser.



With the ability to tune the center wavelength over the

range 700 – 1050 nm, and with individual pulses as

short as 10-fs duration



A commercial version of this laser readily delivers

50-nJ pulses of duration 10 fs and peak power 1 MW, at a

repetition rate of 80 MHz.



Mode-locked lasers find applications including

time-resolved measurements, imaging, metrology,

communications, materials processing, and clinical

medicine.

Mode locking

 _{Mode locking is the most important technique for the generation of}

repetitive, ultrashort laser pulses.

 _{The principle of mode locking is not based on the transient}

dynamics of a laser. Instead, a mode-locked laser operates in a dynamic steady state.

 _{A laser can oscillate on many longitudinal modes, with frequencies}

that are equally separated by the Fabry-Perot intermodal spacing

_q = c/2nd.

 _{Although these modes normally oscillate independently (they are}

then called free-running modes), external means can be used to couple them and lock their phases together.

 _{The modes can then be regarded as the components of a}

Fourier-series expansion of a periodic function of time of period T_F = 1/_q = 2nd/c, which constitute a periodic pulse train.

72

Mode locking

 The mode-locking operation is accomplished by a nonlinear optical element known as the mode locker that is placed

inside the laser cavity, typically near one end of the cavity if the laser has the configuration of a linear cavity.

 In the frequency domain, mode locking is a process that generates a train of short laser pulses by locking multiple longitudinal laser modes in phase.

 _{The function of the mode locker in the frequency domain is}

thus to lock the phases of the oscillating modes together through nonlinear interactions among the mode fields.

72

Light output (laser)

Mode locking

 _{In the time domain, the mode-locking process can be}

understood as a regenerative pulse-generating process by which a short pulse circulating inside the laser cavity is formed when the laser reaches steady state.

 The action of the mode locker in the time domain resembles that of a pulse-shaping optical shutter that opens periodically in synchronism with the arrival at the mode locker of the laser pulse circulating in the cavity.

Consequently, the output of a mode-locked laser is a train of regularly spaced pulses of identical pulse envelope.

74

Mode locking: two modes

 _{The simplest case of multimode oscillation is when there are}

only two oscillating longitudinal modes of frequencies ₁ and

₂.

 _{The total laser field at a fixed location is}

where E₁ and E₂ are the amplitudes of the field amplitudes and ₁ and ₂ are the phases.

 With all the phase information included in ₁ and ₂, E₁ and E₂ are positive real quantities.

 The intensity of the laser is

74

E

(

t

)



E

₁

e

i1(t)

e

i1t



E

2

e

i₂(t)

e

i2t

I

(

t

)



E

(

t

)

2



E

1 2



E

₂₂2



2

E

₁

E

₂

cos (





₁





₂

)

t





₁

(

t

)





₂

(

t

)



Mode locking: two modes

 _{In general, the phases can vary with time.}

 If ₁(t) and ₂(t) vary randomly with time on a characteristic time scale that is shorter than 2/(₁-₂), the beat note of the two frequencies cannot be observed. In this case, the output of the laser has a constant intensity that is the incoherent sum of the intensities of the individual modes.

 This situation represents the ordinary multimode oscillation

76

Mode locking: two modes

 If ₁ and ₂ are time independent, the laser intensity becomes periodically modulated with a period of 2/(₁-₂) defined by the beat frequency.

 _{The modulation depth of this intensity profile depends on the}

ratio between E₁ and E₂. When E₁ = E₂, the modulation depth is 100% with I_min = 0.

 In this case, I(t) resembles a train of periodic “pulses” that have a duty cycle of 50% and a peak intensity of twice the incoherent sum of the intensities.

 This is coherent mode beating between two oscillating modes.

Coherent mode beating between two modes

time Intensity Incoherent sum of the intensities Max. =(1+1)2

78

Properties of a mode-locked pulse train

 _{If each of the laser modes is approximated by a uniform plane}

wave propagating in the z direction with a velocity c/n, we may write the total complex wavefunction of the field in the form of a sum:

where _q = ₀ + q_q, q = 0, ±1, ±2, … is the frequency of mode q, and A_q is its complex envelope.

 Here we assume that the q = 0 mode coincides with the central frequency ₀ of the atomic lineshape.

 The magnitude |A_q| may be determined from knowledge of the spectral profile of the gain and the resonator loss.

 As the modes interact with different groups of atoms in an

inhomogeneously broadened medium, their phases arg{A_q}

are random and statistically independent. 

U

(

z

,

t

)



A

_q

exp

q



i

2



_q

(

t



nz

c

)









Properties of a mode-locked pulse train

 Substituting the _q = ₀ + q_q into U(z, t), we obtain

where the complex envelope A(t)

 _{The complex envelope A(t) is a periodic function of the period TF,}

and A(t-nz/c) is a periodic function of z of period (c/n)T_F = 2d.

 _{If the magnitudes and phases of the complex coefficients Aq are}

properly chosen, A(t) may be made to take the form of periodic

U

(

z

,

t

)



A

(

t



nz

c

)exp

i

2



0

(

t



nz

c

)









A

(

t

)



A

_q q



exp

iq

2



t

T

_F













T

_F



1





_q



2

nd

c

80

Properties of a mode-locked pulse train

 _{Consider, for example, M modes (q = 0,} ±_{1, …} ±_{S, s.t. M =}

2S+1), whose complex coefficients are all equal, A_q = A, q = 0, ±1, …, ±S.

 The optical intensity I(t, z) = |A(t-nz/c)|2

80

A

(

t

)



A

exp

iq

2



t

T

_F













qS S





A

x

q qS S





A

x

S1



x

S

x



1



A

x

S 1 2



x

S 1 2

x

1 2



x

 1 2

A

(

t

)



A

sin(

M



t

/

T

F

)

sin(



t

/

T

_F

)

I

(

t

,

z

)



A

2

sin

2

[

M



(

t



nz

/

c

) /

T

_F

]

sin

2

[



(

t



nz

/

c

) /

T

_F

]

Intensity of periodic pulse train

time intensity Max. = 202 T_F T_F/M Incoherent sum M

82

Properties of a mode-locked pulse train



The shape of the mode-locked laser pulse train is

therefore dependent on the number of modes M, which

is proportional to the atomic linewidth



or



.



If M

≈



/



_q

, then



_pulse

= T

_F

/M

≈

1/



.



The pulse duration



_pulse

is therefore inversely

proportional to the atomic linewidth



.



Because can be quite large, very narrow

mode-locked laser pulses can be generated

.



The ratio between the peak and mean intensities is

equal to the number of modes M, which can also be

83

Properties of a mode-locked pulse train



The

period of the pulse train

is T

_F

= 2nd/c. This is just

the time for a single round trip of reflection within the

resonator.



The

repetition rate of the pulses

= 1/T

_F

= c/2nd =



_q



The light in a mode-locked laser can be regarded as a

single narrow pulse of photons reflecting back and

forth between mirrors of the resonator

.



At each reflection from the output mirror, a fraction of

the photons is transmitted in the form of a pulse of

light.

84

Properties of a mode-locked pulse train



Characteristic properties of a mode-locked

pulse train

84 Temporal period Spatial period Mean intensity Pulse duration Pulse length Peak intensity 2nd/c 2d I _pulse=T_F/M=1/ d_pulse= 2d/M I_p = MI

Properties of a mode-locked pulse train

 _{The mode-locked laser pulse reflects back and forth between}

the mirrors of the resonator. Each time it reaches the output mirror it transmits a short optical pulse. The transmitted

pulses are separated by the distance 2d and travel with

velocity c. The switch opens only when the pulse reaches it and only for the duration of the pulse. The periodic pulse train is therefore unaffected by the presence of the switch. Other wave patterns suffer losses and are not permitted to oscillate.

86

Properties of a mode-locked pulse train

 _{E.g. Consider a Nd}3+_{:glass laser operating at} _{0 = 1.05} _{m. It has}

a refractive index n = 1.5 and a linewidth  = 7 THz.

 The pulse duration _pulse = 1/ ≈ 140 fs and the pulse length d_pulse

≈ ₄₂ _m.

 If the resonator has a length d = 15 cm, the mode separation is 

= c/2nd = 1 GHz, which means that M = _q = 7000 modes.

 _{The peak intensity is therefore 7000 times greater than the average}

intensity.

 _{In media with broad linewidths, mode locking is generally more}

advantageous than Q-switching for obtaining short pulses.

 _{Gas lasers generally have narrow atomic linewidths, s.t. ultrashort}

pulses cannot be obtained by mode locking.

Methods of mode locking

 We consider active mode locking and passive mode locking.

 Suppose that an optical switch controlled by an external applied signal is placed inside the resonator, which blocks the light at all times, except when the pulse is about to cross it, whereupon it opens for the duration of the pulse.

 As the pulse itself is permitted to pass, it is not affected by the presence of the switch and the pulse train continues uninterrupted.

 In the absence of phase locking, the individual modes have different phases that are determined by the random conditions at the onset of their oscillation.

 If the phases happen, by accident, to take on equal values, the sum of the modes will form a giant pulse that would not be affected by the presence of the switch.  Any other combination of phases would form a field distribution that is totally or

partially blocked by the switch, which adds to the losses of the system. Therefore,

in the presence of the switch, only when the modes have equal phases can lasing

occur.

88

Methods of mode locking

 A passive switch such as saturable absorber may also be used to attain mode locking.

 _{A saturable absorber is a medium whose absorption coefficient decreases} as the intensity of the light passing through it increases.

 _{It thus transmits intense pulses with relatively little absorption while} absorbing weak ones.

 _{Oscillation can therefore occur only when the phases of the different} modes are related to each other in such a way that they form an intense pulse that can then pass through the switch.

 Semiconductor saturable-absorber mirrors, which are saturable absorbers operating in reflection, are in widespread use. The more intense the light, the greater the reflection. They work for 800 – 1600nm wavelengths, fs to ns pulse durations, and power levels from mW to hundreds of W.

 Saturable absorbers can also produce Q-switched modelocking, in which the laser emits collections of modelocked pulses within a Q-switching envelope.

Methods of mode locking

 Passive mode locking can also be implemented by means of Kerr-lens mode locking, which relies on a nonlinear-optical phenomenon in which the refractive index of a material changes with optical intensity.

 _{A Kerr medium, such as the gain medium itself, or a material placed} within the laser cavity, acts as a lens with a focal length inversely

proportional to the intensity. (refractive index change n  light intensity I)

 By placing an aperture at a proper position within the cavity, the Kerr lens reduces the area of the laser mode for high intensities s.t. the light passes through the aperture.

 Alternatively, the reduced modal area in the gain medium can be used to increase its overlap with the strongly focused pump beam, thereby

increasing the effective gain.

 The Kerr-lens approach is inherently broadband because of the parametric nature of the process.

 _{The rapid recovery inherent in passive mode locking generally leads to} shorter optical pulses than can be attained with active mode locking.

90

Diode-Pumped Solid-State Ultrafast laser

-Coherent Vitesse 800

Specification



Starter: initiate mode locking by perturbing the cavity. (changing the cavity length by shaking a piece of glass to initiate lasing for a set of cavity longitudinal modes）



Self-mode-locking: Ti:Sapphire itself serves as both the laser medium and Kerr-lens



Long cavity and angle-cut crystal (usually brewster angle): prevent etalon effects; preserve large M (number of longitudinal modes); increase peak power; shorten pulse width



Slit: blocks the CW wide beam and forces energy into mode-locked lasing.- shorter pulse(the CW components beam size is wider than the pulse (mode-locked) beam size. )



NDM: negative-dispersion mirror serves as additional dispersion compensation，to prevent pulse broadening