chapter 4----laser 2015-6-9fundamentals of photonics 1 chapter 4 laser

CHAPTER 4----LASER

23/4/18Fundamentals of Photonics 1

Chapter 4Laser

CHAPTER 4----LASER


LASERS

In 1958 Arthur Schawlow, together with Charles Townes, showed how to extend the principle of the maser to the optical region. He shared

the 1981 Nobel Prize with Nicolaas Bloembergen. Maiman demonstrated the first successful operation of the ruby laser in 1960.

CHAPTER 4----LASER


Two conditions for an oscillation:1. Gain greater than loss: net gain2. Phase shift in a round trip is a multiple of 2π

LASERS

an oscillator is an amplifier with positive feedback

CHAPTER 4----LASER


An oscillator comprises:

Stable condition: gain = loss

◆ An amplifier with a gain-saturation mechanism ◆ A feedback system ◆ A frequency-selection mechanism ◆ An output coupling scheme

If the initical amplifier gain is greater than the loss, oscillation may initiate. The amplifier then satuates whereupon its gain decreases. A steady-state condition is reached when the gain just equals the loss.

0Steady-state

power

Gain

Loss

Power

CHAPTER 4----LASER


Light amplifier with positive feedback

inPout inP gP

Gain medium (e.g. 3-level system w

population inversion)

When the gain exceeds the roundtrip losses, the system goes into oscillation

g

+

+

CHAPTER 4----LASER


LASERS

Light

Amplification through

Stimualted

Emission

Radiation

Gain medium (e.g. 3-level system w

population inversion)

Initial photonAmplified once

Reflected

Amplified twice

Reflected

Output

Amplified Again

Partially reflecting

Mirror

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CHAPTER 4----LASER


MirrorActive medium

d

Partially transmitting

mirror

Laser output

A laser consists of an optical amplifier (employing an active medium) placed within an optical

resonator. The output is extracted through a partially transmitting mirror.

CHAPTER 4----LASER


★ Gain medium The laser amplifier is a distributed-gain device characterized by its gain coefficient

Optical amplification and feedback

2

0 0 0( ) ( ) ( )8 sp

N N gt

0 ( )( )

1 / ( )s

5.1-43 Small signal Gain Coefficient

5.1-42 Saturated Gain Coefficient

CHAPTER 4----LASER


Figure 5.1-5 Spectral dependence of the gain and phase-shift coefficients for an optical amplifier with Lorentzian lineshape function

0( ) ( )

Phase-shift Coefficient (Lorentzian Lineshape)

CHAPTER 4----LASER


A Fabry-Perot resonator, comprising two mirrors separated by a distance d, contains the medium (refractive index n). Travel through the medium introduces a phase shift per unit length equal to the wavenumber

Feedback and Loss: The optical resonator

Optical Feedback-Optical Resonator

In traveling a round trip through a resonator of length d, the photon-flux density is reduced by the factor R1R2exp(-2sd). The overall loss in one round trip can therefore be described by a total effective distributed loss coefficient r, where

2k

c

1 2exp( 2 ) exp( 2 )r sd R R d

CHAPTER 4----LASER


Photon lifetime

r represents the total loss of energy (or number of photons) per unit length, arc represents the loss of photons per second

Loss coefficient 1 2

11

22

1 1ln

2

1 1ln

2

r s m m

m

m

d R

d R

1 21 2

1 1ln

2m m m d R R

1p

rc

CHAPTER 4----LASER


2F

c

d

q 1q 1q

Resonator response

Resonator modes are separated by the frequency

and have linewidths . / 2F c d / 1/ 2F pF

, 1, 2,...,q Fq q

, / 2FF c d

F

2 p Fr

Fd

CHAPTER 4----LASER


Conditions for laser oscillation

Gain condition: Laser threshold

orwhere

0 ( ) r Threshold Gain Condition

0 tN N

( )r

tN

1

( )tp

Nc

2

8 1

( )sp

tp

tN

c g

Threshold Population Difference

CHAPTER 4----LASER


For a Lorentzian lineshape function, as

If the transition is limited by lifetime broadening with a decay time tsp

2

22 spt

p

tN

c

0( ) 2 /g

2 2

22 rt

p

Nc

As a numerical example, if m, p=1 ns, and the refractive index n=1, we obtain Nt=2.1×107 cm-3

CHAPTER 4----LASER


Phase condition: Laser Frequencies

or

Frequency Pulling

Conditions for laser oscillation(2)

2 2 ( ) 2 , 1,2kd d q q ……

0 ( )2 q

c

0 ( )2q

c

0' ( )2

qq q q

c

'q q

CHAPTER 4----LASER


'0( )q q q

Laser Frequencies

CHAPTER 4----LASER


The laser oscillation frequencies fall near the cold-resonator modes; they are pulled slightly toward the atomic resonance central frequency

CHAPTER 4----LASER


Characteristics of the laser output

Internal Photon-Flux Density

Gain Clamping

0 ( ) /[1 / ( )]s r

CHAPTER 4----LASER


Determination of the steady-state laser photon-flux density

The smaller the loss, the greater the value of

0

Photon-flux density

Time

( )

10s ( )s 10 ( )s

0 ( )

Laser turn-on

Steady state

r Loss coefficient

Gain coefficient

CHAPTER 4----LASER


Steady Photon Density

0 0( ) ( )N

00

0

( )( )[ 1], ( )

0, ( )

s rr

r

00

0

( )( 1),

0,

s tt

t

NN N

N

N N

Steady-State Laser Internal Photon-Flux

Density

( )r tN Since and

CHAPTER 4----LASER


Output photon-flux density

Optical Intensity of Laser Output

Steady-state values of the population difference N, and the laser internal photon-flux density , as functions of N0 (the population difference in the absence of radiation; N0, increases with the pumping rate R).

0 2

T

0 2

h TI

CHAPTER 4----LASER


From

We obtain

When,

Then

Optimization of the output photon-flux density

11

1 1 1ln ln(1 )

2 2m Td R d

2

1ln(1 )

2r s m Td

00 0 0 2

1[ 1], 2 ( ) , 2( )

2 ln(1 )s s m

gT g d L d

L T

1/ 20( )opT g L L

ln(1 )T T

1T

use the approximation

CHAPTER 4----LASER


CHAPTER 4----LASER


Spectral Distribution

Determined both by the atomic lineshape and by the resonant modes

Linewidth ?

F

BM

Number of Possible

Laser Modes

Schawlow-Townes limit

CHAPTER 4----LASER


Figure 5.3-3 (a) Laser oscillation can occur only at frequencies for which the gain coefficient is greater than the loss coefficient (stippled region). (b) Oscillation can occur only within of the resonator modal frequencies (which are represented as lines for simplicity of illustration ).

0 ( )

r Loss

Gain

B

Allowed modes

……

Resonator modes

F

CHAPTER 4----LASER


Homogeneously Broadened Medium0

1

( )( )

1 / ( )M

j s ji

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画

CHAPTER 4----LASER


Figure 5.3-6 (a) Laser oscillation occurs in an inhomogeneously broadened medium by each mode independently burning a hole in the overrall spectral gain profile. (b) Spectrum of a typical inhomogeneously broadened multimode gas laser.

q

0 ( )

s

( )

r

1q 1q

Frequency

2

c

d

(a) (b)

Inhomogeneously Broadened Medium


CHAPTER 4----LASER


Hole burning in a Doppler-broadened medium


CHAPTER 4----LASER


Spatial distribution and polarization

Spatial distribution

Spherical mirror

Spherical mirror

Laser intensity

x,y

The laser output for the (0,0) tansverse mode of a spherical-mirror resonator takes the form of a Gaussian beam.

CHAPTER 4----LASER


Figure 5.3-8 The gains and losses for two transverse modes, say (0,0) and (1,1), usually differ because of their different spatial distributions.

0,0

00

B00

Laser output

1,1

11

B11

(0,0) modes(1,1) modes

TEM0,0

TEM1,1

CHAPTER 4----LASER


Two Issues: Polarization, Unstable Resonators

CHAPTER 4----LASER


Mode Selection

Selection of

1. Laser Line

2. Transverse Mode

3. Polarization

4. Longitudinal Mode

CHAPTER 4----LASER


Laser output

Aperture

Active medium

Output mirrorHigh reflectance

mirror

Prism

Unwanted line

Figure 5.3-9 A paticular atomic line may be selected by the use of a prism placed inside the resonator. A transverse mode may be selected by means of a spatial aperture of carefully chosen shaped and size.

CHAPTER 4----LASER


Figure 5.3-10 The use of Brewster windows in a gas laser provides a linearly polarized laser beam. Light polarized in the plane of incidence (the TM wave) is transmitted without reflection loss through a window placed at the Brewster angle. The orthogonally polarized (TE) mode suffers reflection loss and therefore does not oscillate.

High reflectance

mirror

Output mirror

Polarized laser

output

B

Brewster window

Brewster window

Active midum

CHAPTER 4----LASER


Figure 5.3-11 Longitudianl mode selection by the use of an intracavity etalon. Oscillation occurs at frequencies where a mode of the resonator coincides with an etalon mode; both must, of course, lie within the spectral window where the gain of the medium exceeds the loss.

High reflectance mirror

Output mirror

Active midum

Etalon

d1

d

Resonator loss

Laser output

c/2d Resonator mdoes

Etalon mdoes

c/2d1

Sel

ecti

on

of

Lo

ng

itu

din

al M

od

e

CHAPTER 4----LASER


Multiple Mirror Resonators

Figure 5.3-12 Longitudinal mode selection by use of (a) two coupled resonators (one passive and one active); (b) two coupled active resonators; (c) a coupled resonator-interferometer.

(a)

(b)

(c)

CHAPTER 4----LASER


Characteristics of Common Lasers

Solid State Lasers: Ruby, Nd3+:YAG, Nd3+:Silica, Er3+:Fiber, Yb3+:Fiber

Gas Lasers: He-Ne, Ar+; CO2, CO, KF;

Liquid Lasers: Dye

Plasma X-Ray Lasers

Free Electron Lasers

CHAPTER 4----LASER


CHAPTER 4----LASER


Pulsed LasersMethod of pulsing lasers External Modulator or Internal Modulator?

t t

Average power

CW power

ModulatorModulator

(a) (b)

Figure 5.4-1 Comparison of pulsed laser outputs achievable with (a) an external modulator, and (b) an internal modulator

1. Gain switching 2. Q-Switching

3. Cavity Dumping 4.Mode Locking

CHAPTER 4----LASER


Gain Switching

CHAPTER 4----LASER


Q- Switching

Figure 5.4-3 Q-switching.

t

Modulator

t

Loss

Gain

t

Laser output

CHAPTER 4----LASER


Cavity Dumping

Figure 5.4-4 Cavity dumping.

CHAPTER 4----LASER

• Laser modes coupling together

• Lock their phases to each other

Mode locking

CHAPTER 4----LASER


photon lifetime

Probability density for induced absorption/emission

From

We have

Rate equation for the photon-number density

ip

dn nNW

dt

p( ) ( )iW cn

( ) 1/ p tc N

p t p

dn n N n

dt N Photon-Number

Rate Equation

CHAPTER 4----LASER


For a three level system

Note

Then

Where the small signal population difference

Substituting

We have

Rate equation for the Population Difference

2 22 1( )i

sp

dN NR W N N

dt t

1 2 2 1( ) / 2, ( ) / 2,a aN N N N N N N N N

0 2 isp sp

NdN NW N

dt t t

0 2 sp aN Rt N

0 2sp sp t p

NdN N N n

dt t t N Population-difference rate

equation (Three-level system)

/i t pW n N

CHAPTER 4----LASER


Situation for the gain switching

CHAPTER 4----LASER


Situation for the Q-switching


CHAPTER 4----LASER


Dynamics of a Q-Switching process

Dynamics of the Q-switching process

Note the time relationship between the photon density and thepopulation inversion variations!

CHAPTER 4----LASER


Determination of the peak power, energy, width and shape of the optical pulse

( 1)t p

dn N n

dt N

2t p

dN N n

dt N

Dividing 1

( 1)2

tNdn

dN N

1 1ln( ) tan

2 2tn N N N cons t

CHAPTER 4----LASER


Power

Peak pulse power

It is clear So

The larger the initial population inversion, the higher the Q-switched pulse peak power.

1 1ln ( )

2 2t ii

Nn N N N

N

0 0

1

2 2

cP h A h cTAn h T Vn

d

1(1 ln )

2t t t

p ii i i

N N Nn N

N N N

2p p

cP h T Vn

d

When Ni>>Nt

1

2p in N 1

2 2p i

cP h T VN

d

CHAPTER 4----LASER


c. Pulse energy:

The final population difference Nf

( ) ( )2 2

f f

i i

t N

t N

c c dtE h T V n t dt h T V n t dN

d d dN

1

2 2

i

f

N

t p N

c dNE h T VN

d N

1ln

2 2i

t pf

NcE h T VN

d N

ln i fi

f t

N NN

N N

1( )

2 2 p i f

cE h T V N N

d

CHAPTER 4----LASER


d. Pulse width:

A rough estimation of the pulse width is the ratioof the pulse energy to the peak pulse power.

When Ni>>Nth and Ni>>Nf

ppulse The shorter the photon life time,the shorter the Q-switched pulses.

/ /

/ ln( / ) 1i t f t

pulse pi t i t

N N N N

N N N N

CHAPTER 4----LASER


CHAPTER 4----LASER


Techniques for Q-switching

1. Mechanical rotating mirror method:

Q-switching principle: rotating the cavity mirror results inthe cavity losses high and low, so the Q-switching is obtained.

Advantages: simple, inexpensive.Disadvantages: very slow, mechanical vibrations.

CHAPTER 4----LASER


2. Electro-optic Q-switching

Pockels effect: applying electrical field in a uniaxial crystal results in additional birefringence, which changes the polarization of light when passing through it.

Q-switching principle: placing an electro-optic crystal between crossedpolarizers comprises a Pockels switch. Turning on and off the electrical field results in high and low cavity losses.

Advantages:very fast and stable.

Disadvantages:complicate and expensive

CHAPTER 4----LASER


Electro-optic Q-switch operated at (a) quarter-wave and (b) half-wave retardation voltage

CHAPTER 4----LASER


3. Acousto-optic Q-switching

CHAPTER 4----LASER


Bragg scattering: due to existence of the acoustic wave, lightchanges its propagation direction.

Q-switching principle: through switching on and off of the acousticwave the cavity losses is modulated.

Advantages: works even for long wavelength lasers.Disadvantages: low modulation depth and slow.

Sound Transmitted light

Diffracted lightIncident light

RFPiezoelectric transducer

CHAPTER 4----LASER


4. Saturable absorber Q-switchingWhat’s a saturable absorber?

s

0

II

1

Saturable absorber Q-switching:

Absorption coefficient of the material is reversely proportional to the light intensity.Is: saturation intensity.

Insertion a saturable absorber in the laser cavity, the Q-switching will beautomatically obtained.

CHAPTER 4----LASER


General characteristics of laser Q-switching

• Pulsed laser output:– Pulse duration – related to the photon lifetime.– Pulse energy - related to the upper level lifetime.

• Laser operation mode:– Single or multi-longitudinal modes.

• Active verses passive Q-switching methods:– Passive: simple, economic, pulse jitter and intensity fluctuations.

– Active: stable pulse energy and repetition, expensive. • Comparison with chopped laser beams:

– Energy concentration in time axis.

• Function of gain medium– Energy storage

CHAPTER 4----LASER


Laser mode-locking

Aims:1. Familiarize with the principle of laser mode-locking.2. Familiarize with different techniques of achieving laser Mode-locking.

Outlines:

1. Principle of laser mode-locking.2. Methods of laser mode-locking.3. Active mode-locking.4. Passive mode-locking.5. Transform-limited pulses.

CHAPTER 4----LASER


Principle of laser mode-locking

1. Lasing in inhomogeneously broadened lasers:

iii) Multi-longitudinal mode operation.

ii) Cavity longitudinal mode frequencies.

i) Laser gain and spectral hole-burning.

CHAPTER 4----LASER


2. Laser multimode operation:Single mode lasers: ttEtE 00 cos

Multimode lasers:

N

iiii ttEtE

1

cos

Mode-frequency separations:nL

c~

Phase relation between modes: Random and independent!

Total laser intensity fluctuates with time !

The mean intensity of a multimode laser remainsconstant, however, its instant intensity varies withtime.

CHAPTER 4----LASER


3. Effect of mode-locking:

(i) Supposing that the phases of all modes are locked together:

00 ti

(ii) Supposing that all modes have the same amplitude:

0EEi

(iii) Under the above two conditions, the total electric field of the multimode laser is:

L

cNi

eEtE

cci

N

i

tii

i

2

1

Re

0

1

where

purely for the convenience of the mathematical analysis

CHAPTER 4----LASER


Calculating the summation yields:

t

2t

2t

NEtE 0

c

c

0

cossin

sin

0 is the frequency of the central mode, N is the number of modes in the laser, c is the mode frequency separation.i is the frequency of the i-th mode.

Note this is the opticalfield of the total laser Emission !

The optical filed can be thought to consist of a carrier wave of frequency 0 that amplitude modulated by the function

xNx

xAN sin

sin

CHAPTER 4----LASER


The intensity of the laser field is:

2sin

2sin

2

2

20 t

tN

EtIc

c

4. Characteristics of the mode-locked lasers:

The output of a mode-locked laser consists of a series of pulses. The time separation between two pulses is determined by RT and the pulse width of each pulse is tp.

CHAPTER 4----LASER


5. Properties of mode-locked pulses:

i) The pulse separation RT:

c

L

cRT

22

The round-trip time of the cavity!

02

tSin c2

2tc

ii) The peak power:

N times of the average power. N: number of modes.

Considering sin when is small,

20

22 EN0E

The more the modes the higher the peak power of the Mode-locked pulses.

CHAPTER 4----LASER


iii)The individual pulse width:

Narrower as N increases.N

t RTp

c

aN

aa

pt

12

02

tNsin c

cp N

2t

The mode locked pulse width is reversely proportional tothe gain band width, so the broader the gain profile, the shorterare the mode locked pulses.

a: bandwidth ofthe gain profile.

CHAPTER 4----LASER


Techniques of laser mode-locking

Active mode-locking:Actively modulating the gain or loss of a laser cavity in a periodic way,usually at the cavity repetition frequency c/2nL to achieve mode-locking.

Amplitude modulation:

A modulator with a transmission function of

RT

tT

2

cos11

is inserted in the laser cavity to modulate the light. Where is the modulation strength and < 0.5. Under the influence of the modulationphases of the lasing modes become synchronized and as a consequencebecome mode-locked.

Operation mechanism of the technique:

CHAPTER 4----LASER


Shu

tter

loss

es

Time domain analysis:Consider the extreme case where a shutter is inside the cavity, and the opens only for a short time every second. Is the cavity round trip time. In this case only a pulse with pulse width narrower than the opening time can survive in the cavity, all the CW type of operation will be blocked by the shutter. To have a pulse moving in the cavity the phase of all lasing modes must be synchronized. The laser modes will arrange themselves to realize such a state.

It is a natural competition and the fittest will survive.

CHAPTER 4----LASER


Frequency domain analysis:

Amplitude modulation Sidebands generation

mmmm tsintE tcos10m

mmmmmm0m tsin

2tsin

2tsintE

Sidebands are generated by the modulation

Amplitude of each mode is modulatedElectrical filed of each mode

mm+m-m

Without modulation After amplitude modulation

CHAPTER 4----LASER


In the case of a multimode laser

As all modes are modulated by the same frequency, thesidebands of one mode will drive its adjacent modes, andas a consequence, all modes will oscillate with locked phase.

From both the time domain and the frequency domain analysis it is easy to understand why the modulation frequency must be exactly the cavity longitudinal mode separation frequency.

CHAPTER 4----LASER


A typical passive mode locking laser configuration:

laser medium saturable absorber

Passive mode-locking:

Inserting an appropriately selected saturable absorber inside the lasercavity. Through the mutual interaction between light, saturable absorberand gain medium to automatically achieve mode locking.

Mechanism of the mode-locking:

i) Interaction between saturable absorber and laser gain: Survival takes all! ii)Balance between the pulse shortening and pulse broadening:

Final pulse width.

CHAPTER 4----LASER


Transform limited pulses Gaussian pulses:

A Gaussian gain line shape function a Gaussian mode-locked pulse intensity variation, namely a Gaussian pulse.

In our analysis we have assumed that En=E0

The real gain line has a Gaussianprofile,which results in that the lasingmode amplitudes have a Gaussian distribution.

tit

EtE 0

2

2

10

2

1

0 exp2ln2

exp2ln2

CHAPTER 4----LASER


If the product of pulse width and spectral bandwidth of a Gaussianpulse equals 0.441, then the Gaussian pulse is called a transformlimited pulse as in this case the pulse width is purely determined by the Fourier transformation of the pulse spectral distribution.

Transform limited pulses:

For transform limited Gaussian pulses!

tp: pulse width, L: spectral bandwidth

441.0 Lpt

Intensity of the pulse:

2

2

1

2

0 2ln2exp

2ln2

tI Gaussian intensity profile!

CHAPTER 4----LASER

返回激光振荡器

CHAPTER 4----LASER

均匀加宽激光器模式竞争返回

CHAPTER 4----LASER

非均匀加宽激光器模式竞争返回

CHAPTER 4----LASER

多普勒加宽增益饱和返回

CHAPTER 4----LASER

兰姆凹陷返回

CHAPTER 4----LASER

脉冲泵浦调 Q

返回

chapter 4----laser 2015-6-9fundamentals of photonics 1 chapter 4 laser

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