Chapter 6. Laser: Theory and...

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Chapter 6. Laser: Theory and Applications Reading: Sigman, Chapter 6, 7, and 26 Bransden & Joachain, Chapter 15

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  • Chapter 6.

    Laser: Theory and Applications

    Reading: Sigman, Chapter 6, 7, and 26Bransden & Joachain, Chapter 15

  • Laser Basics

    Stimulated emission01 EEh −=ν1E

    0Eνh

    Population inversion (N1 > N0) ⇒ laser, maser

    Light Amplification by Stimulated Emission of Radiation

    Transition rate for stimulated emission

    )()()( 012

    0101201

    01

    0

    2

    2

    2

    01 44 ωω

    ωω

    πεπ IMIe

    cmW ∝⎟⎟

    ⎞⎜⎜⎝

    ⎛=

    Incident light intensity

    Gain mediumHigh reflector Out coupler

    Optical cavity

    Pumping (optical, electrical, etc.) for population inversion

  • Longitudinal Modes in an Optical Cavity

    Boundary condition:

    ::

    L

    EM wave in a cavity

    mcmcmLωπτλ

    ===22

    mLcm

    Lc

    mL πωνλ ===⇒ ,,

    22

    m = 1, 2, 3, …

    Round-trip time of flight: τmcLT == 2

    Typical laser cavity: L = 1.5 m, λ = 0.75 µm

    nsec secsec/m

    m 1010103

    32 88 ==×

    == −cLT

    !! milion m.

    m 410410750

    32 66 =×=×

    == −λLm

    MHz Hz 100101 8 ===⇒TR

    ν

  • Single mode22

    0 tttI coscos)( == ω

    -100 -50 0 50 100Time

    Inte

    nsity

    Frequency1

    Inte

    nsity

    Spectrum

  • Cavity Quality Factors, Qc

    End mirrorR ≈ 1

    Out couplerT = 1 ~ 5 %

    L

    Energy loss by reflection, transmission, etc.

    t

    tiTt eeE c 0/0ωδ −−

    TiE

    dteeEE

    c

    tiTtc

    /)(

    )(

    0

    0

    0

    /0

    0

    δωω

    ω ωδ

    +−=

    = ∫∞ −−

    2220

    202

    /)()(

    TEE

    cδωωω

    +−=

    Emission spectrum

    ωω0c

    c

    QTωδ

    =~

    2)(ωE Lorentzian

  • Energy of circulating EM wave, Icirc(t)

    ,exp)()( ⎥⎦⎤

    ⎢⎣⎡−×=

    TtItI ccirccirc δ0

    Number of round trips in t

    : round-trip time of flightcLT 2=

    ⎥⎦

    ⎤⎢⎣

    ⎡−×=⇒ t

    QItI

    ccirccirc

    ωexp)()( 0

    wherecc

    cLTQ

    δλπ

    δω 14

    ==

    Q-factor of a RLC circuit

    RLQ ω

    ωω

    =∆

    =

    Typical laser cavity: L = 1 m, λ = 0.8 µm, δc = 0.01 (~1% loss/RT)9

    6 10610101

    10804

    ×≈×

    = − ...π

    cQ Hz610≈∆ω

  • [ ] )()()()()()(

    )()(

    2212112

    22121112

    21

    tNtNtNWtNWtNW

    dttdN

    dttdN

    γγ+−−=

    ++−=

    −=

    Two Level Rate Equation

    Two Level Rate Equations and Saturation

    1E

    2E

    221NW 221Nγ

    2N

    1N

    112NW

    : non-radiative decay rate1

    211T

    Total number of atoms: constant)()( 21 =+= tNtNN

    Population difference: )()()( 21 tNtNtN −=∆

    [ ][ ] [ ])()()()()()(2

    )(2)()(2)(

    2121212112

    2212112

    tNtNtNtNtNtNW

    tNtNtNWtNdtd

    −−−−−−=

    +−−=∆

    γ

    γ

    112

    )()(2)(T

    NtNtNWtNdtd −∆

    −∆−=∆

  • Steady-State Atomic Response: Saturation

    112

    )()(20)(T

    NtNtNWtNdtd −∆

    −∆−==∆

    TWNNN ss

    1221+=∆=∆

    112 21,

    /11

    TW

    WWNN

    satsat

    ss ≡+

    =∆

    0.01 0.1 1 100.0

    0.2

    0.4

    0.6

    0.8

    1.0

    1.2

    NNss∆

    TW122Signal strength,

    SaturationIntensity, I=Isat

    Gain coefficientin laser materials

    Nm ∆∝α

    sat

    mm II

    I/1

    )( 0+

    =αα

  • Transient Two-Level Solutions

    1112

    112 )(

    12)()(2)(TNtN

    TW

    TNtNtNWtN

    dtd

    +∆⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛+−=

    −∆−∆−=∆

    ,12exp)(1

    12 ⎥⎦

    ⎤⎢⎣

    ⎡⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛+−+∆=∆ t

    TWANtN ss

    TWNNss

    1221+=∆

    Initial population difference at t = 0: ANN ss +∆=∆ )0(

    ( ) ⎥⎦

    ⎤⎢⎣

    ⎡+−⎥

    ⎤⎢⎣

    ⎡+

    −∆++

    =∆1

    1121212

    21exp21

    )0(21

    )(TtTW

    TWNN

    TWNtN

    0 1 2 30.0

    0.5

    1.0

    NtN )(∆

    1/Tt

    Transient saturation behavior following sudden turn-on of an applied signal

    02 112 =TW

    5.0124

  • Two-Level Systems with Degeneracy

    1E

    2E 2g

    1N1g

    2N

    )()()( 211

    2 tNtNggtN −⎟⎟

    ⎞⎜⎜⎝

    ⎛≡∆Population difference:

    Stimulated transition rates: 212121 WgWg =

    )()()()()( 2212111221 tNWtNWdttdN

    dttdN γ++−=−=

    Rate equation:

    Effective signal-stimulated transition probability: ( )211221 WWWeff +≡

    1

    )()(2)(T

    NtNtNWtNdtd

    eff−∆

    −∆−=∆Rate equation:

    Atomic time constants: T1 and T2T1 : longitudinal (on-diagonal) relaxation time

    population recovery or energy decay timeT2 : dephasing time, transverse (off-diagonal) relaxation time

    time constant for dephasing of coherent macroscopic polarization

  • Rate equation for level 4

    pWWW == 4114

    ,/)(

    )()(

    4441

    4414243414

    τ

    γγγ

    NNNW

    NNNWdt

    dN

    p

    p

    −−=

    ++−−=

    41424344

    1 γγγγτ

    ++=≡where

    pump transition probability,

    Steady-State Laser Pumping and Population Inversion

    Four-level pumping analysis

    E1

    E2

    E3

    E4 τ43 (fast)

    τ32(slow)

    τ21 (fast)

    Wp

    pumping

    Steady state population:

    ,1 1414

    44 NWNW

    WN p

    p

    p ττ

    τ≈

    += if 14

  • 21

    2

    32

    3

    42

    4221332442

    2

    τττγγγ NNNNNN

    dtdN

    −+=−+=

    33342

    2143

    32

    212 NNN βττ

    ττττ

    =⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛+=

    342

    2143

    32

    21

    ττττ

    ττβ +≡where

    If β < 1, 32 NN < : population inversion on the 3 → 2 transition

    In a good laser system, 43 NN >>433 ττ >> so that

    Rate equations for level 2 and 3

    3

    3

    43

    433132443

    3 )(ττ

    γγγ NNNNdt

    dN−=+−=

    at steady state

    443

    33 NN τ

    τ=

    E1

    E2

    E3E4 τ43 (fast)

    τ32(slow)

    τ21 (fast)

    Wppumping

    In a good laser system, 042 ≈γ so that the level 4 will relax primarily into the level 3.

    32

    21

    ττβ ≈ condition for population inversion 1

    32

    21

    3

    2

  • Fluorescent quantum efficient

    rad

    rad

    ττ

    ττ

    γγ

    γγη 3

    43

    4

    34

    43 ×=×=

    The number of fluorescent photons spontaneously emitted on the laser transition divided by the number of pump photons absorbed on the pump transitions when the laser material is below threshold

    Fraction of the total atoms excited to level 4 relax directly into the level 3

    Fraction of the total decay out of level 3 is purely radiative decay to level2

    Four level population inversion

    [ ] radpradradp

    WW

    NNN

    τηττβτηβ

    /211)1(

    43

    23

    +++

    −=

    4321 NNNNN +++=

    In a good laser system, 43ττ >>rad

    radp

    radp

    radp

    radp

    WW

    WW

    NNN

    τητη

    τηβτηβ

    +≈

    ++

    −≈

    −1)1(1

    )1(23

    , β → 0.3-level system

    0 1 2 3

    -N

    0

    N

    ∆N

    Wpτrad

    4-level system

    ∆Nth

  • Laser gain saturation analysis

    E0

    E1

    E2

    E3 γ32

    γ21

    γ1

    Wp

    pumping

    N0≈N

    N1

    N2

    N3

    Wsigstimulated

    Pumping transition

    NWNWNNWdt

    dNppp

    pump

    ≈≈−= 0303 )(

    0NWR ppp η=Effective pumping rate: quantum efficiency E3 → E2

    22122 )( NNNWR

    dtdN

    sigp γ−−−=

    Rate equations for laser levels 1 and 2:

    11221121 )( NNNNW

    dtdN

    sig γγ −+−=

    20212 γγγ +=

    psig

    sig RW

    WN

    21201

    211 )( γγγγ

    γ++

    +=

    psig

    sig RW

    WN

    21201

    12 )( γγγγ

    γ++

    +=

  • Gain saturation behavior

    [ ]

    effsig

    sigp

    WN

    WRNNN

    τ

    γγγγγγγγ

    +∆=

    ++×⎟⎟

    ⎞⎜⎜⎝

    ⎛ −=−=∆

    11

    /)(11

    0

    2120121

    2111221

    Small signal or unsaturated population inversion

    221

    1

    21

    2110 1 ττ

    τγγγγ

    pp RRN ×⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛−=⎟⎟

    ⎞⎜⎜⎝

    ⎛ −=∆

    Effective recovery time201

    211γγ

    γγτ +

    =eff

    ⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛+=

    20

    12 1 τ

    τττ effor

    If γ20 ≈ 0, γ2 ≈ γ212

    1221 11)(

    τττ

    sigp W

    RN+

    ×−=∆

    • Population inversion requires τ21 > τ1.

    •• If τ1 → 0, τeff ≈ τ2.

    • The saturation intensity of the inverted population is independent of Rp.

    )/1( 2120 τττ −×∝∆ pRN

  • Wave Propagation in an Atomic Medium

    Wave equation in a laser medium

    ( )[ ] 0),,(/122 =−++∇ zyxEiat ωεσχµεωOhmic lossatomic

    susceptibility

    ( ) 0)(/1222

    =⎥⎦

    ⎤⎢⎣

    ⎡−++ zEi

    dzd

    at ωεσχβ

    Plane wave approximation

    “Free-space” propagation constant:λπωµεωβ nn

    c2

    ===

    Propagation factor:

    ( )ωεσχβ /122 iat −+−=Γ

    zeEzE Γ−= 0)(

    ωεσωχωχβωεσχβ /)()(1/1 iiiii at −′′+′+=−+=Γ

  • 1/,

  • atomic phase-shift contribution zm )(ωβ∆

    aωω

    Propagation factors

    [ ]zz mtot )(),( ωββωφ ∆+=

    π2total phase shift

    “free-space”Contributionβ(ω) z

    ω

    ( )( )

    [ ]( )[ ]2

    0

    22222

    22

    /21/)(2

    )(

    Γ−+Γ−′

    Γ+−

    −=′∝∆

    a

    a

    a

    am

    a

    ωωωωχ

    ωωωωωωχβ

    atomic gain zm )(ωα

    ω( )[ ]2

    0

    /21)(

    Γ−+

    ′′≈′′∝

    am ωω

    χωχα

  • Single-Pass Laser Amplification

    Gain medium0=z Lz =

    Laser gain formulas

    Complex amplitude gain: [ ] [ ]{ }LLiE

    LEg mm 0)()(exp)0()()( αωαωββω −+∆+−=≡

    Total phase shift amplitude gain or loss

    Power or intensity gain: ( ) [ ]{ }LgI

    LIG m 02 )(2exp

    )0()()( αωαωω −==≡

    )(21 ωχβ ′′=

    Lorenzian transition line shape: [ ]20

    /)(21)(

    aa ωωωχωχ

    ∆−+

    ′′=′′

    Midband value

  • Power gain:[ ] ⎭

    ⎬⎫

    ⎩⎨⎧

    ∆−+×

    ′′= 2

    0

    /)(211exp)(

    aavLG

    ωωωχωω

    Power gain in decibels (dB):

    )(34.4)(ln34.4)(log10)( 10 ωχωωωω ′′==≡

    vLGGG adB

    Amplification bandwidth and gain narrowing

    3-dB amplifier bandwidth:3)(

    33 −

    ∆=∆adB

    adB G ωωω

    Amplifier phase shift

    [ ] )(2

    )(),( ωχβωωββωφ ′+=∆+= LvLLz mtotTotal phase shift:

    [ ]210 /)(21/)(2

    log20)()(2)(

    aa

    aaadBm

    a

    am e

    GLLωωω

    ωωωωωαωωωωβ

    ∆−+∆−

    ×=×⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛∆−

    =∆

    Atomic transition phase shift:

  • Saturation Intensities in Laser Materials

    Saturation of the population difference

    sateff IIN

    WNN

    /11

    11

    00 +×∆=

    +×∆=∆

    τ

    INIdzdI

    m σα ∆== 2Traveling wave:Stimulated transition cross-section

    Population difference:

    sat

    mm IzI

    zdz

    zdIzI /)(1

    2)(2)()(

    1 0+

    ==αα σα 002 Nm ∆≡where

    ∫∫ =⎥⎦

    ⎤⎢⎣

    ⎡+

    =

    =

    L

    m

    II

    IIsat

    dzdIII

    out

    in 002

    11 α

    00 ln2ln GLIII

    II

    msat

    inout

    in

    out ==−

    +⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛α where )2exp( 00 LG mα≡

    unsaturatedpower gain

  • ⎥⎦

    ⎤⎢⎣

    ⎡ −−×=⎥

    ⎤⎢⎣

    ⎡ −−×=

    ⎥⎦

    ⎤⎢⎣

    ⎡ −−×=≡

    sat

    out

    sat

    in

    sat

    inout

    in

    out

    GIIGG

    IIGG

    IIIG

    IIG

    )1(exp)1(exp

    exp

    00

    0

    Overall power gain:

    ⎟⎠⎞

    ⎜⎝⎛

    −=

    GG

    GII

    sat

    out 0ln1

    1⎟⎠⎞

    ⎜⎝⎛

    −=

    GG

    GG

    II

    sat

    out 0ln1

    and

    Power extraction and available power

    ⎟⎠⎞

    ⎜⎝⎛×=−≡

    GGIIII satinoutextr 0ln

    ( ) satmsatsatGavail ILGIGGII ×=×=⎥

    ⎤⎢⎣

    ⎡⎟⎠⎞

    ⎜⎝⎛×≡

    → 000

    12lnlnlim α

  • Specific Laser SystemsLaser media: gas, dye, chemical, excimer, solid-state, fiber, semiconductor, free-electron

    http://en.wikipedia.org/wiki/Image:Laser_spectral_lines.png

    http://upload.wikimedia.org/wikipedia/commons/f/f9/Laser_spectral_lines.png

  • Ruby Laser

    http://web.phys.ksu.edu/vqm/laserweb/Ch-6/C6s2t1p2.htmEnergy level diagram of Ruby laser

    • This system is a three level laser with lasing transitions between E2 and E1. • The excitation of the Chromium ions is done by light pulses from flash lamps (usually Xenon). • The Chromium ions absorb light at wavelengths around 545 nm (500-600 nm). As a result the ions are transferred to the excited energy level E3. • From this level the ions are going down to the metastable energy level E2 in a non-radiative transition. The energy released in this non-radiative transition is transferred to the crystal vibrations and changed into heat that must be removed away from the system. • The lifetime of the metastable level (E2) is about 5 msec.

  • He-Ne Laser

    Energy level diagram

    Schematics http://en.wikipedia.org/wiki/Helium-neon_laser

    http://upload.wikimedia.org/wikipedia/en/a/af/Hene-1.pnghttp://upload.wikimedia.org/wikipedia/commons/4/4b/Laser_DSC09088.JPG

  • Excimer Lasers

    Applications• Marking • Micromachining • Laser Ablation • Laser Annealing • Surface Structuring • Laser Vision Correction • Optical Testing and Inspection • Pulsed Laser Deposition • Fiber Bragg Gratings

    • Gain medium: inert gas (Ar, Kr, Xe etc.) + halide (Cl, F etc.)

    • Excited state is induced by an electrical discharge or high-energy electron beams.

    • Laser action in an excimer molecule occurs because it has a bound (associative) excited state, but a repulsive (disassociative) ground state.

    Excimer WavelenthFe 157 nm

    ArF 193 nmKrF 248 nm

    XeCl 308 nm

    http://tftlcd.khu.ac.kr/research/poly-Si/chapter4.html

  • Semiconductor Lasers – laser diodes

    Vertical cavity surface emitting lasers (VCSEL) structure

    Band structure near a semiconductor p-n junction

    e h e h

    forward bias off forward bias on

    Diode laser structure

    en.wikipedia.org/wiki/Laser_diode, www.mtmi.vu.lt/pfk/funkc_dariniai/diod/led.htm

    http://upload.wikimedia.org/wikipedia/en/d/d9/Diode_laser.jpghttp://upload.wikimedia.org/wikipedia/en/c/c8/Simple_vcsel.svg

  • Laser Q-SwitchingQ-switched laser output: short and intense burst of laser output dumping all the accumulated population inversion in a single short laser pulse (~10 ns)

    High initial cavity lossloss

    gain

    loss

    loss

    lossgain

    gain

    laseroutput

    t

    t

    tPumping process builds up a large population inversion

    t

    Cavity loss is suddenly “switched” to low value

    “giant pulse” laser action takes place

  • Laser Q-switching techniques

    Rotating mirror

    Electro-optic

    Acousto-optic

    Saturable absorber

  • Active Q-switching: rate-equation analysis

    2E

    1E01 ≈N

    2N)(tn

    Coupled rate equations

    )()()()( tNtKntNRdt

    tdNp −−= 2γ)()()(

    )( tntntKNdt

    tdncγ−=

    N(t) = N2 − N1: inverted population difference

    Rp : pumping rateγ2 : decay rate for N

    n(t) : cavity photon numberγc = 1/τc : total cavity decay rate

    K : coupling coefficient between photons and atoms

  • Pumping interval, and population build-up

    gaint

    lossPumping process builds up a large population inversion

    n(t) = 0 and a pump pulse with constant intensity Rp is turned on at t = 0.

    [ ])/exp()(/)()( 222 1 τττ tRtNtNRdttdN

    pp −−=⇒−=

    Pump Pulse Durationτ2 2τ2

    2τpR

    N(t)

    Energy storage during the pumping interval for a fixed pumping rate

    Typical solid state lasers: τ2 ≈ 200 µs

  • Pulse build-up time, Tb

    Initial inversion ratio:th

    i

    NNr ≡ Ni : population inversion just after switchingNth : threshold inversion just after switching

    [ ]

    [ ],/)(exp)(

    )()()()()()(

    ci

    cthi

    c

    trntn

    tnrtnNNKtntntKNdt

    tdn

    τττ

    1

    11

    −=⇒

    −≈−≈−=

    loss

    gaint

    Cavity loss is suddenly “switched” to low value

    Tb

    Tb (~100 ns)

  • Build-up time, Tb from the initial photon density ni to a photon density nss

    ⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛−

    =⇒⎥⎦

    ⎤⎢⎣

    ⎡ −=

    i

    sscb

    c

    b

    i

    ss

    nn

    rTTr

    nn ln)(exp

    11 τ

    τ

    Ratio of initial to final photon numbers:128 1010 to≈

    i

    ss

    nn ±20 % in ⎟⎟

    ⎞⎜⎜⎝

    i

    ss

    nnln

    ( )c

    bT

    rT

    δ×

    −±

    ≈1525

    cc

    τ =wherecLT 2=

    Fractional power loss per round trip

    Therefore, and

    Examples:

    1. Flash-pumped Nd:YAG laserL = 60 cm (T = 4 ns), R=0.35 (δc = ln(1/R)=1.05), r = 3 ⇒ Tb ≈ 50 ns

    2. cw-pumped Nd:YAG laser or low-gain CO2 laserL = 2 m (T = 12 ns), R=0.8 (δc = ln(1/R)=0.22), r = 1.5 ⇒ Tb ≈ 3 µs

  • Pulse output interval

    [ ] )()()()()()( tnNtNKtntntKNdt

    tdnthc −=−= γ

    loss

    gainlaseroutput

    t )()()()()()( tNtKntNtKntNR

    dttdN

    p −≈−−= 2γ

    at the switching time t = tiInitial conditions: thi rNNN == 1== innand

    )(ln)()(

    ))(()(ln)()(

    )()(

    tNN

    rNtNNtn

    NtNN

    tNNdNN

    Ntn

    dNN

    NdnN

    NNdNdn

    iii

    ii

    th

    tN

    Nth

    tN

    Nthtn

    nth

    i

    ii

    −−≈⇒

    −−=⎟⎠⎞

    ⎜⎝⎛ −≈⇒

    ⎟⎠⎞

    ⎜⎝⎛ −=⇒

    −=

    ∫∫

    1

    1

    th

    i

    NNr =where

  • Time evolution of n(t) and N(t) during the pulse output interval

    Ni

    Nf

    Nth

    ni ≈ 0 nf ≈ 0

    np

    t

    )(tN

    )(tn

  • Mode-Locking

    Mode-locking is a technique in optics by which a laser can be made to produce ultrashort pulses with the pulse width of the order of subpicoseconds (< 10-12). The basis of the technique is to induce a fixed phase relationship between the modes of the laser's resonant cavity. The laser is then said to be phase-locked or mode-locked. Interference between these modes causes the laser light to be produced as a train of pulses. Depending on the properties ofthe laser, these pulses may be of extremely brief duration, as short as a few femtoseconds. - Wikipedia

    General description of electric field in a laser cavity

    ∑ +=m

    tim

    mmeEtE )()( φω

    The laser is said to be “mode-locked” when the initial phases are equal.

    φm : initial phase of the mth mode

    φφ =m : constant

  • Single mode22

    0 tttI coscos)( == ω

    -100 -50 0 50 1000

    20406080

    100120

    Time

    Inte

    nsity Frequency

    1

    Inte

    nsity

    Spectrum

  • Three modes in phase21190 ttttI .coscos.cos)( ++=

    -100 -50 0 50 1000

    20406080

    100120

    Time

    Inte

    nsity Frequency

    1

    Inte

    nsity

    Spectrum0.9 1.1

    πδω

    πδω

    20210

    ==

    =

    T

    .

  • Eleven modes with random phases2

    51016050 51016050 ).cos(...).cos(...).cos().cos()( .... φφφφ ++++++++= tttttI

    -100 -50 0 50 1000

    20406080

    100120

    Time

    Inte

    nsity Frequency

    1

    Inte

    nsity

    Spectrum

    0.5 1.5

  • Eleven modes with the same phase, φ = 025141016050 ttttttI .cos.cos....cos....cos.cos)( ++++=

    -100 -50 0 50 1000

    20406080

    100120

    Time

    Inte

    nsity Frequency

    1

    Inte

    nsity

    Spectrum

    0.5 1.5

  • In the time domain, EM field is a periodic repetition: )()( TntEtE +=

    ∑∑ ==m

    tL

    cmi

    mm

    tim eEeEtE m

    πω)(

    )()()(

    tEeeEeEeEnTtE mnim

    tL

    cmi

    mm m

    cnLt

    Lcmi

    m

    nTtL

    cmi

    m ====+ ∑∑ ∑⎟⎠⎞

    ⎜⎝⎛ ++ π

    πππ2

    2 =1

    dtetET

    ET ti∫= 0

    1 ωω )()( )( mm EE ω=Spectrum for one period: Note

    Normalized spectrum for N periods:

    )()(

    ')'(')'(

    )()()(

    ')'(

    )(

    ωω

    ω

    ω

    ωω

    ωωω

    ωω

    Eee

    NEe

    N

    dtetET

    eN

    dtenTtENT

    dtetENT

    dtetENT

    E

    Ti

    NTiN

    n

    nTi

    N

    n

    T tinTiN

    n

    T nTti

    N

    n

    Tn

    nT

    tiNT tiN

    −−

    =⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛=

    =+=

    ==

    ∑ ∫∑∫

    ∑∫∫

    =

    =

    =

    +

    =

    +

    1111

    111

    11

    1

    0

    1

    00

    1

    00

    1

    0

    1

    0

  • Normalized power spectrum for N periods:

    )()/(sin)/(sin

    )()()(

    ωωω

    ωωω ωω

    ITTN

    N

    Eee

    NEI Ti

    NTiNN

    221

    111

    2

    2

    2

    22

    2

    2

    =

    −−

    ==

    Periodic comb

    Tπδω 2=

    NTf 1≈δ

    When N goes to infinity, ∑∞

    −∞=∞→⎟⎠⎞

    ⎜⎝⎛ −⇒

    nN Tn

    TTN πωδ

    ωω 2

    22

    2

    2

    )/(sin)/(sin

    Dirac comb

  • Periodic repetition of EM wave in a laser cavity in the time domain

    )()( TntEtE +=

    T

    Laser power spectrum )(ωI

    pτω 1≈∆

    Lc

    Tππδω == 2

  • -100 -50 0 50 100

    Time

    020406080

    100120

    Inte

    nsity

    020406080

    100120

    Inte

    nsity

    020406080

    100120

    Inte

    nsity

    020406080

    100120

    Inte

    nsity

    Frequency1

    Inte

    nsity

    0.5 1.5

    1

    Inte

    nsity

    0.5 1.5

    Inte

    nsity

    Inte

    nsity

    Spectrum

    10.5 1.5

    10.5 1.5

    Eleven modesrandom phases

    Eleven modessame phase

    Singlemode

    Threemodesin phase

  • Net gain and cavity modes

    Total electric field with phase-locked modes:

    If the gain spectrum is a Gaussian distribution,

    ⎥⎥⎦

    ⎢⎢⎣

    ⎡⎟⎠⎞

    ⎜⎝⎛

    ∆−=

    ⎥⎥⎦

    ⎢⎢⎣

    ⎡⎟⎠⎞

    ⎜⎝⎛

    ∆−

    −==2

    0

    20

    0 2424 ωδω

    ωωωω nEEEE nnn lnexplnexp)(

    ∑∑ ⎥⎥

    ⎢⎢

    ⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛ −−==

    m p

    ti

    n

    tinn

    ti mTteEeEetE2

    0 2200 τω ωδωω lnexp)()(

    ∑=n

    tin

    neEtE ω)(

    pτπω 1222 ln=∆

    Lc

    Tππδω == 2

    ω1ω 3ω2ω 4ω1−ω2−ω4−ω 3−ω

    Net gain spectrum

    δωωω nn += 0……

    T0ω

    pτt

  • For a laser cavity, L = 1.5m:

    pτπω 1222 ln=∆

    Lc

    Tππδω == 2

    ω

    Net gain spectrum

    δωωω nn += 0……

    m0ω 1ω 3ω2ω 4ω1−ω2−ω4−ω 3−ω

    ωπλλ ∆≈∆

    c2

    2

    Gain Medium(wavelength)

    Bandwidth∆λ (nm)

    Number of modesm = ∆ω / δω

    Minimum Pulse duration, τp

    Ar+-ion (520 nm) ~ 0.007 ~ 80 ~ 150000 fs

    Rubi (694.3 nm) ~ 0.2 ~ 1300 ~ 6000 fs

    Nd:YAG (1064 nm) ~ 10 ~ 25000 ~ 120 fs

    Dye (620 nm) ~ 100 ~ 8×105 ~ 12 fs

    Ti:Sapphire (800 nm) ~ 400 ~2 ×106 ~ 3 fs

    ,.. GHzHzLc

    T6280102862 8 =×=== ππδω

  • Passive and Hybrid Mode-LockingPassive mode-locking is accomplished by interplay between saturable absorber and gain medium without any external modulation.

    Saturableabsorber

    Saturablegain medium

    T0

    1

    Intensity (I)Isabs

    1/2

    Tran

    smis

    sion

    (T)

    1

    G0

    Intensity (I)Isamp

    G0/2G

    ain

    (G)

  • Pulse shape modification by saturable absorber and gain medium

    Saturable absorber

    Leading edgeabsorption

    Before After

    Leading edgeamplificationSaturable

    gain medium

    Before After

  • Pulse shortening process due to saturation of the absorption and the gain

    Loss

    Gainnet gain > 0

    Pulse peak is amplified

    Pulse tail is attenuated

    Mode-lockedpulse

    t

  • Colliding-pulse mode-locked dye laser

    GSA

    Ar+

    lase

    r, 51

    5 nm Pulse

    compressor

    GainRh6G

    Saturable absorberDODCI

    620 nm~50 fs

    symmetric amplification

    absorber gain dispersion

    t

  • Kerr-Lens Mode-Locking (KLM)

    Ti:Sapphire

    High power

    Low power

    self-focusing:

    SlitM7 OC

    Ti:Sapphire femtosecond oscillator

    76 MHz100 fs13 nJ

    Band width~ 200 nm

    Innn 20 +=

  • High power

    Low power Hard aperturenonlinear medium + hard aperture= Instantaneous saturable absorber :Transmission is lower at low intensity compared to high intensity.Nonlinear

    mediaNote: Sometimes a hard aperture is not necessary, because larger overlap between the pump beam and the cavity mode in the gain medium at high intensity leads to larger modal gain.

    Time

    020406080

    100120

    Inte

    nsity

    020406080

    100120

    Inte

    nsity

    020406080

    100120

    Inte

    nsity 1. Mechanical shocks (e.g., vibrating starter,

    jolting mirror) initiate mode-locking.

    2. The pulse regime is favored over the continuous regime.

    3. Spectral broadening by SPM supports the pulse shortening.