Optoelectronic 2010, K6 FOLIEN part 2 final -

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Electronics Laboratory: Optoelectronics and Optical Communication 10.04.2010

6-73

Solutions of the Phase- and Gain-Condition: Threshold pumping and Oscillation-Resonances (07/05/2010)

From the gain-condition we can calculate the threshold gain gth(nth) or the corresponding threshold carrier density nth, for a loop gain =1 (T→∞).

Threshold-Gain gth(nth): (field gain)

( )( )

( )

1 2 th th WG

th th res WG WGMI

2

R

1MIR

definition

l

ln r ln r g n 2L 0

nr lnr2L

g n ,α

α

ω α αα

+ + −

++

−

= →

= = + at resonance ω=ωres !!! (g, α field gain/loss)

( ) ( )( )i i 0res2Ln / i ; c i / L nλ λ ω ω π ω= = = Resonance-Frequencies, Wavelengths The mode separation ( )m,m 1 m 1 m 0c / Lnω ω ω π+ +Δ = − = between adjacent modes are equal in a non-dispersive (n=const.) resonator.

Threshold current Ith of laser diodes: from power gain gth(nth)=a(nth-ntr) Ith

From the relation g(n) and assuming for simplicity the resonant mode to be at gain-maximum ωres=ω0 gmax=a(n-ntr), then gth=g(nth) th th trn g / a n= −

th th active spontstat . rate eq. for nI en V / τ

−⎯⎯⎯⎯⎯⎯⎯→ =

Planar LD have long resonators L (∼200μm) >> λ(∼1.5μm) for fabrication reasons (an exception are ultra short vertical emitting VCSEL), there exist many longitudinal standing waves (resonator-modes) close to the weakly selective gain-maximum at ω0.

The active medium provides gain g(ω) close to the threshold gain for a limited number of longitudinal modes around the gain maximum (ωi ~ωo,max ) and is not very gain-selective tendency for multimode-oscillations.

Ideally in a homogeneously broadened gain-medium only one single mode can reach its threshold first and resulting gain-saturation (n=nth ≠f(I)) prevents all other possible modes from lasing (gain-clamping)!

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6-74

6.4.4 Light-Current Characteristic of Diode Lasers

1) Threshold current ITH and Differential Efficiency ηd

For an efficient electro-optical conversion in LASERS the threshold pump rate, resp. Ith should be low (“loss” by spont. emission) and the differential efficiency /D optP Iη = Δ Δ should be high. The gain-condition at resonance ωi of the FP-Laser is for mode i:

( ) ( )( )2 2

0

11

th ,i th ,i res ,i m,i WG max th,ires ,i

g n , g n/

ω α αω ω ω

= + =+ − Δ

optical gain of the i-th mode gth,i(nth,i,λi) to reach loop-gain 1 can be obtained

2) Concept of Gain-saturation, resp. “carrier clamping” at the Threshold of the first oscillating mode

The mode i with the lowest threshold gain gth,i (lowest losses) oscillates first and saturates (Rstim~Rspont) the gain at gth and the carrier population at n=nth,i for loop-gain 1 even if the pump current increases I>Ith.

resulting gain-saturation gth=gth,i prevents any other mode with higher losses or less gain from lasing.

Abrupt Carrier clamping is an excellent simplification, but neglects power in all other amplified modes !

For the stationary stable state the loop gain of the LD must be 1, therefore g(n, λi) can statically not be larger than the threshold gain gth(nth, λi), resp. the carrier density n can, after the oscillation of the first oscillating mode, not become larger than nth.

Popt

IIth

LED-operation

g<gth

amplified spont. emission

Lasing Threshold

g=gth , Popt~0 n=nth

Lasing-operation

g=gth ; Popt>>0 emission from lasing mode

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6-75

1) n < nth sub-threshold-region (LED-Operation for I < Ith, low Pout because ηex<<1)

→ Spontaneous emission dominates, incoherent Noise Emission, Rstim<< Rspont because sph is small

To relate the laser current I to the carrier density n we use the carrier rate equation for LEDs, with R21,net<< Rspont: g<gth

( )

spont 21,net spontspont

spont th spontth opt i i

spont

th thth

spont spontthickness of active layer

I I nn 0 R with R Rt eV eV

I I nVn n P I / eeV eV

n eV n edwLI THRESHOLD CURRENT d

τ

τ τη η

τ

τ τ

∂= = − = − <<

∂

= < = → = =

= = ≈

Assumption: constant carrier lifetime τspont for the spontaneous emission

For minimizing the threshold current Ith , the thickness d and the width w of the actively pumped layer should be minimized. The resonator length L must be long enough to provide sufficient gain to reach threshold.

2) n=nth Lasing-region (LASER-operation I ≥ Ith , high Pout due to directional output of lasing FP-mode)

→ Stimulated Emission dominates, build-up of a monochromatic and coherent wave in the FP-resonator, Rstim~Rspont

For I>Ith we have n=nth=constant>ntr with R21,net~Rspont because sph is large:

( ) ( )( )

( ) ( )

trans

trans trans th

1 2th th th trans WG

th WG

n

For a linear approximation between n and maximum power gain g n a n n

with the n transparency carrier density g n 1

Oscillation condition :lnr lnrg g n a n n

L1 lnrna

= −

= Γ = ⎯⎯→

+Γ = Γ = Γ − = −

= −Γ

α

α 1 2 1 2trans th WG trans

spont

lnr edwL ln r ln rn I anL a L+ +⎧ ⎫⎡ ⎤ ⎡ ⎤+ → = − + Γ⎨ ⎬⎢ ⎥ ⎢ ⎥Γ⎣ ⎦ ⎣ ⎦⎩ ⎭

ατ

Popt

IIth

LED- operation

Threshold current for diode lasers

Popt

IIth

LASER- operation

optD

PI

ηΔ

=Δ

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6-76

The threshold current Ith can be reduced by: Threshold current vers. active layer thickness:

• a thin active layer d large confinement factor Γ loss of confinement Γ~dα α>1 • low WG- and mirror losses, αWG, αmirror

• long carrier lifetime τspont (suppressed spont. emission ?) • low transparency carrier density ntr (narrow gain spectrum, eg. Quantum Dots)

Graphical representation of Gain-saturation and carrier-clamping:

Bild n vers I

n(I) n(I) ≅ n(Ith) ≅ const

Remark: Increase in differential efficiency in lasing

Strong increase of emitted power in lasing operation is caused by the lasing mode which guides the generated photon efficiently through the mirrors to the outside. Below threshold photons are radiated and “lost” in all directions by spontaneous emission.

Gain-Saturation by stimulated emission

Ith is needed to maintain the carrier density at nth in the laser diode. I > Ith provides the carrier rate ~(I-Ith) for feeding the stimulated emission Pout

by Rstim >> Rspont.

Explanation: for I>Ith the photon density sph in the lasing mode increases in such a way to keep n~nth by increasing R21,net.

n nth

Γ~d Γ~1

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6-77

6.4.4.1 Photon Lifetime of a passive Resonators:

Local photon emission at the mirror (R1, R2) have been considered formally as an equivalent distributed loss αm mirror losses can be associated with a distributed loss rate Rloss (similar to g and R21,stim)). Lossy resonators can be considered as a Photon-Reservoir where photons have a finite life-time before being lost.

Resonator modes behave as “reservoirs for photons” with an average life time of the photon population sph of τph.

determined by the photon in- and outfluxes of the resonator mode: - stimulated emission and absorption - a fraction β of the “total” spontaneous emission (β=spont. emission factor) Rsp,mode=βRspont - photon losses through the mirrors and scattering at the WG-imperfections

( ) ( )( ) ( )22

0grm WG

loss

ph

ph

/ v LL

Definition R

s

ph

average loss rate ; s average potondensity

Roundtrip losses mirrors, waveguide in the passive g resonantor :

e e ;

using the photondensity - intensity relation : s phot

+− + = =

− =

=

=

α α loss

loss

RR

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )12

1

0 0

1 10

gr m G phW

ph

opt gr

ph phloss loss ph gr ph gr m WG ph gr m WG

ph

v t t /ph ph ph

ph loss pgr m WG

definition

on density I / v

s sR R / s v / s v s v t solution

t t s

s t s e s e

photon lifetime ~ s for LDs R sv

−

− + −

=

∂ ∂= → = = − + → ∂ = + ∂ ⇒

∂ ∂

= =

= = =+

∫α α τ

τ

ω

α α α α

τα α h ph/ τ

r1 r2

αWG

αm αm

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6-78

6.4.4.2 Carrier and Photon Rate-equations for particle flux calculation

As a particle model the LASER is described by a system of 2 coupled particle reservoirs connected by particle fluxes. - charge carriers n=p distributed to the energy-states in the C- and V-band of the active crystal volume Vactive - photons si in the resonator mode i with a mode volume Vmode

The particle numbers in the reservoirs are described by continuity or rate equations:

Carrier Rate-Equation: + rate of inflowing carriers (pump, absorption) - rate of outflowing carriers (stimulated- spontaneous emission) electron rate-equation

( )

( )

pump 12,abs 21,stim spont pump 21,net spont

spont spontspont

pump 21,net phspont

n R R R R R R R pt t

n p assumes charge neutrality in the active regiont t

nsimplifying we approximate R : R n

nn R R n, st

T

τ

τ

∂ ∂= + + − − = + − − =

∂ ∂∂ ∂

=∂ ∂

≅

∂= + − −

∂

activehe geometrical dimensions of the active medium are: w, d, L, A =wd

( )( ) gr

21,netspontg n v s

I nn Rt e wdL

∂= + − −

∂ τ

VmodeRpump

Vaktiv

S

L

nRstimRspont

nt

∂=

∂

αWG αm

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6-79

Photon Rate-Equation for a resonating mode i (standing wave in FP-resonator): Rspont is the spontaneous emission from Vactive into all existing modes in the laser volume Vmode; Rspont,mode=βRspont is the fraction of the spontaneous emission into the single lasing mode.

The photon rate-equation are formulated first for the total photon numbers S=siVmode instead of the density si to develop the role of the optical confinement factor Γ.

+ rate of inflowing photons (stimulated emission, spontaneous emission per mode) - rate of outflowing photons (absorption, mirror and WG-losses) phs

t∂

=∂

( )

opt ph gr

i i mode

i 21,net i active spont active loss ,tot mode

activei

mo

mode

Photon number per resonator mode i

going back to the photon density s by dividing the equation by V i

I s v

S s V

S R s ,n V R V R Vt

Vst V

→

=

=

∂= + −

∂

→

∂=

∂

ω

β

( )

)

spont i photongr ì

active21,net spont loss ,tot 21,net spont loss ,tot

de mode

active

n/ s /g n v

m d

s

o e Confinement-Factor overlap-integral between mode- and gain-crossection <

VR R R R R RV

with1 V / V =

⎛ ⎞ ⎛ ⎞+ − = Γ + Γ −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Γ =

τ τ

β β

)

) ( )loss ,tot loss , WG loss , mirror i ph

-2 -4

mod e spont laser

1

2 R R R s /

13 ~ spontaneous emission factor typ. 10 -10 for LDV

= + =

=Δ

τ

βρ ω

Vmode

Rloss

Rstim Rspont,mode

Vaktiv

S

L

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6-80

6.4.4.3 Multimode (MM) Rate Equation Systems for M uncorrelated modes

Generalizing the 2 rate equation for single mode lasers to the case of multi-mode lasers with M FP-modes interacting incoherently (with random phases) with the frequency-dependent common gain g(ωi,n) of the active layer:

( ) ( )

M

ii 1

i i21,net 21,net ,i

Mspont spont

i 21,stim,i gr i

n ps s ; Quasi neutrality of the active layert t

I n I nn R R Rate equation for the conduction band electronst e wdL e wdL

with g =R /v /s and the linear gain

=

∂ ∂= = −

∂ ∂∂

= + − − = − −∂

∑

∑η ητ τ

( ) ( ) ( ) ( )( )

( )

( )( ) ( )

( )

ii i trans i trans

M

i ii=1

Mi i

trans i gr ,ii 1 spont

i ii tra

g approximation g n, n n a f n n

n

f(ω )=spectral depence of g , we obtain with the total photon density s= s :

gI nn n n s vt e wdL n

gs n n

t n

=

∂= − = −

∂

∂∂= + − − −

∂ ∂

∂∂= Γ −

∂ ∂

∑

∑

ωω ω

ω

ωτ

ω ( )ns gr ,i i loss ,totspont

nv s R i 1,2,...M+ Γ − =βτ

by using the photon lifetime:

( )( ) ( )

( ) ( )

Mii

trans i gr ,ii 1 spont

i ii trans gr ,i i

spont ph,i

gI nn n n s vt e wdL n

g n ss n n v s i 1....Mt n

=

∂∂= + − − −

∂ ∂

∂∂= Γ − + Γ − =

∂ ∂

∑ωη

τ

ωβ

τ τ

(the nonlinearity are the n si-product terms of the stimulated emission)

System of nonlinear coupled differential equations for the carrier- and photon rate equations of a multi-mode (M) diode laser

Nonlinear carrier rate equation Nonlinear i-th photon rate equation for mode i

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6-81

The optical power Popt emitted through the mirrors of all M modes is obtained under the assumption that all M modes have random phases (non-correlated) Addition of power of each mode:

m,i 21,stim,i i grmode i mirror loss: R / s / vα =

∑∑==

==M

1iMIRRORgrimode

M

1ii,optopt vsVPP αω MM-optical power output (see chap.5)

Remark: even modes that do not reach threshold can contain a substantial amount of amplified spontaneous emission.

Schematic representation of the carrier- and photon rate equations of a single mode laser:

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6-82

Abbreviations:

Rpump= pump rate ηi = pump efficiency (carrier loss eg. due to weak carrier confinement) Rspont= Rsp=total rate of spontaneous emission Rspont,mode= Rsp’= γRspont = rate of spontaneous emission / laser mode R21,net= rate of net stimulated emission R21= R21,stim= rate of stimulated emission R12= R21,abs= rate of stimulated absorption Rnr= rate of non-radiative recombination V= active volume = Vaktiv Vp=mode volume = Vmode

Remarks:

• in the derivation we implicitly assumed by using a single carrier rate equation that the medium is homogeneously broadened all modes couple to all injected e-h-pairs, resp carrier density n

• Effects based on the wave nature of light are not represented by the rate equations because the phase of the light field is not included (phase, interference, polarization)

N=n= electron density Np=sph= photon density τp=τph= photon lifetime Γ= optical confinement factor vgr= group velocity τspont= spontaneous carrier lifetime β=spontaneous emission factor e=q= elementary electrical charge

Additional definitions:

ηi=pump efficiency (carrier losses) V=active volume Vp=mode volume Np=sph τp=τph η0=mirror losses (eg. VCSEL)

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6-83

6.4.4.4 Static solution of rate equations for P-I- and I-V-characteristics

Solving the static coupled, nonlinear rate equation gives n and sph,i ; i=1-M light power – diode current (P-I-) characteristic

we consider 1) that at and above threshold the carrier density n is clamped at nth and

2) that below threshold the LASER behaves like an LED 3) the 1 lasing mode only above threshold, defining the output power

the P-I-characteristic of both operation ranges can be calculated relatively easily (assuming the concept of carrier clamping) a) Sub-Threshold I < ITH , n< nth , LED-operation

( ) ext intout spont ,out D D DP R I I R I

eωη ηω= = =

b) Lasing-range I > ITH , n= nth , LASER-operation (single mode)

( ) ( )

( ) ( )

( ) ( ) ( ) ( )( )

thth

spont

th thph

spont

thth

th trans ph

th trans ph gr ph

t

g

tr

r

h

I n0 n ~ ne wdL

I n0 at threshold s 0e wdL

subtracting the second from the first equationI II

g n n s v a

I g0 n n s v

bove thresholdn

s I ge wdL n e wdL n nn

= + − −

= + − =

→

−− ∂= + − − → =

∂−

∂

∂∂ −∂

τ

τ

( )ans grv

Derivation of Pout(I) assuming a static sph(I):

( ) ( )( )

( )( )mode gr MIRROR MIRROR

out ph mode gr MIRROR th th

th trans gr th trans

V vP s V v I I I Ig ge wdL n n v e n n

n n

ωα ω αω α= = − = −∂ ∂− Γ −∂ ∂

LED LASER

( ) ( ) ( )MIRRORout th th

th trans

P I I ~ I Ie a n n

ωα= − −

Γ −

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6-84

in lasing operation I >Ith the photon density sph(I) and the optical power Pout(I) depend linearly on ~ (I-Ith). Remark: For the calculation of the lasing power Popt emitted through the mirrors, we have to consider that the mirror losses αmirror were formally assumed as distributed along the cavity.

As a consequence we have to formally integrate the “distributed” mirror losses over the length L of the resonator:

Per volume dVmode of the mode the „mirror“-power is lost as

out mod edP ds / dt dVω=

MIRROR gr MIRRORs s v s /t

α τ∂= − = −

∂ this gives us:

( )mode

MIRROR gr MIRROR

Vph ph

out mode mode ph gr MIRROR mode mode phMIRROR 0

out ph gr MIRROR mode

1 / v

s sd P dV dV s v dV integrating dV and assuming s f(z)

tP s v V

τ α

ωω ω α

τω α

=

∂= = = → ≠

∂

=

∫

( ) ( )( ) ( )

( )( ) ( )

( )

( )( )

thph

th trans gr

thmodeout MIRROR MIRROR th

gainth trans th trans

MIRROR th

th trans

inserted in:I I

s I ge wdL n n vn

I IV 1 1P I Ig ge V en n n nn n1 I I qed !ge n n

n

ω ωα α

ω α

−= →

∂ −∂

−= = − =

∂ ∂Γ− −∂ ∂

= −∂Γ −∂

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6-85

( ) ( ) ( )

th trans

th th trans WG MIR WG1 2

gusing the threshold gain condition to eleminate the material parameters n , n , a= :n

g 1g n n n follows:n 2Lln r r

α α α

∂∂

∂Γ = Γ − = + = +

∂

( ) ( )

i

MIRRORout i th D th th

MIRROR WG

internal quantum efficiency=photons/injected e-h-pair

P I I I I for I Ie

η

ω αη ηα α

=

⎛ ⎞= − = − ≥⎜ ⎟+⎝ ⎠

MIRRORD i

MIRROR WGeω αη η

α α⎛ ⎞

= ⎜ ⎟+⎝ ⎠

external differential efficiency ηD (wave guide and mirror losses only. No non-radiative carrier losses)

MIRRORD i

MIRROR WG

PI e

ω αη ηα α

⎛ ⎞Δ= = ⎜ ⎟Δ +⎝ ⎠

..... exact, nonlinear solution

LED LASER

P-I-Characteristic (only with mirror and wave guide scattering losses)

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6.4.5 Multimode-Operation of Diode Lasers

In a long resonator with a large number of modes, only the mode with the largest net-gain (closest to the gain-maximum) should start oscillating. If the medium is homogeneously broadened then the first oscillating mode clamps the carrier density at nth.

above threshold only the main-mode (m=0) increases the power with current (I-Ith), the side-modes (m>0) saturate their power at the threshold values.

Power of Main- and Side-Modes vs. pump current I: (simulation) ← Main-mode m=0 LASING , Pmode,0(I)~I-Ith for I>Ith Side-modes ImI>0 SPONTANEOUS LED , Pmode,I(I)~ Pmode,I(Ith) for I>Ith (fig. )

a) b) a) „Multi-Mode”-Spectrum, I~Ith, b) „Single-Mode”-Spectrum, I>>Ith weak gain saturation I~Ith strong gain saturation I>>Ith

ω ω

⎫⎪⎪⎬⎪⎪⎭

LED LASING

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6-87

lasing modeclose to lasing ( ) ( )WG m

lasing :g ω α α ωΓ − =

( ) WGg ω αΓ −

mα

lasing modeclose to lasing lasing modeclose to lasing ( ) ( )WG m

lasing :g ω α α ωΓ − =

( ) WGg ω αΓ −

mα

lasing modeclose to lasing lasing modeclose to lasing ( ) ( )WG m

lasing condition :g ω α α ωΓ − =

( ) WGg ω αΓ −

mαdifference betweenrequired and provided gain

6.4.5.1 Wavelength selections and single frequency techniques: Ideally for a homogeneous broadened gain-medium the mode i with the lowest losses αi and the highest gain g(ωi) would clamp the gain g at its threshold gain g(nth, ωi) and no other mode could reach threshold anymore.

Single frequency operation would result.

Diode are not ideally homogenous broadened and show undesired multi-frequency operation (broad spectrum ~1000 GHz ) !

Static Gain Condition: ( )loop WG m

waveguide mirror

g g n, a 1ω α= Γ − − = loopg 1<

P-I-characteristic and MM-spectrum: Gain-loss situation budget in LD with cleaved facets (constant mirror losses αm): - the WG and mirror losses αWG and αm are not frequency dependent - the gain g(ω) is only weakly dependent on ω (parabolic around ω0) the threshold gain gth(ωi) differences of different modes are small !

Solution for mode selection: Increasing selection, resp. threshold gain difference (α(ωi±1)-g(ωi)) between the modes by frequency dependent mirror reflectivity R(ω) am(ω)

WG

dominant main mode side modes

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6-88

lasing modefar from lasing

Λ= grating periodm= number of periodsr= interface reflection

Lg= grating length

2 1

2 1

n nrn n

−=

+

lasing modefar from lasing lasing modefar from lasing lasing modefar from lasing

Λ= grating periodm= number of periodsr= interface reflection

Lg= grating length

2 1

2 1

n nrn n

−=

+

• DBR- and DFB increase mode-selection by stronglyfrequency dependent, but current independent losses.

• difficult to fabricate because the grating period Λ~λ/2~200nm

• epitaxial embedding of the grating requires difficult overgrowth

( ) ( )( )

( )0

0

0 8

2

1 1

2

g,max g,max

g g g

g,max g

g

for moderate max. reflections r : mr < ; r ~ .

the approximations hold : r sin L / L

max. reflectance : r tanh mr m L /

" zero" bandwidth : from L /m

Bragg wavelength : with nn

π

δ δ

λ δ πλ

λ

<

=

= = Λ

Δ− = =

Λ− = = 1 2

1 2

n nn n

( center wavelength )+

• DBR- and DFB increase mode-selection by stronglyfrequency dependent, but current independent losses.

• difficult to fabricate because the grating period Λ~λ/2~200nm

• epitaxial embedding of the grating requires difficult overgrowth

( ) ( )( )

( )0

0

0 8

2

1 1

2

g,max g,max

g g g

g,max g

g

for moderate max. reflections r : mr < ; r ~ .

the approximations hold : r sin L / L

max. reflectance : r tanh mr m L /

" zero" bandwidth : from L /m

Bragg wavelength : with nn

π

δ δ

λ δ πλ

λ

<

=

= = Λ

Δ− = =

Λ− = = 1 2

1 2

n nn n

( center wavelength )+

Distributed Bragg-Reflector (DBR)- and Distributed Feedback (DFB) Lasers: Schematic DBR-LD structure: Narrow stop-band DBR-reflection coefficient (chap.4):

Gain-loss budget in LD with frequency dependent DBR-mirrors αm(ω): Low mirror losses (stop band) strong frequency discrimination of modes

High reflectivity stop-band → Low mirror losses

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6.4.6 Thermal properties of Diode Lasers

The inversion of a SC requires high threshold carrier densities ~nth~1018cm-3. Because the spontaneous recombination time constant τspont is in the range of 0.1-1ns, high threshold current densities of 1000-100A/cm2 are necessary.

Because the internal quantum efficiency ηi is <100% (typ.60–90%), ohmic losses and because part of the carriers recombine by phonon emission, heat is generated. Also the ohmic resistance of the contacts (a few Ω for planar FP-diode lasers) produces heat.

The optical gain g(n,ω) is reduced and broadened strongly with increasing temperature T strong increase in Ith

( ) ( ) ( ) 0/temperature sensitivity parameter;refT T T

th th ref oI T I T e T−= = (empiric approximation)

increases exponentially with T, requiring efficient heat-sinking of the laser diodes. The high threshold current density jth and excessive heating were a main obstacle for the application of early laser diodes in systems.

The double heterojunction diode and thin Quantum Well active layers (d<7nm) and better material processing in the 70-ties reduced the threshold current density jth by 4 orders of magnitudes

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Temperature dependence of the optical gain is caused by

a) a reduction of the optical gain g and

b) a broadening of the gain spectrum Δω caused by:

Gain-broadening by the temperature dependence of the Fermi-distribution F(E,T)

Carrier losses from the active medium by thermal emission over the heterojunction barriers ΔEc, ΔEv in conduction- and valence bands

Temperature dependence of the P-I-characteristic of a InGaAsP (1.3μm) Diode Laser:

Conclusions:

Lasers for long distance communication are temperature stabilized at RT by hybrid Peltier coolers for highly stable operation.

If the threshold current Ith~1mA as eg. for modern VCSELs then I>>Ith and the temperature sensitivity of Ith is less critical.

( ) ( ) ( )r 0T T / TTH TH r

0 0

r

I T I T e

with :T temperature sensitivity parameter, T parameterT reference temperature, often room temperature 300K

−=

= −

=

• Exponential increase of Ith with T

• Decrease of the external ηD efficiency with increasing T

• Reduction of the maximum optical power (thermal roll-over) with increasing T

temperature

threshold Ith

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6.4.7 Small Signal Current-Light Modulation Transfer Characteristics

The optical output of LD can be efficiently and fast modulated by analog or digital current modulation.

The rate-equation are nonlinear and can not be solved analytically (digital large signal modulation) but can be linearized for analytical solutions (resonant 2.order system) for small signal sinusoidal modulation around the OP ( optn , s , I , P ):

We determine the Modulation-Transferfunction ΔPout=power amplitude at modulation frequency ω, ΔI=current amplitude at ω sinusoidal current modulation

Solution by small signal approximation and linearization (optional):

using the nonlinear single mode rate-equations for linearization: Digital Bias Point nonlinear ns-products Analog Bias Point

BiasBiasBias

( ) ( )( )

outPM

Iω

ωω

Δ=

Δ

( )( )

j t

j topt opt opt opt

I t I I e cc. with I I

P t P P e cc. with P P

ω

ω

= + Δ + Δ <<

= + Δ + Δ <<

( )( )

( ) ( ) ( )

( )

j t

j t

j t j tmod emod eopt opt opt

ph ph

WG

n t n ne cc.

s t s se cc.

s Vs VP t P P e cc. e cc.

if for simplicity 0

ω

ω

ω ωωω

τ τ

α

= + Δ +

= + Δ +

Δ= + Δ + = + +

→

( ) ( )

( )

trans grspont

trans gr loss ,totspont

I g nn n n s vt e wdL n

g ns n n v s Rt n

τ

βτ

∂ ∂= + − − −

∂ ∂

∂ ∂= Γ − + Γ −

∂ ∂

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( ) ( )

( )

2: , ; 0

n from the DC-equations leads t

trans gr grspont

trans gr grph

AC Solution j linearized carrier and photon rate equationt

I g g nj n n n v s n v se wdL n n

g g sj s n n v s nv sn n

replacing

ω

ωτ

ωτ

⎛ ⎞∂− = − − Δ →⎜ ⎟∂⎝ ⎠

Δ ∂ ∂ ΔΔ = + − − Δ − Δ −

∂ ∂

∂ ∂ ΔΔ = Γ − Δ + Γ Δ −

∂ ∂

→

( )

( )

)

o:

.2

grph spont

grph ph

grph spont

gr

I s g nj n nv se wdL n

s g sj s nv sn

I s g nj n nv se wdL n

gj s v s n eqn

ωτ τ

ωτ τ

ωτ τ

ω

Δ Δ ∂ ΔΔ = + − − Δ −

Γ ∂

Δ ∂ ΔΔ = + Γ Δ − →

∂

Δ Δ ∂ ΔΔ = + − − Δ −

Γ ∂

∂Δ = +Γ Δ

∂

Nonlinear eq. for Operation Points OP (Rspont,mode neglected for I>>Ith)

Assume the solutions for ( ) ( )n I , s I are determined, we linearize the system at the operation point by ( )x x x t= + Δ with Δω2 0:

Separating static and time-dependent parts linearized carrier- and photon rate-equations

Observe the 900-phase shift (jω) between Δn and Δs !

( )

( ) ( )

( )1

0,:

0

0

trans grspont

trans gr gr ph transph

nonlinear equationsDC Solution

I g nn n v se wdL n

g s gn n v s n v nn n

τ

ττ

−

Δ =−

∂= + − − −

∂

∂ ∂⎛ ⎞= Γ − − → = Γ +⎜ ⎟∂ ∂⎝ ⎠

Operation far above threshold I >> Ith means that we can neglect the spontaneous emission into the laser mode compared to stimulated emission ( )trans gr spont

g n n v s n /n

β τ∂Γ − >>

∂:

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( )

( ) ( ) ( )

( )

opt mode gr MIRROR

For the calculation of M ω we eliminate Δn:

P V v sM

I Iwithout the elimination details we obtain for M :

ω ωα ωω

ω

Δ Δ= ≈

Δ Δ

( ) ( ) ( )( )mod e gr MIRROR gr

mod e gr MIRROR

2gr gr

ph spont

r gr optph ph

1 gV v v sV v s e wdL n

M f s I ,I 1 g g 1v s j v s

n n

1 g g 1Resonance frequency in resonance frequency: v s ~ s ~ P ~ ~ recorn n

ω αω α

ω ω

ω ωτ τ

ωτ τ

∂ΓΔ ∂

= = =Δ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂− + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∂ ∂=

∂ ∂( )d 40GHz, intrinsic 60-70GHz

Modulation transferfunction

( )

( ) ( ) ( )

( )

opt mode gr MIRROR

For the calculation of M ω we eliminate Δn:

P V v sM

I Iwithout the elimination details we obtain for M :

ω ωα ωω

ω

Δ Δ= ≈

Δ Δ

( ) ( ) ( )( )mod e gr MIRROR gr

mod e gr MIRROR

2gr gr

ph spont

r gr optph ph

1 gV v v sV v s e wdL n

M f s I ,I 1 g g 1v s j v s

n n

1 g g 1Resonance frequency in resonance frequency: v s ~ s ~ P ~ ~ recorn n

ω αω α

ω ω

ω ωτ τ

ωτ τ

∂ΓΔ ∂

= = =Δ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂− + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∂ ∂=

∂ ∂( )d 40GHz, intrinsic 60-70GHz

Modulation transferfunction

1r r ph

r spont

Re sonance damping : γ ω ω τω τ

⎛ ⎞= +⎜ ⎟⎜ ⎟

⎝ ⎠

( )2

211

D

MIRRORr ph

WG MIRROR r r r spont

M je

η

αω ω ωω ω τα α ω ω ω τ

⎡ ⎤⎛ ⎞⎛ ⎞= − + +⎢ ⎥⎜ ⎟⎜ ⎟+ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

( )2

211

D

MIRRORr ph


M je

η


⎡ ⎤⎛ ⎞⎛ ⎞= − + +⎢ ⎥⎜ ⎟⎜ ⎟+ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

_______________________________________________________________________________________________________________________________


( )2

211

D

MIRRORr ph


Resonance damping

M je

η


⎡ ⎤⎢ ⎥⎛ ⎞⎛ ⎞⎢ ⎥= − + +⎜ ⎟⎜ ⎟+ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

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Summary:

Because diode lasers contain 2 energy reservoirs (particle reservoir), excited carriers n, p in the bands of the SC and photons s stored in the FP-resonator → the system behaves like a resonant 2.order low pass filter with relaxation oscillations ~ωr

To increase the resonance frequency ωr and the –3dB-bandwidth ω-3dB , 1) s must be large, that means the laser has to be operated at high bias power Popt , I>>Ith 2) the photon lifetime τph must be short 3) the differential gain g / n∂ ∂ must be high

4) keep resonance damping γ critical

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6.4.8 Intensity Noise of Diode Lasers The light intensity I(t)= I+in(t) of lasers contains random fluctuations in(t), so called Intensity Noise, excited by internal rate-fluctuation processes such as the spontaneous emission and rate noise of the particle currents of carriers and photons.

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

( ) ( ) ( )n opt opt opt ,n n

2 2i n opt ,n mod e

I t I i t , P t P p t , n t n n t

with the intensity noise power density spectrum S i p / A of the intensity noiseω ω ω

= + = + = +

= =

For the characterization of the intensity-noise we define as a:

Noise measure ( ) 22opt ,rn

2opt

p niRIN

PIω

= = Normalized Intensity-Noise Density Spectrum RIN (Relative Intensity Noise)

• The frequency behaviour of the RIN(ω) is the same as the modulation-transfer function M(ω) with a resonance at the resonance frequency ωr.

• The RIN-noise at low frequencies becomes a maximum at the lasing threshold and decreases above threshold.

The spectral dependence S(ω) is very similar to M(ω).

RIN(I,ω)

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6.4.9 Large Signal Modulation of Diode Lasers In the dominating digital data transmission the laser diode is large signal modulated by rectangular current pulses. Unfortunately exact solutions are only possible numerically. Four detrimental effects are often observed:

Turn-on delay td of the light Popt(t) relative to the current pulse I(t) Relaxation oscillations ωr in the optical output Dynamic bandwidth reduction by the dynamic operation point Dynamic modulation of the optical spectrum → frequency chirp and dynamic multi-mode operation

Numerical simulation of the carrier and photon response on a current pulse:

Time Domain Transients: Current Step Response

numerical simulation of the carrier and photon response on a current pulse

(current step 0.2 Ith → 1.2 Ith) (current step 0.8 Ith → 2.0 Ith)

Turn-on delay between current step and light:

THmax

minmax0d II

IIlnt−−

=τ

Relaxation oscillation

Current step

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Turn-on Delay td

For digital modulation a high Signal Extinction-Ratio ( ) ( )opt max opt min on offP I / P I P / P= is desirable. Choosing Imin < ITH and Imax > ITH. leads to large turn-on delays td because the pump current I has to build-up the carrier density n to nth before lasing can start.

Choosing Imin>Ith minimizes td , but reduces the extinction-ratio → trade-off. Dynamic Spectrum Shift (Wavelength-Chirping) (self-study)

The dynamic simulation of the carrier density n(t), resp. g(t) following a current step Iσ(t) shows, that n(t) oscillates around the static average n with a small amplitude Δn(t), the same holds for Δg(t). The time-dependent gain- and carrier dynamic Δn(t) causes changes in the refractive index Δn(t) (with Kramer-Kronig relation), resulting in a change of optical resonator length Δn(t)L=ΔL(t)

a dynamic wavelength-shift of the gain-maximum of the active medium → λmax(t)= λmax+Δλmax(t)

because Δg(t) also changes the optical index of refraction nopt(t), leading to a change of the resonance frequency of the FP-resonator.

→ nopt(t)= nopt+ Δnopt(t) → λ(t)

Dynamic Multimode-Operation. Static single mode lasers (M=1) can degenerate to multi-mode operation (M>1) due to internal carrier dynamics

THmax

minmax0d II

IIlnt−−

=τTurn-on delay td

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All these effects lead to a complex time-variable optical spectrum λ(t) resp. to an effective widening of the modulated optical spectrum

Frequency-, resp. Wavelength Chirping The total spectral chirp Δλ can lead to massive pulse broadening in fiberoptic communication. Frequency-Chirp and current pulse: Dynamic Evolution of the temporal spectrum during a current step: (transient multi-mode spectrum, static “quasi-single-mode” spectrum)

1

2

3 4

Current step: 0.3Ith → 1.43Ith

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Dynamic Multimode-Spectrum for different modulation frequencies:

Modulation-Frequency

100% modulation of the light

Ibias=1.2 Ith

In real systems for Gb/s data rates and transmission distances of 100km and more, current modulated diode lasers are hardly used any more. Diode Lasers are used as single frequency DC-light source modulated by external modulators with low chirp.

DFB- or DBR-diode laser with built-in single mode selective filters improve the dynamic single-mode operation achieving low chirp.

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6.4.10 Fiber-LD-Coupling, Packaging

Demanding optical packages for diode lasers have to fulfill a large range of functions and can be a substantial cost factor:

• efficient mechanical and optical coupling structure (fiber alignment) between LD and fiber with tolerances from 0.1 – 1μm.

• efficient temperature stabilization, active heat-sinking by thermo-electric Peltier-cooler for stable P-I-characteristics

• Monitor-Photodiode for monitoring of the optical power, resp. operation point (OP depends critical on precise current levels)

• Optical Isolation (eg. magneto-optic Faraday-Isolators ~40-60dB) against external back-reflections into the laser leading to excess-noise and spectral instabilities

• High frequency package for signal coupling of the high frequency modulation signal (up to 40 Gb/s)

Butt-Coupling: Laser – Fiber: Monitor- Photodiode Dioden-Laser Peltier-cooler Fiber fixture

Documents

Optoelectronic 2010, K6 FOLIEN part 2 final -