Contemporary Photonics Serie2

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Lecture 2Parametric amplification and oscillation:

Basic principles

David HannaOptoelectronics Research Centre

University of Southampton

Lectures at Friedrich Schiller University, Jena

July/August 2006



utline of lecture

! "o# to calculate parametric gain via the coupled #ave e$uations

! %&pressions for small'gain and large'gain cases

! %ffect of phase'mismatch on gain, hence find signal gain'(and#idth

! )omparison of threshold of S* and +*

! )omparison of longitudinal mode (ehaviour of S* and +*

! )alculation of slope'efficiency

! Focussing considerations



Calculation of parametric gain

Assume plane #aves

Assume c# fields

eglect pump depletion

)oupled'#ave e$uations for signal and idler are then solu(le,

calculate output signal and idler fields for

given input pump, signal and idler fields



Coupled equations

Fields

Intensity

d - effective nonlinear coefficient

[ ]cct r k ir E t r E +−= ).(exp),(2/1),( ω ω 2

0 ),(2/1 ω ε r E nc I =

)exp(*

2311

kz i E E idz

dE

∆= κ

)exp(*

1322 kz i E E i

dz

dE ∆= κ

cnd j j j /ω κ =

123 k k k k −−=∆



.anley'*o#e relations

ntegrals of the coupled e$uations

n1%3412/5 n21%23412/52 7 const

n1%3412/5 n81%83412/58 7 const

n21%23412/52 9 n81%83412/58 7 const

um(er of pump photons annihilated in L medium e$uals the

num(er of signal photons created, #hich also e$uals the

num(er of idler photons created

:hese imply

n1%3412 n21%23412 n81%83412 7 const

i;e; conservation of po#er flo# in propagation

direction



Solution to coupled equations: (1)

[ ] 2/122 )2/( kL g ∆−Γ = 2

321

2 E κ κ =Γ

gLkLi g

E E i gL

g

k i gLkLi E L E sinh)2/exp(

)0(sinhcosh)2/exp()0()(

*

23111 ∆+

∆−∆= κ

#here and

gLkLi g

E E i gL

g

k i gLkLi E L E sinh)2/exp(

)0(sinhcosh)2/exp()0()(

*

13

222 ∆+

∆−∆= κ



Solution to coupled equations: (2)

f only one input %2, %804 7 04 <amplifier or S*=

Single'pass po#er gain increment4 is,

)orresponding multiplicative po#er gain, 4

( ) 2

222

2

2

2

2

2

sinh1

)0(

)()(

gL

gL L

E

L E LG Γ =−=

L LG Γ =

2

2 sinh)(

L LG LG x Γ =+= 2

22 cosh)(1)(

For e&act phase'match, g 7 > , so



Plane!ave" p#asematc#ed" parametric gain

f gain is small, ?2L4 @@ 84 , gain increment is

ote- incremental gain proportional to pump intensity

proportional to 52

3

0321

3

2

2122 2

cnnn

I d L

ε

ω ω =Γ

proportional to d2 / n

#idely $uoted as L Figure f .erit4



Plane!ave" p#asematc#ed parametric gain(multiplicative)

ote- since >

2

∝ pump po#er Bthe gain e&ponent depends on $B

unliCe *aman gain, #here e&ponent ∝ B4

)2exp(4/1cosh)( 2

2 L L LG Γ ≅Γ =

%or #ig# gain" &L '' 1

Dery high gain is possi(le #ith ultra'short pump pulses,

since gain is e&ponentially dependent on peaC pump intensity



P#ase relation et!een pump" signal" idler

2/1)(exp 123123 π φ φ φ φ φ φ −=−−⇒=−−ii

[ ]).(exp),(2/1),( φ ω ω +−= t r k ir E t r E

Suppose (oth signal and idler are input;

Assuming EC 7 0 , then

Adds, ma&imally, to gain if

?ain ma&imised if phase of nonlinear polarisation at 52

leads (y π/24 the phase of e;m; #ave at 52

ote- Fields are

L

E

E E i L

E

L E Γ

Γ +Γ = sinh

)0(

)0(cosh

)0(

)(

2

*

132

2

2 κ



P t#res#old: S* vs D* (1)

f EC 7 0 , threshold condition

assuming pump, signal idler phases G 9 G2 9 G8 7 ' π/2 at input to crystal4

2/1

2

2/1

1

2/1

21 )(1cosh

R R

R R L

++

=Γ

*epresent round'trip po#er loss (y one cavity mirrorhaving reflectance *8 idler4, *2 signal4

:hreshold H round'trip gain 7 round'trip loss

for signal only, S*, for signal and idler, +*4

*8,2



P t#res#old: S* vs D* (2)

1cosh 2

2 =Γ L R

222 1 R L −=Γ

4/)1)(1( 12

22 R R L −−=Γ

For S*, *8 7 0

S*

+*

Advantage of +* is lo# threshold

f 8' *8,2 @@ 8

S*threshold

+*threshold

7 200 for 8 9 *8 7 0;02 2I4



Parametric gain and!idt#

For plane #aves, ma& parametric gain is for fre$uencies

50 7 520 580 that achieve e&act phase'match, C 7 C2 C8

f the signal fre$uency 52 is offset (y

there is a phase'mismatch

For small gain, the signal gain is reduced to ma& for ECLK

2022 ω ω δω −=

1232 )( k k k k −−=∆ δω

EC 7 0 , 52 7 520

π δω =∆ Lk )( 2

52

?ain

52' 520

Solve for 52

, 52

'

"ence gain (and#idth 52 ' 52

'

Mand#idth reduces #ith greater L



Parametric gain and!idt#: small gain

For small gain >L @@ 84, gain'half'ma&imum is appro&imately given

(y 1EC1 7 π/L , hence independent of > therefore of intensity4;

For high gain >L NN 84, po#er gain is O e&p2>L4, hence NN>2L2

22 )2/( k g ∆−Γ =

22)2/( Γ −∆=′ k g

Po!er gain (increment) vs +,

2/122 ])/([2 Lk π +Γ =∆

2

22

)(

sinh)(

gL

gL LΓ

L g c L ′Γ

22

sin)(

sinh2>L

>L42

0 EC72>1EC17 π/L

gPL 7 π , hence

EC



Parametric gain and!idt#: large gain

dM gain reduction for ECL42 / Q>L 7 ln 2 R EC 7 2>ln2/L4

EC (and#idth high gain4

EC (and#idth lo# gain4

Q ln 2 >L4

7 0;T >L4

π

EC @@ >4

sinh2>L

0 EC72>

half ma& ],)4/)((2exp[

))2/(2exp(4/1)2exp(4/1~

2

22

Lk L

Lk gL

Γ ∆−Γ ≅

∆−Γ =

EC

>2L2

For >LNN8, ?ain is-



Pump acceptance and!idt#

( ) ( )1

1

1

313

3// −− −

=∂∂−∂∂

= g g vvk k

π

ω ω

π δω

Assumes first term in :aylor series dominates4

hat range of pump fre$uencies can pump a single signal fre$uencyV

Lo! gain case: #alf!idt#"



Signal gain and!idt# (1)

?ain peaC- phase'matched 50 7 520 580 , C0 9 C20 9 C80 7 0

For same pump, 50 , calculate

corresponding to signal 520 δ 52 idler 580 ' δ 524

( ) ...21 2

222

2

21

2

221 +

∂∂+∂∂−

∂∂−∂∂=∆ δω

ω ω δω

ω ω k k k k k

:aylor series-

)()( 1102201230 k k k k k k k k −+−=−−=∆

Solve for 52



Signal gain and!idt# (2)

)( 1

2

1

1

2 −−

−

=∴ g g vv L

π δω

For small gain, ECL/2 7 π/2 defines the half'ma&; gain condition

provided 8st; :aylor series term

dominates

At degeneracy, use second :aylor term note δ5 ∝ EC∝L' 4

For accuracy, use Sellmeier e$un; rather than :aylor series

For high gain find EC (and#idth via ( )[ ] 1sinh/1 222 =Γ + gL g R

"alf'#idth



S* tuning range !it#in gain profile

2/122 ))/((2 Lk π +Γ =∆

Wero gain incremental4 for

f >L NN 8 then 1EC1, hence

tuning range, ∝ <I=

[ ]{ }[ ]

1)2/(

)2/(sinh1

22

2/12222

=

∆−Γ

∆−Γ Γ +

k

Lk R

A more e&act treatment calculates the EC that maCes

f >L @@ 8 then 1EC1, and hence

tuning range, independent of >P

sinh2>L

0 EC



Consequences of p#ase relation et!een pump" signal" idler-

f more than one #ave is fed (acC in an B,

then phases may (e over constrained

+ou(le' or multiple pass amplifiers can also suffer similar pro(lems

:he fi&ed value of relative phase X0'X2'X8, can (e e&ploited to achieve self'sta(ilisation of carrier envelope phase )%B4

n a S*, relative phase of pump and signal is not determined, hence signal

selects a cavity resonance fre$uency;



Stailit.: comparison of S* and D*

S*- o idler input; ?ain does not depend on pump/signal relative phase;

Signal fre$uency free to choose a cavity resonanceR

dler free to taCe up appropriate fre$uency and phase;

Signal fre$uency sta(ility depends on cavity sta(ility

and pump fre$uency sta(ility;

+*- )avity resonance for (oth signal idler generally not achievedR

verconstrained;

Signal/idler pair seeCs compromise (et#een cavity resonance and

phase'mismatchR

large fluctuation of fre$uency result;



P: Spectral e#aviour of c! S*

o analogue of spatial hole'(urning in a laser

scillation only on the signal cavity mode closest to gain ma&imum

Use of a single'fre$uency pump typically results in single fre$uencyoperation signal idler4;

.ulti fre$uency pump can give multiple gain ma&ima, possi(ly multiplesignal fre$uencies, certainly multiple idler fre$uencies

Signal fre$uency #ill mode'hop if B cavity length varies, or if pump

fre$uency changes

Additional signal modes possi(le #hen pumping far a(ove

threshold 9 due to (acC conversion of the phase'matched mode,

allo#ing phase'mismatched modes to oscillate



C/ singl.resonant Ps in PPL0

%irst c! S*: Mosen(erg et al; ;L;, 28, Y8 8ZZ64

8# d[A? pumped T0mm \L, # threshold, N8;2# ] ;µm

C! singlefrequenc.: van "erpen et al; ;L;, 2^, 2QZY 2004

Single'fre$uency idler, ;Y H Q;Y µm, 8# H 0;8#

Direct diodepumped: _lein et al; ;L;, 2Q, 88Q2 8ZZZ4

Z2Tnm .BA diode, 8;T# thresh;, 0;T# ] 2;8µm 2;T# pump4

%irelaserpumped: ?ross et al; ;L;, 2Y, Q8^ 200248;Z# idler ] ;2µm for ^;# pump



Calculation of conversion efficienc. (1)

Bro(lem- pump is depleted, hence need all three coupled

e$uations; :hreshold calculation avoids this4;

Solve appro&, assuming constant signal field

i;e; solve t#o coupled e$uations, for pump and idler;

?enerated idler photons 7 generated signal photons

ncrease gain4 in signal photons 7 loss of signal photons

"ence calculate pump depletion, and hence signal/idler o/p



Calculation of conversion efficienc. (2)

[ ])(sincos)0(

)(2/112

2

3

2

3 −−

= N c

E

L E

2

,3

2

3 )0(/)0(threshold

E E N =

For S*, #ith EC 7 0 and plane #ave, find for pump

hen 7 π/242

2;T , find %L4 7 0 i;e; 800I pump depletion

nitial slope efficiency at threshold, defined as

dsignal photons generated4/dpump photons annihilated4,

is i;e; 00I `4



.pical P conversion efficiencies

?enerally high conversion efficiency N T0I4

is o(served at 2' & threshold

nitial slope efficiency N 800I is typical

Bumping a(ove 'Q & threshold typically results in reduced efficiency (acC'

conversion of signal/idler to pump4

UnliCe lasers, Bs do not have competing path#ays for

loss of pump energy



Analytical treatment of B

#ith pump depletion

! Armstrong et al;, Bhys *ev ,82Y, 8Z8^, 8Z624

! Mey and :ang, %%% J uantum %lectronics, % ^, 68,

8ZY24

! *osencher and Fa(re, JSA M, 8Z, 880Y, 20024



Y RY s )1( −=′

( )

( ) p pth

s s

P

P Y

ω

ω

/

/= pth p P P X /=

( )[ ] s s s

R X Y R X Ri sn /11//1cosh 12

−=−−

Y

Y X

′

′=

2

sin

( )16 −= X Y

nput \4, output [4 relation for phase matched S*B

f 8'*s@@8 then-

f, also, \'8@@8,

then-

%&actR given \, *s, find [

*osencher and Fa(re JSA M,8Z, 880Y, 2002 4



ormalised signal output versus

normalised pump input

ps is normalised pump threshold intensity4

*osencher Fa(re, JSA M, 8Z, 880Y, 2002



B #ith focussed ?aussian pump (eam;

! Seminal paper:

bBarametric interaction of focussed ?aussian light (eams

Moyd and _leinman, J; Appl; Bhys; Z, TZY, 8Z6^4

! 3tension to nondegenerate P- *elates treatments for plane!ave"collimated 4aussian and focussed 4aussian:

bFocussing dependence of the efficiency of a singly resonant B

?uha, Appl; Bhys; M, 66, 66, 8ZZ^4



ptimum 4aussian Beam %ocussing to5a3imise parametric gain6pump po!er

Confocal parameter 72π!8

2n6

4ain is ma3imised(degenerate P"no doulerefraction)for L6 7 2-9

Some!#at smaller L6 can e more convenient (11-)"!it# onl. small gain reduction ut a (usefull.) significantreduction of required pump intensit.-

L

n

#0

#02#02

(

Boyd&Kleinman, J. Appl. Phys. 39, 3597, (1968)



ffect of tig#t focus on ,L value for optimum gain

∆C C'C2'C8 is phase'mismatch

for colinear #aves;

Focussed (eam

introduces non'colinearity;

C2 C8

C

C2 C8

C

)losure of C vector triangle,

to ma&imise parametric gain,

re$uires C2C8NC, negative ∆C

ig#ter focus" or #ig#erorder pumpmode(greater noncolinearit.) needs more negative ,



:%.00 to :%.08 mode change via tuning over the

parametric gain (and

"anna et al, J; Bhys; +, Q, 2QQ0, 20084



Summary- Attractions of Bs

! ;er. !ide continuous tuning from a single device" via tuningt#e p#asematc# condition

! Hig# efficienc.

! 0o #eat input to t#e nonlinear medium

! 0o analogue of spatial#oleurning as in a laser" #encesimplified singlefrequenc. operation

! ;er. #ig# gain capailit.

! ;er. large and!idt# capailit.



+emands posed (y Bs

! Signal fre$uency mode'hops caused (y B cavity length change,

as in a laser4, A+ (y pump fre$uency shifts

! Single'fre$uency idler output re$uires single'fre$uency pump

! "igh pump (rightness is re$uired, i;e; longitudinal laser'pumping4R

no analogue of incoherent side'pumping of lasers

! ?ain only #hen the pump is present

! Analytical description of B more comple& than for a laser

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Contemporary Photonics Serie2