P O L I T D i p a r t i Motivation h E 2 E - 1 - - PowerPoint PPT Presentation

Nonlinear optics in the short pulse regime: basics and practice

• M. Marangoni

Physics Department, Politecnico di Milano (Italy) Institute of Photonics and Nanotechnology of CNR (Italy) e-mail: marco.marangoni@.polimi.it

P O L I T D i p a r t i

Motivation

• MOLECULES

ATOMS

1 2 E

• E

h

• GOLDEN

RULE

Optical frequency comb synthesizers

Ti: Sapphire = 0.8 m SC = 0.4-1.2 m frep = 0.07-10 GHz P < 3 W Er: fibre = 1.55 m SC = 0.8-2.2 m frep = 0.1-0.25 GHz P < 1 W Yb: fibre = 1.05 m SC = 0.6-1.6 m frep = 0.1-1 GHz P < 80 W

How to change spectral range ?

1 2

SECOND ORDER NONLINEAR OPTICS !!

(2)

Optical rectification

21 22 1+ 2 1- 2

cc t i A t i A t E

• )

exp( ) exp( ) (

2 2 1 1

• ]

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

2 1 * 2 1 2 1 2 1 2 2 2 1 2 1 ) 2 ( * 2 2 * 1 1 ) 2 ( 2 ) 2 ( ) 2 (

cc t i A A t i A A t i A t i A A A A A t E t P

• Sum frequency generation (SFG)

Difference frequency generation (DFG) Second harmonic generation (SHG)

OUTLINE

Equations governing a cw second order parametric process The problem of phase matching The equations of linear pulse propagation Parametric processes in the femtosecond pulse regime Examples: analytical and numerical discussion

The photons picture

• 2

(2)

• 2

SHG 1 2 3= 1+ 2 (2) SFG 1 2 3 3 1 2= 3- 1 (2) DFG 3 1 2

1 + 2 = 3 k1 + k2 = k3

Optical parametric amplification (OPA) & optical parametric generation (OPG): what are they ?

• They are the same process as DFG, but differ in the initial conditions

3 1 2= 3- 1 (2) 3 1 2

• In DFG, 3 and 1 have comparable energies and you look for an intense 2
• In OPG, 1 photons come from vacuum noise and you are looking for

extreme parametric gains (10 nJ > 1011 photons !!)

pump signal idler

• In OPA, 1 has an energy 100-10000 times lower than 3 and you look for a

strong amplification of 1 (1 acts as a seed)

• You may enclose your crystal in an optical cavity
• (2)

HR@1 T@1

1

3

OPTICAL PARAMETRIC OSCILLATOR

Femtosecond OPOs vs. OPAs:

Femtosecond OPOs

are pumped by simple laser oscillators provide high repetition rates (100 MHz) have low output energy (nJ level) require matching of the OPO cavity length to pump laser large yet not huge oscillation bandwidth

Femtosecond OPAs

require pumping by amplified laser systems provide low repetition rates (1-100 kHz) have high output energy (J-mJ level) are easy to operate (no length stabilization) ultrabroad bandwidth, up to the few-cycles regime

The wave equations for second order parametric processes

The wave equation for nonlinear optical media

charges and currents, we get the wave equation

The polarization of the medium is made of a linear and a nonlinear

contribution

P = PL + PNL

For a continuous wave, the linear polarization is PL = 0 (r -1 ) E Making the scalar approximation and considering a plane wave, the

propagation equation becomes

2 2 2 2 2 2 2

1 1

• t

c t c

• P

E E

• 2

2 2 2 2 2 2

1 1

• t

P c t E c E

NL

The slowly varying envelope approximation

Starting from the scalar propagation equation

we look for a solution with

By substitution, we get the equation Assuming

(slowly varying envelope approximation, SVEA) we get the equation

2 2 2 2 2 2

1

• t

P t E c E

NL

• t

kz i z A t z E

• exp

) ( ) , (

• t

z k i z P t z P

P NL

• exp

) ( ) , (

• z

k k i P A c A k dz dA ik dz A d

p

• exp

2

2 2 2 2 2 2

• dz

dA ik dz A d 2

2 2

• z

k k i P k i dz dA

p

• exp

2

2

The nonlinear polarization in second-order parametric interactions

Consider the superposition of three waves at frequencies 1, 2 and 3

with 1+ 2 = 3

By second order nonlinear effect, the following polarizations are generated

at the three frequencies where deff is an effective second order nonlinear coefficient

• t

z k i z A t z k i z A t z k i z A t z E

3 3 3 2 2 2 1 1 1

exp ) ( exp ) ( exp ) ( ) , (

• t

z k k i e xp ) z ( A ) z ( A d ) t , z ( P

* e ff NL 1 2 3 3 2 1

2

• t

z k k i z A z A d t z P

eff NL 2 1 3 3 * 1 2

exp ) ( ) ( 2 ) , (

• t

z k k i z A z A d t z P

eff NL 3 2 1 2 1 3

exp ) ( ) ( 2 ) , (

Three-frequency interaction in a second order nonlinear medium

Consider three waves at 3 (pump) , 1 (signal) and 2 (idler) , with

1+ 2 = 3 . We obtain the following equations

where k = k3 - k2 - k1 is the wave vector mismatch between the three waves

• kz

i A A c n d i z A

eff

• exp

3 * 2 1 1 1

• kz

i A A c n d i z A

eff

• exp

3 * 1 2 2 2

• kz

i A A c n d i z A

eff

• exp

2 1 3 3 3

• Setting k = 0 is crucial to get highly efficient

energy transfer between the interacting waves

OPA/DFG solution for small pump depletion

By neglecting pump depletion (A3 = cost.) and assuming an input beam at

the signal frequency 1 and no input at the idler frequency 2 (A2(0) = 0) the coupled differential equations admit the solution: with g and given by: the latter representing a figure of merit for the parametric gain. The presence of a phase-mismatch clearly affects such gain.

3 3 3 2 1 2 1

2 I c n n n deff

• gL

sinh g I L I

2 2 2 1 1

1

• 2

2

2

• k

g

• gL

sinh g I L I

2 2 2 1 2 1 2

Parametric gain

In the high gain approximation (L>>1) and under phase-matching (k = 0):

• ne has:
• L

I L I

• 2

exp 4 ) ( ) (

1 1

• L

I L I

• 2

exp 4 ) ( ) (

1 2 1 2

• L

I c n n n d exp L exp I L I G

eff 3 3 3 2 1 2 1 1 1

2 2 4 1 2 4 1

For high gain we need high pump intensity (ultrashort pulses are good!), large nonlinear coefficient deff and high signal and idler frequencies

This allows us to define a parametric gain:

The gain is exponential since the presence of a seed photon at the signal wavelength stimulates the generation of an additional signal photon and of a photon at the idler wavelength. Due to the symmetry of signal and idler, the amplification of an idler photon stimulates in turn the generation of a signal

• photon. Therefore, the generation of the signal field reinforces the

generation of the idler field and viceversa, giving rise to a positive feedback

Parametric gain: examples with BBO

20 40 60 80 100 10 10

1

10

2

10

3

10

4

10

5

10

6

10

7

BBO

p = 0.8 m s = 1.2 m

5 mm 4 mm 2 mm L = 1 mm 3 mm Parametric Gain Pump Intensity (GW/cm

2)

Red-pumped BBO crystal

20 40 60 80 10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

BBO

p = 0.4 m s = 0.6 m

4 mm 2 mm L = 1 mm 3 mm Parametric Gain Pump Intensity (GW/cm

2)

Blue-pumped BBO crystal: higher gain because

2 1

• G. Cerullo and S. De Silvestri,
• Rev. Sci. Instrum. 74, 1 (2003).

Are those gains achievable with frequency combs ?

• L

exp G

• 2

4 1

p i s p eff i s

I c n n n d

3

2

• V

/ pm d ; . n : BO B m . m . ; m .

eff i , s , p i s p i s p

2 6 1 4 2 1 1 1 2 1 8

• 2

2

15700 2 1 10 m w A 60fs nJ; U ; A U I

eff eff p

• m

w mm L

• 100

2

It complies with bandwidth and temporal walk-off issues (see next slides) It complies with spatial walk-off issues

HIGH Ip = 1 GW/cm2 G = 0.89 !!!

Energy conservation in parametric interaction

By manipulation of the previous equations, it is easy to show that

i.e. the sum of the energies of the three waves is conserved (assuming a lossless medium)

In addition, the following relationship (Manley-Rowe) can be proven

stating photon conservation: one photon at 3 is annihilated and two photons at 1 and 2 are simultaneously created

3 2 1

• dz

dI dz dI dz dI dz dI dz dI dz dI

3 3 2 2 1 1

1 1 1

The problem of phase matching

SHG process

Let us consider for simplicity second harmonic generation (SHG)

(1= 2 = , 3 = 2, A1 = A2 = A )

Neglecting pump depletion ( A cost ) After a length L of nonlinear medium

• z

k i A c n d i dz dA

eff

• exp

2

2 2 2

• 2

4 2

2 2 2 2 2 2

kL sin k I kL c sin L I ) L ( I

2 2 2 2 eff

d ) L ( I I ) L ( I

• k

kL sin ) L ( I k L ) L ( I

• 2

2 2 2 2

• t

z k i e xp E

• 2

2 2

Driving wave Generated wave

NL

P

• 2

E

• t

z k i e xp A E P

NL

• 2

2

2 2

Phase mismatch in more detail

k Lc

• 2
• 2

I z

• k

k L k k E P

c NL

2 2

2 2

Phase shift at Lc/2 How to get phase matching ?

• n

n n c n c k k k

2 2 2

2 2 2

?

Propagation in birefringent media

Optical axis

E E//

In the simpler case of uniaxial crystals, propagation may be described recurring to a pair of refractive indices, ne and no (extraordinary and ordinary index, respectively, each one with its own dispersion), and to an index-ellipsoid model:

1

2 2 2 2 2 2

• e
• n

Z n Y n X

• n

E// E

Each propagation direction, which is given by the wave- vector k, defines in the plane perpendicular to k an ellipse whose axes correspond to two polarization eigenstates:

E

• 2

2 2 2 2 2 2

cos

e

• e
• e

n sin n n n n

E//

Ordinary wave Extraordinary wave

Birifringence phase matching

• e

n , n 2

Negative uniaxial crystals: ne < no

• 2
• e

n , n

Positive uniaxial crystals: ne > no

Polar diagram showing the refractive index dependence as a function of the angle between k and the optical axis, at the two frequencies NOTE: the refractive indexes ne and no at

each frequency are obtained by Sellmeier equations

Birifringence phase-matching involves

coupling between orthogonally polarized fields - non diagonal terms of the second-

• rder nonlinear-susceptibility 2 tensor

The spatial walk-off probelm

• In birefringent crystals the pointing vector of the

extraordinary wave Se = E x H, which gives the energy propagation direction, suffers from an angular offset from the k vector. This is referred to as the walk-off angle wo.

Se So

The Pointying vector of the extraordinary wave Se my be shown to be perpendicular to the extraordinary normal index surface at its crossing point with k. This does not happen for the ordinary wave, with So // k.

• Length limitation approximately given by:

wo

• -wave

So 2k

Phase-front

D

k

Se wo

wo

Dtan L

• It seriously limits the interaction length L for a

given input field diameter D:

Birifringence phase matching: examples

BBO

• negative uniaxial crystal (ne<no)
• high-birifringence:

no = 1.672 @ 633 nm FF ordinary ne = 1.549 @ 633 nm SH extraord dNL ~2.3 pm/V rather LOW

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 50 100 FF Wavelength[micron] Phase-matching angle [°] 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

• 5

5 FF Wavelength[micron] Walk-off angle [°] 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 40 60 80 100 FF Wavelength[micron] Phase-matching angle [°] 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

• 4
• 2

2 FF Wavelength[micron] Walk-off angle [°]

LiNbO3

• negative uniaxial crystal (ne<no)
• small-birifringence:

no = 2.283 @ 633 nm FF ordinary ne = 2.203 @ 633 nm SH extraord dNL ~4 pm/V LOW-MEDIUM

Wide phase-matching bandwidth Quite high spatial walk-off

Phase matching bandwidth: calculation

kL c sin ) L ( I

• 2

2 2

• 2

2 2

• n

n k k k

• Let us assume phase-matching satisfied at a given

fundamental frequency (FF) 0: and let us determine the FWHM spectral width of the I2 curve. This implies evaluating k for a given frequency shift from the phase-matching frequency , while taking into account that a frequency shift at the fundamental frequency is doubled at the second harmonic:

• 2

2

2

d dk d dk k

• The FWHM bandwidth at the second harmonic, SH = 2FF, becomes (see figure):
• L

. k k

SH

783 2 2 2

• L

k k ,

SH SH

2 886 2

• I2

L .783 2

Phase matching bandwidth & dispersion

• Recalling that:

c d d ; d d d dn d dn ; n c k

• 2

2

• 0 2

2 1 886

• L

n n c .

SH

• ne may easily refer the SH bandwidth to the crystal dispersion at FF & SH:

SHG in the visible range L= 2.5 cm BBO Birifringence phase-matching PPSLT Quasi-phase- matching

EFF EFF ESH ESH

FF SH

• we may describe the SHG process between these two pulses as follows:

Phase matching bandwidth: an insight

• Anticipating a result of the short-pulse regime, i.e. the fact that FF and SH pulses
• 1
• k

v FF

, g

• 2

1

• k

v SH

, g

SH

GROUP DELAY MISMATCH

GDM

SH , g FF , g SH

• 2

1 1 1 1 1

• k

k L v L v L GDM

FF , g FF , g SH , g FF , g SH SH

• According to Fourier theory we could figure out that:

• t

z k i e xp E

• 2

2 2

• t

z k i e xp A P

NL

• 2

2

2 2

Quasi-phase matching (QPM)

k Lc

• 2
• 2

I

z 2 2 2 2

c

mL

• It occurs in special crystals that exhibit a

periodic change of the sign of 2, with a period:

• The quasi-phase-matching

condition is thus:

• 2

2 2

• k

k

• This period allows a periodic re-

phasing of the driving field (PNL) with the generated SH field, resulting in a quadratic dependence of I2 with L with an effective nonlinear 2:

2 2

2

• m

eff ,

QPM: pros and cons

CONS PROS

• Just need to change the poling period to adjust phase matching (the grating

provides the momentum you need to get phase matching

• You may phase match fields with parallel polarization direction and exploit

extremely high nonlinear coefficients

• Absence of any spatial walk-off because interacting fields may be set parallel to the

crystal optical axis.

• Few crystals lend themselves to QPM since you need ferroelectric crystals (e.g.

LiNbO3, KTP, LiTaO3) or semiconductors (GaAs)

• The fabrication procedure is rather complex for ferroelectrics periodic poling

needed and very complex for semiconductors orientation patterning

• Pretty hard to get phase-matching at short wavelengths due to the technological

barrier of m-level poling periods

• Optical damage at high fluence, especially for LiNbO3.

k 2k K = 2/

QPM: example

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 20 40 FF wavelength [micron] Poling period [micron] °]

PPLN: periodically-poled lithium niobate

FF extraordinary SH extraordinary dNL,eff ~15-20 pm/V HIGH

EFF ESH Optical axis

Absence of spatial walk-off allows for confocal focusing: interaction length only limited by diffraction !

n w L

• 2

2

Efficient nonlinear processes with frequency combs & without resonant cavities need for QPM nonlinear crystals !!

L=2mm

PPLN w0,min =15 m BBO w0,min =120 m wo = 3.5

Phase matching in a parametric interaction

k1 + k2 = k3

• r

1n(1) + 2n(2) = 3n(3)

In a medium with normal dispersion (dn/d > 0)

n(1) < n(2) < n(3)

if 1< 2 < 3

• 3

2 2 1 1 3

• n

n n

• 3

1 2 1 2 3

• n

n n n

• Types of possible birefringence phase matching:

negative uniaxial (ne < no) positive uniaxial (ne > no)

TYPE I n3

e 3 = n1

• 1 + n2
• 2 (o+oe)

n3

• 3 = n1

e 1 + n2 e 2 (e+eo)

TYPE II n3

e 3 = n1 e 1 + n2

• 2 (e+oe) n3
• 3 = n1

e 1 + n2

• 2 (e+oe)

n3

e 3 = n1

• 1 + n2

e 2 (o+ee) n3

• 3 = n1
• 1 + n2

e 2 (o+ee)

Example: type I phase matching

The phase matching condition is

ne3(m) 3 = no1 1 + no2 2

• 3

2 2 1 1 3

• m

e

n n n

• 2

2 2 2 2 2 2

cos

e

• e
• e

n sin n n n n

• 2

3 2 3 2 3 2 3 3 3

) (

e

• m

e

• m

e e m

n n n n n n sin

• QPM in a periodically-poled crystal

Birifringence phase-matching in a negative uniaxial crystal

3 2 1

2 k k k

• 3

3 2 2 1 1

2 n c n c n c

• each wavelength being
• btained by Sellmeier

equations

In a uniaxial crystal, the extraordinary index for propagation along is

giving which gives

Phase matching curves of a near-IR OPA

20 25 30 35 40 1,00 1,25 1,50 1,75 2,00 2,25 2,50

BBO OPA

p = 0.8 m

Type I Type II (os+ei ep) Type II (es+oi ep)

Wavelength (m) Phase matching angle (degrees)

Phase matching curves of a visible OPA

20 25 30 35 40 45 50 0,5 1,0 1,5 2,0 2,5

BBO OPA

p = 0.4 m

Type I Type II (os+ei ep) Type II (es+oi ep)

Wavelength (m) Phase matching angle (degrees)

The equations of linear pulse propagation

2 2 2 2 2 2 2

1 t P t E c z E

• z

k t i e xp t , z A t , z E

• t

z P t z P t z P

NL L

, , ,

• z

k t i t z p t z P

L L

exp , ,

• the polarization on the r.h.s. acts as a driving term.

The electric field is a plane wave The polarization can be decomposed in linear and nonlinear parts: we consider only the linear component:

The polarization as a driving term

By introducing the Fourier transform we get: Recalling the derivative rule for the Fourier transform:

Switching to the Fourier domain

• dt

t i t z E t z E z E

• exp

, , , ~

• z

ik z A z E

0 exp

, ~ , ~

• z

ik z p z P

L L 0 exp

, ~ , ~

• F

i dt t F d

n n n

~

• we obtain:

L

P E c z E ~ ~ ~

2 2 2 2 2

We express the second derivative as: We assume: The Slowly Varying Envelope Approximation (SVEA) neglects variations of the envelope over propagation of the order of wavelength.

The slowly varying envelope approximation

With this assumption we obtain:

• z

ik A k z A ik z A z E

2 2 2 2 2

exp ~ ~ 2 ~ ~

• z

A k z A

• ~

~

2 2 L

p A c A k z A ik ~ ~ ~ ~ 2

2 2 2 2

For a monochromatic wave: recalling that:

The frequency-dependent polarization

which simplifies to:

• E

P

L ) 1 (

~

• )

1 (

1

• L

n

We obtain:

• A

n c A c A k z A ik

L

~ 1 ~ ~ ~ 2

2 2 2 2 2 2

• A

k k z A ik ~ ~ 2

2 2

Propagation in a dispersive medium (I)

Starting from the propagation equation:

• A

k k z A ik ~ ~ 2

2 2

• We expand k() in a Taylor series around the carrier frequency 0 :
• ...

6 1 2 1

3 3 3 2 2 2

• d

k d d k d d dk k k An expansion up to the third order (or to the second order for moderate pulse bandwidths) is sufficient. By approximating:

• 2

2

2 k k k k k k k k k

• we obtain:
• A

k A k A k z A i ~ ' ' ' 6 1 ~ ' ' 2 1 ~ ' ~

3 2

Propagation in a dispersive medium (II)

where vg0 is the group velocity of the carrier frequency

• A

k A k A k z A i ~ ' ' ' 6 1 ~ ' ' 2 1 ~ ' ~

3 2

• 1

'

g

v d dk k

• GVD

d k d k

• 2

2

' '

• is known as Group Velocity Dispersion

(GVD)

Propagation in a dispersive medium (III)

We now Fourier transform back to the time domain. Recalling the derivative rule: we obtain:

• n

n n n

dt t F d i F ) ( ~

1

• '

' ' 6 1 ' ' 2 1 ,

3 3 2 2

• t

A k t A k i t A v z t z A

g

Which, neglecting third order dispersion ( ) becomes:

' ' ' k

• '

' 2 1 ,

2 2

• t

A k i t A v z t z A

g

The parabolic equation captures the main physics of linear propagation of ultrashort pulses in dispersive media.

In the absence of dispersion

g

v z t t ; z z

• 1

1

• t

A v t A v z A

g g

The original equation takes the form:

t A t z z A t t t A t A t A v z A z t t A z z z A z A

g

• 1

By transformation of derivatives in the new reference frame :

• z

t , z A

The pulse envelope propagates without distortion at a speed vg0 taking a time g0 to cross the crystal

g g

v L

• Let us set it in a new reference-frame

moving at vg0, with space/time variables:

• 1
• t

t , z A v z t , z A

g

• ne gets:

In the presence of dispersion

• GVD

L k L k L v L GDD

g g

• The pulse gets more and more broadened while propagating, with a pulse

broadening per unit bandwidth given by the GDD (group-delay-dispersion) parameter (expressed in fs2) : If the dispersion-induced pulse broadening is far in excess

• f the input pulse duration, at the crystal output one has:

B GDD

• ut
• where B is the angular-frequency bandwidth B

The equations of nonlinear pulse propagation

Propagation in a nonlinear medium (I)

We start from the equation: where:

2 2 2 2 2 2 2 2 2

1 t P t P t E c z E

NL L

• z

k t i t z p t z P

p NL NL

• exp

, ,

• We emphasize that the wavenumber kp of the nonlinear polarization

at 0 is different from that of the electric field k0. We express:

• z

k t i p t p i t p t P

p NL NL NL NL

• 2

2 2 2 2

exp 2

• NL

NL NL

p t p t p

2 2 2

,

• assuming that the envelope pNL varies slowly over the timescale of

an optical cycle:

Propagation in a nonlinear medium (II)

From the equation: By the same procedure applied to the linear propagation equation, we obtain:

• z

k t i p t P t E c z E

p NL L

• 2

2 2 2 2 2 2 2

exp 1

• which can be rewritten as:
• kz

i p t A k k t A v ik z A ik

NL g

• exp

' ' 2 2

2 2 2

• kz

i p n c i t A k i t A v z A

NL g

• exp

2 ' ' 2 1

2 2

• where k = kp-k0 is the -between the

nonlinear polarization and the field

The nonlinear polarization in second-order parametric interaction (I)

Consider the superposition of three waves at frequencies 1, 2 and 3 with 1+ 2 = 3 impinging on a medium with a second order nonlinear response: The nonlinear polarization has components at several frequencies, such as 21, 22 etc. We assume that the phase-matching condition selects only the interaction between the three fields at 1, 2 and 3 to be efficient.

• .

. exp , exp , exp , 2 1 ,

3 3 3 2 2 2 1 1 1

c c z k t i t z A z k t i t z A z k t i t z A t z E

• t

z E t z P

NL

, ,

2 2

The nonlinear polarization in second-order parametric interaction (II)

• .

. exp 2 ,

2 3 2 3 3 * 2 ) 2 ( 1

c c z k k t i A A t z PNL

• .

. exp 2 ,

1 3 1 3 3 * 1 ) 2 ( 2

c c z k k t i A A t z P NL

• .

. exp 2 ,

2 1 2 1 2 1 ) 2 ( 3

c c z k k t i A A t z P NL

• We derive the following terms:

Which we plug into the nonlinear propagation equations:

• kz

i p n c i t A k i t A v z A

NL g

• exp

2 ' ' 2 1

2 2

The nonlinear coupled propagation equations (I)

thus deriving the three coupled equations:

• z

k k k i A A d n c i t A k i t A v z A

eff g 1 2 3 3 * 2 1 1 2 1 2 1 1 1 1

exp 2 ' ' 2 1

• z

k k k i A A d n c i t A k i t A v z A

eff g 2 1 3 3 * 1 2 2 2 2 2 2 2 2 2

exp 2 ' ' 2 1

• z

k k k i A A d n c i t A k i t A v z A

eff g 3 2 1 2 1 3 3 2 3 2 3 3 3 3

exp 2 ' ' 2 1

• 2

1 3

k k k k

• 2

) 2 (

• eff

d

These are coupled nonlinear partial differential equations which are in general not amenable to an analytic solution and must be treated numerically. with

The nonlinear coupled propagation equations (II)

As a first simplification we neglect the GVD terms. This is justified by considering that the three interacting pulses are propagating at very different group velocities vgi. The effects of this group velocity mismatch are more relevant than those of GVD between the different frequency components of a single pulse.

• kz

i A A i t A v z A

g

• exp

1

3 * 2 1 1 1 1

• kz

i A A i t A v z A

g

• exp

1

3 * 1 2 2 2 2

• kz

i A A i t A v z A

g

• exp

1

2 1 3 3 3 3

• where the nonlinear coupling constants are defined as:

i eff i i

cn d 2

The nonlinear coupled propagation equations (III)

By moving to a frame of reference translating with the group velocity of the pump pulse: . is the Group Velocity Mismatch (GVM) between signal/idler and pump waves, typically expressed in ps/mm. It gives the group delay accumulated by the two pulses per unit length.

• kz

i A A i t A z A

• exp

3 * 2 1 1 13 1

• kz

i A A i t A z A

• exp

3 * 1 2 2 23 2

• kz

i A A i z A

• exp

2 1 3 3

• 3

'

g

v z t t

• 2

, 1 1 1

3 3

• i

v v

g gi i

• where

Phase matching bandwidth in OPA/DFG

• gi

gs i s

v v k k k 1 1

• si

/ gi gs / /

L v v L ln

• 1

1 1 1 2 2

2 1 2 1 2 1

Introducing k in the expression for the gain G and looking for a solution at 50% of the maximum gain, one gets a FWHM bandwidth:

2 2

2

• k

g

It may be estimated from the results obtained in the cw regime under the high gain approximation:

• gL

exp G 2 4 1

• For a given fixed pump frequency p, if the signal frequency s increases to s+,

by energy conservation the idler frequency decreases to i-. The wave vector may thus be written as:

i s p

k k k k

• High gain bandwidth

demands for group- velocity matching between signal and idler

with k = 0 for a given (p s i) set

Few general rules for ultrashort-pulse interactions

PUMP (p, 3) Ap(0) 0 SIGNAL (s, 1) As(0) 0 IDLER (i, 2) Ai(0) = 0

OPA/DFG REGIME

Input pump duration > input signal

duration

Interaction length limited by temporal

walk-off

Length of the crystal primarily chosen as

a function of ps

Signal delayed from the pump Exponential gain only as long as the

three pulses remain superimposed

Pulse distortion without temporal overlap High gain for vgi < vgp < vgs

Low si for broadband amplification

.

vgp < vgs< vgi vgi < vgp < vgs

The starting point: the GVM curves (I)

BBO: Ti-sapphire pumped OPA

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

Signal wavelength [micron] deltaps (b) e deltapi (r) [ps/mm]

ep os + oi p = 0.8 m

This determines L for given pump/signal durations This fixes the bandwidth HUGE BANDWIDTH AT DEGENERACY

Type I interaction

vgp > vgs> vgi

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

• 0.3
• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

0.05 Signal wavelength [micron] deltaps (b) e deltapi (r) [ps/mm]

The starting point: the GVM curves (II)

BBO: Ti-sapphire pumped OPA ep os + ei p = 0.8 m

MAXIMUM BANDWIDTH POINT

Type II interaction

vgi > vgp > vgs

HIGH GAIN ALSO BEYOND THE TEMPORAL WALK-OFF LIMIT ps ~ 50 fs/mm over the whole tuning range

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

• 0.05

0.05 0.1 0.15 0.2 Signal wavelength [micron] deltaps (b) e deltapi (r) [ps/mm]

Generating a frequency comb above 5 m (I)

GaSe: Er:fiber pumped DFG

Due to high pi failure of exponential gain after few hundreds m

ep os + oi p = 1.55 m Type I interaction p = 70 fs

TUNING REGION 1.7 m s 2.2 m ps < 80 fs/mm over the whole tuning range L = 1 mm

vgp < vgs < vgi

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 10 20 Idler wav. [um] 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 10 15 PM angle [°] 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

• 5
• 4
• 3

Signal wav. [um] WO angle [°]

Generating a frequency comb above 5 m (II)

GaSe: Er:fiber pumped DFG

TUNING POSSIBLE FROM 5 TO 12 m SMALL CRYSTAL ROTATION NEEDED: HIGH BIRIFRINGENCE SEVERE SPATIAL WALK-OFF: BEAM DIAMETER ~ 60 m

The first frequency comb above 5 m (I)

fR p=1.55 m <P>=250 mW =65 fs s = 1-2.2 m <P> = 160 mW

The first frequency comb above 5 m (II)

Pump (p) Signal (s)

extremely broad tunability: 5 -16 m fceo-free comb synthesis absence of 2-photons absorption

mid-IR radiation generated as idler beam GaSe

The first frequency comb above 5 m (III)

Tunability through: angle tuning chirp tuning

Spectrum limited to > 5 m Comb mode power: ~ 1-2 nW

• A. Gambetta et al, Opt. Lett. 33, 2671 (2008)

A more recent experiment with a more powerful Er:fiber oscillator

Menlo Systems @ 250 MHz Raman fiber for signal pulse generation

Second experiment: results

Tunability through: angle tuning power tuning

Spectrum limited to > 7 m Comb mode power: ~ 100-200 nW

• A. Gambetta et al, Opt. Lett. 38 1155 (2013)

GaSe