9. Third-order Nonlinearities: Four-wave mixing Third-harmonic

generation

Induced gratings

Phase conjugation

Nonlinear refractive index

Self-focusing

Self-phase modulation

Continuum generation

Third-harmonic generation

3 3 3exp[ (3 3 )] *( , )

othe

e

r te

xp[ (3 3 )

m

]

r s

E i t kz E i t kr zt

E

*exp( [ ( )] exp[ (, ) )]E i t kzr t E i t kz

E

3

THG Medium

We must now cube the input field:

Third-harmonic generation is weaker than second-harmonic and sum-frequency generation, so the third harmonic is usually generated using SHG followed by SFG, rather than by direct THG.

Noncollinear Third-Harmonic Generation

We can also allow two different input beams, whose frequencies can be different.So in addition to generating the third harmonic of each input beam, the medium will generate interesting sum frequencies.

22 1 2

21 2 1 2 1

1 2

2

1

3

exp{ [(2 ) (2 ) )]}

exp{ [(2 ) (2 ) )( , )

other ter

}

ms

]

E E i t

E E i t k r

k k r

t kr

E

THGmedium

Signal #1

Signal #2

Nonlinearmedium

Signal #1

Signal #2

Third-order difference-frequency generation: Self-diffraction

2

2 *2 1 2 1 2 1

*31 2 1 2 1 2

exp{ [(2 ) (2 ) )]}

exp{ [(2 ) (2 ) )( , )

other ter

}

ms

]

E E i t

E E i t k r

k rk

t kr

E

Consider some of the difference-frequency terms:

Samplemedium

Signal

The excite-probe geometry

2

1 2 13

1exp[ (( , )

other ter

)]

ms

E E i rr t t k

E

One field can contribute two factors, one E and the other E*. This will involve both adding and subtracting the frequency and its k-vector.

This effect is automatically phase-matched!1 1 2 2

1 1 2 2k k k k

The excite-probe beam geometry has many applications, especially to ultrafast spectroscopy. The signal beam can be difficult to separate from the input beam, E1, however.

Nonlinearmedium

Wave plateyielding 45˚polarization

Signal

Polarization gating

*1 2 2v 1 1

3 ˆ( ) ex( , ) + .p[ ..( )] vh hE E E i tr t k r

E

Here field #2 contributes two factors, one E and the other E*. But one is vertically polarized, while the other is horizontally polarized. This yields a signal beam that’s orthogonally polarized to the input beam E1.

If E1 is horizontally polarized, the signal will be vertically polarized:

This arrangement is also automatically phase-matched. It’s also referred to as polarization spectroscopy due to its many uses in both ultrafast and frequency-domain spectroscopy.

The input beam is the signal beam direction is rejected by polarizer!

x

1k

2k

The irradiance of two crossed beams is sinusoidal, inducing a sinusoidal absorption or refractive index in the medium––a diffraction grating!

An induced grating results from the cross term in the irradiance: *1 2E E

*1 2Re exp ( cos sin exp ( cos sin

Re exp 2 sin cos(2 sin )

E i t kz kx E i t kz kx

ikx kx

Many nonlinear-optical effects can beconsidered as induced gratings.

xx x x

2 ˆˆcos sink k z k x

1 ˆcos

ˆsin

k k z

k x

Time-independent fringes

Diffraction off an

induced grating

*0 0 1 2 3E P E E E

A third beam will then diffract into a different direction.This results in a beam that’s the product of E1, E2*, and E3:

This is just a generic four-wave-mixing effect.

x

1k

2k

Induced gratings

3 2

1 2exp{ [ (( , ] ...}) )pr ex pr ex ex prE E i t k k rr t k

E

Phase-matching condition:

1 2

1 2 ˆ2 sin

sig ex ex pr pr sig pr

sig ex ex pr ex ex pr

k k

k k k k k x k

Assume: 1 2ex ex ex

The diffracted beam has the same frequency and k-vector magnitude as the probe beam, but its direction will be different.

Nonlinearmedium

Diffractedsignal

ex1

ex2

sigprpr

ex

z

pr

sig1 2ex ex exE E E

1 2ex exk k

but

Phase-matching induced gratings

Phase-matching condition:

ˆ2 sin

sig pr sig pr

sig ex ex pr

k k

k k x k

ˆ ˆcos cospr sig pr prk z k z

ˆ ˆsin [ 2 sin sin ]pr pr ex ex pr prk x k k x

2 sin 2 sinpr pr ex exk k

z-component:

x-component:

ex

ex

pr

ex

z

pr

sig

x

2 2sin sinpr ex

pr ex

sig pr

The minus sign is just the excite-probe effect.

sin sinprpr ex

ex

The “Bragg Condition”

Induced gratingswith different frequencies

Nonlinearmedium

Diffractedsignal

*1 2 3 1 2 3 1 2 3

3 exp{ [( )( ) ...( ) ]}E E E i t k k k rt

E

Phase-matching condition:0 1 2 3

0 1 2 3k k k k

This effect is called “non-degenerate four-wave mixing.” In this case, the intensity fringes sweep through the medium: a moving grating.

The set of possible beam geometries is complex. See my thesis!

Induced gratings with plane waves and more complex beams (of the same frequency)

A plane wave and avery distorted wave

A plane wave and a slightly distorted wave

All such induced gratings will diffract a plane wave, reproducingthe distorted wave:

Two plane waves A plane wave and aslightly distorted wave

3 *1 2 3 1 2 3( ) [ ( , ) ] exp{ ( ) } ...t E x y E E i k k k r

E

E2 and E3 are plane waves.

One of the write beams has a complex spatial pattern that is the image. Different incidence angles correspond to different fringe spacings, so different views of the object are stored as different fringe spacings.

A third beam (a plane wave) diffracts off the grating, acquiring the image information. In addition, different fringe spacings yield different diffraction angles––hence 3D! The light phase stores the angular info.

Holography is an induced-grating process.

When a nonlinear-optical effect produces a light wave proportionalto E*, the process is called a phase-conjugation process. Phaseconjugators can cancel out aberrations.

exp[ , , ]i k z i x y i x y

exp ,ikz i x y

A normal mirrorleaves the sign of the phase unchanged

A phase-conjugatemirror reverses the sign of the phase

The second traversal through the medium cancels out the phase distortion caused by the first pass!

exp ikz ,x y

exp ,i k z i x y

exp ,i k z i x y

exp 2 ,i k z i x y

Phase conjugation

Phase Conjugation = Time Reversal

E(x,y,z, t) E0(x, y,z) exp[i(kz t)] E0* (x,y,z) exp[ i(kz t)]

A light wave is given by:

Epc(x,y,z,t) E0*(x, y, z) exp[i( kz t)] E0 (x, y, z) exp[ i( kz t)]

E0*(x,y,z) exp[ i(kz { t})] E0 (x, y,z) exp[i(kz { t})]

E0 (x, y, z) exp[i(kz { t})] E0*(x, y,z) exp[ i(kz { t})]

Epc(x,y,z, t) E(x, y,z, t)

If we can phase-conjugate the spatial part, we have:

Thus phase conjugation produces a time-reversed beam!

Degenerate Four-Wave Mixing

Consider only processes with three input frequenciesand an output frequency that are identical.

Degenerate four-wave mixing gives rise to an amazing variety of interesting effects. Some are desirable. Some are not.Some are desirable some of the time and not the rest of the time.

P (3) 0 (3)E1 E2

* E3

k 0

k 1

k 2

k 3

Because the k-vectors can have different directions, we’ll distinguish between them (as well as the fields):

Single-Field Degenerate Four-Wave Mixing

Single-field degenerate four-wave mixing gives rise to “self” effects. These include:

Self-phase modulationSelf-focusing (whole-beam and small-scale)

Both of these effects participate in the generation of ultrashort pulses!

P(3) 0 (3) E

2E

k 0

k 1

k 2

k 3

k

If just one beam is involved, all the k-vectors will be the same,as will the fields:

E1 E2 E3 E

So the polarization becomes:

Degenerate 4WM means a nonlinearrefractive index.

So the refractive index is:

P 0 (1)E (3) E2

E

Recall the inhomogeneous wave equation:

and the polarization envelope (the linear and nonlinear terms):

2 2 2

02 2 2 20

1E E P

z c t t

0 (1) (3) E2 E

Substituting the polarization into the wave equation (assuming slow variation in the envelope of E compared to 1/):

2 2 22(1) (3)

0 02 2 2 20

1E E EE

z c t t

2(1) (3)2 2

2 2 20

10

EE E

z c t

since2

0 0 01/ c

n 1 (1) (3) E2

Nonlinear refractive index (cont’d)

Usually we define a “nonlinear refractive index”, n2:

The refractive index in the presence of linear and nonlinear polarizations:

Assume that the nonlinear term << n0:

Now, the usual refractive index (which we’ll call n0) is:

So:

n 1 (1) (3) E2

n0 1 (1)

n n02 (3) E

2

n0 1 (3) E2

/ n02

So:

n n0 1 (3) E2

/ 2n02

n n0 (3) E2

/ 2n0

0 2n n n I

since:

I E2

The nonlinear refractive index magnitude and response timeA variety of effects give rise to a nonlinear refractive index.Those that yield a large n2 typically have a slow response.

Thermal effects yield a huge nonlinear refractive index through thermal expansion due to energy deposition, but they are very very slow. As a result, most media, including even Chinese tea, have nonlinear refractive indices!

Whole-Beam Self-Focusing

This is precisely the behavior of a lens! But one whose focal power scales with the intensity.

If the beam has a spatial Gaussian intensity profile, then any nonlinear medium will have a spatial refractive index profile that is also Gaussian:

Near beam center:

The nonlinear refractive index, , causes beams to self-focus.0 2n n n I

n n0 n2 I0 exp( 2r 2/ w02 )

n0 n2 I0 (1 2r 2/w02 )

2 20 0 0 2 0 0 0( ) (1 2 / )r n k L n k L n k L I r w

The phase delay vs. radial co-ordinate will be:

2 22 0 0 0( ) 2 ( / )r constant n k L I r w

Small-Scale Self-Focusing

Such filaments grow exponentially with distance.

And they grow from quantum noise in the beam, which is always there.

As a result, an intense beam cannot propagate through any mediumwithout degenerating into a mass of tiny highly intense filaments,which, even worse, badly damage the medium.

If the beam has variations in intensity across its profile, it undergoessmall-scale self-focusing.

Each tiny bump in the beam undergoes its own separate self-focusing, yielding a tightly focused spot inside the beam, called a “filament.”

Inte

nsity

Position

The self-phase-modulated pulse develops a phase vs. time proportional to the input pulse intensity vs. time.

Self-Phase Modulation & Continuum Generation

0 0 0 2( , ) (0, ) exp (0, )E z t E t ik n I t z

The further the pulsetravels, the moremodulation occurs.

A flat phase vs. time yields the narrowest spectrum. If we assume thepulse starts with a flat phase, then SPM broadens the spectrum.

This is not a small effect! A total phase variation of hundreds can occur! A broad spectrum generated in this manner is called a “Continuum.”

0 2( , ) (0, )z t k n I t z That is:

Pulse intensity vs. time

The instantaneous frequency vs. time in SPM

0 2( , ) ( )z t k z n I t

0 2

( , ) ( )( )inst

z t I tt k z n

t t

A 10-fs, 800-nm pulse that’s experienced self-phase modulation with a peak magnitude of 1 radian.

Self-phase-modulated pulse in the frequency domain

The same 10-fs, 800-nm pulse that’s experienced self-phase modulation with a peak magnitude of 1 radian.

It’s easy to achieve many radians for phase delay, however.

A highly self-phase-modulated pulse

A 10-fs, 800-nm pulse that’s experienced self-phase modulation with a peak magnitude of 10 radians

Note that the spectrum has broadened significantly. When SPM is very strong, it broadens the spectrum a lot. We call this effect continuum generation.

Continua created by propagating 500-fs 625nm pulses through 30 cm of single-mode fiber.

The Supercontinuum Laser Source, Alfano, ed.

Low

Energy

Medium

Energy

High

Energy

Broadest spectrum occurs for highest energy.

Experimental Continuum spectrum in a fiber

Continuum Generation Simulations

Input Intensity vs. time (and hence output phase vs. time)

Output spectrum:

Instantaneously responding n2; maximum SPM phase = 72π radians

The SupercontinuumLaser Source, Alfano, ed.

Original spectrum is negligible in width compared to the output spectrum.

Oscillations occur in spectrum because all frequencies occur twice and interfere, except for inflection points, which yield maximum and minimum frequencies.

Continuum Generation Simulation

Output phase vs. time (≠ input intensity vs. time,due to slow response)

Output spectrum:

Noninstantaneously responding n2; maximum SPM phase = 72π radians

The SupercontinuumLaser Source, Alfano, ed.

Asymmetry in phase vs. timeyields asymmetry in spectrum.

Experimental Continuum Spectra

Inputwavelength

625-nm (70 fs and 2 ps) pulses in Xe gas

L = 90 cm

The SupercontinuumLaser Source, Alfano, ed.

Data taken by Corkum, et al.

p = 15 & 40 atm

Ultraviolet continuum

4-mJ 160-fs 308-nm pulses in 40 atm of Ar; 60-cm long cell.

Lens focal length= 50 cm.

Good quality output mode. The SupercontinuumLaser Source, Alfano, ed.

308 nm input pulse; weak focusing with a 1-m lens.

The SupercontinuumLaser Source, Alfano, ed.

UV Continuum in Air!

Continuum is limited when GVD causes the pulse to spread, reducing the intensity.

Continuum Generation:Good news and bad news

Good news:

It broadens the spectrum, offering a useful ultrafast white-light source and possible pulse shortening.

Bad news:

Pulse shapes are uncontrollable.Theory is struggling to keep up with experiments.In a bulk medium, spectral broadening is not really that great—

you need a log scale to see it…In a bulk medium, continuum can be high-energy, but it’s a mess

spatially.In a fiber, continuum is clean, but it’s low-energy.In hollow fibers, things get somewhat better.

Main problem: dispersion spreads the pulse, limiting the spectralbroadening.

Microstructure optical fiber

Microstructure optical fibers modify dispersion.

The spectrum extends from ~400 to ~1500 nm and is relatively flat (when averaged over time).

This continuum was created using unamplified Ti:Sapphire pulses.J.K. Ranka, R.S. Windeler, and A.J. Stentz, Opt. Lett. Vol. 25, pp. 25-27, 2000

Cross section of the microstructure fiber.

The continuum from microstructure optical fiber is ultrabroadband.

Continuum is quite beautiful!

Other third-order nonlinear-optical effects

Raman scattering

Two-photon absorption