1

2) Optical Parametric Amplifiers and Oscillators

• Optical Parametric Generation (OPG)• Nonlinear Optical Susceptibilities• Continuous Wave OPA• Theory of Optical Parametric Amplification• Phase Matching• Quasi Phase Matching• Ultrashort Pulse Parametric Amplifiers (OPA)• Optical Parametric Amplifier Designs• Ultrabroadband Optical Parametric Amplifiers• Using Noncollinear Phase Matching• Optical Parametric Chirped Pulse Amplification (OPCPA)

[5] Largely follows the review paper of Cerullo et al., “Ultrafast Optical ParametricAmplifiers” Rev. Sci. Instr. 74, pp 1-17 (2003)

Ultrafast Optical Physics II (SoSe 2019) Lecture 10, June 21

1) Second-order nonlinear optical effects

Wave equation: Source term describing light-matter interaction

2

Second-order nonlinear optical effects

][ 2)2()1(0 EEP cce +=

)2(c 2nd order susceptibility

Input electric field

The total polarization at frequency ω is:

)cos(10 tAAE w+=Example: Pockels Effect

)cos(]2[ 10)2()1(

0)( tAAP wccew +=

New refractive index:0

)2()1( 21 An cc ++=

The Pockels effect is used to make optical switch (or modulator) using an electrical field to control the interaction between an optical crystal and the optical field propagating in it.

Second-harmonic generation (SHG)

Sum-frequency generation (SFG)

Difference-frequency generation (DFG)

Optical rectification

Mixing of two sine wavesInput electric field

)2(c may be both complex and frequency dependent.

),:()2( baba wwwwc + keeps track of the input and output frequencies involved in a particular interaction. 3

SHG in daily life: green laser pointer

4

Wavelength conversion using 2nd order nonlinear optics

Optical frequency

1w2w 13 2ww =23 2ww =

213 www +=

213 www -=

Shorter wavelength

Longer wavelength

DFG SHG SHG

SFG

Energy conservation

213 www -=

13 2kk =

213 www +=SHG

SFG

DFG

Momentum conservation

13 2ww =

213 kkk +=

213 kkk -=

Phase matching condition

11

33 2 n

cn

cww

= 13 nn =

221133 nnn www +=

221133 nnn www -=5

How to achieve phase matching?

221133 nnn www +>

For the frequency (wavelength) far away from absorption resonance, refractive index increases with increasing frequency, which leads to

Dispersion prevents phase matching.

6

Phase matching in birefringent media

Birefringent materials have different refractive indices for different polarizations. Ordinary (o-wave) light has its polarization perpendicular to the optical axis and its refractive index, !" , does not depend on propagation direction, θ. Extraordinary (e-wave) light has its polarization in the plane containing optical axis and propagation vector, and its refractive index, !" , depends on propagation direction, θ.

7

§ In an isotropic medium, normal dispersion always results in

( ) (2 )n nw w<§ In birefringent uniaxial crystal there are ordinary wave and extraordinary

wave.

0.4 0.6 0.8 11.5

1.55

1.6

1.65

1.7

1.75

wavelength [um]

refra

ctiv

e in

dex

of B

BO

BBO crystal is a typical negative uniaxial crystal with no>ne. If red light is set as the ordinary beam and the SHG the extraordinary one, angle tuning the BBO crystal permits achieving phase matching condition.

Angle tuningthe BBO

Phase matching in birefringent media

8

)(won

)(wen

Phase matching: type I Vs. type IIIn general, second-order nonlinear effects involve three waves with frequencies linked by the equation

Here is the highest frequency of the three.

Type I phase matching:

wave and wave have the same polarization; that is, they are both ordinary waves or extraordinary waves:

321 www =+

3w

1w 2weoo ®+ oee ®+or

Type II phase matching:

wave and wave have different polarization:1w 2w eeo ®+

eoe ®+ ooe ®+oeo ®+

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

1.54

1.56

1.58

1.6

1.62

1.64

1.66

1.68

1.7

1.72

wavelength [um]

refra

ctiv

e in

dex

of B

BO

q=340q=240

0(34 ,350 ) (700 )e on nm n nm= 0(24 ,500 ) (1000 )e on nm n nm=

10

Type I phase matching SHG

Linear susceptibility is a matrix for optically anisotropic media

11

EP )1(0ce=

xx EP )1(0ce=

yy EP )1(0ce=

zz EP )1(0ce=

Only true for optically isotropic media

å=j

jiji EP )1(0 ce

][ )1()1()1(0 zxzyxyxxxx EEEP ccce ++=

][ )1()1()1(0 zyzyyyxyxy EEEP ccce ++=

][ )1()1()1(0 zzzyzyxzxz EEEP ccce ++=

For optically anisotropic media, linear susceptibility is a 3X3 matrix (a second-rank tensor):

),,(),( zyxji =

2nd-order susceptibility is a 3rd-rank tensor

12

)()(),:()( 21,

213)2(

03)2( wwwwwcew k

kjjijki EEP å=

)()()()()()([)( 21)2(

21)2(

21)2(

03 wwcwwcwwcew zxxxzyxxxyxxxxxx EEEEEEP ++=

Take sum frequency generation(SFG) as an example:

),,(),,( zyxkji =

)()()()()()( 21)2(

21)2(

21)2( wwcwwcwwc zyxyzyyxyyxyxyx EEEEEE +++

)]()()()()()( 21)2(

21)2(

21)2( wwcwwcwwc zzxzzyzxzyxzxzx EEEEEE +++

321 www =+

We can represent the lengthy expression using tensor notation:

is a 3rd-order tensor with 27 (3X3X3) elements. According to the crystal symmetry, most of them are zeros.

),:( 213)2( wwwc ijk

w1

w1

w3

w2 = w3 - w1

Parametric Down-Conversion(Difference-frequency generation)

Optical Parametric Oscillation (OPO)

w3

w2

"signal"

"idler"

By convention:wsignal > widler

Difference-frequency generation: optical parametric generation, amplification, oscillation

w1

w3 w2

Optical Parametric Amplification (OPA)

w1

w1

w3

w2

Optical Parametric Generation (OPG)

Difference-frequency generation takes many useful forms.

mirror mirror

Frequencywp

Spec

trum

0 wi ws

12. 8 Optical Parametric Amplifiers and Oscillators12.8.1 Optical Parametric Generation (OPG)

Energy Conservation:

Momentum Conservation:

Degeneracy: wi=ws=wp/2

14

pump

Signal and idler resonant

Optical Parametric Oscillator (OPO)

Double resonant:

Single resonant: Only Signal resonant

Advantage: Widely tunable, both signal and idler can be used!

For OPO to operate, less gain is necessary in contrast to an OPA.

15

Total field: Pump, signal and idler:

Nonlinear Optical Susceptibilities

Drives polarization in medium:

Polarization can be expanded in power series of the electric field:

Defines susceptibility tensor:

16

Special Cases

17

Continuous Wave OPA

Wave equation (2.7) :

Include linear and second order terms:

Changes group and phase velocities of waves

Nonlinear interaction of waves

Wave amplitudes

z-propagation only:

18

Slowly varying amplitude approximation:

Separate into three equations for each frequency component:

Introduce phase mismatch:

and eff. nonlinearity and coupling coefficients:

19

Coupled wave equations:

20

Manley-Rowe Relations:

Intensity of waves:

X

21

Theory of Optical Parametric AmplificationUndepleted pump approximation:

with:

Gain Max. gain, when phase matched

22

Maximum Gain

General solutions:

Here:

For large gain:

23

Figure of merit:

Exact

Ip/Is(0)

I s(z)/I s(0)

Fig. Exact solution for signal gain, plotted together with hyperbolic secant and exponential function solutions, approximate solutions derived by assuming the pump is undepleted. The exact solution for the pump intensity is also shown.

Fig. 12.24 Parametric gain for an OPA at the pump wavelength lp = 0.8 µm and the signal wavelength ls = 1.2 µm, using type I phase matching in BBO (deff = 2 pm/V).

25

Fig. 12.25 Parametric gain for an OPA at the pump wavelength lp = 0.4 µm and the signal wavelength ls = 0.6 µm, using type I phase matching in BBO (deff = 2 pm/V).

26

27

Phase Matching

Type I: noncritical

Uniaxial Crystal: ne < no

Type I: critical

Fig. 12.27 Type I critical phase matching by adjusting the angle θ between wave vector of the propagating beam and the optical axis.

Fig. 12.26 Type I noncritical phase matching.

28

Phase Matching

Critical Phase Matching

Fig. 12.28 Angle tuning curves for a BBO OPA at the pump wavelength λp=0.8 μm for type I phase matching (dotted line), type II (os + ei → ep) phase matching (solid line), and type II (es + oi → ep) phase matching (dashed line).

29

12.8.5 Phase Matching

30

Fig. 12.28 Angle tuning curves for a BBO OPA at the pump wavelength λp=0.4 μm for type I phase matching (dotted line), type II (os + ei → ep) phase matching (solid line), and type II (es + oi → ep) phase matching (dashed line).

31

Quasi Phase Matching

Fig.12.30: Variation of def f in a quasi phase matched material as afunction of propagation distance.

32

Ultrashort Pulse Optical Parametric Amplification

Pulse envelopes

33

Temporal walkoffGroup Velocity Mismatch (GVM)

Pump pulse width

Fig. 12.31: Pump-signal (δsp) and pump-idler (δip) group velocity mismatch curves for a BBO OPA at the pump wavelength λp=0.8 μm for type I phase matching (solid line) and type II (os + ei → ep) phase matching (dashed line).

34

Fig. 12.32: Pump-signal (δsp) and pump-idler (δip) group velocity mismatchcurves for a BBO OPA at the pump wavelength λp=0.4 μm for type I phase matching (solid line) and type II (os + ei → ep) phase matching (dashed line).

35

Figure 12.34: Signal pulse evolution for a BBO type I OPA with λp = 0.4 μm, λs = 0.7 μm, for different lengths L of the nonlinear crystal. Pump intensity is 20 GW/cm2. Time is normalized to the pump pulse duration and the crystal length to the pump-signal pulse splitting length. [5]

36

Figure 12.35: Signal pulse evolution for a BBO type II OPA with λp = 0.8 μm, λs = 1.5 μm, for different lengths L of the nonlinear crystal. Pump intensity is 20 GW/cm2. Time is normalized to the pump pulse duration and the crystal length to the pump-signal pulse splitting length. [5]

37

OPA Bandwidth

Bandwidth limitation due to GVM

For vanishing dispersion:

Figure 12.35: Phase matching bandwidth for a BBO OPA at the pump wavelength λp=0.8 μmfor type I phase matching (solid line) and type II (os + ei → ep) phase matching (dashed line). Crystal length is 4 mm and pump intensity 50 GW/cm2.

Figure 12.36: Phase matching bandwidth for a BBO OPA at the pump wavelength λp=0.4 μm for type I phase matching (solid line) and type II (os + ei → ep) phase matching (dashed line). Crystal length is 2 mm and pump intensity 100 GW/cm2.

40

Optical Parametric Amplifier Designs

Figure 12.37: Scheme of an ultrafast optical parametric amplifier. SEED: seed generation stage; DL1, DL2: delay lines; OPA1, OPA2 parametric amplification stages; COMP: ompressor.

41

Near-IR OPA

Figure 12.38: Scheme of a near-IR OPA DL: delay lines; WL: white light generation stage; DF: dichroic filter. [5]

42

Noncollinear Optical Parametric Amplifier (NOPA)

Figure 12.39: a) Schematic of a noncollinear interaction geometry; b) representation of signal and idler pulses in the case of collinear interaction; and c) same as b) for noncollinearinteraction.

Phase Matching Condition: Vector Condition:

p

p

43

Variation on phase matching condition by

X

X

And addition

Only possible if:

gs giCorrectindex

Figure 12.40: Phase-matching curves for a noncollinear type I BBO OPA pumped atpumped at λp=0.4 μm, as a function of the pump-signal angle a. [5]

Figure 12.41: Scheme of a noncollinear visible OPA. BS: beam splitter; VA: variable attenuator; S: 1-mm-thick sapphire plate; DF: dichroic filter; M1 ,M2 , M3 , spherical mirrors.[5]

NOPA Layout

Figure 12.42: a) Solid line: NOPA spectrumunder optimum alignment conditions;

dashed line: sequence of spectra obtained by increasing the white light chirp; b) points: measured GD of the NOPA pulses; dashed line: GD after ten bounces on the ultrabroadband chirped mirrors.

47

Figure 12.43: Reconstructed temporal intensity of the compressed NOPA pulse measured by the SPIDER technique. The inset shows the corresponding pulse pectrum.[5]

MgO:PPLN31.0µm

MgO:PPLN13.1µm

DCM

Nd:YLF regen amp@1047nm

120ps, 1 mJ @1kHz

OPA 1

YDFA

2pJ, 2.0µm, 5.0 ps100µJ

Si

30 mm

>200 µJ

suprasil

300 mm

10 nJ, 6.7 ps

MgO:PPSLT31.4µm

OPA 2

3.5mJ

AOPDF Si

30 mm

BBO

OPA 3

400µJ

2 Nd:YLF-MPS modules120ps, 6mJ @1kHz

12 ps, 4 mJ800 nm OPCPA 800 nm OPCPA

YDFA

120ps

Grating Pair Compressor

CFBG stretcherCirculator

Ti:Soscillator

2-µm OPCPA

48

Optical Parametric Chirped Pulse Amplifier (OPCPA)

Optical Synthesis from OPAs

49

DOPA

Combination of light from broadband Optical Parametric Amplifiers.

χ(2)optical process in nonlinear crystals Broadband phase-matching

D. Brida et al., Journal of Optics A 12, 013001 (2010)

Multi-millijoule Pulse Synthesis with OPAs

50

broadband DCMs

• Two-octave-wide waveform synthesis from OPAs at the multi-mJ energy level

• passively CEP-stable WLG seed [G. Cerullo et al., Laser Photonics Rev. 5, 323 (2010)]

• WLG seed split into 3 wavelength channels and amplified in 3 OPA stages each

• Three channels are individually compressed and coherently recombined

• relative timing is tightly locked using balanced optical cross-correlators (BOCs)

51

Optical Pulse Synthesizer

51

VIS NOPA NIR DOPA IR DOPA0.17 mJ signal 0.20-0.25 mJ signal 1.7 mJ octave-spanning

signal20% (0.8 mJ pump)

pump-signal conversion efficiency

12-15% (1.7 mJ pump)pump-signal conversion

efficiency

22% (7.7 mJ pump)pump-signal conversion

efficiencyTL 5.6 fs TL 5.2 fs TL 5.2 fs

2.9 optical cycles @ lc=573nm

2.1 optical cycles @ lc=750nm

1.1 optical cycle @ lc=1.4um