Nonlinear Optics (WiSe 2018/19) Lecture 11: January 11, 2018 11 - - PowerPoint PPT Presentation

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Nonlinear Optics (WiSe 2018/19)

Lecture 11: January 11, 2018 11 Terahertz generation and applications 11.1 Auston switch 11.2 Optical rectification 11.2.1 Optical rectification with tilted-pulse-fronts 11.2.2 Optical rectification by Quasi-Phase Matching (QPM)

11 Terahertz generation and applications

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• 0.3 – 30 THz

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Current Sample THz Transmitter THz Detektor Femtosecond Laser Ti:sapphire or Cr:LiSAF Delay Dielectrics, Tissue, IC - Packaging etc. LT-GaAs Substrate

V

50 - 100 fs Laser Pulse

• THz Time Domain Spectroscopy

THz Time Domain Spectroscopy

• Fig. 11.1: THz pulses generated (a) and received (b) with photoconductive switches.

THz Time Domain Spectroscopy using optical rectification in GaAs

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.

Quantum- Cascade Laser THz - OPOs

• OPAs

E2 E1

GaInAs/GaAs/AlAs

12 fs Ti:sapphire Oscillator Delay

QW

15 fs ~ 5 - 15 m ~ 60 THz 6 fs > 100 THz Optical Rectification in GaAs FIR - Probe:

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Time Domain THz Spectroscopy

Figure 11.2: Terahertz waveforms modified by passage through (a) a 10 mm block

• f stycast and (b) a chinese fortune cookie. The dashed lines show the shape of

the input waveform multiplied by 0.5 in (a) and by 0.1 in (b). In (a) the transmitted pulse exhibits a strong ”chirp” due to a frequency-dependent index, while in (b), pulse broadening indicates preferential absorption of high

• frequencies. [7]

Attosecond diffraction and spectroscopy of biomolecules

Undisturbed electronic structure Damage-free structure

All laser driven, intrinsic attosecond synchronization Only pico-second lasers at 1J-level necessary -> kHz operation

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à All optical driven fully coherent attosecond X-ray source: à has its own science case à seeding of large scale FELs à resolve access problem to large facilities

Dielectrically Loaded Circular Waveguide

• Traveling wave structure is best for coupling broad-band single cycle pulses
• Phase-velocity matched to electron velocity with thickness of dielectric

Dispersion Relation w/ dielectric w/o dielectric Copper Inner Diameter = 940 µm Fused Silica Inner Diameter = 400 µm ~1-5 cm

L.J. Wong et al., Opt. Exp. 21, 9792 (2013).

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THz Acceleration

Terahertz-driven Linear Electron Acceleration

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THz Off THz On

11.2 Optical rectification

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Figure 11.3: THz generation by DFG from two cw lines or from intrapulse spectral

• components. Once intense enough THz has been generated it acts back on the

generating lines and creates additional down-shifted lines, which themselves again generate THz by DFG. This cascaded DFG process leads to a continuous down- shifting of the center of the optical spectrum.

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crystal ZnTe LiNbO3 LiTaO3 GaP

• pt. wl (µm)

0.8 1.06 1.06 1.06

• pt. ref. index

2.85 2.16 2.14 3.11 THz ref. index 3.2 5.2 6.5 3.21 ∆n = nTHz − ng,opt 0.35 3.0 4.32 0.1 THz abs. (cm−1) 9.9 21.7 95 3.3

• transp. range (µm)

0.55-30 0.4-5.2 0.4-5.5 0.55-10 band gap (eV) 2.26 3.7 5.65 2.25

• nonlin. coeff. (pm/V)

d14=23.1 d33=152.4 d33=145.2 d14=21.7

• nonlin. ref. index n2

10-15cm2/W at λ (µm)

120 at 1.06 71 at 0.8 0.91 at 1.06 0.37 at 1.06 20 at 0.78 FOM1, long pulses 0.03 1 0.21 0.06 FOM2, ultrashort pl. 0.74 1 0.64 1,67 FOM3, Kerr-limited 0.00045 1 0.416 0.005

Table 11.1: Linear and nonlinear properties, and figures of merit (normalized to LiNbO3) of crystals transparent in the 0-4 THz range and most widely used for

• ptical THz generation according to Ref. [10].

THz Materialproperties I

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THz Materialproperties II

crystal GaSe GaAs ZGP CdSiP2

• pt. wl (µm)

1.06 2.1 2.1 2.0

• pt. ref. index

2.8 3.33 3.15 3.0 THz ref. index 3.26 3.6 3.37 3.05 ∆n = nTHz − ng,opt 0.34 0.18 0.17 0.05 THz abs. (cm−1) 2.5 1 1 <0.1

• transp. range (µm)

0.65-18 0.9-15 0.75-12 0.5-9 band gap (eV) 2.1 1.424 2.34 2.45

• nonlin. coeff. (pm/V)

d22=24.3 d14=46.1 d36=39.4 d36=85

• nonlin. ref. index n2

10-15cm2/W at λ (µm)

45 at 1.06 150 at 2.1 40 at 2.1 ? at 2.1 FOM1, long pulses 0.13 0.83 0.68 FOM2, ultrashort pl. 0.13 0.64 0.55 FOM3, Kerr-limited 0.004 0.014 0.047

Table 11.2: Linear and nonlinear properties, and figures of merit (normalized to LiNbO3) of crystals transparent in the 0-4 THz range and most widely used for

• ptical THz generation according to ref. [10]

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Three Wave Interaction

d ˆ E (ω1) dz = −jκ1 ˆ E (ω3) ˆ E∗ (ω2) e−j∆kz, d ˆ E (ω2) dz = −jκ2 ˆ E (ω3) ˆ E∗ (ω1) e−j∆kz, d ˆ E (ω3) dz = −jκ3 ˆ E (ω1) ˆ E (ω2) e+j∆kz,

κi = ωideff/nic0, and ∆k = k3 − k1 − k2.

nteresting to look at the wav e ω3 = ω0 + Ω and ω2 = ω0,

Difference Frequency Generation

= ω0 + Ω and ω2 = ω0 d Ω is a THz frequency. collinear interaction

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∆k = ∂kopt(ω) ∂ω

• ω0

Ω − kTHz (Ω) =

• 1

vg,opt − 1 vp,THz

• Ω

= Ω c (ng,opt − np,THz) .

Phase Mismatch (for collinear interaction)

For Lithium Niobate 2 5 à Broadband non collinear phase matching by tilted pulse fronts à Quasi-phase matching

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11.2.1 Optical rectification with tilted-pulse-fronts

Figure 11.4: (a) Noncollinear phase matching for THz generation. Note, the THz phase index in lithium niobate is more than twice as large as the optical group index. (b) Broadband implementation of the noncollinear phase matching using a grating and imaging system that leads to the generation of pulses with a tilted pulse front.

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Tilted pulse front technique

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Figure 11.6: Noncollinear phase-matching condition for pulse-front-tilted optical rectification.

∆kz(ω) = cos γ k(ω + Ω) − cos(γ + θ(ω)) k(ω) − kTHz (Ω) = cos γ ∂kopt(ω) ∂ω Ω + sin γ

• − ∂θ

∂ω

• Ω k(ω) − kTHz (Ω) = 0,

∆ky(ω) = sin γ k(ω + Ω) − sin(γ + θ(ω)) k(ω) = sin γ ∂kopt(ω) ∂ω Ω − cos γ ∂θ ∂ωΩ k(ω) = sin γ ∂kopt(ω) ∂ω Ω − cos γ

• − ∂θ

∂ω

• Ω k(ω) = 0.

z - component y - component

Non-collinear phase matching

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∂kopt(ω) ∂ω Ω − cos γkTHz (Ω) = 1 vg,opt − 1 vp,THz cos γ = ng,opt = np,THz cos γ,

∂θ ∂ω = − tan γ vp,opt ω vg,opt = − tan γ ng,opt ω np,opt .

k(ω) = 1 cos γ ω n(ω) c + (ω − ω0)2 2 k

′′

AD

k

′′

AD

= −n2

g,opt(ω0)

ω0 c (ω0) tan2 γ.

1D – spatial Model Necessary angular spread Tilt angle

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1D - Model

d ˆ ETHz (Ω, z) dz = −αTHz(Ω) 2 ˆ ETHz (Ω, z) (11.13) −j Ω deff c np,THz ∞ ˆ Eopt (ω + Ω, z) ˆ Eopt (ω, z)∗ ej∆k(ω)zdω . which also includes the THz absorption. For the optical field, we obtain d ˆ Eopt (ω, z) dz = −αopt(Ω) 2 ˆ Eopt (ω, z) −j ω deff c np,opt ∞ ˆ Eopt (ω + Ω, z) ˆ ETHz (Ω, z)∗ dΩe−j∆k(ω)z −j ω deff c np,opt ∞ ˆ Eopt (ω − Ω, z) ˆ ETHz (Ω, z) e−j∆k(ω)zdΩ +F

• j ε0ω0 n2

p,opt n2deff

2 |Eopt (t, z)|2 Eopt (t, z)

• (11.14)

+F

• j ε0ω0 n2

p,opt n2deff

2

• |Eopt (t − t′, z)|2 ⊗ hr(t′)
• Eopt (t, z)
• ,

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Figure 11.7: Comparison of experimental and simulated optical spectra for different amounts of generated THz.

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Figure 11.8: Conversion efficiencies as a function of effective length are calculated by switching on/off various effects. Material dispersion and absorption are considered for all cases. The pump fluence is 20 mJ/cm2, for a crystal temperature of 100 K. (a) Gaussian pulses with 500-fs FWHM pulse width with peak intensity of 40 GW/cm2 are used. Cascading effects together with GVD-AD leads to the lowest conversion efficiencies. The drop in conversion efficiency is attributed to the enhancement of phase mismatch caused by dispersion due to the large spectral broadening caused by THz generation (See

• Figs. 11.7(b)-(c)). However, since group velocity

dispersion due to angular dispersion (GVD-AD) is more significant than GVD due to material dispersion at optical frequencies in lithium niobate, cascading effects in conjunc- tion with GVD-AD is the strongest limitation to THz generation. SPM effects are much less detrimental since they cause relatively small broadening of the optical pump spectrum (see 11.7 (a)). (b) Cascading effects along with GVD-AD are most detrimental even for a 150-fs Gaussian pulse with 3× larger peak intensity. [19]

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2D - Simulation

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11.2.2 Optical rectification by Quasi-Phase Matching (QPM)

∆k = ∂kopt(ω) ∂ω

• ω0

Ω − kTHz (Ω) + m2π Λ =

• 1

vg,opt − 1 vp,THz

• Ω +

= ng,opt − np,THz c

• Ω + m2π

Λ = 0 → Λ = m λTHz np,THz − ng,opt .

) + m2π Λ

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Figure 11.10: Schematic illustration of collinear THz-wave generation in a nonlin- ear crystal with periodically inverted sign of χ(2). (a) Optical rectification with femtosecond pulses, (b) difference-frequency generation with two picosecond pulses (Ω = ω3 − ω2) [10].

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Plane-wave analysis of optical-to-THz conversion in QPM crystals with ultrashort pulses

Eopt(t) = Re{E0 e−t2/τ 2 ejω0t} = 1 2{E0 e−t2/τ 2 ejω0t + c.c.}, ˆ Eopt(ω) = Eoτ 2√π exp

• −τ 2ω2

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• d ˆ

ETHz (Ω, z) dz = −j Ω dQPM

eff

c np,THz E2

• τ 2

4π +∞

−∞

exp

• −τ 2 (ω + Ω)2

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• exp
• −τ 2ω2

4

• dω

= −j Ω dQPM

eff

c np,THz E2

• τ

2 √ 2π exp

• −τ 2Ω2

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• ej∆k(Ω)z.

(11.23)

• ˆ

ETHz (Ω, z)

• 2

= Ω2 dQPM,2

eff

c2 n2

p,THz

E4

• τ 2

8π exp

• −τ 2Ω2

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• L2sinc2(∆k(Ω)L

2 ) (11.24) with ∆k(Ω) = ∆k = ng,opt − np,THz c + m2π Λ (11.25)

dQPM

eff

=

2 πdeff.

e approximation is

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ηPW

THz =

Fluence THz Fluence (pump)

Optical to THz conversion effficiency

Fpump = cε0np,opt 2 +∞

−∞

|Eopt (t, 0)|2 dt = π 2 cε0np,opt 2 E2

• τ

and THz fluence FTHz = cε0np,THz 2 +∞

−∞

|ETHz (t, L)|2 dt = cε0np,THz 2 2π +∞

−∞

• ˆ

ETHz (Ω, z)

• 2

dΩ (11.28) where we used Parceval’s theorem +∞

−∞

|f(t)|2 dt = 2π +∞

−∞

• ˆ

f (Ω)

• 2

dΩ. (11.29) Thus, we obtain ηPW

THz

= Ω2 dQPM,2

eff

E2

• τ

2 √ 2πc2 np,optnp,THz L2 +∞ exp

• −τ 2Ω2

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• sinc2(π

2 Ω − Ω0 ∆Ω ) dΩ, (11.30) with Ω0 = 2π c Λ (ng,opt − np,THz) and ∆Ω = 2π c L (ng,opt − np,THz) = Λ LΩ0 (11.31)

and is equa ds N = L

Λ

ained in th

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ηPW,sp

THz

= Ω2 dQPM,2

eff

E2

• c

2 √ 2πc2 np,optnp,THz L2g1(Ω0) 2 ∆Ω = = Ω2 dQPM,2

eff

E2

• τ

c np,optnp,THz √ 2π (ng,opt − np,THz) L g(Ω0) = 2Ω2 dQPM,2

eff

L ε0 c2 n2

p,optnp,THz (ng,opt − np,THz) g1(Ω0) Fpump (11.32)

with g1(Ω0) = exp

• −τ 2Ω2

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• = exp
• − (πfTHzτ)2

and (11.33) 2 ∆Ω = +∞

−∞

sinc2(π 2 Ω − Ω0 ∆Ω ) dΩ (11.34)

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Influence of optical bandwidth

Figure 11.11: Relative THz generation reduction due to g1(Ω0).

ηPW,lp

THz

= Ω2 dQPM,2

eff

E2

• τ

2 √ 2πc2 np,optnp,THz L2 +∞ e

• − τ2(Ω−Ω0)2

4

• sinc2(π

2 Ω − Ω0 ∆Ω ) dΩ, (11.35) = Ω2 dQPM,2

eff

E2

• τ

2 √ 2πc2 np,optnp,THz L2 +∞ e

• −

τ2(Ω′)2 4

• sinc2(π

2 Ω′ ∆Ω) dΩ′.(11.36)

For long pulses

lw = √πcτ (ng,opt − np,THz),

ηPW,lp

THz

= Ω2 dQPM,2

eff

E2

• 2

√ 2πc2 np,optnp,THz L2 +∞ exp

• − y2

4

• sinc2(

√π 2 L lw y) dy, ηPW,lp

THz

= Ω2 dQPM,2

eff

E2

• √

2c2 np,optnp,THz lwL π g2(2lw L ) g2(x) = +∞ exp

• −x2µ2

π

• sinc2(µ) dµ, with x = 2lw

L .

Walk-off

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ηPW,lp→∞

THz

= Ω2 dQPM,2

eff

E2

• 2

√ 2πc2 np,optnp,THz L2 = Ω2 dQPM,2

eff

πε0c3 n2

p,optnp,THz

Fpump τ L2,

Very long pulses

Figure 11.12: Relative THz generation reduction due to g2(2lw

L ).

Influence of walk-off

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Optimal length of the EO crystal

ηTHz(L) ∼ 1 αTHz

• 1 − e−αT HzL

= Leff

ηTHz(L) ∼ g3L, with g3 = 1 αTHzL

• 1 − e−αT HzL

For L = 1/αTHz, g3 = 0.63.

Optimal focusing

ηTHz = UTHz Upump = g1 g3 2Ω2 dQPM,2

eff

L ε0 c2 n2

p,optnp,THz (ng,opt − np,THz)

Upump πw2 ,

DFG case, we can characterize the focusing strength by r ξ = (λ L/2πnTHzw2), where λ is the THz wavelength Ratio between crystal length and THz Rayleigh range

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Enhancement factor

Figure 11.13: Enhancement factor h as a function of the focusing parameter ξ. Solid curve is based on Ref. [26]. Dashed curve – plane-wave approximation. Dots represent calculations based on the Green’s function method. Inset: far-field THz intensity profiles at different ξ for a 1-cm-long GaAs.

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Cascading and red shift

ation argument, it follows (if we neglect losses) that the center the optical pulse spectrum will be red-shifted by ∆ω/ω0 ∼ ηTHz, e central optical frequency. When optical-to-THz photon conve

as N = 0.5 × (acceptance bandwidth) / (terahertz frequency). tance bandwidth is with respect to the pump frequency and can

d∆k dω = Ω c dng,opt (ω) dω = Ω c dng,opt (λ) dλ dλ dω = Ω c dng,opt (λ) dλ λ2 2πc

d∆k dω L∆ωacc = 2π ∆ωacc = 2πcω LΩ

• λdng,opt (λ)

dλ −1 .

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Figure 11.14: Number of THz cascading cycles as a function of THz frequency and pump wavelength for GaAs, L = 1 cm.

Cascading and red shift

FOM1= d2

eff

np,optαTHz (11.45)

• r

FOM2= d2

eff

n2

p,opt(np,THz − ng,opt).

(11.46) If the maximum propagation distance is limited by the Kerr effect, the critical FOM becomes FOM3= λopt d2

eff

n2

p,optnp,THzαTHzn2

. (11.47)

Summary

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