THz Essentials - Appunti

E

(2) 0

2 (1 + cos 2ωt)

P = χ E = χ

2 2

N L 2

which are the optical rectification and second harmonic generation. The same effects

appear for the case of two overlapping fields, which will resolve into the sum and difference

frequency generation. THz radiation production is based on low difference frequency

generation. In the same way, the generation of THz pulses is based on the difference

mixing of all frequencies within the bandwidth of a femtosecond laser. Critical features

in the production of high intensity THz pulses are the damage threshold of the medium,

the phase-matching condition and a high absorption coefficient.

The modification of the polarization (nonlinear) at the same frequency of the incident

field is the fingerprint of an electrooptic effect. In an experimental setup, the nonlinear

(3)

polarization is induced by the electric field of a short laser pulse. The χ term doesn’t

cause a production of THz and is therefore excluded. This can arise from a second order

→

treatment in the limit of ω 0. Thus we have

2 Z (2)

0 0 0 0 0

NL − −

(ω) = dω χ (ω, ω , ω ω )E (ω )E (ω ω )

P 0 j k

i ijk − −ω −

with the integral developing through the intervals (−ω ∆ω, + ∆ω) and (ω

0 0 0

N L

∆ω, ω + ∆ω). Therefore, if ∆ω is smaller than ω , the polarization P is non zero

0 0 ±2ω ±2ω

only in three distinct spectral ranges: two centred at (SHG) and one within

0 0

which causes optical rectification, and is the source of THz. If ω is not near a resonance

0

(2)

frequency of the material under consideration, the dispersion of χ is generally small and

0 0

OR (2)

≡ −ω

we can substitute it with an effective value relevant for OR χ (ω, ω ) χ (ω, ω , ω )

0

78 CHAPTER 6. TERAHERTZ SOURCES

0

|ω − |

if ω ω . Within this definition we can write

0 0 OR

1 χ (ω, ω )

0

OR ∗

P (ω) = χ (ω, ω )(E E )(ω) = I(ω)

OR 0 0 0 0

2 n(ω )c

0

where the relation between the convolution of the electric field and its intensity has been

introduced. The relationship between the linear electrooptic coefficient r and the

jki

optical rectification effective tensor is

2 OR

− χ (ω, ω )

r (ω , ω) = 0

jki 0 ijk

2 2 (ω )

n (ω )n 0

0

j k

Using Kleinman’s symmetry, the coefficient can be used to compute the deformation in

the index ellipsoid as 3 3

1 1

n

X ≈ −

∆ ∆

= r E ∆n

ij j

2 2

n 2 n

i j=1

In applications, usually the largest tensor element r is used.

33

The bandwidth of an electrooptic crystal for THz generation and detection is determined

by the coherence length and optical phonon resonances in the material, these latter

limiting the generation due to absorption. If the refractive indices of the near infrared

laser excitation frequency and the THz radiation are identical, then the bandwidths of the

THz radiation depend only on the pulse width of the incident femtosecond laser. Increas-

ing crystal thickness, instead, will cause a growth in the output signal intensity. However,

the refractive indices are not equal, consequently, the two signals travel at different speeds

in the medium. The distance over which the slight velocity mismatch can be tolerated is

called the coherence length. Crystal thickness is bounded by this latter parameter

πc 1 ∂n opt

−

l (ω ) = n = n (ω) λ

c T Hz opt ef f opt opt

|n −

ω (ω ) n (ω )| ∂λ

T Hz opt ef f 0 T Hz T Hz λ opt

The THz emission bandwidth increases with thinner crystals, but its emission strength

6.2. THZ FROM OPTICAL RECTIFICATION [4][5][6] 79

decreases. For efficient generation of THz radiation by optical rectification, it is important

to select the proper orientation of the crystal with respect to the linear polarization of

the laser beam.

In the case of detection with the longitudinal linear electrooptic effect, the electric

field is applied parallel to the direction of propagation of the optical probe laser beam. In

case of transverse LEE, the field is applied perpendicular to the direction. Detection of

THz pulses is generally performed using the geometry of the transverse effect. The detec-

tion of amplitude and phase of the subpicosecond THz pulse are measured by recording

the phase change of femtosecond near infrared laser pulses travelling collinearly and si-

multaneously with the THz pulses through the electrooptic crystal. The material becomes

birefringence due to the THz pulses, and the phase retardation Γ between the ordinary

and extraordinary ray after propagation is proportional to the amplitude and phase of

the THz electric field, the crystal thickness, the linear electrooptic coefficient r , the

14

index of refraction of the crystal at the near infrared frequency n , and the near-infrared

0

wavelength λ :

0 l 30

n r E

Γ = 2π 41 T Hz

λ

The field is mapped, just like for photoconductive antennas, through a delay of the NIR

pulse. The polarization change induced by the THz electric field is not influenced by the

thermal noise, which allows for sensitive detection of THz pulse, even at room temper-

ature. The detection of the polarization changed signal form the electrooptic crystal is

achieved through the use of a Wollaston prism. This separates light into two separate

linearly polarized outgoing beams with orthogonal polarization.

80 CHAPTER 6. TERAHERTZ SOURCES

6.2.1 Generation and Detection Equations [6]

The conversion efficiency of THz generation in a nonlinear optical crystal depends on

the linear and nonlinear optical properties and thickness of the crystal and on laser pa-

rameters. For a theoretical calculation of the conversion efficiency, the nonlinear wave

equation for the generated THz signal is typically solved in the frequency domain. An

exact solution can be found if plane wave and a non-depleted pump wave conditions are

assumed and cascaded order and higher order nonlinear optical effects are excluded. The

nonlinear wave equation in the frequency domain takes the form

2 2 2

∇ − ∇[∇ · − −ω

E(r, ω) E(r, ω)] + ω µ (ω)E(r, ω) iωµ σ(ω)E(r, ω) = µ P (r, ω)

0 0 0 0 N L

If we consider the above approximations, we can write the intensity as

−α −α

z i(ωn /c)z z

− →

I(t, z) = I (t z/v )e I(ω, z) = I (ω)e e

g

0 0

0 g 0

where we have introduced the absorption coefficient α at frequency ω , the input pulse

0 0

entering the crystal I (t), and the group velocity and index

0 ∂n

c n (ω ) = n(ω ) + ω

v (ω ) = g 0 0 0

g 0 n (ω ) ∂ω

g 0 ω

0

Taking the propagation direction along z and substituting I(ω) in the nonlinear OP po-

larization relation, we obtain a solution of the form

OR

µ χ (ω, ω )ωI (ω)

0 0 0 ×

E(ω, z) = h i

α (ω)

c T

n(ω ) + α + i(n(ω) + n )

0 0 g

ω 2 ωng

αT (ω)

ωn(ω)

−i − −i −α

z z z z

−

e e e e 0

2 2 c

× α (ω) ωc

− −

T α + i [n(ω) n ]

0 g

2 µ c

0

where the THz absorption coefficient α (ω) = σ(ω) and the effective generation

T n(ω)

length have been introduced

−α −2α −(α ωc

(ω)z z (ω)/2+α )z

− −

e + e 2e cos [n(ω) n ]z

0 0

T T g

2

L (ω, z) =

gen 2

2

α (ω) ωc 2

− −

T α + [n(ω) n ]

0 g

2

the latter being the modulus of the second fraction, which reaches its maximum value for

zero absorption and equal propagation velocities of the THz wave and probe pulse. If we

use a Gaussian trial intensity with a characteristic time τ = 64 f s, we get a FWHM for

the THz pulse of 150 f s and a bandwidth ∆ν = 5 T Hz.

6.2. THZ FROM OPTICAL RECTIFICATION [4][5][6] 81

For the coherent detection of the THz transients one can employ conventional EO sam-

pling. Here the THz electric field induces a change ∆φ in the phase of a copropagating

optical probe pulse (or one of its polarization components). A measure of ∆φ as a func-

tion of the time delay t between the THz and the probe pulse allows the determination

d

of E (t).

T Hz

If we consider a probe pulse with the normalized amplitude A(t) propagating through

the crystal in the same way as in the generation process, and taking the group velocity

to be unaffected by the THz field (the induced relative index change is small), the total

phase shift will be

L 3

Z Z n

0

− − − rE (z, t)

∆φ(t ) = k ∆n(z, t)A[t (n z/c) t ]dtdz ∆n(z, t) = T Hz

d 0 g d 2

0 R

averaged over the probe pulse duration and accumulated over the propagation through

ω

the EO detection crystal length. The parameters introduced are k = and the active

0

0 c

Pockels coefficient r. Taking the THz field to be ωn(ω)

−α(ω)z/2 z

i

E (ω, z) = E (ω)e e c

T Hz i

the phase shift in Fourier space will be α(ω) ω

− z i [n(ω)−n ]z

30 −

1 e e

n g

2 c E (ω)

∆φ(ω, z) = k rA(ω) i

0 α(ω)

2 ω

− −

− i [n(ω) n ]

g

2 c

where the fraction modulus can be written as a detection length in analogy with the

the generation process αT (ω)

−

−α ω

z

(ω)z −

− cos (n(ω) n )z

e + 1 2e

T 2 g

2 c

L (ω, z) =

det 2

2

α (ω) ω 2

−

T + [n(ω) n ]

g

2 c

The spectrum that is measured by EO sampling is thus proportional to that of the THz

pulse, filtered y two multiplicative factors. The the time domain, the filtering given by

L corresponds to a distortion of the THz waveform. A maximum for the characteristic

det

lengths with respect to z cannot be computed analytically, bu