THz Essentials - Appunti

UF and THz Spectroscopy Essentials

Tomarchio Luca

July 24, 2019

2

Contents

1 Theory of Lasers [1] 5

1.1 Gain and Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Amplifier Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Pumping Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Amplifier Non-linearity and Noise . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Amplifier Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Theory of Laser Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Characteristics of the Laser Output . . . . . . . . . . . . . . . . . . . . . . 12

1.5.1 Spectral Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5.2 Spatial Distribution and Polarization . . . . . . . . . . . . . . . . . 14

1.6 Common Types of Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6.1 Chirped Pulse Amplification . . . . . . . . . . . . . . . . . . . . . . 16

2 Ultrafast Optics [2] 17

2.1 Principles of Mode-Locking . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.1 Active Mode-locking . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.2 Passive Mode-Locking . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.3 Absorber and Gain Medium . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Pulse Measurement Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Electric Field Autocorrelation . . . . . . . . . . . . . . . . . . . . . 23

2.2.2 Cross-Correlation Measurements . . . . . . . . . . . . . . . . . . . . 25

2.2.3 Intensity Correlation Measurements . . . . . . . . . . . . . . . . . . 26

2.2.4 Chirped Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.5 Frequency Resolved Optical Gating (FROG) . . . . . . . . . . . . . 29

2.3 Dispersive Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Ultrafast Nonlinear Optics [2] 35

3.1 Ultrashort-Pulse SHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Three-Wave Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1 Sum-Frequency Generation . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Optical Parametric Amplification (OPA) . . . . . . . . . . . . . . . 40

3.3 Propagation for Nonlinear Refractive Media . . . . . . . . . . . . . . . . . 41

3.3.1 Self-Phase Modulation (SPM) . . . . . . . . . . . . . . . . . . . . . 43

3

4 CONTENTS

3.3.2 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Continuum Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Ultrafast Time-Resolved Spectroscopy 47

4.1 Degenerate Transmission Measurements [2] . . . . . . . . . . . . . . . . . . 48

4.1.1 Orientational Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Quantum-Statistical Approach . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Terahertz Spectroscopy 53

5.1 Fourier-Transform Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.1 Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Time-Domain Spectroscopy [7] . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2.1 Thick Medium Insights [12] . . . . . . . . . . . . . . . . . . . . . . 59

5.2.2 THz-TDS Apparatus Properties [11] . . . . . . . . . . . . . . . . . 61

5.3 Terahertz Detectors [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3.1 Semiconductor Quantum Wells . . . . . . . . . . . . . . . . . . . . 66

5.3.2 Photoconductive Antennas [4] . . . . . . . . . . . . . . . . . . . . . 68

6 Terahertz sources 73

6.1 Photoconductive THz Generation [4] . . . . . . . . . . . . . . . . . . . . . 74

6.1.1 Terahertz Beam Propagation . . . . . . . . . . . . . . . . . . . . . . 76

6.2 THz from Optical Rectification [4][5][6] . . . . . . . . . . . . . . . . . . . . 77

6.2.1 Generation and Detection Equations [6] . . . . . . . . . . . . . . . . 80

6.2.2 Nonlinear Materials for THz Generation and Detection [4] . . . . . 82

6.3 Two-Colors THz Plasma Generation [9, 10] . . . . . . . . . . . . . . . . . . 82

6.4 CW THz Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.4.1 Photomixers and Beating Sources [5] . . . . . . . . . . . . . . . . . 85

6.4.2 Quantum Cascade Lasers [4][3] . . . . . . . . . . . . . . . . . . . . 86

6.5 Free Electron Laser [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Chapter 1

Theory of Lasers [1]

A coherent optical amplifier is a device that increases the amplitude of an optical field

while maintaining its phase. A basic physical phenomenon that can be used to obtain such

an effect is stimulated emission. Its main application is the construction of LASERs (light

amplification by stimulated emission od radiation). Real coherent amplifiers deliver a gain

and phase shift that are frequency dependent. Furthermore, real amplifiers produce noise,

so that a random fluctuating component is present at the output, and sufficient large input

amplitudes might encounter a saturation boundary, caused by non-linear behaviours. An

amplifiers is thus characterized by the following features:

ˆ Gain

ˆ Bandwidth

ˆ Phase Shift

ˆ Power Source

ˆ Non-linearity and gain saturation

ˆ Noise

The main physical model to understand the basic principles will be a monochromatic

optical plane wave travelling in the z direction with frequency ν, electric field E(z) =

2

<{E |E

(z) exp(iνt)}, intensity I(z) = (z)| /2η and photon-flux density φ(z) = I(z)/hν.

0 0

This will interact with an atomic medium, provided that the atoms have two energy levels

whose energy difference nearly matches the photon energy hν. The numbers of atoms in

the lower and upper energy level will be N and N respectively. Furthermore, the wave is

1 2

amplified with a gain coefficient γ(ν) (per unit length) and undergoes a phase shift ϕ(z).

1.1 Gain and Bandwidth

Spontaneous emission will be characterised by a specific transition cross section σ(ν),

meaning that the probability per unit time of emission will be W (ν) = φ(ν)σ(ν). The

i

5

6 CHAPTER 1. THEORY OF LASERS [?]

average density of emitted and absorbed photons will be given by the same probabil-

ity density, meaning that the number of photons gained per second per unit volume is

−

(N N )W = N W . Given a cylinder (shape of the laser) composed of the atomic

2 1 i i

medium, a differential equation can be written for the gain (or lose) of flux

dφ(z) = γ(ν)φ(z) γ(ν) = N σ(ν)

dz

The solution is an exponentially growing function.For an interaction length d, the gain of

the laser amplifier is defined as φ(d) = exp[γ(ν)d]

G(z) = φ(0)

Of course, a medium at thermal equilibrium cannot generate N > 0, therefore a population

inversion (out of equilibrium) is mandatory, obtained by a pumping process.

The dependence of the gain coefficients from the frequency is contained in the line-

shape function g(ν), such that ∞

2 Z

λ

σ(ν) = g(ν)dν = 1

g(ν)

8πt

sp 0

where t is the effective spontaneous lifetime for stimulated emission, meaning the time

sp

for radiative decay from 2 to 1. The width of g is known as transition linewidth ∆ν,

∝

usually defines as the FWHM, and, due to the normalization condition, ∆ν 1/g(ν ),

0

−

where ν = (E E )/h is the frequency associated to the function peak.

0 2 1

1.1.1 Phase Shift

Because the gain of the resonant medium is frequency dependent, the medium is disper-

sive and a frequency dependent phase shift must be associated to its gain. The electric

field amplitude can be rewritten as

E (z) = E (0) exp[γ(ν)z/2] exp[−iϕ(ν)z]

0 0

from which ' −

E (z + ∆z) = E (z) exp[γ(ν)∆z/2] exp[−iϕ(ν)∆z] E (z)[1 + γ(ν)∆z/2 iϕ(ν)∆z]

0 0 0

Therefore ∆E 1

0 −

= E (z)H(ν) H(ν) = γ(ν) iϕ(ν)

0

∆z 2

1.2. AMPLIFIER PUMPING 7

where H(ν) is the transfer function of a linear causal system, such that its real and imag-

inary part are related by the Hilber transform

∞ 00

Z

1 H (s)

0

H (ν) = ds

−

π s ν

−∞ ∗

from which Kramers-Kronig relations are obtained while H(−ν) = H (ν). For a Lorentzian

shape of g(ν), the phase-gain relation becomes

−

ν ν 0 γ(ν)

ϕ(ν) = ∆ν

1.2 Amplifier Pumping

The mechanics of pumping requires the presence of ancillary energy levels. For instance,

the population inversion between state 1 and 2 can be obtained more easily through the

pumping from 1 to 3, and then relying on the natural decay to 2. The pumping might

be achieved optically, electrically or chemically. For continuous-wave (CW) operation,

the rates of excitation and decay of the various energy levels must be balanced to maintain

→

the steady-state inverted population on the 2 1 transition.

The equations that describe the rates of change of population densities N and N are

1 2

known as rate equations. In general we define two rates, R and R , for the transport

1 2

of atoms out of level 1 and into level 2 respectively. Therefore the rate equations are

straightforward N dN N N

dN 2 2 1 1 2

− −R −

= R = +

2 1

dt τ dt τ τ

2 1 21

where the total and partial decaying times have been used

−1 −1 −1

−1 −1 −1

+ τ τ = t + τ

τ = τ sp nr

21 20 21

2

with τ a non-radiative spontaneous emission component, due to collisions of the atom.

nr

Under steady state conditions, the steady-state population difference N , in the absence

0

8 CHAPTER 1. THEORY OF LASERS [?]

of amplifier radiation, takes the form

τ 1

−

N = R τ 1 + R τ

0 2 2 1 1

τ

21

≈ ≈ ≈

Ideally one seeks τ t τ such that τ t and τ t . Under these conditions

21 sp 20 2 sp 1 sp

≈

N R t + R τ

0 2 sp 1 1

where the latter contribution in cancelled in the absence of depumping. Once we add the

amplifier radiation, the equations become more complex

dN N dN N N

2 2 1 1 2

− − −R −

= R N W = + + N W

2 i 1 i

dt τ dt τ τ

2 1 21

and, under steady-state conditions, the population difference takes the form

N τ

0 2

−

N = τ = τ + τ 1

s 2 1

1 + τ W τ

s i 21

where τ is known as saturation time constant, which is always positive.

s

1.2.1 Pumping Schemes

The key feature when creating a population inversion is to have the high energy state

2 to live long enough with respect to the others. In particular, we distinguish four and

three-level pumping schemes, pictured in the image below

The former is characterised by an instant decay from level 3 to 2, and R = 0 since

1

the pumping is made from a ground state. Under general assumptions, given a pumping

transition probability W , we can prove the results

t N W t

sp a sp

≈ ≈

N τ , N = N + N + N + N

0 s a 1 2 3 gs

1 + t W 1 + t W

sp sp

1.3. AMPLIFIER NON-LINEARITY AND NOISE 9

from which we see that, at increasing W , the maximum possible population difference is

achieved.

The three-level scheme, instead, under the same general assumptions on the decaying

rates used for the four-level, will show relations of the form

−

N (t W 1) 2t

a sp sp

N = τ =

0 s

1 + t W 1 + t W

sp sp

from which we see that a slight worst performance is obtained due to the multiplicative

factor of 2 in the saturation time constant.

1.3 Amplifier Non-linearity and Noise

Since the atomic medium won’t necessarily be at equilibrium, we have to take in mind

the flux growth as a cause of modification of the parameters introduced in the previous

chapters, thus exploiting its contribution to them. In particular, the gain coefficient can

be written as 1

γ (ν)

0 γ (ν) = N σ(ν), = τ σ(ν)

γ(ν) = 0 0 s

1 + φ/φ (ν) φ (ν)

s s ≈

where φ is known as the saturation photon-flux density. When τ t the interpretation

s s sp

of φ is straightforward: roughly one photon can be emitted during each spontaneous

s

emission time into each cross sectional area. Analogously, a given lineshape will cause a

saturated bandwidth s φ

1+

∆ν = ∆ν

s φ (ν)

s

A more real representation of the gain curve will be defined through the modified equation

−

dφ γ φ φ(z) φ(z) φ(0)

0 →

= ln + = γ z

0

dz 1 + φ/φ φ(0) φ

s s

from which we distinguish two regimes:

ˆ If both φ(0)/φ and φ(d)/φ are much smaller than unity, we obtain the simple

s s

exponential relation obtained before, but with γ instead of γ, meaning an indepen-

0

dence from the photon flux ≈

φ(d) φ(0) exp(γ d)

0

10 CHAPTER 1. THEORY OF LASERS [?]

ˆ

When φ(0) φ , instead, we can write

s N d

0

≈

φ(d) φ(0) + τ s

which is a heavily saturated condition, where the atoms of the medium are ”busy”

emitting a constant photon-flux density N d/τ .

0 s

ˆ Intermediate condition can only be solved numerically.

1.3.1 Amplifier Noise

Whereas the amplitude signal has a specific frequency, direction and polarization, the

noise associates is broadband, multidirectional and unpolarized. One way to accommo-

date this spontaneous emission noise is to replace the differential equation governing the

growth of photon-flux density 1 dΩ

dφ = γ(ν)φ + ξ (ν) ξ (ν) = N g(ν)B

sp sp 2

dz t 2π

sp

where Ω is the solid angle and B is a filter centred about ν.

1.4 Theory of Laser Oscillation

The laser is an optical oscillator, meaning that it comprises a resonant optical amplifier

whose output is fed back to the input with matching phase. As we have already shown,

the amplifier bandwidth is determined by the linewidth of the atomic transition, or by

an inhomogeneous broadening mechanism such as the Doppler effect. The amplified

frequency band will therefore be amplified till a saturation condition is reached, where

1.4. THEORY OF LASER OSCILLATION 11

the difference of occupation decreases till N = N /[1 + φ/φ (ν)] for an homogeneously

0 s

broadened medium.

Optical feedback is achieved by placing the active medium in an optical resonator.

A Fabry-Perot resonator, for example, consists of two parallel mirrors separated by a dis-

tance d, containing the medium (refractive index n) in which the atoms of the amplifier

reside. Travel through the medium introduces a phase shift per unit length equal to the

wavenumber πν

k = c

The resonator also include an absorption contribution which causes the photon flux to be

reduced by a factor R R exp(−2α d), where R and R are the reflectances of the two

1 2 s 1 2

mirrors, and α the attenuation coefficient. This total attenuation can be written through

s

a distributed loss coefficient α such that

r 1 1

−2α −2α

d d ln

e = R R e α = α + α , α =

r s

1 2 r s m m 2d R R

1 2

Thus, the photon lifetime in the cavity is τ = 1/α c. The resonator only sustains fre-

p r

quencies that correspond to a round/trip phase shift that is multiple of 2π. For an active

atom medium, this condition corresponds to modes of frequency

ν = qv q = 1, 2, 3, ...

q F

with v = c/2d and c = c /n.

F 0

The FWHM spectral width of these modes is ν F

≈

δν F

F

where is the finesse of the resonator, such that, under the approximation of large values

and small losses, can be written π

F≈ = 2πτ v

p F

α d

r

Having seen the above features, two conditions arise for the laser to oscillate. The former

12 CHAPTER 1. THEORY OF LASERS [?]

is the threshold gain condition, which requires the gain coefficient to be

γ (ν) > α

0 r

The second condition, instead, is a phase match equation of the form

2kd + 2ϕ(ν)d = 2πq q = 1, 2, ...

which frequency peaks differ slightly from the ones written above. For instance, under a

steady state condition and a slight deviation from the atomic media result with Lorentzian

lineshape, we can write δν

0 ≈ − −

ν ν (ν ν )

q q 0

q ∆ν

1.5 Characteristics of the Laser Output

Once the amplification and the phase match conditions are satisfied, the intensity starts

to grow up in the cavity, thus decreasing the gain coefficient till it is no more greater than

the absorption one, reaching a steady state condition. The smaller the loss, the greater

the value of φ. Figure 1.1: The result is then provided by

equating the large-signal (saturated) gain

coefficient to the loss one, which provides

γ (ν) −

0

φ = φ (ν) 1

s α r

Since this is the mean number of pho-

tons per second crossing an unit area in

both directions, the flux in a single direction

will be φ/2.

The steady state number of photons per unit volume inside the resonator is simply given

by n = φ/c. Only a portion of the steady-state internal photon-flux density determined

p

above leaves the resonator in the form of useful light. This output φ is that part that

0

propagates toward mirror 1 (φ/2) and is transmitted by it. Given the transmittance T

hνT φ

Tφ → I =

φ =

0 0

2 2

There exists an optimal transmittance which