Frequency Selection

Consider the OPO cavity is stable at single-mode emission. Now it is convenient for spectroscopic measurements, to tune the lasing frequency. A coarse way to change the emission frequency is increasing or lowering the temperature of the PPLN crystal. One approximation of the phase matching gain profile is

I = sinc²

2πLc

Λ(λ_p, λ_s, T)− 2πLc

Λ₂

. (4.46)

The following calculations assume the bow tie cavity design of Section 4.4.1, a PPLN crystal of length L_c = 5 cm and a period length Λ₂ = 31µm. The temperature and wavelength dependent period length Λ(λp, T) is calculated with Equation (4.45). Thus, the resulting phase matching gain profile has a full width at half maximum (FWHM) around 100 GHz (Figure 4.16a). A stable single-mode OPO lases at the cavity mode closest

4.4. Optical Parametric Oscillator

Figure 4.16: (a) Schema of the frequency selectivity inside an OPO cavity. The phase matching profile of the PPLN crystal allows a coarse tuning of the lasing wavelength. (b) A more precise frequency selectivity is supplied by a thin etalon. The material and the thickness of the etalon determines the final gain profile.

to the highest gain. The cavity modes are separated by the free spectral range

FSR = c

nd_total = 428.46 MHz, (4.47)

with the refraction index in air n ≈ 1 and the total optical round trip path length d_total = 699.7 mm.

The phase matching condition of the PPLN supplies nearly no frequency selectivity (Figure 4.16b). Without frequency selective elements inside the cavity, the lasing mode is defined by parasitic etalon effects or absorption features caused by impurities of the nonlinear medium. These spurious frequency selective losses modify the gain profile and create local maxima and minima^[106]. The lasing mode can stay at the absorption related maximum of the gain medium and resist temperature fluctuations of the PPLN crystal of as much as 100 mK^[106]. Thus, thermal-locking cancels noise fluctuations induced by temperature changes, but also prevents tuning of the OPO signal frequency. However, frequency tuning can be achieved by adding an etalon into the OPO cavity^[117]. The low etalon thicknessdallows a significant spacing between the transmitted interference fringes.

The second important parameter of a Fabry-Perot interferometer like the etalon or OPO cavity is the coefficient of finesse F = 4R/(1−R)², which is a measure of the spectral width of the modes inside the interferometer and increases with the reflectivityR. A higher reflectivity of the cavity mirrors leads to a narrower mode profile. The reflectivity at the etalon surfaces is dependent on the refraction indexn of the material and the polarization state of the light wave. In the case of p-polarized light, the reflection coefficient is given by the Fresnel equation

Chapter 4. Nonlinear Optics

Snell’s law sinθ_o =nsinθ_i describes the relation between the angle outside the medium θ_o and the angle inside the medium θ_i. Since the reflectivity of the etalon surfaces is much lower than the reflectivity of the cavity mirrors, the finesse of the etalon is several orders of magnitude smaller than that of the cavity. It is convenient to calculate the transmission profile of an etalon to illustrate the influence of different etalon materials and thicknesses.

The transmission of an etalon is described in approximation by the Airy function^[118]

I = 1

1 +F sin²(φ/2) with φ= 4π

λ ndcos(θ_i). (4.49) However, to study the etalon in more detail, it is necessary to calculate the transmission for a Gaussian electromagnetic field (Equation (4.32)). First, the incident beam gets refracted at the etalon surface. The propagation direction inside the etalon changes according to Snell’s law. Some fraction of the beam experiences a back and forth reflection inside the etalon, which introduces the walk-off X = 2dsin(θ_t) relative to the initial beam. The magnitude of the walk-off depends critically on the etalon thickness d and the internal reflection angleθ_i of the etalon. Multiplem ∈Nof this steps lead to the total transmitted electric field^[118]

E_t(x, y, z) = X∞ m=0

(1−R)R^mE_m(x_m, y, z_m), (4.50) with x_m =x−mX and z_m =z+ 2mdcos(θ_i). As an example, the transmissions curves have been calculated for a 800µm thick silicon etalon and a 3 mm thick YAG etalon at a zero incidence angle (Figure 4.16a). The higher refractive index n_Si ≈3.4^[119] of silicon relative to the refractive index n_YAG ≈ 1.8^[120] of YAG leads to a higher finesse of the silicon etalon. Thus, transmission fringes are more pronounced (Figure 4.16a). However, the smaller FSR_YAG=c/(2nd) = 27 GHz of the YAG etalon relative to the silicon etalon FSRSi≈54 GHz partly compensates for it. The resulting mode selection inside the cavity improves with both etalons (Figure 4.16b). The silicon etalon is here superior to the YAG etalon.

Assuming the lasing mode is successfully selected with the etalon, the next step is to find a way to tune the frequency of the etalon’s transmission maximum. One possibility is tiling the etalon, which changes the optical pathway inside. For example, the calculated frequency tuning of the YAG etalon is around 8 GHz/degree², and the frequency tuning of the silicon etalon is around 2.2 GHz/degree². Therefore, the silicon etalon is slightly better suited for angle tuning. However, with increasing incidence angle the transmission of the etalon decreases. The beam walk-off inside the etalon causes the lower transmission.

It becomes critical for thick etalons and small beam waists. A one percent reflection level is reached at an incidence angle of 2.4°for the YAG etalon, while the thinner silicon etalon reaches the same reflectivity at 6.8°. Considering a power of around 100 W is circulating

4.4. Optical Parametric Oscillator inside the cavity, a one percent reflection at the etalon would result in an additional Watt of power being coupled out of the cavity. Additionally, the etalon tilt distorts the beam profile and slightly changes the alignment of the cavity. Thus, it is convenient to avoid potential large reflections and alignment issues by changing the etalon temperature instead of the tilt angle. This calculation requires consideration of the temperature dependent change of thickness give by the refraction index dn/dT of YAG^[121] and silicon^[119], respectively.

Equally important is the change of the thermal expansion coefficient α(T) for YAG^[121]

and silicon^[122], respectively. The approximate temperature tuning coefficient for the YAG etalon is around 2 GHz/K and for the silicon etalon around 9.4 GHz/K. The frequency selectivity of the OPO, in this thesis, relies on the 3 mm thick YAG etalon (Section 9.5).

Chapter 5

Frequency Stability and Stabilization

The frequency stability of the laser system relies on the neodymium-doped yttrium alu-minum garnet (Nd:YAG) reference laser (Chapter 3) and the various other frequency references used to stabilize it further. In this chapter, I will first define stability and then introduce the various frequency standards required for stabilizing and monitoring of the laser frequency. Transferring the stability from a frequency standard onto the Nd:YAG laser depends on the bandwidth of the used feedback loop. Although this bandwidth limitation is no issue for the inherent stable reference laser, it is important for a further stability transfer onto the optical frequency comb (OFC). Two techniques of measuring the feedback bandwidth are presented at the end of this chapter.

5.1 Definition of Stability

Almost any oscillator suffers an irregular variation of the amplitude or frequency. Under-standing the origin of potential noise sources is a cornerstone of building a high-resolution measurement system. In general, the output signal of a modulated real oscillator is ex-pressed as^[123]

U(t) =U₀(t) cosϕ(t) = [U₀ + ∆U₀(t)] cos (2πν₀t+φ(t)], (5.1) with the nominal amplitude U₀ and its small perturbation ∆U₀(t). For simplicity, the perturbation of the amplitude ∆U₀(t) is assumed to be zero. The derivative of the phase ϕ(t) determines the instantaneous frequency^[123]

ν(t) = 1 2π

dϕ(t) dt = 1

2π 1

dt [2πν₀t+φ(t)] = ν₀+ 1 2π

dφ(t)

dt =ν₀+ ∆ν(t). (5.2) Thus, the instantaneous frequency deviation from the ideal oscillator at frequency ν₀ is

∆ν(t). It is convenient to replace the small value of ∆ν(t) with the instantaneous fractional

5.1. Definition of Stability

A perfect, instantaneous measurement ofy(t) is generally not possible. A frequency counter, for example, must measure multiple periods of the waveform, resulting in a sample time τ. Therefore, the fractional frequency deviation becomes a discrete series of consecutive measurements

The individual measurements differ from each other and are, for simplicity, uncorrelated.

The mean value is

and the N-sample variance is given by^[124]

σ²(N, τ) = 1 with the ensemble size N of consecutive measurements y_i, corresponding to an measure-ment timeτ of each sample. The dead time between individual measurements y_i is zero.

However, if the measurement time τ is shorter than the correlation time between two consecutive measurementsy_i, the calculation of the mean value and the variance leads to difficulties. Concerning the description of a frequency standard, consecutive measurements are potentially correlated due to a frequency drift, which increases the variance with the sample size. In contrast, the correlation of two consecutive measurements reduces the variances for short data sets. Thus, the sample size N is crucial in describing the stability of the system.