Thermal Effects

The previous consideration of the OPO stability left out absorption inside the nonlinear medium. This effect causes a rise in temperature along the propagation axis of the Gaussian laser beam. Regions of the crystal exposed to a higher field intensity have a higher temperature than those exposed to a lower intensity. This temperature gradient

Chapter 4. Nonlinear Optics

leads to a radially changing index of refraction. Instead of a free propagation inside the nonlinear medium, the beam gets focused by the thermal lens. The ABCD matrix associated with this duct is^[105] The ray propagating along the z axis experiences the refraction indexn₀, while a ray at an off-axis transverse position r experiences a different index of refraction, which depends on the second derivative n⁰⁰₀ = n₂. In general, it is challenging to determine the axial temperature dependence of this matrix. Thus, some authors^[106] prefer instead to use a thin lens matrix (Equation (4.35)) in the middle of the crystal with the focal length^[106,107]

f_T = πK_c αP_c(dn_s/dT)

w²

L_c. (4.44)

Crystal heating measurements give an estimation for the absorption coefficientα≈0.08 % of a PPLN crystal at λs = 1611 nm^[106,108]. The thermal conductivity of a MgO-doped (5 mol%) lithium niobate crystal isK_c= 4.02 W/mK^[106]and the temperature dependence of the crystal refractive index is dn_s/dT = 5·10⁻⁶/K^[109]. Assuming a signal power of Pc = 50 W and a crystal length ofLc= 50 mm results in a focal length about fT ≈4 mm.

In the case of linear cavities, this might cause bi-stability^[106]. A precaution against the thermal lens is a tightly focused beam (ξ ≈2) for higher pump depletion^[106]. This method is counter-intuitive since a large waist decreases the power density and increases the focal length of the lens (Equation (4.44)). Additionally, the dimension d₁ might be chosen slightly larger relative to the cold cavity stability center, to consider the thermal lens in advance^[110]. However, a thermal lens is of less importance for ring cavities, due to a lower temperature increase. This statement is valid as long as the absorption of the idler beam is negligibly small. Consider a MgO-doped (5 mol%) PPLN for spectroscopy in the mid infrared between 2.5µm and 4µm, with a discrete set of poling periods of different length Λ_n. A typical set of seven poling periods for a mid-IR OPO goes, for example, from Λ₁ = 31.5µm to Λ₇ = 28.5µm in steps of 0.5µm (Figure 4.14a). Estimating the phase matching condition for planes waves with Equation (4.30) yields

1

Λ_n = n(λp, T)

λ_p − n(λs, T)

λ_s −n(λi, T)

λ_i , (4.45)

where n(λ, T) is the wavelength and temperature dependent refraction index of a MgO-doped (5 mol%) PPLN crystal^[111]. The pump wavelengthsλ_p = 1064 nm leads to a signal wavelength of λs= (1/λp−1/λI)⁻¹. For producing the 2.7µm to 2.9µm light needed to excite the fundamental vibrational transitions of OH, only the two longest poling periods with Λ₁ = 31.5µm and Λ₂ = 31.0µm satisfy the phase matching condition at reasonable

4.4. Optical Parametric Oscillator

Figure 4.14:(a) Theoretical temperature dependence of the phase matching condition of a MgO-doped (5 mol%) PPLN crystal^[111]. The congruent poling periods are Λn= 32µm−n·0.5µm withn∈[1,2,··,7]. (b) Only two poling periods are suited for nonlinear conversion to target wavelengths around 2.7µm and 2.9µm driving a 1 photon and 2 photon transition in the OH, respectively. In the vicinity of the target wavelengths, there is also an absorption feature in the crystal around 2.829µm^[106].

temperatures (Figure 4.14b). Concerning the idler absorption inside the PPLN crystal, two wavelength regions are worth mentioning, the phonon absorption above 4µm and an absorption peak around 2.829µm^[106]. At the peak of the absorption feature, a 5 cm long crystal absorbs more than 80 %^[106]. Without the MgO-doping of the crystal the absorption peak would overlap with the wavelength needed for two-photon vibrational spectroscopy on OH^[112](Figure 4.14b). Fortunately, the MgO-doping shifts the absorption line in between the two wavelengths of interest for OH spectroscopy.

4.4.2.1 Spectral Instabilities

The temperature distribution inside the crystal plays an important role in selecting the phase matching bandwidth. Additionally, a temperature rise caused by absorption leads to spectral instabilities. Above a critical pump level, the OPO signal line width might experience a broadening^[113]. Increasing the pump power further, the OPO starts to emit multiple modes over a frequency range larger than the bandwidth of the gain profile.

One reason is spontaneous Raman scattering, caused by phonons inside the crystal^[114,115]. At high powers, this leads to a stimulated Raman scattering of the signal wave, adding satellite peaks to the spectrum^[113]. The frequency shift of Raman lines relative to the signal frequency is constant and independent of the poling period. However, some modes change monotonically with the poling period of the PPLN crystal. These modes are associated with cascade optical parametric oscillations^[116]. The ideal OPO operation describes the conversion of a pump wave λ_p into an idler λ⁽¹⁾_i and a signal wave λ⁽¹⁾s

(Section 4.0.3). The wave vectors of the pump and the idler wave are both in the forward

Chapter 4. Nonlinear Optics

direction (Figure 4.15a). At high powers, however, the signal acts as pump wave itself,

(a) (b)

Figure 4.15:(a) Schema of the quasi wave vector mismatch for the ideal OPO operation. (b) Conversion to an additional wave withk⁽²⁾s caused by backward oscillation of an idler wave withk⁽²⁾_i (adapted from^[116]).

which is converted into a second signal wave with λ⁽²⁾s (Figure 4.15b). The frequency difference between the initial and the additional signal waves depends on the phase matching condition. In contrast to the previous wave vector orientations, the idler can also propagate in the direction opposite to the signal and pump, denoted as a parametric backward oscillation (Figure 4.15b). As an example, forλ⁽¹⁾s = 1.7µm and a period length of Λ₂ = 31µm, the secondary wavelengths of signal and idler are aroundλ⁽²⁾s = 2.8µm and λ⁽²⁾_i = 4.3µm. Those wavelengths are in general unwanted. A simple solution to this issue along with the Raman scattering is the reduction of the intra-cavity signal power. It is convenient to couple the signal power out with one partly transmissive mirror. Optimum OPO operation of a bow tie cavity has been observed at a signal wave output of around 3 %^[108]. The suggested signal output of a linear cavity is 4 % or larger, to ensure reliable single mode operation^[106]. The limit of the circulating power inside the cavity additionally increases the idler power, as well enhances the beam quality factor M², corresponding to an ideal Gaussian beam^[108].