UV Laser Source
4.5 Set-up of the Cavity
4.5.6 Impedance Matching
Impedance matching describes the optimization of the transmission of the incoupling mirror to maximize the energy circulating in the cavity.
The specific choice of the resonator’s dimensions (l,d,as) leads to a
longitudinal mode spacing, the free spectral range ∆νFSR(Tab. 4.3). Free spectral range, resonance width, finesse Due to the losses in the cavity (“resonator lifetime”) the resonator’s
resonance width δν is non-zero. The ratio between free spectral range and resonance width is called the cavity’s finesse
F = ∆νFSR Hereinα contains all possible losses e. g. absorption of the crystal, diffraction, conversion, and losses of the mirrors, except that of the incoupling mirror M1 which is described by its transmission T. The transmission of the incoupling mirror is of essential importance for the optimization of the power circulating in the cavity.
Maxi-mum enhancement is obtained, if the round trip losses of the cavity, Meaning of impedance matching α, are compensated by the transmissionT of the incoupling mirror.
This situation is calledimpedance matching
T =α . (4.61)
In the situation of optimal mode- and impedance matching, the light from the pump laser reflected on the backside of the incoupling mirror interferes with light escaping from the cavity destructively.
With respect to sum-frequency generation, due to conversion, the losses of the green light depend on the circulating power of the red light and vice versa (4.11) & (4.46). This leads to coupling equa-tions, which are discussed in this section. We will give both, an exact numerical treatment of the equations, and a handy approxi-mation to quickly find the optimal transmissions of the incoupling mirror.
Look at the situation given in Fig. 4.9. We introduce the complex electric field amplitudes given by|Ei|2 =Pi and|Ec|2 =Pc, and the amplitude transmission and reflection coefficients |t|2 = T, |r|2 = 1−T, and |r0|2 = 1−α. The circulating electric field amplitude is then described by
Ec =tEi+eiφr0rEc, (4.62) whereφ is the phase of the light wave after one round trip.
With P =|E|2 we find the enhancement factor between the circu- Power enhancement of the pump waves lating power and the power before the cavity
Pc
Pi
= |t|2
(1−r0r)2+ 4r0rsin2(φ/2) = |t|2
(1−r0r)2 , (4.63)
a Pc Pi
T
Figure 4.9: Schematic situation of impedance matching.
where in the last step resonance was assumed, sin (φ/2) = 0. Re-placing the amplitude variables with those corresponding to the intensity, we find
Pc
Pi = T
³1−p
(1−α)p
(1−T)´2 ≈
(T,α¿1)
4T
(α+T)2 . (4.64)
Assuming impedance matching (T =α), we find the relation PPc
i = F/π. However, this is not directly applicable to our situation, where the lossesαfor the one wave depend on the circulating power of the other wave, and vice versa.
The mirrors of our resonator (Fig. 4.7) were produced by Layertec GmbH. The mirrors M2, M3, and M4 have the same customized HR HR HT coating: high-reflection for the fundamental waves at 532 nm and 760 nm, and high-transmission at the sum-frequency 313 nm. All data necessary to calculate the cavity’s round-trip losses are collected in Tab. 4.4.
Internal losses in the cavity
532 nm 760 nm 313 nm
THR 0.16 % 0.13 % 92.8% Transmission of the mirrors M2, M3, M4
RHR &99.84 % &99.87 % Reflection, respectively
RcA 0.6 % 0.14 % <2 % Reflection from the crystal, side A
RcB 0.75 % 0.09 % <2 % ” , side B
αc 0.035. . .0.31 Absorption in the crystal (Tab. 4.1) ENLSFG ≈0.375/kW Conversion (4.51), Tab. 4.3
Table 4.4: Data of internal losses in the cavity, i.e. all losses except those of the incoupling mirror. The mirrors were measured in transmis-sion and 13◦ incidence. The crystal’s reflection at 313 nm could not be measured and is specified by the manufacturer.
As above, we now use the subscripts 1,2 to distinguish different values of a specific variable for the two fundamental waves. With the data from Tab. 4.4 we then find the wavelength dependent loss Loss rates
4.5 Set-up of the Cavity 87
rates for our specific cavity α1,2 = R3HR 1,2 ¡
In this equation, all losses are collected in RR1,2, except those of conversion. The loss due to conversion is derived from (4.46) and the fact that for each UV photon one photon from each fundamen-tal wave is required. The latter leads to the factor ω1,2/ω3 in the conversion loss term, as also seen from (4.11).
The circulating powers of the fundamental waves in the cavity are Circulating powers now given by the coupled equations
Pc1 = κ001 ·Pi1 ·T1
where the additional parameterκ001,2 quantifies the mode matching of the fundamental laser light with the corresponding TEM00cavity mode. For simplicity we assumeκ001,2 = 1, and correct the measured laser power before the cavity by the effective incoupling efficiency to determine Pi1,2.
From (4.46) we see, that the output power of the sum-frequency is
proportional to the productPc1·Pc2. For impedance matching we Impedance matching need to optimize the transmissions of the incoupling mirror T1,2 to
maximize the productPc1·Pc2.
In general, the circulating powers can be found by numerical solu-tion of the two coupled equasolu-tions (4.66). The SFG outputP3 can then be calculated. Before quantitative analysis is presented, some handy approximations are derived in the following to evaluate the optimal T1,2 analytically without using computers.
Impedance Matching for Small Losses and Small Internal Efficiencies
We rewrite the circulating powers (4.66) in the form rT1Pi1
and a corresponding expression holds for Pc2 after subscript ex-change 1↔2.
The optimum values can be calculated explicitly in the case of small losses αi ¿1 and small internal efficiencies ENLSFGPc1,2 ¿1, when the square root in equation (4.67) can be approximated.
We define the normalized powers εi and the function f(ε1, ε2) as
is emitted when the input transmissions are chosen as follows at . . .
We see that the impedance matched output power P3opt is crucially dependent on the productENLSFG/(α1α2). In the regime of low input powers, ε1,2 ¿1, we have the simple result
P3opt =ENLSFGPi1Pi2/(α1α2), (4.71) in the limit of one weak and one strong fundamental wave, say ε1 À1, ε2 ¿1, we have full conversion of the weak wave and find P3opt =Pi,2ω3/ω2. (4.72) Results of Impedance Matching
To evaluate the transmissions of the incoupling mirror we calculate the losses in the cavity by the specification of each optical element given in Tab. 4.4 and find the cavity losses RR532 = 1.9% and RR760 = 0.67%. The fundamental laser powers coupled into the cavity are assumed to be equal to their actual values used in the experiment Pi532 = 0.88 W and Pi760 = 6.4 mW (→4.7).
The resulting dependence of the sum-frequency output power as a function of the incoupling mirror transmissions is shown in Fig. 4.10.
The optimal incoupling transmissions are found to beT532opt = 1.88%
andT760opt = 1.39%, giving a theoretical maximum UV output power of P313opt =8.1 mW.
Further discussion of the impedance matching and comparison with the experimentally achieved output power follows in (→4.7).
4.6 Stabilization 89
Figure 4.10: SFG output power versus the incoupling mirror trans-missions for the assumed pumping power of Pi532 = 0.88 W and Pi760= 6.4mW. The plots shows a point of optimal transmission and a steep drop of the output power for small transmissions. Left: 3d plot of the SFG output power (z-axis) as a function of the transmissions of the incoupling mirror (x−y plane). Right: SFG output power as function of a single transmission of the incoupling mirror, while the transmission of the other fundamental wave is matched for maximum output power.
4.6 Stabilization
The resonance condition in a running wave cavity – like ours – is given by n·λ = U, where n is an integer value: the cavity length is an integer multiple of the resonated wavelength. This can be achieved either by adjusting the length of the cavity or by tuning the wavelength of the laser. In our doubly resonant set-up, where the resonance condition has to be fulfilled for two waves, we exploit both possibilities of resonance adjustment.
To obtain spectrally pure and frequency-stable 313 nm light suit-able for high-resolution spectroscopy, we make use of ultra-narrow linewidth and the high intrinsic frequency stability of the 1064 nm master laser in the MOPA and transfer it to the UV by using the
doubly-resonant cavity as a transfer resonator. To this end, the Using the cavity as a transfer resonator cavity is frequency-locked to the 532 nm laser, and the 760 nm
diode laser is frequency-locked to the cavity. This transfers the fre-quency stability and other spectral properties to the cavity, thus to the red diode laser which is stabilized to the cavity, and finally to the generated sum-frequency light.