Comparison of simulation with experiment

A comparison of the count rates for Cs and Mg reveals a difference of up to four orders of magnitude between the experiment and the simulation. The reason for this difference may be attributed to over estimated experimental parameters entering the calculations and systematic effects which are not taken into account, but reduce the count rate. Possible effects are discussed in the following.

Excitation geometry and alignment

Misalignments in the pulse overlap in form of phase front tilting result in destructive interference of the back and forth propagating electric fields. To get an expression for the associated signal decrease consider the two-photon transition rate which is proportional to the squared absolute value of the product of the forth and back propagating electric field

Rge ∝ |E1(t, rrr)E2(t, rrr)|2 (6.15) Assuming two plane waves of same frequency and amplitude but with different phases

ϕ1(rrr) and ϕ2(rrr), E1,2(t, rrr) = E(t) exp [−iϕ1,2(rrr)], equation (6.15) can be rewritten to

Rge ∝ I2(t) cos4 ϕ1(rrr) +ϕ2(rrr) 2 , (6.16)

where _I(t) = (1/2)ε0cE(t) is the time dependent intensity in one beam and ϕ1,2(rrr) =

k

kk1,2rrr is a real spatial phase. The wave vectors may be written in spherical coordinates

k

kk1 = (k1sin(θ1) cos(φ1), k1sin(θ1) sin(φ1), k1cos(θ1)) ,

kkk2 = (−k2sin(θ2) cos(φ2),−k2sin(θ2) sin(φ2),−k2cos(θ2)) .

and rrr = (x, y, z). Assume a by an angle θ2 tilted back propagating wave (φ2 = 0) while the forth propagating beam travels along the z direction (θ1 =φ1 = 0). On laser axis (x =y = 0) the total phase amounts toϕ1(rrr) +ϕ2(rrr) = −kz(1−cos(θ2)) setting

z =c/(2νr) yields Rge ∝ I2(t) cos4 − πc 2νrλ2ω (1₋cos (θ2)) (6.17) According to this relation a small angle detuning of 0.5 mrad results in a signal decrease by two orders of magnitude. Furthermore a misalignment in the position of the pulse

overlap with respect to the focus position results in a reduced laser intensity due to larger beam diameter. This will further reduce the count rate. These effects may partially explain the strong deviation between the simulation and the experiment in table 6.2.

Using a symmetrical spectroscopy resonator of appropriate length, as is the case in Mg and H spectroscopy, the counter propagating pulses overlap at the center of the cavity. However, while propagating in the cavity the two laser beams acquire a spatial phase. The corresponding maximum tilt angle of the laser beam can be approximated byθmax =λ2ωπ/(2wF) where 2wis the beam diameter assuming a collimated beam and F is the cavity finesse which can be approximated by_F = 2πU withU the enhancement of the cavity. With a measured enhancement ofU = 2 the maximum tilt angle amounts to θmax = 0.1 mrad resulting in a signal decrease according to equation (6.17) by only 1 %.

Number of atoms

Uncertainties in the calculation of the number of atoms interacting with the laser also affect the calculated count rates. Magnesium, for example, is heated in an oven to gen- erate an atomic beam (see chapter 5), so only the temperature of the oven is accessible here, while the actual temperature of the Mg atoms might be somewhat different. De- pending on the quality of the contacts different temperatures are measured while the heating current is the same. Here variations in the temperature by only 1.6 % result in particle density variations by around 60 %.

For hydrogen a big uncertainty arises from fluctuations in the degree of dissociation which strongly depend on the treatment of the copper nozzle and the PTFE tube. Imperceptive differences in the handling have big impact on the recombination rate. But once cleaned properly the parts can be used for weeks of measurements until they are exposed to air for a longer time.

Envelope position and fluctuations

As discussed in section 2.4 and 4.3.1 the count rate is maximized for a centered spectral pulse envelope. Detuning the spectrum with respect to the transition frequency results in an exponential decrease of the signal with squared detuning. Using an OSA the spectral envelope at the fundamental wavelength can be adjusted within 0.003 nm. This small uncertainty results in a signal decrease by less than 1 % at the fundamental as well as at the second and fourth harmonic and thus can be neglected.

By monitoring the position of the spectral pulse envelope at 820 nm small jumps and drifts of the envelope could be observed from time to time during the operation. These effects may probably be attributed to atmospheric turbulences and convections inside the laser resonator. Albeit one can manually compensate for the drift, the jumps cannot be compensated and could cause additional statistical fluctuations in the signal.

Another serious problem comes along with the SHG in the BBO crystal. Due to the high GVM of 6.5 ps in a 5 mm long crystal only a part of the incident spectrum at 410 nm can be frequency doubled. Approaching the phase matching cut-off of the nonlinear crystal, which is at 204.7 nm for BBO, the nonlinear coefficient decreases rapidly. Thus, while maximizing the output power of the UV pulses by adjusting the phase matching angle of the crystal their spectrum might be shifted to somewhat higher wavelengths. This effect can be avoided by using a UV spectrometer while aligning the resonator.

Residual chirp

As discussed in section 2.3 and 4.3.2 the signal strength drops with the inverse time bandwidth product if the pulses posses a symmetric phase shift around the two-photon transition frequency. A characterization of the fundamental, SH and FH pulses in sec- tion 3.3 revealed close to transform limited pulses. A residual chirp of the SH and FH pulses may be approximated by an upper limit for the TBP of 0.34 resulting in a signal decrease by around 8 % for Mg and H.

Fresnel losses

The detection efficiency given in table 6.2 accounts only for the quantum efficiency, trans- missivity of the filters and detected solid angle while Fresnel, scattering and absorption losses of the imaging optics, the gas cell and the detection windows are neglected. In Cs and Mg spectroscopy the interaction region is focused onto the cathode of the PMT using two AR coated focusing lenses. Due to the AR coating the Fresnel reflections at their surfaces are below 0.5% resulting in a maximum loss of around 2 %.

A much higher contribution is expected to come from the vapor-cell / detection window of the spectrometer. Cs spectroscopy is carried out in a gas-cell at room temperature. Cs depositions can be observed at the inner surface of the cell which reduce the transmis- sivity of the cell for the fluorescence light at 456 nm. In Mg spectroscopy the emitted fluorescence light is detected through a CaF2 window in the vacuum chamber. After some time a thin Mg layer appears on this window reducing continuously its transmis- sivity during a measurement run. Between the runs this window is cleaned. In contrast, in H spectroscopy the PMT’s are placed inside the vacuum chamber and the imaging optics are omitted to avoid additional losses.

In Cs and Mg spectroscopy the laser beam suffers losses through focusing lenses and laser windows of the vapor-cell and the vacuum chamber. Due to these losses the back and forth reflected beams are of unequal intensity. To avoid these losses the 1S₋3S

spectroscopy resonator is built inside the vacuum chamber and only reflective optics are used.