A highly stable monolithic enhancement cavity for SHG generation in the UVS. Hannig,1 J. Mielke,2 J. A. Fenske,1 M. Misera,1 N. Beev,1, a) C. Ospelkaus,1, 2 and P. O. Schmidt1, 2, b)1)Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig,Germany2)Institut fur Quantenoptik, Leibniz Universitat Hannover, Welfengarten 1, 30167 Hannover,Germany

(Dated: 28 September 2017)

We present a highly stable bow-tie power enhancement cavity for critical second-harmonicgeneration into the UV using a Brewster-cut -BaB2O4 (BBO) nonlinear crystal. The cavitygeometry is suitable for all UV wavelengths reachable with BBO and can be modified to ac-commodate anti-reflection coated crystals, extending its applicability to the entire wavelengthrange accessible with non-linear frequency conversion. The cavity is length-stabilized usinga fast general purpose digital PI controller based on the open source STEMlab 125-14 (for-merly Red Pitaya) system acting on a mirror mounted on a fast piezo actuator. We observe130 h uninterrupted operation without decay in output power at 313 nm. The robustness ofthe system has been confirmed by exposing it to accelerations of up to 1 g with less than10% in-lock output power variations. Furthermore, the cavity can withstand 30 minutes ofacceleration exposure at a level of 3 grms without substantial change in SHG output power,demonstrating that the design is suitable for transportable setups.

PACS numbers: 42.60.Da, 42.60.Lh, 42.60.Pk, 42.62.Eh, 42.65.Ky, 42.79.Nv

I. INTRODUCTION

Todays quantum optics experiments require a broadrange of laser frequencies. Usually, several continuouswave laser systems are required per experiment13 whichimposes tight bounds on the reliability and cost per sys-tem. Recent progress in the development of quantum sen-sors has led to systems outperforming classical devices,such as quantum gravimeters4,5, and clocks exceeding1017 accuracy2,6,7. New applications such as relativisticgeodesy8 using transportable optical clocks911 or takingadvantage of the long interaction times in atom inter-ferometry experiments in a microgravity or even spaceenvironment1214 have emerged. For these applications,quantum optics experiments need to be operated out-side highly-specialized laboratories, increasing the de-mands in terms of mechanical robustness of the opti-cal setups. Small and at the same time reliable lasersources are available only for a restricted wavelengthrange15,16. To reach other wavelengths, non-linear con-version processes, such as sum- or difference frequencyor harmonic generation1725 are typically employed. Acommon approach to generate the desired wavelength issecond harmonic generation26 (SHG) of External Cav-ity Diode Lasers (ECDLs) or fiber lasers. The frequencydoubling of infrared (IR) lasers has been demonstrated insingle-crystal monolithic ring cavities2729. With the ad-vent of commercially available periodically-poled wave-guide doublers, e.g. based on Lithium Tantalate orLithium Niobate nonlinear crystals, wavelength conver-sion into the blue spectral range has been achieved incompact setups30. Monolithic ring cavities have beendemonstrated to produce blue light down to wavelengths

a)current address: CERNb)Corresponding author: Piet.Schmidt@ptb.de

of 429 nm3133. However, to our knowledge neither single-crystal monolithic ring cavities nor modules are availablefor UV generation below 350 nm. For these UV wave-lengths SHG in nonlinear crystals such as Beta-Barium-Borate (BaB2O4, BBO), placed in an optical enhance-ment resonator, are typically employed34. Commerciallyavailable systems usually come with restricted flexibility,e.g. lacking access to the SHG light internally reflectedat the output facet of Brewster-cut nonlinear crystals.In contrast, self-built systems made from off-the-shelvecomponents usually lack mechanical stability and relia-bility.

Traditionally, the length of the cavity is kept interfer-ometrically stable by displacing one of the cavity mir-rors in a proportional-integral (PI) feedback loop usu-ally implemented using analog electronics. With theavailability of fast general-purpose digital hardware suchas Field Programmable Gate Arrays (FPGAs) and/ormicrocontroller-based systems it has become popular toemploy digital feedback controllers instead3539. Thesedevices are more universally applicable, feature higherusability, and can easily be linked to existing experimen-tal control infrastructure.

Here we report on a mechanical monolithic bow-tiecavity design40 for critically phase-matched SHG gener-ation in a BBO crystal and its general purpose digital PIlocking electronics based on a modified STEMlab 125-14application41,42. The cavity is implemented in a robustmonolithic mechanical support frame and equipped witha minimal set of high-quality adjustment screws that areaccessible from outside. The cavity geometry is suitablefor the generation of all wavelengths accessible with BBOin SHG ooe-processes, starting at 204.8 nm43. The inter-nal reflections of the pump light (PL) at the fundamentalfrequency and SHG on the crystals facets are accessiblethrough two additional windows. In order to preventmoisture-induced fogging of the crystal, a dry purginggas, such as nitrogen or oxygen can be applied inside the

arX

iv:1

709.

0718

8v1

[ph

ysic

s.in

s-de

t] 2

1 Se

p 20

17

mailto:Piet.Schmidt@ptb.de

2

sealed cavity.

We measure a locking bandwidth of 17 kHz and demon-strate 130 h continuous operation in lock for the conver-sion from 626 nm to 313 nm without substantial decayin the output power. Moreover, the cavity remains inlock while being exposed to accelerations in the verti-cal direction of around 1 g. A comparison of the outputpower before and after a 30 min high-acceleration shak-ing in one dimension demonstrates that the cavity canwithstand a truck transport according to ISO13355:2016without significant change in alignment.

The paper is structured as follows: Section II brieflysummarizes the theory of SHG and bow-tie power en-hancement cavities. In Section III we describe the mono-lithic cavity design and in Section IV the setup of thecomplete system including the STEMlab-based lockingelectronics is described. The results on acceleration test-ing and long term stability are presented in Section V.

II. THEORETICAL BACKGROUND

When light passes through a nonlinear medium, SHGlight of twice the frequency can be generated. The non-linear conversion process scales with the square of thePL intensity. In the case of low conversion efficiencyand therefore undepleted PL, the SHG power PSHG scaleswith the square of the pump light power PPL:

PSHG = P2PL, (1)

where is the conversion coefficient. Efficient SHG gen-eration is only achieved if the wavevectors of the pump

light ~kPL and the SHG ~kSHG fulfill the phasematchingcondition

~kSHG = 2~kPL. (2)

In BBO this can be achieved by taking advantage of thecrystals birefringence with different refractive indices noand ne for different incident polarisations. By choosingthe appropriate phase-matching angle pm within the re-sulting index ellipsoid, nSHG = nPL can be achieved

44.For SHG of 626 nm in BBO the resulting phase-matchingangle is

pm = 38.4. (3)

A. SHG power optimization

The single-pass conversion efficiency of cw light in aBBO crystal of a few mm length general is rather low(on the order of 104 1/W). Therefore, an enhance-ment cavity for the PL is built around the nonlinearcrystal26,45,46. The optimization of the cavity SHG out-put power can be divided into two steps: first the opti-mization of the power generated per single pass of thePL through the crystal47 and second the optimization ofthe power enhancement48.

1. Single pass optimization

As mentioned above, the SHG power generated per in-finitesimal crystal volume is proportional to the squareof the PL intensity. Tighter focusing of the PL enhancesthe intensity at the focus, at the expense of lower in-tensity away from the focus. This tradeoff for a givencrystal length l has been investigated for the propagationof a circular Gaussian beam with minimum waist w0 byBoyd and Kleinman47, who derived an optimum focus-ing ratio of l/b = 2.84 in the absence of birefringence,where l is the crystal length and b = w20k the confocalparameter with k = 2/. However, in the presence of

birefringence, the phase vector ~k and Poynting vector ~Sof the extraordinary SHG wave are in general not paral-lel, which results in a walk-off angle % between SHG andPL beam of

% = 4.6 (4)

for SHG of 626 nm in BBO. This effect is quantified bythe walk-off parameter B47:

B = %lk/2 (5)

A strong walk-off (such as in BBO), requires weaker fo-cusing of the pump light to optimize the spatial over-lap between PL and SHG. Furthermore, a focused Gaus-sian beam exhibits a phase deviation k from an idealplane wave that changes along the propagation direction,known as Gouy effect49. It is typically quantified by theparameter = 12bk. For the SHG considered here,this means that the degree to which the phasematch-ing condition is fulfilled changes over the crystal length.Taking all these effects into account through integrationover infinitesimal contributions of the PL to the SHGlight when propagating through the crystal, the Boyd-Kleinman theory provides optimum parameters l/b and for single-pass SHG output power. For SHG from 626 nmto 313 nm in a 10 mm long BBO crystal, we numericallyobtain l/b = 1.42, = 0.75, and 0 = 19m with asingle-pass conversion efficiency of = 1.1 104. A sim-lar analysis can be performed in case of an elliptical focusinside the crystal50, as is the case for astigmatic cavitiesand/or Brewster-cut crystals.

2. Cavity geometry

A nonlinear medium with low conversion efficiencyplaced in an optical cavity in which the PL power is en-hanced by a factor on the order of 50, can provide up to2500 times more SHG output power than is possible insingle pass configuration.

A common design for an optical enhancement cavity,referred to as bow-tie, consists of four mirrors and isschematically shown in Fig. 1. Compared to a linearcavity made of two mirrors, this geometry avoids the for-mation of a standing PL wave and thus reduces the like-lyhood of photo-refractive effects. Furthermore, it allowsfor astigmatism compensation of Brewster-cut crystalsby choosing a suitable distance between the two mirror

3

FIG. 1. Schematic layout of a bow-tie cavity. The resonatoris pumped through mirror (1) with light focused on the in-coupling waist ic. A lightweight mirror (2) on a piezoelectricactuator is used to lock the length of the cavity. (3) and (4)focus the light to the waist 0 inside the nonlinear crystal andmodematch the beam for the next roundtrip.

pairs involved (see below). The incoupling mirror (1)fulfills the impedance matching condition (see below).Mirror (2) is mounted on a piezoelectric actuator witha maximum stroke sufficient to change the cavity lengthby more than one free spectral range (FSR) to enablelength stabilization of the cavity to the PL wavelength.It is lightweight to achieve high mechanical resonance fre-quencies and thus a high feedback bandwidth. Mirrors(3) and (4) are concave, and their distance is chosen toprovide a focus of the appropriate size at the center ofthe crystal. The fourth mirror also acts as outcoupler forthe SHG light. A fully monolithic bow-tie cavity designsimilar to the Nd:YAG NPRO design27 would suffer fromdetrimental PL loss due to the high linear absorption ofPL 0.01 cm1 for = 626 nm in BBO51.

3. Crystal shape

For maximum PL power enhancement and therefore ahigh SHG output power (cf. Eq. (1)), low PL loss insidethe cavity from absorption or scattering is required. Onemajor source of loss in circulating PL is reflection at theinterfaces of the nonlinear medium and the surroundinggas. These reflections can either be avoided by the ap-plication of an antireflective coating on the surfaces orby cutting and mounting the crystal under Brewstersangle to the incident beam. While AR coatings protecthygroscopic crystals like BBO against moisture in the en-vironment, they are prone to damage induced by UV orpump light. Furthermore, residual reflections on surfacesorthogonal to the PL beam can cause a backward trav-eling PL wave. Weighting these properties and given theoption to operate the crystal in a dry environment, itseems advantageous to choose a crystal cut under Brew-sters angle B for the PL. For SHG of 626 nm in BBOthe resulting angle is

B = arctan(nPL) = 59.0, (6)

where nPL is the index of refraction for the PL. SincePL and SHG have orthogonal polarization in ooe SHG,

the Brewster condition can only be fulfilled for the pumplight. The corresponding fractional internal reflection ofthe SHG at the crystal-gas-interface is given by Fresnelsequations for s-polarized light that simplify to

RSHG = (sin(/2 2B))2 22% (7)

4. Impedance matching

Assuming ideal mirrors and input beam mode match-ing, the entire pump light can be coupled into the cavity,if the incoupling mirror transmission is chosen in such away that coupled pump light compensates all losses in-side the cavity during a round trip. In the case of a SHGcavity, the latter consist of two parts: PL loss due to con-version and parasitic loss due to imperfect reflection in-side the resonator and linear absorption in the nonlinearcrystal and surrounding gas with absorption coefficientPL.

The total round trip transmission of the cavity as afunction of the single cavity mirror reflectivity RM andthe crystal facet transmission TC without consideringconversion into SHG is

t = R3MT2C(1 2lPL) (8)

Taking into account the loss through conversion witha factor P 2PL, the optimal incoupling mirror reflectivityRin for impedance matching is

52

Rin = 1

(1 t

2+

(1 t)2

4+ tPPL

), (9)

where the small correction factor t from the incouplingmirror has been applied.

5. Cavity parameters

Both, the non-normal incidence of the cavity mode onthe curved mirrors and crystal facets cut under Brew-sters angle, introduce astigmatism in the cavity whichneeds to be accounted for in the design. The sys-tem forms two effective cavities in the xy (sagittal)and xz (tangential) planes which both need to be sta-ble. Optimizing for simultaneous stability and conversionefficiency23,50 for a 10 mm BBO crystal and a 50 mmradius of curvature of the mirrors yields a mechanicalcavity footprint which is inconveniently large to realizeexperimentally in a compact way. Instead, we choose thecavity parameters outlined in Tab. I. w0,xy and w0,xz arethe saggital and tangential waist, respectively, which arelarger than the optimum waist 0 from Boyd-Kleinmann.This has the added advantage of reducing crystal dam-age from high intensities at the crystal surfaces. In theend the actual cavity geometry is defined by l, B, themirror to crystal distance dmc, cm, and the geometricround-trip length lgeo.

For this choice of parameters, both the sagittal andtangential cavity are stable simultaneously and the inputcoupling waist (between the two flat mirrors) is nearly

4

TABLE I. Geometric parameters of the doubling cavity usinga 10 mm BBO-crystal pumped with PPL = 0.5 W at PL =626 nm and r = 50 mm mirrors

Phasematching angle pm = 38.4

Waist saggital 0,xy = 25.3m

Waist tangential 0,xz = 39.6m

Brewster angle B = 59.0

Angle on curved mirrors cm = 15.7

Distance mirror to crystal dmc = 25.1 mm

Geometric round trip length lgeo = 304.8 mm

Input coupling waist ic = 166m

circular with a waist of 166m. Compared to the ge-ometry for optimal efficiency23,50, the expected loss inconversion efficiency is only about 12%. This moderateloss is outweighed by the gain in mechanical stabilityand ruggedness due to the smaller footprint. The waistsand Brewsters angle have been calculated for SHG of = 626 nm, but their dependence on the PL wavelengthis rather weak. For = 534 nm PL for instance, thewaists change by 1.9m and 2.9m. Brewsters an-gle increases by 0.1. Therefore, the cavity geometry ispractically suitable for frequency doubling of a spectralrange of several hundred nm, as long as a 10 mm longBBO-crystal cut under the corresponding phasematchingangle is used. In conclusion, the shape of the monolithichousing (see below) is universal for SHG to the UV.

B. Length stabilization and its limit

During operation, the length of the cavity is stabilizedto the PL wavelength via a voltage applied to the piezo-electric actuator using a PI controller. The required er-ror signal is generated using the Hansch-Coulliaud (HC)locking scheme53 that generates a dispersion-shaped er-ror signal from the phase shift upon reflection of lightfrom the cavity. P-polarized light is coupled into the cav-ity and experiences a varying phase shift near the cavityresonance, while s-polarized light does not couple into thecavity and thus obtains a constant phase shift upon re-flection. Compared to the Pound-Drever-Hall54 lockingscheme, no sidebands are imprinted on the PL and thusare also absent in the SHG, which is an advantage forcertain applications.

The locking bandwidth is fundamentally limited by themechanical resonance frequency f of the piezoelectricactuator and the attached mirror. It is given as55

f = f0

m3

m3 +M

(10)

where f0 is the mechanical resonance frequency of thepiezoelectric actuator, m its mass and M the mass of theattached mirror.

For the monolithic cavity design presented here, weexpect only relatively small changes of the round-trip

FIG. 2. Schematic cross-section of the SHG cavity. Allmechanical components are mounted directly to the mono-lithic housing. The incoupling and outcoupling mirror aretiltable using micrometer screws while the other mirrors canbe translated only. The temperature-stabilized BBO crystalis mounted on a 5-axis aligner. Ar-coated windows allow foroptical access to the sealed cavity.

length. Therefore a short piezoelectric actuator with lowweight and high resonance frequency seems to be suffi-cient to keep the cavity length stabilized under moderateenvironmental perturbations.

III. CAVITY DESIGN

The main design goals of the cavity are turn-key op-eration after being exposed to mechanical environmentalconditions typical for long distance truck and plane trans-portation. Furthermore, robust locking and operation ina non-lab environment is important to be a reliable partof a transportable quantum optics experiment.

Therefore, the main body is milled from a single Alblock shown in Fig. 2. Two mirror holder front platesmounted via three micrometer screws (150m displace-ment per turn) and two springs each, all mounted di-rectly on the main body, feature the four degrees of free-dom (DOF) required to close the beampath. In com-bination with the two other mirrors that are mountedon fine-threaded aluminum cylinders, the cavity roundtrip length can be adjusted on the few mm-scale whilekeeping the angles between the beams and therefore theastigmatism compensation constant. The mirrors aremounted using stress-free retaining rings to avoid bire-fringence caused by mechanical stress. This does notapply for the mirror behind the incoupler which is glued(Thorlabs 353NDPK Epoxy) on a piezoelectric actuator(Thorlabs AE0505D08F) to enable cavity length stabi-lization. In crystals used for SHG into the UV, crystalinhomogeneities and degradation effects have been ob-served. Compensation of these effects requires two trans-lational DOF parallel to the front surface of the crystal.With a third DOF the waist position in the crystal alongthe direction of the PL propagation is adjusted. Twoadditional rotational DOF allow the fine adjustment of

5

the phasematching and Brewsters angle. All these DOFare provided by a commercially available 5-axis-aligner(NEWPORT 9081-M) that is mounted directly in themain body. The crystal is mounted in two shells thatare mounted to a lever connected to the aligner. Theseshells and the lever are equipped with channels that leadoxygen from a supply connected to the main body di-rectly to the two facets of the crystal to prevent light-induced damage56. Additionally, the tip of the levercan be activly temperature stabilized using a thick-filmresistor (Vishay Sfernice RTO 20) and a sensor (TDKB57861S) connected via a standard D-sub feedthroughto a suitable PI controller to prevent condensate forma-tion on the crystal facets that would likely cause degra-dation of hygroscopic crystals like BBO. Finally, the cav-ity is equipped with FKM sealings to prevent unwantedsubstances from entering the enhancement cavity andcause degradation of the crystal and/or mirrors. There-fore, a gas outlet in the main body ensures a moderategas exchange rate and overpressure inside the housing.The oxygen concentration can be measured by a sensor(Greisinger GOX 100) located in the lid of the main body.

IV. SETUP

UV-Detector

626 nm

313 nm

Woll.-

prism

Di .

photod.

2

STEMlab-Box

PBS

Block

SFG

PC

O2

1%

Pump-

Detector

Closed box on breadboard

Fiber lasers

Block

ND

2

4

L1

CL

L2 M2

M1

M3

M4

M5

L3

L4

M6

Power

photod.

1550 nm

1051 nm

FIG. 3. Schematic overview of the complete setup for 313nm generation. FCi : fiber coupler, Li : lens, Mi : mirror,PBS : polarizing beam splitter cube,

2/

4half/quarter wave

plate, ND: neutral density filter. Left: pump light generationat 626 nm by SFG of two IR fiber lasers. Center: 313 nmgeneration in the resonant enhancement bow-tie cavity lockedby the Hansch-Coulliaud locking scheme (dashed beam pathand digital multi purpose PI controller STEMlab-Box). Foroutput power stability monitoring, a fraction of the pump andSHG light is picked off and measured via photo diodes.

Fig. 3 shows a schematic overview of the completesetup for SHG at 313 nm. It consists of a sum frequencygeneration (SFG) setup for the 626 nm pumplight, theSHG cavity setup, the locking electronics, and a com-puter for power recording. The general approach is sim-ilar to Wilson et al.23.

A. Pumplight setup

Up to 1.1 W pump light (PL) at 626 nm are generatedby sum frequency generation (SFG) of two fiber lasersat 1550 nm and 1051 nm in a PPLN crystal describedin57 and coupled in a polarization maintaining fiber atFC1. This fiber leads to a 12.7 mm thick (450 mm)2 alu-minum breadboard with all the optical components forthe 313 nm SHG generation. This setup as well as theaforementioned SFG are covered by two boxes made ofaluminum. Both are mounted on a standard air dampedoptical table in a laboratory temperature stabilized at the1 K level. The breadboard holding the cavity is placed on5 mm thick viscoelastic damping rubber (Sorbothane R).

B. SHG setup

Behind the PL fiber coupler FC2 a telescope consistingof two lenses L1 and L2 with focal lengths f = +50 mmand f = +30 mm, respectively, mounted on single axistranslation stages is used for the modematching with thePL cavity mode. A 1% pickup beam is used to mon-itor the PL power. The two highly stable mirrors M1and M2 provide the four DOF required for the couplingthe PL into the cavity. Both mirrors are mounted usingretaining rings to avoid birefringence due to mechanicalstress. A polarizing beamsplitter cube (PBS) cleans thepolarization of the PL and a subsequent /2 waveplaterotates it into the plane required for the SHG. Behind theoutcoupling mirror of the SHG cavity a cylindrical lensCL with focal length f = +100 mm is employed for cor-recting the astigmatism arising from the walk-off of theSHG light generated inside the BBO crystal. A secondmirror (M3) of the same type as the outcoupler is usedto filter the transmitted PL from SHG beam. For thetest presented here, a thermal powermeter measurementhead is used to collect the SHG light. The PL reflectedby the cavity is attenuated by a 3.0 absorptive neutraldensity filter (ND), focussed by lens L3 with focal lengthf = +250 mm and send through a polarization analyzerconsisting of a half-wave plate, a quarter-wave plate, anda Wollaston prism to generate two orthorgonally polar-ized beams of similar intensity carrying the informationabout the phase relation of the PL entering the cavity53.These two beams hit a differential photodiode. The re-sulting signal is the error signal for the digital PI con-troller. The small fraction of PL reflected on the frontsurface of the crystal due to imperfect polarization andscatter is focused on a photodiode to generate a signal asa measure for the circulating power.

C. Digital PI controller design

The error signal generated by the differential photodi-ode is fed into a digital PI controller based on a modifiedSTEMlab 125-14 (formerly: Red Pitaya). A real-timedigital PID controller algorithm is implemented in theFPGA as part of the hardware conguration. It oper-ates independently from the embedded software on the

6

STEMlab. The parameters of the digital controller areset either through a remote network connection, or usinga standalone user interface including a touchscreen androtary encoder inputs. For the purpose of cavity locking,the D-part of the controller is set to zero. Besides thePID controller, a function generator was implemented. Itcan output sine, square or triangular waveforms withinfrequency range of 0 - 50 MHz.

1. Hardware

STEMlab58 hardware is not open-source, but its pro-cessing core and peripherals give many options for low-level customization. The hardware platform is builtaround the Xilinx Zynq7010 All Programmable System-on-Chip (SoC)59,60, which combines a dual-core ARMCortex A9 processor, and a field programmable gatearray (FPGA) in one chip. Its peripherals include 16general-purpose input/output (GPIO) lines, various dig-ital interfaces such as I2C, SPI and UART, and four12-bit ADCs and DACs operating at 100 kS/s. All ofthese signal lines are accessible on the extension connec-tors of the board. In addition, there are two high-speedcommunication interfaces based on Ethernet and USB.The STEMlab provides two fast analog input and out-put channels, operating at sampling rate of 125 MS/s andhaving 14-bit nominal resolution. They are implementedusing fast ADCs (LTC2145)61 and DACs (AD9767)62,plus various analog signal conditioning circuits. Fig. 4shows an overview of the components in the STEMlabsystem that are most relevant for the implemented digi-tal controller. The blue boxes indicate the two principaldomains of the Zynq, where custom hardware and soft-ware is implemented.

FIG. 4. Component overview of the STEMlab box cf.63.

The main characteristics of the fast analog frontendchannels are given in Tab. II. They are the most criticalinterfaces, because their conversion errors determine theachievable performance of the digital controller. Apartfrom the effects of quantization that would be presenteven in ideal converters, various other error sources inthe ADCs and DACs have to be considered. Among themare static gain and offset errors, as well as temperature-dependent drifts.

To obtain a stand-alone controller unit based on theSTEMlab, we added a user interface consisting of an LCD

display with touchscreen and a rotary encoder. The volt-age range of the DAC outputs was increased using non-inverting amplifiers with optional external low-pass filtersfor control applications with reduced bandwidth. Usingampliers with Gain G=8, the output range of 2 Vpp wasscaled to a 16 Vpp, sufficient to cover two free spectralranges of the monolithic doubling cavity, when applied tothe cavity piezo-electric actuator. To reduce the outputnoise of the STEMLab, we implemented the modificationdescribed in64, which eliminates the excess noise couplingfrom the digital supply rail to the analog output. Thisimprovement led to a noise reduction from 3-4 LSB peak-to-peak to 1-2 LSB peak-to-peak. A certain variation ofexcess noise was observed between different units. In ad-dition to the described circuits, the standalone box con-tained separate power supplies for the STEMlab board,its cooling fan, and the output amplifiers. The piezo-electric actuator inside the cavity was connected withan series resistor of 10 , which, together with the ca-pacitance of the actuator, forms a low-pass filter with acut-off frequency of 21.2 kHz, preventing noise-inducedoscillations at high frequency.

2. Software and FPGA configuration

The STEMlab embedded system uses the Linux oper-ating system (presently the Ubuntu 16.04 distribution).The OS image is loaded from an SD card, which alsocontains the customized software and the FPGA config-uration.

The open-source web applications provided with theSTEMlab, as well as the PID controller application, canbe accessed from any web browser via a network connec-tion. The digitized input signals can be monitored in realtime. All parameters of the digital PID controller can beset remotely through the web application, via serial con-nection, or using the local user interface when operatingin standalone mode.

To achieve low-latency digital control, the calculationsfor the PID algorithm are implemented in the FPGAhardware. Hard real-time processing is possible due tothe deterministic timing behavior of the FPGA. The orig-inal STEMlab PID application contains a Multiple In-put Multiple Output (MIMO) PID controller which con-sists of four standard PID controllers with P, I and Dparameter settings and integrator reset control65. ThePID parameters can be set in the user interface, and arethen sent to the FPGA registers described in the registermap66. The implementation of the four discrete PID con-trollers can be approximately described by the equationsfor a continuous-time control system:

m(t) = KP e(t) + KIe(t) dt + KD

de(t)

dt(11)

G(s) = KP +KIs

+ KDs, (12)

where m(t) is the actuating variable, e(t) the error signal,and G(s) the transfer function of the controller. KP, KI,and KD are the scaling factors for the P, I, and D part

7

of the controller. All three summands are calculated inparallel.

To match the specific requirements for quantum op-tics experiments, the standard PID controller was modi-fied. The source code of the new implementation is freeand can be downloaded from67. Its structure is shown inFig. 5. A master gain term, a second integral part, andan offset setting were added to the basic PID. The rangesof the control parameters were extended, and the originalSTEMlab signal generator application was integrated inthe controller.

FIG. 5. Block diagram of the modified STEMlab - PID in afeedback control loop

A number of additional functions were implemented:the settings can be saved, the output voltage can be lim-ited digitally, and an auxiliary input analog signal can beused to enable the controller. The latter function was im-plemented to allow automatic re-locking of the controller.For this the auxiliary intensity signal from the photode-tector Power diode in Fig. 3 is used to discriminatebetween actual resonances of the PL TEM00-mode andparasitic signals. Moreover, a automatic integrator resetand a sample-and-hold mode were implemented. All ad-ditional functions are optional and run in parallel withthe PID algorithm, without affecting its real-time perfor-mance.

To allow finer adjustment of the integrator settings, anoptional decimation scheme was implemented, in whichonly every N th sample (N = 2 to 1000) is summed.

D. Characteristics of the digital controller

1. Amplitude frequency response

Fig. 6 shows the amplitude response with all rangesand step sizes of the control parameters, as well as theircharacteristic frequencies and slopes. The individually

adjustable control parameters are the master gain G, theP-, I-, 2nd I- and D- contributions of the control loop.

FIG. 6. Amplitude response of the PID versus frequency. Theranges for the different control parameters are shown togetherwith their effect on the response curve.

The physical meaning of the control parameters forthe I- and D-parts as 0 dB gain crossover frequencies be-comes evident when we convert from angular to linearfrequencies (Eq. (13) and (14)):

fi =Ki2

(13)

fd =1

2 Kd(14)

2. Limits

The closed-loop bandwidth of the digital controller isfundamentally limited by its group delay = 155 ns.The phase shift from the delay increases linearly withfrequency, and cannot be compensated in any real-timecausal system. Its effect is visible in the Bode plots ofFig. 10. The additional phase shift is approximately 55.4

at 1 MHz, measured in-loop. This suggests a maximumclosed-loop control bandwidth of a few MHz, determinedby the total phase shift in the loop.

Another limitation exists due to the dynamic rangebottlenecks at the ADC and the DAC. The effective res-olution of the converters is lower than their nominal res-olution of 14 bits. This was partially explained earlierfor the DAC, where an excess digital noise contributionwas observed. Fig. 11 in the Appendix shows the volt-age noise spectral density measured at the output of twoSTEMlab boards, and two standalone boxes with andwithout an output amplifier. The integrated noise volt-age within 1 Hz to 1 MHz is measured to be 76.89V forthe unmodified STEMlab board without amplifier stageand deactivated PID controller (A, red curve). With theoffset resistors removed, 32.44V is measured (B, blackcurve). For the modified STEMlab installed inside thestandalone box with activated PID controller, 120.12Vis obtained (C, green curve). Finally, an integrated noisevoltage of 392.66V is measured for the STEMlab boardwith offset resistors removed inside the standalone boxand with amplifier stage added.

8

V. EXPERIMENT

During all experiments the cavity was flushed with99.95% pure oxygen at a rate of 0.022 l/min. A fi-nesse Fmeas = 317 was measured with the Brewster-cutBBO crystal in place and mirrors with reflectivities ofR1 = 98.50% and R2 = R3 = R4 = 99.95%.

A. Long term stability

For many applications in quantum optics a power-stable UV source is required that reliably stays in lockover many hours. In order to demonstrate that oursetup fulfills those requirements, it has been locked over130 h during which the PL power and SHG Power havebeen recorded every 1 s using two power meters (Thor-labs PM100D with S130VC Sensor and Ophir Vega withPD300-UV Sensor, cf. Fig. 3). Fig. 7 shows the mea-sured SHG output power scaled to the average SHGpower (red) and to the instantaneous PL power (blue).The SHG power stayed on a level of about 186 mW, whilethe cavity was pumped with about 530 mW PL. The SHGpower fluctuates by about 7% peak-to-peak (3% rms)around its mean value on a few hour timescale, while theSHG power rescaled to the PL varies only by 3% peak-to-peak (1% rms). This suggests that the SHG powerfluctuations were mainly caused by pump power fluctu-ations, indicating that the alignment of the cavity didnot change on relevant length or angular scales over theduration of the measurement.

To our knowledge, this is the longest published in-lockoperation of a resonant SHG ring cavity. The outputpower of the SHG cavity fully agrees with the correspond-ing measurement and calculation presented in23.

B. Lock performance

The performance of the cavity lock has been investi-gated by recording the power spectral density of the errorsignal generated by the differential photo diode (s. Fig. 3)using a vector signal analyzer (Agilent 89441A). Fig. 11shows the error signal for a loose lock achieved by choos-ing gain setting just sufficient enough to lock (blue) andfor optimized settings (black). With the optimized lock asuppression by up to almost one order of magnitude foracoustic frequencies is achieved. A broad resonance atabout 17 kHz sets the upper limit for the locking band-width in this setup. This frequency is likely to be the firstmechanical resonance of the small mirror, the piezoelec-tric actuator, and its holder. In a future setup, it couldpossibly be improved by applying the mount design pre-sented in6870, since Equ. 10 results in an upper limit off 120 kHz for the piezo electric actuator and mirrorused here.

C. Acceleration sensitivity tests

The monolithic cavity described here is a prototype tobe used for generating the clock transition probe lightof a transportable Al+ clock to be used in relativisticgeodesy campaigns. In order to prove the suitability ofthe doubling cavity for such an application, the entirebreadboard was accelerated in vertical direction while thecavity was locked. The acceleration was measured usingan analog MEMS sensor (ADXL345) glued on the toplid of the cavity. Fig. 8 shows the output power and ver-tical acceleration versus time. The measurement showstemporary SHG power drops of about 10% synchronousthe acceleration acting on the system. While an absoluteacceleration of up to 1 g was obtained, the cavity stayedin lock and fully recovered the initial SHG power afterthe acceleration stopped.

In order to demonstrate the reliable operation of thesystem after transport, it was exposed to an accelera-tion profile typical for on road transportation. There-fore, the system was optimized first and then the SHGbreadboard was placed on a multi-component accelera-tion exciter71 and exposed to a vibration power spectrumin the vertical direction according to ISO 13355:2016 witha total acceleration of 0.604 grms for 30 minutes. Af-terwards, the SHG breadboard was reconnected to therest of the system and locked without further realign-ment or optimization. The results are shown in Fig. 9.The obtained SHG power normalized to the pump powerdropped to about 60%, while the SHG power normal-ized to the squared pump power circulating in the cavityremained almost constant. Since this indicates a dropin incoupling efficiency, the alignment of the mirrors M1and M2 in Fig. 4 was optimized, resulting in approxi-mately the same SHG power as before the accelerationexposure test. This strongly indicates that the alignmentof the cavity itself was not substantially affected by theshaking test. Therefore the same test was repeated withfive times the original acceleration (3.020 grms), also re-sulting in the same SHG output power after shaking andalignment correction of the mirrors M1 and M2. Fromthese measurements we conclude that the optical align-ment of the cavity itself can withstand typical on roadtransport situations without any deterioration, while themechanical stability of the input coupling optics and/orfiber connector for the PL needs to be improved.

VI. SUMMARY AND OUTLOOK

A mechanically stable monolithic enhancement cavityfor SHG generation in the UV was demonstrated, includ-ing in-lock SHG power measurements during accelerationexcitation. Less than 10% SHG power reduction duringexposure of up to 1 g were observed with full recovery ofthe initial SHG power after acceleration stopped. It wasshown that the cavity optical alignment can withstand 30minutes of acceleration excitation with 3.020 grms. Thisis five times the acceleration amplitude as specified inthe corresponding ISO13355:2016 norm, demonstratingthe suitability for transportable experiments. 130 h un-

9

FIG. 7. Long term SHG output power measurement. Red: measured SHG power, a power of 1 equals 186 mW. Blue: SHGnormalized to the PL power squared as a measure of the conversion efficiency.

FIG. 8. Cavity SHG output power and acceleration versustime during acceleration excitation in vertical direction. Red:SHG power, blue: vertical acceleration. For accelerations upto 1 g (gravity substracted) the SHG output power fluctuateson the 10% level while the cavity stays in lock.

FIG. 9. SHG power before (left) and right after (middle) be-ing exposed to a 30 min ISO 13355:2016 shaker test. Right:following optimization of the incoupling beam after the shak-ertest. Red: measured SHG power, blue: SHG normalized tothe PL power squared, pink: SHG normalized to the circulat-ing PL power as a measure of the conversion efficiency.

interrupted operation without decay in output power at313 nm was demonstrated. During this time the SHGpower scaled to the pump power fluctuated by 1% rms.The basic design can easily be adapted to other resonatorgeometries in order to install crystals of different mate-rials or shapes. The locking bandwidth of the setup pre-sented here is 17 kHz. To obtain a higher locking band-width in the future, the design of the piezoelectricallyactuated mirror mount demonstrated in6870 could beadopted. To reduce the risk of unwanted substances in-side the cavity causing degradation of the crystal and/ormirrors in the next cavity generation, the housing can beleft unanodized and metal sealings can be employed. Forimproved mechanical robustness and leak tightness, thecrystal aligner can be made part of the cavity housing.

ACKNOWLEDGMENTS

We thank L. Klaus for performing the acceleration ex-citation measurements on the Multi-component acceler-ation exciter and S. A. King for stimulating discussion.We acknowledge support from DFG through grants CRC1128 geo-Q, project A03, CRC 1227 DQ-mat, projectsB03 and B06, and Leibniz Gemeinschaft through grantSAW-2013-FBH-3.

10

VII. APPENDIX

FIG. 10. Bode diagramm open loop. From top to bottom:proportional/integral/differential part.

1T. R. Tan, J. P. Gaebler, Y. Lin, Y. Wan, R. Bowler, D. Leibfried,and D. J. Wineland, Nature 528, 380 (2015).

2C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland,and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010).

3K. Yamanaka, N. Ohmae, I. Ushijima, M. Takamoto, and H. Ka-tori, Phys. Rev. Lett. 114, 230801 (2015).

4M. Hauth, C. Freier, V. Schkolnik, A. Senger, M. Schmidt, andA. Peters, Appl. Phys. B. 113, 49 (2013).

5B. Barrett, P.-A. Gominet, E. Cantin, L. Antoni-Micollier,A. Bertoldi, B. Battelier, P. Bouyer, J. Lautier, and A. Lan-dragin, Proc. Intl Sch. Phys. Enrico Fermi , 493 (2014), arXiv:1311.7033.

6N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, and E. Peik,Phys. Rev. Lett. 116, 063001 (2016).

7T. L. Nicholson, S. L. Campbell, R. B. Hutson, G. E. Marti,B. J. Bloom, R. L. McNally, W. Zhang, M. D. Barrett, M. S.Safronova, G. F. Strouse, W. L. Tew, and J. Ye, Nat. Commun.6, 6896 (2015).

8A. Bjerhammar, Bulletin Godsique 59, 207 (1985).9S. B. Koller, J. Grotti, S. Vogt, A. Al-Masoudi, S. Dorscher,S. Hafner, U. Sterr, and C. Lisdat, Phys. Rev. Lett. 118, 073601(2017).

10J. Grotti, S. Koller, S. Vogt, S. Hafner, U. Sterr, C. Lisdat,H. Denker, C. Voigt, L. Timmen, A. Rolland, F. N. Baynes,H. S. Margolis, M. Zampaolo, P. Thoumany, M. Pizzocaro,B. Rauf, F. Bregolin, A. Tampellini, P. Barbieri, M. Zucco, G. A.Costanzo, C. Clivati, F. Levi, and D. Calonico, arXiv:1705.04089[physics] (2017), arXiv: 1705.04089.

11J. Cao, P. Zhang, J. Shang, K. Cui, J. Yuan, S. Chao, S. Wang,H. Shu, and X. Huang, arXiv:1607.03731 (2016), arXiv:1607.03731.

12H. Muntinga, H. Ahlers, M. Krutzik, A. Wenzlawski, S. Arnold,D. Becker, K. Bongs, H. Dittus, H. Duncker, N. Gaaloul,C. Gherasim, E. Giese, C. Grzeschik, T. W. Hansch, O. Hellmig,W. Herr, S. Herrmann, E. Kajari, S. Kleinert, C. Lammerzahl,W. Lewoczko-Adamczyk, J. Malcolm, N. Meyer, R. Nolte, A. Pe-ters, M. Popp, J. Reichel, A. Roura, J. Rudolph, M. Schiemangk,M. Schneider, S. T. Seidel, K. Sengstock, V. Tamma, T. Valen-zuela, A. Vogel, R. Walser, T. Wendrich, P. Windpassinger,W. Zeller, T. van Zoest, W. Ertmer, W. P. Schleich, and E. M.Rasel, Phys. Rev. Lett. 110, 093602 (2013).

13S. Origlia, S. Schiller, M. S. Pramod, L. Smith, Y. Singh, W. He,S. Viswam, D. wierad, J. Hughes, K. Bongs, U. Sterr, C. Lis-dat, S. Vogt, S. Bize, J. Lodewyck, R. L. Targat, D. Holleville,B. Venon, P. Gill, G. Barwood, I. R. Hill, Y. Ovchinnikov, A. Ku-losa, W. Ertmer, E.-M. Rasel, J. Stuhler, W. Kaenders, andt. S. c. contributors, Proc.SPIE 9900, 9900 (2016).

14J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q.Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong,Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia,G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang,Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng,J.-Y. Wang, and J.-W. Pan, Science 356, 1140 (2017).

15E. Luvsandamdin, C. Kurbis, M. Schiemangk, A. Sahm,A. Wicht, A. Peters, G. Erbert, and G. Trankle, Opt. Express22, 7790 (2014).

16A. Kohfeldt, C. Kurbis, E. Luvsandamdin, M. Schiemangk,A. Wicht, A. Peters, G. Erbert, and G. Trankle, Proc. SPIE9900, 9900 (2016).

17K. Wakui, K. Hayasaka, and T. Ido, Appl. Phys. B. 117, 957(2014).

18J. Hu, L. Zhang, H. Liu, K. Liu, Z. Xu, and Y. Feng, Opt.Express 21, 30958 (2013).

19I. Sherstov, M. Okhapkin, B. Lipphardt, C. Tamm, and E. Peik,Phys. Rev. A 81, 021805 (2010).

20M. Scheid, F. Markert, J. Walz, J. Wang, M. Kirchner, andT. W. Hansch, Opt. Lett. 32, 955 (2007).

21X. Wen, Y. Han, J. Bai, J. He, Y. Wang, B. Yang, and J. Wang,Opt. Express 22, 32293 (2014).

22U. Eismann, F. Gerbier, C. Canalias, A. Zukauskas, G. Trnec,J. Vigu, F. Chevy, and C. Salomon, Appl. Phys. B. 106, 25(2012).

23A. C. Wilson, C. Ospelkaus, A. P. VanDevender, J. A. Mlynek,K. R. Brown, D. Leibfried, and D. J. Wineland, Appl. Phys. B.

http://dx.doi.org/10.1038/nature16186http://dx.doi.org/10.1103/PhysRevLett.104.070802http://dx.doi.org/ 10.1103/PhysRevLett.114.230801http://dx.doi.org/ 10.1007/s00340-013-5413-6http://arxiv.org/abs/1311.7033http://dx.doi.org/ 10.1103/PhysRevLett.116.063001http://dx.doi.org/10.1038/ncomms7896http://dx.doi.org/10.1038/ncomms7896http://dx.doi.org/10.1007/BF02520327http://dx.doi.org/ 10.1103/PhysRevLett.118.073601http://dx.doi.org/ 10.1103/PhysRevLett.118.073601http://arxiv.org/abs/1705.04089http://arxiv.org/abs/1705.04089http://arxiv.org/abs/1607.03731http://dx.doi.org/10.1103/PhysRevLett.110.093602http://dx.doi.org/ 10.1117/12.2229473http://dx.doi.org/ 10.1126/science.aan3211http://dx.doi.org/ 10.1364/OE.22.007790http://dx.doi.org/ 10.1364/OE.22.007790http://dx.doi.org/ 10.1117/12.2231537http://dx.doi.org/ 10.1117/12.2231537http://dx.doi.org/10.1007/s00340-014-5914-yhttp://dx.doi.org/10.1007/s00340-014-5914-yhttp://dx.doi.org/10.1364/OE.21.030958http://dx.doi.org/10.1364/OE.21.030958http://dx.doi.org/ 10.1103/PhysRevA.81.021805http://www.opticsinfobase.org/ol/fulltext.cfm?uri=ol-32-8-955http://dx.doi.org/10.1364/OE.22.032293http://dx.doi.org/10.1007/s00340-011-4693-yhttp://dx.doi.org/10.1007/s00340-011-4693-yhttp://dx.doi.org/ 10.1007/s00340-011-4771-1

11

TABLE II. STEMlab 125-14 analog front- and backend specifications72

Inputs Outputs

Number of channels: 2 Quantity: 2

Bandwidth: 50 MHz Bandwidth: 50 MHz

Sample rate: 125 MS/s Sample rate: 125 MS/s

DAC resolution: 14 Bit ADC resolution: 14 Bit

Full scale voltage: selectable: 1 V, 20 V Full scale power: 9 dBmMinimal Voltage Sensitivity: 0.122 mV, 2.44 mV 1 LSB: 0.122 mVInput impedance: 1 M || 10 pF Load impedance (for full scale power): 50 DC offset error: < 5 % DC offset error: < 5 %

Gain error: LV: < 3 %, HV: < 10 % Gain error: < 5 %

Input noise level: 119 dBm/Hz Slew rate limit: 200 Vs

TABLE III. Limit values of the control parameters

Parameter Maximum Step size Min. frequency Max. frequency Inaccuracy

Gain : 10 0.01 - - < 10 %

Kp : 10 0.01 - - < 10 %

Ki : 36001ms

0.5 1ms

80 Hz 573 kHz < 17 %

Kd : 58 ns 0.01 ns 2.74 MHz 16 GHz < 6 %

FIG. 11. Error signal (=STEMlab input) power spectral density for two different PI settings. A (black): lock with optimizedgain settings, B (blue): low-gain lock.

105, 741 (2011).24S. Vasilyev, A. Nevsky, I. Ernsting, M. Hansen, J. Shen, and

S. Schiller, Appl. Phys. B. 103, 27 (2011).25R. A. Carollo, D. A. Lane, E. K. Kleiner, P. A. Kyaw, C. C.

Teng, C. Y. Ou, S. Qiao, and D. Hanneke, Opt. Express 25,7220 (2017).

26P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys.Rev. Lett. 7, 118 (1961).

27T. J. Kane and R. L. Byer, Opt. Lett. 10, 65 (1985).28W. Kozlovsky, C. D. Nabors, and R. Byer, IEEE J. Quant.

Electron. 24, 913 (1988).29D. C. Gerstenberger, G. E. Tye, and R. W. Wallace, Opt. Lett.16, 992 (1991).

http://dx.doi.org/ 10.1007/s00340-011-4771-1http://dx.doi.org/ 10.1007/s00340-011-4771-1http://dx.doi.org/ 10.1007/s00340-011-4771-1http://dx.doi.org/ 10.1007/s00340-011-4771-1http://dx.doi.org/ 10.1007/s00340-011-4771-1http://dx.doi.org/ 10.1007/s00340-011-4771-1http://dx.doi.org/ 10.1007/s00340-011-4771-1http://dx.doi.org/ 10.1007/s00340-011-4771-1http://dx.doi.org/ 10.1007/s00340-011-4435-1http://dx.doi.org/10.1364/OE.25.007220http://dx.doi.org/10.1364/OE.25.007220http://dx.doi.org/10.1103/PhysRevLett.7.118http://dx.doi.org/10.1103/PhysRevLett.7.118http://dx.doi.org/10.1364/OL.10.000065http://dx.doi.org/10.1109/3.211http://dx.doi.org/10.1109/3.211http://dx.doi.org/10.1364/OL.16.000992http://dx.doi.org/10.1364/OL.16.000992

12

FIG. 12. Output noise (=STEMlab output) power spectral density with terminated input. A (red): unmodified STEMlabwithout amplifier stage, PID deactivated. B (black): the same as A, but with offset resistors removed. C (green): STEMlab inthe box, PID enabled, without amplifier, Kp = 1. D (blue): the same as C, but with amplifier (8x) and 100 KHz lowpass filter.Integrated output voltage noise (bandwidth 1 Hz 1 MHz): A = 76.89V, B = 32.44V, C = 120.12V, D = 392.66V

30N. E. Corporation, Wavelength Conversion Module, (2017),http://www.ntt-electronics.com/en/products/photonics/

conversion-module.html.31W. J. Kozlovsky, W. P. Risk, W. Lenth, B. G. Kim, G. L. Bona,

H. Jaeckel, and D. J. Webb, Applied Physics Letters 65, 525(1994).

32A. Hemmerich, C. Zimmermann, and T. W. Hansch, Appl. Opt.33, 988 (1994).

33D. Skoczowsky, A. Jechow, R. Menzel, K. Paschke, and G. Er-bert, Opt. Lett. 35, 232 (2010).

34C. Zimmermann, R. Kallenbach, and T. W. Hansch, Phys. Rev.Lett. 65, 571 (1990).

35K. Huang, H. Le Jeannic, J. Ruaudel, O. Morin, and J. Laurat,Review of Scientific Instruments 85, 123112 (2014).

36M. R. Dietrich and B. B. Blinov, arXiv:0905.2484 [physics.atom-ph] (2009).

37B. M. Sparkes, H. M. Chrzanowski, D. P. Parrain, B. C. Buchler,P. K. Lam, and T. Symul, arXiv:1105.3795 [physics] (2011),arXiv: 1105.3795.

38D. R. Leibrandt and J. Heidecker, Rev. Sci. Instrum. 86 (2015).39Toptica Photonics AG, DigiLock 110: Digital Laser

Locking, (2017), http://www.toptica.com/products/tunable-diode-lasers/laser-locking-electronics/

digilock-110-digital-locking/.40H. Abitan and T. Skettrup., J. Opt. Pure Appl. Opt 7, 7 (2005).41Red Pitaya Project Website, (2016), http://redpitaya.com/.42J. Fenske, Implementierung eines digitalen PID-Reglers mitdem Entwicklungsboard Red Pitaya, Bechelors thesis, OstfaliaHochschule fur angewandte Wissenschaften, Germany (2015).

43K. Kato, IEEE J. Quant. Electron. 22, 1013 (1986).44D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Sur-vey (Springer, 2005).

45J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Per-shan, Phys. Rev. 127, 1918 (1962).

46A. Ashkin, G. Boyd, and J. Dziedzic, IEEE J. Quant. Electron.2, 109 (1966).

47G. D. Boyd and D. A. Kleinman, Journal of Applied Physics 39,3597 (1968).

48E. Jurdik, J. Hohlfeld, A. F. van Etteger, A. J. Toonen, W. L.Meerts, H. van Kempen, and T. Rasing, JOSA B B 19, 1660(2002).

49L. G. Gouy, C. R. Acad. Sci. 110, 1251 (1890).50T. Freegarde, J. Coutts, J. Walz, D. Leibfried, and T. W.

Hansch, J. Opt. Soc. Am. B 14, 2010 (1997).51M. Watanabe, K. Hayasaka, H. Imajo, J. Umezu, and S. Urabe,

Appl. Phys. B. 55, 11 (1991).52E. S. Polzik and H. J. Kimble, Opt. Lett. 16, 1400 (1991).53T. W. Hansch and B. Couillaud, Opt. Commun. 35, 441 (1980).54R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M.

Ford, A. J. Munley, and H. Ward, Appl. Phys. B.: Lasers andOptics 31, 97 (1983).

55Thorlabs Inc., (2017), Co-Fired Piezoelectric Actuator datasheethttps://www.thorlabs.de.

56J. C. Sandberg, Research Towards Laser Spectroscopy of TrappedAtomic Hydrogen, Ph. d. thesis, MIT (1993).

57A. Idel, Ein kompaktes Lasersystem zum sympathetischenKuhlen einzelner (Anti-) Protonen durch 9Be+ Ionen, Mastersthesis, Leibniz Universitat Hannover, Germany (2016).

58Red Pitaya team, Red Pitaya web page, (2017), http://redpitaya.com/.

59Xillinx, Zynq-7000 All Programmable SoC Overview, (2016),http://www.xilinx.com/support/documentation/data_sheets/

ds190-Zynq-7000-Overview.pdf.60L. H. C. et. al., The Zynq Book - Embedded Processing

with the ARM Cortex-A9 on the Xilinx Zynq-7000 All Pro-grammable SoC, (2014), http://www.analog.com/media/en/technical-documentation/data-sheets/AD9763_9765_9767.

pdf.61L. Technology, LTC2145-14 - 14-Bit, 125Msps Low Power Dual

ADCs, (2011), http://cds.linear.com/docs/en/datasheet/21454314fa.pdf.

62A. Devices, Ad9767 - 14-bit, 125 msps dual txdac+ digital-to-analog converters, (2011), http://www.analog.com/media/en/technical-documentation/data-sheets/AD9763_9765_9767.

pdf.63Red Pitaya team, Red Pitaya: Open instruments for everyone,

(2016), https://www.kickstarter.com/projects/652945597/

http://www.ntt-electronics.com/en/products/photonics/conversion-module.htmlhttp://www.ntt-electronics.com/en/products/photonics/conversion-module.htmlhttp://dx.doi.org/ 10.1063/1.112286http://dx.doi.org/ 10.1063/1.112286http://dx.doi.org/10.1364/AO.33.000988http://dx.doi.org/10.1364/AO.33.000988http://dx.doi.org/ 10.1364/OL.35.000232http://dx.doi.org/10.1103/PhysRevLett.65.571http://dx.doi.org/10.1103/PhysRevLett.65.571http://dx.doi.org/ 10.1063/1.4903869http://arxiv.org/abs/0905.2484http://arxiv.org/abs/0905.2484http://arxiv.org/abs/1105.3795http://dx.doi.org/10.1063/1.4938282http://www.toptica.com/products/tunable-diode-lasers/laser-locking-electronics/digilock-110-digital-locking/http://www.toptica.com/products/tunable-diode-lasers/laser-locking-electronics/digilock-110-digital-locking/http://www.toptica.com/products/tunable-diode-lasers/laser-locking-electronics/digilock-110-digital-locking/http://redpitaya.com/http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1073097&isnumber=23096http://dx.doi.org/10.1103/PhysRev.127.1918http://dx.doi.org/10.1109/JQE.1966.1074007http://dx.doi.org/10.1109/JQE.1966.1074007http://dx.doi.org/10.1063/1.1656831http://dx.doi.org/10.1063/1.1656831http://dx.doi.org/ 10.1364/JOSAB.19.001660http://dx.doi.org/ 10.1364/JOSAB.19.001660http://dx.doi.org/ 10.1364/JOSAB.14.002010url="https://doi.org/10.1007/BF00325475"http://dx.doi.org/10.1364/OL.16.001400http://www.sciencedirect.com/science/article/pii/0030401880900693https://doi.org/10.1007/BF00702605https://doi.org/10.1007/BF00702605https://www.thorlabs.dehttp://redpitaya.com/http://redpitaya.com/http://www.xilinx.com/support/documentation/data_sheets/ds190-Zynq-7000-Overview.pdfhttp://www.xilinx.com/support/documentation/data_sheets/ds190-Zynq-7000-Overview.pdfhttp://www.analog.com/media/en/technical-documentation/data-sheets/AD9763_9765_9767.pdfhttp://www.analog.com/media/en/technical-documentation/data-sheets/AD9763_9765_9767.pdfhttp://www.analog.com/media/en/technical-documentation/data-sheets/AD9763_9765_9767.pdfhttp://cds.linear.com/docs/en/datasheet/21454314fa.pdfhttp://cds.linear.com/docs/en/datasheet/21454314fa.pdfhttp://www.analog.com/media/en/technical-documentation/data-sheets/AD9763_9765_9767.pdfhttp://www.analog.com/media/en/technical-documentation/data-sheets/AD9763_9765_9767.pdfhttp://www.analog.com/media/en/technical-documentation/data-sheets/AD9763_9765_9767.pdfhttps://www.kickstarter.com/projects/652945597/red-pitaya-open-instruments-for-everyone/description

13

red-pitaya-open-instruments-for-everyone/description.64User LNEUHAUS, Red Pitaya DAC performance,

(2016), https://ln1985blog.wordpress.com/2016/02/07/red-pitaya-dac-performance/.

65Red Pitaya team, PID Controller, (2016), http://wiki.redpitaya.com/index.php?title=PID_controller.

66Red Pitaya team, RedPitaya - FPGA memory map, (2016),http://wiki.redpitaya.com/tmp/RedPitaya_HDL_memory_map.

pdf.67(2016), Sourcecode: https://github.com/Julia-F/RedPitaya.68W. Jitschin and G. Meisel, Applied physics 19, 181 (1979).69T. C. Briles, D. C. Yost, A. Cingz, J. Ye, and T. R. Schibli, Opt.

Express 18, 9739 (2010).

70D. Goldovsky, V. Jouravsky, and A. Peer, Opt. Express 24,28239 (2016).

71Multi-component acceleration exciter PTBWorking Group 1.71, (2017), https://www.ptb.de/cms/en/ptb/fachabteilungen/abt1/

fb-17/ag-171/research-and-development-01/

multi-component-acceleration-exciter.html.72Red Pitaya team, Hardware Overview, (2016), http://wiki.redpitaya.com/index.php?title=Hardware_Overview.

https://www.kickstarter.com/projects/652945597/red-pitaya-open-instruments-for-everyone/descriptionhttps://ln1985blog.wordpress.com/2016/02/07/red-pitaya-dac-performance/https://ln1985blog.wordpress.com/2016/02/07/red-pitaya-dac-performance/http://wiki.redpitaya.com/index.php?title=PID_controllerhttp://wiki.redpitaya.com/index.php?title=PID_controllerhttp://wiki.redpitaya.com/tmp/RedPitaya_HDL_memory_map.pdfhttp://wiki.redpitaya.com/tmp/RedPitaya_HDL_memory_map.pdfhttps://github.com/Julia-F/RedPitayahttp://dx.doi.org/10.1007/BF00932394http://dx.doi.org/ 10.1364/OE.18.009739http://dx.doi.org/ 10.1364/OE.18.009739http://dx.doi.org/10.1364/OE.24.028239http://dx.doi.org/10.1364/OE.24.028239https://www.ptb.de/cms/en/ptb/fachabteilungen/abt1/fb-17/ag-171/research-and-development-01/multi-component-acceleration-exciter.htmlhttps://www.ptb.de/cms/en/ptb/fachabteilungen/abt1/fb-17/ag-171/research-and-development-01/multi-component-acceleration-exciter.htmlhttps://www.ptb.de/cms/en/ptb/fachabteilungen/abt1/fb-17/ag-171/research-and-development-01/multi-component-acceleration-exciter.htmlhttps://www.ptb.de/cms/en/ptb/fachabteilungen/abt1/fb-17/ag-171/research-and-development-01/multi-component-acceleration-exciter.htmlhttp://wiki.redpitaya.com/index.php?title=Hardware_Overviewhttp://wiki.redpitaya.com/index.php?title=Hardware_Overview

A highly stable monolithic enhancement cavity for SHG generation in the UVAbstractI IntroductionII Theoretical BackgroundA SHG power optimization1 Single pass optimization2 Cavity geometry3 Crystal shape4 Impedance matching5 Cavity parameters

B Length stabilization and its limit

III Cavity designIV SetupA Pumplight setupB SHG setupC Digital PI controller design1 Hardware2 Software and FPGA configuration

D Characteristics of the digital controller1 Amplitude frequency response2 Limits

V ExperimentA Long term stabilityB Lock performanceC Acceleration sensitivity tests

VI Summary and Outlook AcknowledgmentsVII Appendix