Resonator

As we saw in the previous chapter, the prism can allow a fraction of a light beam in total reflection in the prism to enter the resonator. The amount of light coupling in the resonator controlled by the distance between the prism and the resonator can fluctuate continuously between ~80% to 0% ([92]). If the angle in the resonator is exactly 45◦_{, the light will bounce on every other surface with total internal reflection}

and come back exactly along the same path. It is a resonant cavity (Figure 11.4) with one tunable coupling mirror and 3 mirrors with transmission T = 0, but eventually

some losses. The reflected intensityIr in function of the incident intensityIi is given

by (section 4.2)): Ir =FIi    (r1−(1−Lm)(1−L))2 4r1(1−L) + (1−Lm) sin 2_(kd) 1 +Fsin2_(kd)    (11.7) 126

11.2 Resonator

Where Lm is the losses due to the coupling, L is the losses in the cavity during a

round trip, k is the wave vector in the resonator, d is the round trip length of the

resonator andF is the coefficient of finesse of the resonator given byF = 1/sin2( π 2F)

whereF is the finesse of the cavity. We also sometimes use the linewidth to describe

the resonator which is given by:l = F SR/F where F SR is the free spectral range

of the cavity (19GHz for our cavity).

The losses in the resonator L come essentially from two sources: the scattering

in the mirrors and material absorption. The scattering is due to imperfections in the mirrors. All dust or micro scratches at the interface could make a part of the light experience a different surface angle causing a bit of the light not to be resonant anymore, and even transmitting out. This problem can be solved by carefully polishing the surface of the mirrors to a better quality surface.

The absorption is more of an issue because we can’t really do a lot for it. It is just the matter of getting the most quality material possible to avoid unnecessary impurities absorbing more of the light at the frequency that we are using. Lithium Niobate has an absorption loss at 1064nm of∼0.05%/mmand ∼0.25%/mm at 532nm. For

a resonator of diameter d=2.3mm (total propagation length of 2√2d= 6.5mm), it

corresponds to ∼ 1.5% of losses by round trip for green and ∼ 0.30% for red. (Or

a maximum finesse around ∼ 400 and a minimum linewidth of ~40M Hz for green

and a maximum finesse of ∼2000 and a minimum linewidth of ∼10M Hz for red.)

The linewidth of the light also depends on how much coupling there is between the prism and the resonator, something that we can tune with the distance between the prism and the resonator (Figure 11.3).

0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 scan signal ZOOM b) L + - a) Time (ms) Coupling

Figure 11.3.: (a) Schematic of the resonator with an electrode for applying voltage

and a control of the coupling distanceL. (b) Example of scan obtained by varying

the applied voltage at the electrode (around 600V for 532nm light (green).) The zoom shows the effect of the movement of the coupling distanceLon the resonant

peak.

At resonance, the square sine in Eq. 11.7 is zero, and only the term (r1−(1−Lm)(1−L))2

4r1(1−L) matters. When the prism is far r1 > (1−Lm) (1−L) (under coupling regime

Chapter 11 Resonator Coupling the absorption in the resonator, and the finesse is the largest, but the light directly reflected by the resonator is larger than the light coming from the resonator and they can’t cancel each other perfectly at resonance, the reflected light doesn’t reach zero. When the prism gets closer, eventually, r1 = (1−Lm) (1−L) (critically coupling

regime (Figure 11.4)) and the coupling is the highest possible, only limited by the mode matching. If the mode matching is perfect, the light directly reflected by the first mirror is perfectly canceled by the light coming from the resonator at resonance. When the prism continues to get closer,r1 <(1−Lm) (1−L) (over coupling regime

(Figure 11.4)) the finesse is the smallest, and the light coming out of the resonator at resonance is larger than the light directly reflected leading to a smaller coupling.

Distance (μm) Distance (μm) Coupling Finesse -0.2 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0 500 1000 1500 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 Over Coupling Ereflected Eresonator Ereflected < Eresonator Critical Coupling Ereflected Eresonator Ereflected = Eresonator Under Coupling Ereflected Eresonator Ereflected > Eresonator Finesse

Figure 11.4.: Schematic of the three coupling regimes, over-coupling, critical cou-

pling, and under-coupling, and experimental data of coupling (1- normalised re- flection) and finesse for 1064nm (red) and 532nm (green) in function of the dis- tance between the prism (SF11) and the resonator. The coupling doesn’t reach one because of mode matching.

The design of the system allows another prism to be added in the other side (Figure 11.5). It means adding another coupler to the detector equivalent to a back cavity mirror for a standard cavity (with a tunable reflectivity from 20% to

∼ 100% ). The beam exiting can be sent to a detector or used for alignment pur-

pose. It is easy when there is a first beam exiting the prism, to align another one

11.2 Resonator

in a contra propagating way. The light exiting from the back prism can be consid- ered as additional losses for the resonator, and it is possible to a certain extent to always change the position of the input prism to reach the critically coupling regime (Figure 11.5) whatever the distance of the output prism. That means, for a large range of finesses, having all the light going through the cavity and exiting the other side. a) _b) 0 500 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 Normalized P ower Frequency (MHz)

Figure 11.5.: (a) Schematic of the system to achieve impedance matching for dif-

ferent linewidths of the resonator. By scanning the resonator frequency and by tuning the two distances between the resonator and the prism we demonstrate constant impedance matching of the resonator. (b) shows the normalized power in reflection (red) and the transmission power (blue) during the scan of the fre- quency for different distances between the resonator and the second prism (on the side of the blue detector) for some experimental data for 1064nm.

11.2.0.1. Electro-Optics Tuning

To make the light resonate in the cavity it is possible to change the frequency of the light to match the cavity, but it is not really appropriate when dealing with more than one system. It is much more interesting to be able to directly move the frequency of the resonator. In a standard cavity, we can move the position of the mirrors to change the frequency of the resonator, but this monolithic design doesn’t allow it. Fortunately, the refractive index of Lithium Niobate can be tuned with an applied voltage with the Pockel effect ([85][89]). It provides a similar result then moving a mirror in a standard cavity, allowing us to match the frequency of the resonator to the frequency of the laser.

Chapter 11 Resonator Coupling

∆n = 1

2rn30

V h

with V the applied voltage, h the length of crystal between the two electrodes, n0

the index at V = 0 and r the electro-optic coefficient. The green light is polarized

vertically in the resonator which correspond to the r22 electro-optic coefficient of

33pm/V. The red light is polarized horizontally and correspond to the r33 electro-

optic coefficient of 7pm/V.

With 600V of HV tuning, it is possible to scan a bit more than an FSR for the green (Figure 11.3) which is useful for alignment purposes. Unfortunately the red coefficient is much smaller so it is not possible to observe one full FSR. The initial alignment is harder, but when the TEM00 peak is identified, a change of temperature of the crystal or frequency of the laser will make the peak in the range of the scan. The monolithic aspect of the resonator makes the system very stable, and almost no drift of the peak is experienced. (The resonator is 200µmthick, and allows us to

get a tuning of 40M Hz/V for green and 6M Hz/V for red (FSR = 19GHz)).

Electro Optic tuning also has the advantage to be very fast. It is possible to modulate the cavity frequency at hundredth of M Hz directly by adding a modulation in the

voltage (Figure 11.6) . In Figure Figure 11.6(d) we modulate the voltage at 249MHz and demodulate the signal to create an error signal for the red light without any external EOM on the beam.

Time (ms) Signal ( V) 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 0.4 Signal ( V) 0 0.2 0.4 0.6 0.8 1 Time (ms) -0.4 -0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 Signal ( V) 0 0.2 0.4 0.6 0.8 1 Time (ms) -0.4 -0.2 0 0.2 0.4 a) b) c) Signal ( V) 0 0.1 0.2 -0.1 Time (ms) -0.2 -0.1 0 0.1 d)

Figure 11.6.: (a-c) A few volts of modulation applied directly to the electrode

during the scan of the resonator (with the voltage of this same electode) with 532nm light (green). ((a) is the reference without modulation.) (d) Error signal obtained by modulating the voltage at the electrode at 249MHz and demodulating the reflected signal (magenta) during the scan of the frequency of the light of the laser at 1064nm (red).