Self Locking

Fortunately, it is also possible to use this effect to our advantage. The thermal effect can be a slow active feedback, changing the frequency of resonance to the eventual fluctuation of the frequency of the light, or the change of cavity length due to vibrations (([37], [74]) ).

The transmission power P in an impedance matched Fabry-Pérot cavity without

absorption loss is given by:

Ptrans

Pinc

= _{1 +} 1

F sin2(_kd), (7.3)

where F is the coefficient of finesse, k the wave vector, and d the optical length of

the cavity. The phase kddepends on the temperature of the cavity: kd=ω(t)d0(1 +β∆T)/c

with ω the frequency of the light, c the speed of light, d0 the optical length at

ambient temperature andβ a coefficient accounting for index and length fluctuation

with distance.

If we assume that the temperature in the crystal T has a simple first order impulse

response h(t) of characteristic time constantτ0,

h(t) = ( ₁ τ0 exp(− t τ0) for t ≥0 0 for t <0

Chapter 7 Experimental Method

∆T(t) = αPtrans(t)∗h(t)

with f ∗g the convolution of f and g, and α a positive constant. Substituting in

equation Eq. 7.3 we get the transmission powerPtrans in a Fabry Perot cavity given

by: Ptrans Pinc = 1 1 +F sin2ω(t)_c d0(1 +βαPtrans(t)∗h(t)) (7.4)

By considering v = dω_dt = constant, we can solve the equation Eq. 7.4 numerically

(Figure 7.25).

If we decouple the scan fluctuation and the fluctuation due to self locking, saying that these fluctuations are small compared to the wavelength, we get:

Ptrans Pinc = 1 1 +F sin2ω0+vt c d0 + ω0 c βαd0Ptrans(t)∗h(t) (7.5) 0 0.2 0.4 0.6 0.8 1 -0.001 0 0.001 0.002 0.003 0 0.2 0.4 0.6 0.8 1 -0.001 0 0.001 0.002 0.003 Ptran s / Pin c Time (s) 1) 2) 3) Time (s) 1) 2) 3) a) b)

Figure 7.25.: a) Ptrans_Pinc in function of time for a finesseF = 150, a length of cavity d0 = 5.7mm, a characteristic timeτ = 0.002s, an input power Pinc = 10mW for

a light at 1064nm and αβ = 6∗10−4_W−1_{. The scan speed is 3T Hz/s for (1) and}

−3T Hz/s for (3). For (2) we keep the same scan speed as for (1), but we take αβ = 0. b) is the experimental data for scanning the cavity at the same speed

but positive for 1) and negative for 2), and we reduced a lot the input power for 3) but renormalized the peak to remove thermal effects. In this experiment, the thermal effect is very fast because of the very small waist in the cavity.

When the scan is fast enough, it is possible to completely neglect dynamic thermal effect and we will observe a normal Lorentzian (Figure 7.25 (2)). Changing temper- ature in this condition will simply change the position of the Lorentzian by changing the length of the cavity of the index.

7.5 Locking the System

Now by considering dynamical thermal effect (by scanning slowly enough or increas- ing the power), but when the cavity is far from resonance, there is no power in the cavity, there is no absorption, so no change of temperature. Thermal effects are not doing anything. When the system is approaching resonance, for example during the scan of the frequency of the light in the cavity, when the cavity gets close to reso- nance, the power in the cavity increases, increasing the temperature which changes the “normal position” of the peak. If the scan speed is negative (dν

dt < 0) and the

length and index thermal fluctuation coefficient is positive β >0 (Figure 7.25 (3)),

the change due to thermal fluctuation will go in the opposite direction to the scan, so it will correspond to a “slower scan”, which means the peak looks spread during its positive slope (blue detuning). After resonance, the power decreases and eventually the temperature will also decrease bringing back the peak to its “normal position”, but because of the delay in response of the temperature the scan still looks a bit slower in the negative slope (red detuning).

If the scan speed is positive (dν

dt >0) (Figure 7.25 (1)) the thermal effects and the

scan move in the same direction, corresponding to a faster scan during the positive slope (red detuning) bringing the system to resonance very quickly. The temperature doesn’t have the time to increase a lot, so the negative slope (blue detuning) part of the scan looks pretty much like the case without thermal fluctuations.

Rather than moving the frequency of the light, we can move directly the length of the cavity and observe the same effect (Figure 7.25 (b)). In one direction of the scan, the peak looks bigger, and in the other direction it looks smaller, and this effect increases with the power.

Thermal absorption can be used to lock the cavity at low frequencies. If the cavity is brought almost to resonance from the direction that usually makes the peak bigger (blue detuning if β > 0 and red detuning if β < 0; we suppose here β > 0 ). If

we stop a bit before maximum transmission to avoid reaching the other part which is unstable, it is possible to achieve self locking (Figure 7.26). The temperature of the crystal will increase creating a kind of reservoir to reduce fluctuation. If the frequency of the laser decreases for one reason or another (going from O to A inFigure 7.26), it means that the cavity goes closer to resonance, the power in the cavity increases, decreasing the frequency of resonance of the cavity (going from the curve (b) to the curve (a) inFigure 7.26 and from the point A to B). If the frequency of the light increases, it will reduce the power in the cavity decreasing the temperature, increasing the resonance frequency (going from (b) to (c) and from O to D). In this way, any fluctuation at a frequency low enough will be canceled by the cavity allowing it to stay at a power almost constant. This explanation can be transposed to length fluctuations of the cavity, the temperature fluctuation will help to cancel every fluctuation due to low frequency vibration of the mirrors and the table around the cavity. In practice this locking system can be very efficient and efficiently lock some systems that a PDH locking couldn’t lock easily otherwise. More experiment data will be provided in the next chapter (section 12.6).

Chapter 7 Experimental Method TT T 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 Ptran s / Pin c Frequency (GHz) A B C D a) b) c) O

Figure 7.26.: Frequency response of the cavity for different temperatures (suppos-

ing β < 0). If the system starts in O and moves to A because the frequency

decreases, the temperature in the cavity increases going from (b) to (a) and the power decreases to B. If from O, the frequency increases to D, the temperature in the cavity decreases going from (b) to (c) and the power increases to D. So if the fluctuations in frequency are not too fast, the power stays almost constant.