The 3 He atomic fountain clock

Bz = 803 G

magnetic shielding
dc discharge source laser coll or Zeeman sl

r MCP d

Figure 2.1: The fountain setup: it consists of three parts: 1) a metastable helium source, 2) a cooling section which is formed by a laser collimator, Zeeman slower and magneto-optical trap and 3) the fountain part, containing a microwave cavity and a very homogeneous constant magnetic eld. (Figure taken from [14]).

The clock conguration

The operating principle of atomic fountain clocks is as follows: laser-cooled atoms are launched upwards and fall back due to gravity. The cloud of atoms passes twice through a microwave magnetic eld. At the end of the trajectory, the hyperne state of the atoms is detected. The frequency of the microwave eld is locked to the frequency of the hyperne transition that is probed. This frequency is fed to an readout device that uses the periodic signal for a time display.
Metastable helium beam source
The metastable 3 He atoms are produced in a liquid nitrogen cooled dc discharge source. The atoms are transferred to the metastable S 1 state (see gure 2.2) by a discharge current of a few mA, in a narrow tube that is called a nozzle. Only 0.001% of the atoms are transferred to the metastable state [10]. The metastable atoms are cooled by laser light; all other helium atoms are a source of unwanted background pressure. The beam of metastable atoms enters the collimation section through the nozzle and a skimmer, which provides a rst transverse velocity selection. Using focussed laser beams, the velocity component perpendicular to the

6.7 GHz

$ % %

$# ! " % $ #

ng molasses
2.2 The clock conguration

1083 nm 2 3S1S0

Figure 2.2: The relevant part of the energy level scheme of helium.
curvature of the laser eld is reduced, creating a less divergent beam [11].

Zeeman slower

The average velocity of the metastable atoms in the beam is much too high to catch them in a magneto-optical trap1. The atoms already have transverse velocities that are suciently low, because of the transverse cooling mentioned in the previous section. The longitudinal velocity can be decreased using the Zeeman slowing technique: a red-detuned2 laser beam travels in the direction opposite to the direction of the atoms; because of the Doppler shift, the atoms see the light as resonant with the S 1 to P 2 transition at 1083 nm. The helium atoms absorb photons that have momenta directed along the laser beam and they emit photons in a random direction when falling back to the S 1 state. This procedure generates a net force directed against the movement of the atoms. When the atoms have absorbed enough photons to have been slowed down by a signicant amount, the laser beam will become non-resonant for these atoms. The Zeeman slower is used to compensate for the decreasing Doppler shift experienced by the atoms. The magnetic eld generated by the cone-shaped solenoid (see gure 2.1) shifts the resonance frequency of the atoms (the Zeeman shift). The atomic resonance frequency decreases with the decreasing magnetic eld, thereby compensating for the decreasing Doppler shift. The maximum force exerted on the atoms in the Zeeman slower is limited by the amount of photons that are absorbed per unit time. This amount

The longitudinal velocity is about 1100 m/s when cooled by liquid nitrogen. The velocity of uncooled metastable helium atoms is about 2000 m/s. 2 This red-detuned laserlight has a frequency that is about 500 MHz below the atomic resonance frequency. The light is called blue-detuned if its frequency is higher than the resonance frequency.
depends on the laser intensity and the lifetime of the exited state of the cooling transition. Furthermore, the force depends on the momentum p of the absorbed photons and therefore on the transition frequency. The deceleration of the atoms also involves the mass of the atoms. At laser intensities for which the transition is saturated, the atoms spend half of the time in the upper state, so we have for the number of absorbed photons per 1 unit time N = 2. The force exerted by the laser beam is then given by: F = Np = 1 h 2 c (2.1)
from which the deceleration a of a mass m is easily found to be: a= h 2mc (2.2)
The maximum deceleration can be calculated from equation (2.2) and is 4.m/s2 (!) [12]. In practice, about half of the maximum stopping force is chosen, thereby slowing down the atoms from 1100 m/s to 20 m/s in a distance of 2.4 m.

Magneto-optical trap

Once the atoms are slowed down to suciently small velocities they are caught in a magneto-optical trap (MOT). This device combines laser beams and a magnetic eld to trap atoms both in momentum and real space. Six laser beams, arranged in counterpropagating couples that are orthogonal to each other (see gure 2.1), exert a velocity-dependent force on the atoms that is always directed against the direction of movement of the atoms (just like in the Zeeman slower). This eect is called optical molasses. The magnetic eld, generated by a set of coils in anti-Helmholtz conguration yields a space-dependent force, directed towards the centre of the trap. The two counterpropagating beams of each couple are circularly polarised: one beam is -polarised and the other one is a beam of + light. Both are slightly red-detuned. Because the magnetic eld shifts the energy of the magnetic sublevels of the atom3 , and transitions to the dierent magnetic sublevels are polarisation-dependent4 , either absorption from the + or the beam is favoured. In this way the atoms can be pushed back towards the centre of the trap, even if they started out with velocities low enough not to be resonant with the red-detuned light, thus able to escape from the optical molasses.
In the S 1 state of 3 He these are the J = 1, mJ = 1, 0, 1 levels that are split by the magnetic eld with an amount proportional to mJ. 4 + -polarised light only induces transitions with m = +1 and -polarised light only induces transitions with m = 1.

Time evolution in a rotating frame
In order to study the evolution of the wave functions under the inuence of the Hamiltonian of equation (2.14) one can use the Schrdinger equation: o i d | = (H0 + Hrf ) | dt (2.15)
Solving the Schrdinger equation (2.15) would give us the time evolution o of |. We can get rid of the time-dependence in the Hamiltonian if we make a transformation to a frame rotating with the frequency of the magnetic eld [15]. | R R = eiz t/2 R1 = R (2.16)
The z in the exponential is the Pauli z matrix. The exponential is a symbolic notation of the explicit expression given by the following equation: exp(i A) = 1cos|A| + i A sin|A| |A| (2.17)
We now have: d d (R ) = z R + i R = HR i dt 2 dt d z H = RHR i dt 2

(2.18)

From equation (2.18) we can easily nd the explicit expression of the Hamil tonian in the rotating frame H in the {|2 , |5 } representation: H= 2 R R = z (R x + ) 2 (2.19)
In equation (2.19), = 0 is the detuning of the applied rf-eld with respect to the transition frequency and is the vector of Pauli matrices. The Schrdinger equation becomes: o d 1 = i (R x + ) z dt 2 This equation can be integrated directly with respect to time: R x + z (t) = exp i (t t0 ) 2 (t0 ) (2.20)

(2.21)

Using equation (2.17), the time evolution of the coherent cloud of atoms can be written as follows: z R x + sin( t) (t0 ) (t) = 1cos( t) + i (2.22) in which =
2 R + 2 is called the Rabi opping frequency.

3 t t2 t3

Figure 2.4: The rf magnetic eld experienced by the cloud of atoms in the course of their ight.

Evolution operators

Equation (2.22) can be applied consecutively for each stage in the journey of the cloud of atoms (rst cavity passage, free ight, second cavity passage). Each stage has its own evolution operator that can be derived from (2.22). First cavity passage (0 < t < ): U1 = 1cos( )+i 2

z R x+

sin( ) 2
Free ight ( < t < + T ): U2 = 1cos( 2 T ) + i zsin( 2 T )
Second cavity passage ( + T < t < 2 + T ): U3 = U1 The nal probability that an atom of the cloud of atoms ends up in a nal state |f is given by: f

f U3 U2 U1 (t0 )

= | f | U3 U2 U1 |(t0 ) |2

(2.23)

The last step is allowed because the transformation matrix has only diagonal elements, so the rotation matrices can be pulled through the product of operators. Using R = R1 does the rest of the trick. If we start with all atoms in the lower state: |(t0 ) = |5 , then the probability that the atoms are in the upper state |2 after the Ramsey interrogation becomes: PRamsey = | 2| U3 U2 U1 |5 |2 (2.24)

Stability

The stability of a clock is dened as the Allan standard deviation y of the fractional frequency dierence y(t) [17]: y(t) = (t) (2.29)
The fractional frequency dierence is integrated over a time in order to obtain an average value y k. Subsequently, the root mean square value of the dierence between each pair of average fractional frequency dierences is taken: yk = y ( ) = 1

tk + tk N 1 k=1 1/2

y(t)dt (y k y k1 )

(2.30) (2.31)

1 2(N 1)
For atomic fountain clocks, the Allan deviation is typically given by [16]: y ( ) = 1 Qat Tc + + N + 2 Nat Nat nph Nat

(2.32)

where is the measuring time, Tc the frequency cycle duration, Qat = 0 / is the atomic resonance quality factor and Nat is the number of detected atoms. The second term in parentheses stands for shot noise in the detection signal (in the case of detection by a microchannel plate detector, the photon noise must be replaced by an electron noise contribution), the third term indicates the rms uctuations in the number of detected atoms and the last term is the frequency noise of the interrogation oscillator. All of these terms are expected to be negligible in the helium fountain clock compared to the rst term, that is called quantum projection noise and is related to the
statistical nature of quantum mechanics. If an atom is in a superposition of two states | = |g + |e , the probability of nding the atom in state |e is ||2 = p. If we dene Pe as the projection operator onto state |e , the standard deviation of the quantum uctuations of the measurement of |e is given by [16]: =
2 | Pe | ( | Pe | )2 1/2 1/2

p(1 p)

(2.33)
[18]. In the operation of the clock, This standard deviation scales as Nat the transition probability is alternately measured on each side of the central Ramsey fringe, where p = 1/2. The quantum projection noise therefore contributes to the stability budget of the clock.

Chapter 3

The microwave cavity

3.1 Introduction

The microwave magnetic eld is in a way the heart of the atomic clock. The eld interacts with the atoms and the result of the interaction is monitored, as well as the exact frequency of the eld. The eld acts as an intermediate between the intrinsic structure of the atoms and the output device of the clock. It will be explained how this crucial electromagnetic eld can be constructed inside a cavity, starting from basic electrodynamics and nally calculating the relevant parameters of the cavity.

A nite elements calculation has to be performed to calculate the eect of the cavity holes on the elds. It is possible, however, to use results from other groups, since the dimensions of cavities suited for cesium or rubidium clocks can be scaled to dimensions adapted for the helium clock.
the cut-o waveguides should be smaller than the cut-o radius Rc in order to keep elds of frequency inside the cavity [24]: cxmn (3.29) for transverse electric TEmn elds. This equation provides an upper limit for the size of the holes in the end plates. The cut-o radius for our fountain clock cavity can be calculated to be 2.87 cm, almost as large as the cavity radius of 3.72 cm. Since the magnetic eld amplitude changes too much over such a large distance, the cavity hole radius should be well below the cut-o radius. Rc =
Coupling of elds into the cavity
Now that we know which elds can be contained in a microwave cavity, we need to know how to couple these elds into the cavity. Electromagnetic elds can be coupled into a resonator or wave guide in two distinct ways: electrically and magnetically. In both cases the central conductor of a coaxial transmission line is used as an antenna. In electric coupling, the central conductor has to be aligned parallel to one of the electric eld components of the wanted mode. Magnetic coupling is accomplished by bending the conductor into a loop, which has to be positioned perpendicular to the direction of one of the magnetic eld components of the mode that one wants to excite [24]. Sometimes a combination of both methods is used. Only modes that have a resonance frequency that is within a certain range of the frequency of the coupling signal can be excited (the power that is coupled in obeys a Lorentzian spectrum centered around the cavity resonance frequency). The elds can be coupled directly into the cavity or using a waveguide as an intermediate stage. In the latter case, the waveguide is connected to the cavity via a hole (iris) in the cavity wall. Symmetric coupling from two sides of the cavity is recommended by some groups [25]. Excitation of the TE011 mode yields an interesting problem: this mode has the same resonance frequency as the TM111 mode and it has unwanted radial and azimuthal magnetic components. Calculating the surface currents associated to the both modes, however, shows that for the TM111 mode current ows from the cylindrical wall to the end plates. This is not the case for the TE011 mode. This provides the opportunity to exclude the TM111 mode from the cavity by placing insulating rings between the cylindrical wall and the end plates [7, p.1194].
Amplitude of the magnetic eld
The amplitude of the magnetic eld component directed along the z-axis can be calculated from the height of the cavity, the velocity of the atoms
3.8 Amplitude of the magnetic eld
that are launched through the cavity (ca. 3 m/s) and the conditions for the /2-pulse as derived in section 2.4.5. These conditions can be united in a single formula for the rf magnetic eld amplitude expressed in Gauss: Brf = 9 v v 1.26B d d (3.30)

The magnetic eld dependence of the shift in resonance frequency can be calculated for each given magnetic eld strength. Equation (2.4) yields the following relation between the fractional frequency shift and the C-eld: = |B0 (B B0 ) + | (B B0 )2 (B0 ) B (B0 ) B B0
Equation (4.1) can be applied to two distinct situations: high and low Celd. In each case, the magnetic eld homogeneity required for / = is calculated.

Spatial homogeneity

Low C-eld In conventional fountain clocks, based on bosonic atomic species, low Celds are used, typically of the order of one Gauss. The atomic transition frequency is to rst order independent of the magnetic eld in those clocks. For the helium fountain clock, however, the rst order term of the Taylor expansion (4.1) is dominant for small deviations from B0 in this regime. For a magnetic induction of 0.8 Gauss we get = 9.(B B0 ) Hz/G. At a transition frequency of = 6.74 GHz requiring that / = yields (B B0 ) = B = 7.G. This comes down to a relative homogeneity1 of the eld B/B of 9 109. High C-eld For the situation of the (high) anticrossing eld of 802.6 Gauss, the rst order term of equation (4.1) vanishes2 and we get = 6.(B)2 Hz/G2. We now have = 6.35 GHz, yielding (B)2 = 1.G2 , and resulting in an absolute accuracy of 3.G, or a relative homogeneity of B/B = 4 106. It is clear that the homogeneity requirements imposed on the C-eld are much less demanding for the anticrossing high eld than for low elds. Even though a given absolute accuracy can be much easier reached at low elds than at high elds, the fact that at anticrossing the eld is to rst order independent of the magnetic eld uctuations implies that the clock
1 Since the magnetic eld is proportional to the electric current density through the solenoid, and increasing the magnetic eld comes mainly down to increasing the total amount of current, one can see that inhomogeneities present at low elds are amplied with the increase of current, but without changing the relative homogeneity. This relative homogeneity is therefore the relevant parameter if the technical feasibility is concerned. 2 See section 2.3.
4.2 Homogeneity requirements for high and low eld
Figure 4.1: Dependence of the Zeeman shift in the atomic resonance frequency on the deviations of the magnetic eld around anticrossing eld B.
frequency accuracy requirements can be met more easily with anticrossing eld. In clocks based on bosons, e.g. cesium, the frequency dependence on the deviations of the magnetic eld is also second-order (quadratic), and the absolute accuracy needed is of the same order of magnitude as in the helium clock. The required relative homogeneity, however, is much lower due to the smaller absolute magnitude of the elds. In the remainder of this chapter, 803 G will be used as the value for the C-eld and a relative homogeneity of is the goal that is aimed for.

to be c = 0.9, which comes down to a notch length of 1.8 times the length of the inner radius of the coil (81 mm). Should one be able to construct such an inside notch, this would mean a reduction of the normalised root mean square deviation from the average magnetic eld value from 4 to within the radius of convergence of the expansion. Over the range that is needed for the clock, ca. 45 cm, the homogeneity increases from 9.to 8.9 103. Outside notch The procedure for the outside notch is similar. One has to use a slightly adapted version of equation (4.9) (see appendix C). The results are also quite similar: the outside notch comes down to a cutaway with a depth of 0.17% of the outer radius of the main coil, extending over a length of 1.8 times the outer radius of the main coil. The outer radius of the main coil depends on the number of windings on it, but 69 mm was used in the simulation. Using this outer radius, the depth of the cutaway has to be 117 m. This would lead to an increase in magnetic eld homogeneity to 2.over a region of 90 mm and the outside notch would yield a homogeneity of 8.over a region of 45 cm.

Single coil summation

In this picture one examines the magnetic eld generated by an innitely thin wire, carrying a current I. The eld generated by the solenoid is then approximated as being the sum of the elds that correspond to a number of loops, each a certain distance apart. This view can be extended to solenoids
of nite thickness by summing the elds from several layers of coaxial loops of dierent radius (see gure 4.2). This approach is mainly suitable for estimating the eect of small displacements of some of the current loops, corresponding to possible construction defects that one has to take in account when the clock is built. In principle, this approach could also be used for numerical optimisation of the number of coils at the dierent positions along the z-axis, aiming at the elimination of the nite coil length eect. Some investigations have been made towards this subject, but the observed sensitivity of the eld with respect to mechanical imperfections suggests that the practical implementation of the results of the calculations would be useless. As mentioned in the previous section, the eld on the axis of a single loop is given by: Hz (z) = R2 I 2 (R2 + z 2 )3/2 (4.10)

coil summation model, so no ripple eect is to be expected.

Conclusions

In order to obain a clock with a relative accuracy of 1012 , a C-eld with a homogeneity of is needed. The temporal stability has to be of the same order of magnitude and is completely determined by the stability of the current through the solenoid, provided that the C-eld region is magnetically shielded. Neither the shielding nor the current stability should be limiting factors. The spatial homogeneity was evaluated with two models. The current density integration approach shows that the desired theoretical homogeneity can be reached, in a much too small region (!), by applying rather small corrections to the basic shape of the solenoid. The outside notch is the best candidate for such a correction. The nite coil length eect is a problem if one tries to reach homogeneities below 103. The single coil summation approach, however, yields the insight that even if that systematic problem is solved (which seems no trivial thing to do) random construction errors of sizes that are not unimaginable cause inhomogeneities of the order of 104. These inhomogeneities have to be corrected by shimming techniques. This implies a time-consuming procedure of which even the best results known would not satisfy our needs [28, 29].

Chapter 5

Stabilisation of an extended cavity diode laser at 1083 nm
Cooling and manipulating the helium atoms that are needed for the fountain clock is done by lasers emitting light at 1083 nm. An extended cavity diode laser system was acquired for this purpose. The system features an elaborate control system, including temperature stabilisation and diode-current control. Furthermore, the length of the laser cavity can be scanned, applying a voltage to a piezo crystal on which one of the cavity mirrors is mounted. The measured intensity of the laser beam is fed back into the system in order to control the laser frequency. In this way the laser can be frequencystabilised on a spectral feature, such as a transmission or absorption peak of a Fabry-Perot interferometer or a Lamb dip in a gas cell. The main goal of the experimental work described in the following chapter was to lock the laser output frequency to an absorption line of metastable 3 He.

The width of the velocity distribution of a gas at room temperature gives rise to Doppler broadening of the absorption prole. It is an example of an inhomogeneous line broadening process. As mentioned in chapter 2, the average speed of the helium atoms in the gas cell is of the order of thousands of meters per second. This means that the Doppler eect plays an important role in the absorption of photons by the fast-moving atoms. The absorption prole of a single atom, which displays the absorption intensity versus the laser light frequency, has a Lorentzian shape.[12]: L( 0 ) = /2 ( 0 )2 + (/2)2 (5.1)
with /2 = 1/ the natural linewidth of the transition. Each atom with velocity v sees the laser frequency shifted to a frequency : =kv (5.2)
where k is the wave vector of the photon; |k| = 2/. Assuming a MaxwellBoltzmann velocity distribution, the probability for absorption of a photon in the frequency interval (, + d) by an atom with transition frequency 0 is given by g()d and g() = c 0 m c2 ( 0 )2 exp 2 2kB T 0 2kB T /m (5.3)
This is approximately the shape of the Doppler broadened prole1. The natural linewidth of the transitions in 3 He at 1083 nm is about 1.6 MHz. The Doppler broadened prole has a width of 1 GHz. The Doppler broadened proles of lines that are close to each other tend to overlap to great extend, thus forming a broad absorption prole of a few GHz. It is impossible to stabilise a laser to a few MHz by locking to this prole. One likes to be able to detect the individual underlying transition lines, that are much narrower than the Doppler broadened prole. Saturated absorption spectroscopy provides a means of recapturing the individual lines. A pump beam is used to saturate the transition. At the rear end of the gas cell, the beam is reected and sent back along the path of the pump beam. The retroreected beam is called the probe beam. The intensity of the probe beam is detected by a photodetector. Atoms that have a velocity component along the direction of the laser beams will experience a dierent Doppler shift for the pump and probe beams. Saturation of these
Actually, in order to calculate the Doppler broadened prole, the convolution of this expression with the Lorentzian associated with the natural linewidth should be taken, yielding a Voigt prole. And even the Lorentzian will be broadened by saturation broadening.

Laser frequency control

As mentioned before, the laser frequency can be controlled by adjusting the laser temperature, diode current and piezo voltage. Diode lasers that consist only of a pn-junction have a linewidth of almost 100 MHz [32]. This linewidth can be reduced by forming a larger laser cavity, using the Littrow setup: the light emitted by the diode is collimated onto a reection grating, from which the rst order is sent back into the diode. Since the reectivity of the grating is higher than the reectivity of the diode front facet, the rst order light experiences an extended cavity of a few centimeters, which has a smaller free spectral range and higher nesse than the diode cavity. This yields a linewidth of below 1 MHz4. A micrometer screw can be used to tilt the grating and coarsely tune the frequency that is reected back to the diode. The laser can be tuned over a range of 7 nm (1800 GHz) in this way [32].

Temperature control

The temperature of the laser can be tuned with an accuracy of 0.1o C using a thermo-electric Peltier element. A thermistor temperature sensor is cemented into the base plate of the laser head to provide a feedback signal. The temperature of the laser head can be set in the range of 10o C to 50o C using a maximum current through the thermo-electric element of 2 A [32].

Scan control

The output frequency of the laser can be adjusted by the piezo scan control much faster than by adjusting the temperature. The high voltage version of the laser system provides possible output voltages between 75 V and +150 V. The scan control consists of three parts: the amplitude, oset and frequency tuning. The amplitude potentiometer controls the amplitude
Putting the laser on the slope of a FPI transmission peak (not locked), yielded a measured full range submillisecond variation of about 750 kHz, as measured on the oscilloscope.
of the triangular scan ramp that is sent to the piezo element (maximum peak-to-peak amplitude of 150 V). The oset of this periodic signal can be adjusted between 4 V and 150 V. The voltage is limited to a maximum of 150 V, which causes clipped triangles when the oset is set above 75 V (at full amplitude). The frequency of the scan ramp can be varied over ten orders of magnitude. The maximum mode hop free tuning range using the scan control was just above 4.5 GHz, measured with the transmission from the FPI. According to the manual, the piezo element extends linear with the applied voltage: 33 nm/V, and up to 5 m [32]. Since the piezo actuator is integrated in the grating holder, the length of the cavity is not extended by the same amount: the eect of the piezo length consists of a combination of cavity extention and grating tilting.

Mixing of the photodiode signal and the internal oscillator yields a signal made up of a dc component and a 2 component, since the frequency of the internal oscillator and the frequency dierence between the carrier and the rst-order sidebands are equal. The 2 component is ltered out, using a low-pass lter. The relative phase of the two signals can be adjusted in such a way that the dc signal either becomes the prefactor of the cosine term in equation (5.8), or the prefactor of the sine term. A theoretical error signal can be constructed if the frequency dependence of the transmission coecient associated with the medium is known. In general, the transmission coecient can be expressed as [35]: T () = e()i() (5.11)
where is the amplitude attenuation and is the optical phase shift of the electric eld. The shape of the spectral feature that is to be locked to is Lorentzian in the case of a Lamb dip [35]: () = peak () = peak 1 R2 () +1 R() R2 () + 1 (5.12) (5.13)
where peak is the peak attenuation at the line centre and R() is a normalised frequency scale given by: R() = /2 (5.14)
where and are the line centre frequency and FWHM of the Lorentzian, respectively. The shape of the Pound-Drever error signal depends strongly on the ratio of the modulation frequency and the width of the spectral feature. In our case, the modulation frequency was 20 MHz and the FWHM of the Lamb dips was about 16 MHz. In order to obtain the characteristic Pound-Drever
signal amplitude a.u. 0.075 0.050 0.10 0.025 0.050 0.20 f MHz
Figure 5.3: Pound-Drever error signal: plot of the real part of equation (5.9), for a modulation frequency = 5 MHz and FWHM of the Lorentzian feature of 16 MHz.
signal amplitude a.u. 0.4 0.2 f MHz

25 0.2 0.4

Figure 5.4: Pound-Drever error signal: plot of the real part of equation (5.9), for a modulation frequency = 20 MHz and FWHM of the Lorentzian feature of 16 MHz. This is the situation of the experiment (compare gures 5.8 and 5.9).
signal amplitude a.u. 0.6 0.4 0.0.2 0.4 0.f MHz
Figure 5.5: Pound-Drever error signal: plot of the real part of equation (5.9), for a modulation frequency = 400 MHz and FWHM of the Lorentzian feature of 16 MHz. This is the characteristic Pound-Drever error signal.
error signal, the modulation frequency should be much larger than the width of the Lamb dips. In any case, either the real or imaginary part of expression (5.9) can be selected from the total photodetector signal, in order to serve as the error signal that is fed to the PID section of the system. The real part of (5.9) is plotted for a few ratios of the modulation frequency and the FWHM of the spectral feature. A scheme of the setup that was used to lock to the Lamb dips is shown in gure 5.6. In gures 5.3 to 5.5 theoretical error signals are plotted. Figure 5.3 represents the error signal that can be obtained with low frequency modulation. Figure 5.5 shows the characteristic Pound-Drever error signal for high modulation frequencies. This is the shape of the error signal one would like to achieve. Unfortunately, the modulation frequency of 20 MHz, provided by the local oscillator, was not high enough compared to the width of the Lamb dips to obtain an error signal like gure 5.5. The theoretical shape of the error signal shown in gure 5.4, calculated using the measured Lamb dip width of ca. 16 MHz, does very well resemble the measured error signal shown in gures 5.8 and 5.9.

1 Many authors use the denition = 0 r , which can be confusing; I will only use (and ) as a dimensionless parameter. 2 e.g. E and B are both derived from the vector and scalar potentials A and :
Moreover, E and B appear on equal footing in the Lorentz Force Law: F=E+vB
We now have a magnetic permeability of vacuum 0 and relative permeability (r). The dimensionless parameter m is called magnetic susceptibility. Both the electric and magnetic susceptibilities are in principle dimensionless complex tensors [24]. The polarisation can thus have a dierent direction than the electric eld. In the case of an isotropic medium, the tensor becomes a scalar (which is just a tensor of rank 1). If the medium is lossless, the imaginary parts of the susceptibilities vanish.3

Maxwell equations

The electromagnetic elds that exist in the medium inside the cavity are described by the Maxwell equations. In the MKSA system of units these equations read: D = f H = Jf + E+ (B.5) D t (B.6) (B.7) (B.8)

B =0 t B=0

in which f is the free electric charge density in the medium and Jf is the free electric current density that is caused by the electric eld if free charges are present in the medium: Jf = E in which is called the conductivity of a medium. (B.9)

Electromagnetic waves

We can use the Maxwell equations to obtain equations for the electric and magnetic elds that relate the second derivatives of these elds with respect to space and time [24]. If we consider the case of an isotropic medium, such that the associated electric susceptibility is a scalar and assuming no net charge and current densities are present (f = 0, Jf = 0), we can take the curl of equation (B.7) and get:

(B.10)

The same arguments hold for the conductivity of section B.2
B.3 Electromagnetic waves
Using an appropriate vector identity4 on the left-hand side of this equation and subsequently applying equations (B.5) and (B.2), we obtain 2 E on the left-hand side. On the right-hand side we replace B by 0 H and then use equation (B.6). In the resulting expression we write D = 0 E in order to obtain a wave equation for E:

(B.11)

Similar manipulations yield the wave equation for the magnetic eld:

(B.12)

Equations (B.11) and (B.12) are wave equations for waves travelling with velocity 1/ , if we consider elds in a vacuum we have = 1 and = 1 and the velocity becomes the speed of light in vacuum: c = 1/

(B.13)

Solutions of the wave equations are plane waves, travelling, for instance, in the z-direction in a cartesian coordinate system (this choice is of course arbitrary, we can simply dene the coordinate system such that the z-direction coincides with the direction the wave is travelling). These solutions are then of the form: E = E0 ei(tz) (B.14)
is the angular frequency of the wave and is called the phase constant. If we substitute the electric eld (B.14) into the wave equation (B.11) we obtain an expression for the phase constant: 2 = 2

(B.15)

The phase constant is related to the wavelengh as: = 2/ = 2

c = 0

(B.16)
For all vectors V the following relation holds: V = ( V)

Appendix C

Notches
In section 4.5.1, the Taylor expansion (4.9) for the inside and Helmholtz notches is given. The error coecients that are used in equations (4.9) and (C.7) are given by the following equations [30]:
C1 = + 2 C2 = F = + 2 C3 = 2 + 2 C4 = 2 + 2 (C.1) (C.2) (C.3)
1 sinh1 3/2 3/2 F E2 (, ) = C3 ) (C 1 3/2 3/2 [C (2 + 3C2 + 15C2 ) C3 (2 + 3C4 + 15C4 )] F E4 (, ) = sinh1 3/F E6 (, ) = [C (8 + 12C2 + 15C2 70C2 + 315C2 ) 5 1
4 C3 (8 + 12C4 + 15C4 70C4 + 315C4 )] 3/2

(C.4) (C.5) (C.6)

2.1.0.5 1.002 1.004 1.006 1.008
Figure C.1: Contour plot that shows the parameters of the outside notch for which the second order term (dashed line) and the fourth order term (full line) in the Taylor expansion (C.7) vanishes. The intersection of both curves gives the desired values of c and c for the 6th order outside notch.
The expansion that can be applied to the outside notch reads [30]:
Hz = ja2 F0 (, ) Fc (c , c ) z + [F0 E2 (, ) Fc E2 (c , c )c ] c a2
3 + F0 E4 (, )3 Fc E4 (c , c )c 2

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