an external cavity
Olof Rydberg
Sweden
www.physics.umu.se
Stabilization of an optical frequency comb to an external cavity
Author: Olof Rydberg Supervisor: Amir Khodabakhsh Examiner: Aleksandra Foltynowicz-Matyba
September 22, 2014
Abstract
The subject of this master’s thesis is stabilizing a frequency comb laser to an external cavity using a couple of servo controllers. The aim of the project was to build a pair of servo controllers, replacing parts of the existing commercial and proprietary solution already in use. The system under control is an optical frequency comb, which is locked to an external cavity and is used for trace gas detection and spectroscopy. The comb is a broadband light source and needs to be locked to the external cavity in order to achieve maximum transmission through the cavity. The goal was to replace two of the original controllers and try to improve the locking capabilities of the system. The controllers were also supposed to be customizable and for that reason the control system with all its components was built on breadboards and confined in an aluminium box.
Control circuits were built for the purpose, one for controlling the comb offset frequency by modulating the pump diode current, the other for controlling the repetition rate of the comb laser by altering the length of the laser cavity using a piezo-electric transducer (PZT). A commercial and proprietary servo controller was also in the system, controlling an intra-cavity electro-optic modulator. It was kept for controlling the higher frequency region, for which the PZT no longer worked.
In order to simulate and design the system, Matlab was used with functions described by both theoretically and experimentally obtained mathematical equations. The con- trollers were tested thoroughly in order to make sure they acted according to the intended design, before they were tested with the laser. After an initial lock was obtained, the controllers were optimized further using both experimental and theoretical methods until the lock was optimized and the transmission through the cavity was maximized. The error signals that were used for controlling the system were monitored with both an oscilloscope and a spectrum analyser, the latter producing a spectrum with the power ratio plotted versus frequency. The transmission intensity through the cavity was measured when a good lock had been achieved and the results were analysed by applying a Fourier trans- form to the measured data. This was done with both the old controllers and the new controllers and the resulting plots were compared. Analysis showed that the new control system yielded a transmission signal with a slightly reduced noise level compared to the signal resulting from using the old controllers. The results from the spectrum analyser also showed slightly reduced error signals for the new controllers compared to those of the old controllers.
When summarising this work it can be concluded that the goals set up at the start were achieved with results living up to the expectations. The results also verified that such a control system can be built for locking an optical frequency comb to an external cavity with simple and rather cheap components and with good results.
Sammanfattning
Uppsatsens ¨amne ¨ar stabiliseringen av en optisk frekvenskam mot en extern kavitet med hj¨alp av ett reglersystem. F¨or att uppn˚a maximal transmission genom kaviteten beh¨over den bredbandiga frekvenskammen l˚asas mot den, dess frekvenser stabiliseras och d¨armed m¨ojligg¨ora m¨atningar. F¨or att uppn˚a detta byggdes tv˚a reglerkretsar f¨or att ers¨atta motsvarande delar av den redan existerande propriet¨ara l¨osningen. M˚alet var ocks˚a att om m¨ojligt f¨orb¨attra reglersystemet och f˚a en m¨atsignal med l¨agre brusniv˚a. Reglerkretsarna skulle ¨aven vara modifierbara och d¨arf¨or byggas p˚a kopplingsplattor som placerades i en aluminiuml˚ada.
Reglerkretsar byggdes f¨or ¨andam˚alet, den f¨orsta kretsens uppgift var att reglera kam- mens offsetfrekvens genom att modulera str¨ommen till en diodlaser i kammen. Den andra kontrollerade kammens repetitionsfrekvens genom att ¨andra l¨angden p˚a dess kavitet med hj¨alp av ett piezoelektriskt element. Ett existerande kommersiellt och propriet¨art regler- system anv¨andes sedan tidigare f¨or att styra en elektro-optisk modulator inne i kammen.
Detta system beh¨olls f¨or att korrigera fel inom de frekvensomr˚aden som ligger utanf¨or det piezoelektriska elementets verkningsomr˚ade.
F¨or att simulera och designa systemet anv¨andes Matlab. Kretsarna testades grundligt f¨or att m¨ota de tidigare n¨amnda m˚alen innan n˚agra tester genomf¨ordes tillsammans med lasern. Efter att en f¨orsta l˚asning hade uppn˚atts optimerades kretsarna tills dess att transmissionen genom kaviteten var maximerad. Felsignalerna som anv¨andes f¨or att kon- trollera systemet m¨attes med hj¨alp av b˚ade ett oscilloskop och en spektrumanalysator, d¨ar den senare anv¨andes f¨or att m¨ata felsignalens relativa storlek inom det kontrollerade frekvensomr˚adet. Efter att en, enligt de n¨amnda premisserna bra l˚asning hade uppn˚atts m¨attes det transmitterade ljusets intensitet, m¨atdatan fouriertransformerades, plottades och analyserades. Det nya och gamla systemet kunde d¨arefter j¨amf¨oras. Resultaten av m¨atningarna visade att det nya reglersystemet gav en n˚agot l¨agre brusniv˚a i m¨atningarna.
D˚a arbetet summerades kunde det konstateras att de uppsatta m˚alen hade uppn˚atts.
Resultaten visar s˚aledes att ett reglersystem av den h¨ar typen kan byggas med enkla och billiga elektriska komponenter och med goda resultat.
Acknowledgements
First, I would like thank my supervisor, Amir Khodabakhsh, for his support during the work on this thesis. For his guidance, his patience, and for reading my report. Secondly, I would like to thank my examiner, Aleksandra Foltynowicz-Matyba, who gave me the opportunity to work with this project and for correcting my thesis. I would also like to thank Joakim Paulsson and Ragnar Seton for their great advices during the work, and for keeping me company during an otherwise long summer. Special thanks to Alexandra Johansson for reading my report, for great friendship and for all the advices and discussions related to the subject. Last I would like to thank all of the spectroscopy group for the support and for the willingness to answer all the questions I came up with during the work.
Abbreviations
AC - alternating current
BNC - Bayonet Neill–Concelman, a special type of coaxial connector cw - continuous wave
DC - direct current
EOM - electro-optic modulator FSR - free spectral range LED - light-emitting diode
LFGL - low frequency gain limited PBS - polarizing beam splitter PD - proportional derivative PDH - Pound-Drever-Hall PI - proportional integral
PID - proportional integral derivative PZT - piezo-electric transducer WP - wave plate
Symbols
νm - optical frequency ω - angular frequency C - capacitance f0 - offset frequency f - frequency
fc- corner frequency
frep - repetition rate (frequency) G - system gain (transfer function) H - transfer function of controller i - current
j - complex number R - resistance Z - impedance
Contents
1 Introduction 1
1.1 Background . . . 1
1.2 The aim . . . 1
1.3 Structure of the thesis . . . 2
2 Theory 3 2.1 Optical frequency combs . . . 3
2.2 External cavity . . . 4
2.3 Laser-cavity stabilization . . . 5
2.3.1 Transfer functions and block diagrams . . . 6
2.3.2 Basic control theory . . . 6
2.4 Bode plots . . . 8
2.5 PID controller . . . 8
2.5.1 Building blocks . . . 11
3 Experimental setup 15 3.1 The laser . . . 15
3.1.1 Laser actuators . . . 17
3.1.2 Current driver . . . 17
3.1.3 Piezoelectric transducer . . . 17
3.1.4 Electro-optic modulator . . . 17
4 Experimental procedure 18 4.1 Modelling . . . 18
4.2 Building the controllers . . . 21
4.3 Testing the controllers . . . 22
5 Results 24 5.1 Experimental results . . . 24
5.1.1 Theoretical - open loop transfer functions . . . 24
5.1.2 Experimental - transfer function of controllers . . . 26
5.1.3 Theoretical - closed loop transfer functions . . . 28
5.1.4 Experimental - closed loop transfer functions . . . 29
5.2 The product . . . 35
6 Discussion 39 7 Conclusions 41 References 42 A Circuit derivations 43 A.1 Amplifying stage . . . 43
A.2 Differentiating stage . . . 43
A.3 Integrating stage . . . 44
A.4 Low-pass filtering stage . . . 44
B Controller schematic 45
C Simplified circuit layout 46
D Current controller layout 47
E PZT controller layout 48
1 Introduction
The subject of this thesis is stabilizing an optical frequency comb to an external cavity.
A simple and general explanation of this would be that a light source is stabilized with respect to certain frequencies (wavelengths) in order to achieve optimized transmission through an external cavity. The cavity contains a gas that is analysed and the role of the cavity is to increase the interaction length between gas molecules and laser light, compared to the exposure that would result from just sending light directly through the gas. The increased exposure is due to the increased travelling length of light in the cavity. The light that is transmitted through the cavity can be measured and the resulting spectrum analysed. The targeted optical system is used for research, which makes the project more interesting as an improvement of the locking mechanism may help the research group to have cleaner measurements with less noise and more efficient compensation for drifts.
1.1 Background
The spectroscopy group at the Department of Physics in Ume˚a is using an optical fre- quency comb for trace gas detection with cavity enhanced techniques. The comb laser is locked to a cavity and three servos are used for the purpose of stabilizing the comb for certain frequencies, maximizing the transmission through the cavity. In order to fully control the laser, each laser actuator needs a separate controller. For that purpose, three proprietary and commercially available controllers were used where each of them was con- trolling different aspects of the comb. These controllers are known for their stability and can be used for many different setups due to the large number of parameters that may be altered through knobs and buttons. The controllers are however not specialized for any specific setup and are limited in the sense that they act in a pre-specified way that cannot be changed. Due to the limitations of the controllers, the group wished to replace two of the controllers with new home-built ones that would be optimized for the specific setup. The third controller was kept for controlling the high frequency region where high stability of the controller is needed. The concept of locking a laser to an external cavity using custom built servo controllers was well known to the group because it has earlier been done in a different setup with a continuous wave fiber laser instead of an optical frequency comb [1].
1.2 The aim
The aim of this project was the following:
• Build and test two replacement controllers, one for controlling the Piezo-electric Transducer and the other for controlling the current fed to the pumping diode in the laser.
• Optimize the new controllers, if possible improve the control system.
• Make the control system customizable and make it possible to duplicate.
The main goal was to build a replacing solution that performs at least equally well com- pared to the old control system. The new controllers should be able to replace the old controllers without losing any significant performance. If possible, the system should also be improved in the sense that the transmission through the cavity gets better, i.e. more transmitted light and less noise. The controllers should be built on protoboards and be customizable such that further improvements and optimizations may be implemented.
This requires a thorough documentation and a structured building technique, both for making changes easier and for making it simple to duplicate the controllers, in case more of them are needed later for other projects.
1.3 Structure of the thesis
This thesis contains the following sections:
Theory - the most important parts of the theory are explained, most weight on control theory and circuit analysis but some shorter parts concerning the optics are also included.
Experimental setup - a much simplified description of the experimental setup in which the controllers are used.
Experimental procedure - the working progress of the project is described, the meth- ods that were used and how the controllers were tested.
Results - the results are presented, both results from simulations and experiments where the controllers are tested and their behaviour is verified. The product, i.e. the controllers, is also presented in a separate section in which the physical product is described.
Discussion - the results and the limitations of the model and the control system are discussed.
Conclusions - the conclusions of the work are presented.
2 Theory
The purpose of this section is to introduce the reader to the most important parts of the theory related to this thesis. The first part of the theory section is dedicated to optics and the frequency comb with the purpose of introducing the reader to the subject and to explain why stabilising mechanisms are needed. The intention is to keep this part as simple as possible. This thesis and the work related to it is about controlling the optical frequency comb laser, not about the system itself. The second part of the theory, from section 2.3.1 to the end of the chapter, is the electrical part and it is important for understanding the rest of the report. For that reason the second part is far more thorough and detailed compared to the optical part.
2.1 Optical frequency combs
The optical frequency comb is a laser that produces periodic pulse trains with pulses of extremely short duration. The name frequency comb comes from the structure of the pulse trains in the frequency domain, where the optical spectrum has equally spaced spikes that, as can be seen in Figure 1, very much resemble that of a comb. These pulse trains are created using a technique called mode-locking, a technique where many different longitudinal modes of a laser cavity with different phases are combined through interference to form a pulse train with a fixed spacing between the pulses [2].
I(ν)
ν E(t)
t
f0
τ
frep
ΔΦ 2ΔΦ
Figure 1: The pulse train of the comb in time domain is seen in the upper picture and the corresponding spectrum in frequency domain is seen in the lower picture.
With this technique the laser can produce powerful pulses of extremely short duration, down to the level of a few femtoseconds. In the frequency domain, the optical frequency of the comb can mathematically be described as
νm = m · frep+ f0, (1)
where m is an integer, νm is the optical frequency of the mth laser line in the comb, frep is the repetition rate and f0 is the offset frequency. The time between each pulse in time
domain is denoted τ and the repetition rate is the inverse of the time between two pulses, i.e.
frep = 1
τ, (2)
where the repetition rate is the frequency spacing between adjacent lines in the frequency domain. The offset frequency f0 is given by
fo= ∆φ
2πfrep, (3)
where ∆φ is the difference in carrier-envelope phase. By inspecting the equations of this section, it can be seen that the entire comb can be controlled by controlling only the two frequencies frep and f0. Note that this section is only meant to brief the reader into the subject, for more reading see [3].
2.2 External cavity
The external cavity contains the gas that is analysed and depending on the properties of the gas, the intensity of the light that passes through the cavity will differ. The external cavity is a cavity with a pair of highly reflective mirrors. Light bounces back and forth between the mirrors and for the light to be transmitted out from the cavity, the length of the cavity needs to be an integer number of half the wavelength of the light, in order to create standing waves in the cavity. The light will then interfere constructively and produce a ray with sufficiently high intensity, otherwise the light will interfere destructively with itself and no significant amount of light will pass [4]. The wavelengths of the light must be matched very precisely with the cavity length for this to happen. The number of modes, q, allowed in the cavity follows the relation
L = λ
2q, (4)
where λ is the wavelength of the light and L is the cavity length [2]. Using that the frequency is equal to the speed of the light in the media vlight divided by the wavelength λ, where vlight= c/n the allowed frequencies are
νq= c
2Lnq, (5)
where νq is the frequency of a mode and n is the index of refraction of the medium in the cavity. The free spectral range of the cavity is given by
F SR = c
2nL, (6)
and the allowed frequency modes of the cavity can be written as
νq= q · F SR. (7)
The half width of a mode Γc in the cavity follows the relation Γc= F SR
2F , (8)
where F is the finesse and depends on the properties of the cavity [1]. The finesse, F , can be determined by the relation
F = π√ R
1 − R, (9)
where R =√
r1r2 and r1, r2 are the reflectivities of the mirrors in the cavity [2].
2.3 Laser-cavity stabilization
Generally, a pulsed laser that is manually matched with an external cavity does not gen- erate a sufficiently stable and maximized transmission through the cavity. This is due to disturbances such as drift and mechanical disturbances, for example vibrations [5]. For that reason, a stabilizing process is needed in order to keep the laser lines in the cavity as stable as possible. This is absolutely necessary if the laser is to be locked to a high finesse external cavity and as mentioned in section 2.2, the wavelengths and frequencies must be very exact. In order to lock the frequency comb to the external cavity, the frequency spacing of the comb needs to be equal to that of the external cavity (frepcomb = F SRcavity), a situation illustrated in Figure 2.
FSR
2Γc Cavity
Comb
Frequency domain
frequency
fref
ferr Comb
frep
(mode matched)
Figure 2: Modes of the comb together with cavity in frequency domain. The upper part is the cavity, middle part is a mode matched comb with the repetition rate equal to the FSR of the cavity and zero offset. The lowest picture is the comb with some offset but the same repetition rate as in the mode matched comb.
The figure illustrates two scenarios, both with a correct repetition rate of the comb that is equal to the cavity free spectral range. The upper comb illustration in the figure shows comb lines well matched to the modes of the cavity. The lower comb illustration shows a comb with correct repetition rate but with a small frequency offset denoted ferr in the picture. In that case the offset has to be corrected in order to achieve maximum transmission through the cavity. If the repetition rate of the comb is not equal to the F SR of the cavity, some modes may be matched and the offset may be correct in which case the repetition rate needs to be corrected for a perfect match.
The actual stabilization is needed both in order to maximize the transmission in the first place but also for compensating for drifts and changes related to the cavity and the system in general. In order to correct for any small mismatches, an error signal is usually used which gives information not only about how large the offset is but also whether the frequency needs to be decreased or increased, thus providing a kind of direction for the correction of the signal.
2.3.1 Transfer functions and block diagrams
The function that relates the output signal to the input signal of a system is called a transfer function. If the input signal is Xin and the output signal is Xout, then the relation between the input and the output signal can be written as Xout= H · Xin, where H is the transfer function. It follows that the transfer function, H, can be written as
H = Xout Xin
, (10)
where H represents the gain (or loss) of the system. The transfer function can be unit-less but is not required to be so, it depends entirely on what the system takes as input and the corresponding output. The transfer function is often used together with something called a block diagram, which is a schematic picture for describing a system. It is used to describe a more or less complex system without going into details about how different parts of the system work. A block in the diagram describes how the output from the block is related to the input and the diagram itself describes how the blocks are linked together. The blocks are treated as black boxes and are very useful for describing, deriving, designing and understanding a system.
2.3.2 Basic control theory
In basic control theory there is usually a system that needs to be controlled for the process to work properly and that is done using negative feedback. A simple description of how this works would be that the output signal of the system is compared to a reference signal, which is done by subtraction.
Figure 3: Block diagram for a simple control loop where G(ω) is the transfer function of the controlled system and H(ω) is the transfer function of the controller [1].
The resulting signal, which is called the error signal, is fed through a controller and back into the system under control. Some disturbances may also add to the signal in the control loop, which are modelled as an additional signal after the controller and can be seen in the derivations that is to follow in this section. A schematic of the system in the form of a block diagram can be seen in Figure 3, where G(ω) is the transfer function of the controlled system, H(ω) is the transfer function of the controller, X is the output signal from the system, Xref is the reference signal, Xerris the error signal, Xcorr is the correction signal and Xdist is some kind of disturbance signal that can be interpreted as noise and interference. Below follows a simple derivation of a very important and central relation
for this thesis, a relation that can also be seen in [6] where the matter is discussed more thoroughly. The error signal is created by subtracting a reference signal from the output signal as seen below.
Xerr= X − Xref (11)
The reference signal is here the desired value, i.e. the value that the output signal ideally should have and the signal that one is striving to achieve. The error signal is fed through a servo controller with a transfer function H and the resulting signal is
Xcorr = H(ω)Xerr, (12)
where Xcorr is the correction signal from the output of the controller. The controlled systems output is then the input signal (to the system) multiplied with its own transfer function, G(ω), as seen below.
X = G(ω)[Xdist− Xcorr] (13)
The input signal in Eq. (13) is the correction signal subtracted from external disturbances.
By inserting Eq. (11) into Eq. (12) and inserting the result into Eq. (13), the output signal become that of Eq. (14) below.
X = G(ω)[Xdist− H(ω)(X − Xref)] (14) This can be rewritten as
X = G(ω)H(ω) 1 + G(ω)H(ω)
| {z }
Hclosed
Xref + G(ω) 1 + G(ω)H(ω)
| {z }
Hdist
Xdist, (15)
where the term before Xref is the closed loop transfer function Hclosed of the system and the term before Xdist is the error propagation function Hdist. These two functions are important and are displayed below.
Hclosed= G(ω)H(ω)
1 + G(ω)H(ω) (16)
Hdist = G(ω)
1 + G(ω)H(ω) (17)
The output signal X can also be rewritten as in Eq. (18) below which is advantageous for reasons that will become apparent in the following part of this section.
X = 1
1 +G(ω)H(ω)1
Xref+ 1
H(ω)Xdist
(18)
The open loop transfer function is another important relation, maybe even the most important in this section and is seen below.
Hopen= G(ω)H(ω) (19)
The open loop transfer function is often modelled when designing control loops. The open loop transfer function is, as it sound, the relation between the output and the input signal when the control loop is open. From the results of the derivation for the signal of the control system (Eq. (18)) a couple of important properties of the loop can be seen. First of
all, for H(ω)G(ω) >> 1 the expression in Eq. (18) is reduced to X ≈ Xref+ Xdist/H(ω).
It can further be seen that the relatively small disturbance in the control loop will be significantly reduced and may ultimately be neglected if H(ω) >> Xdist, which reduces the signal to X ≈ Xref. This will make the error signal close to zero and the system will be stable and will converge to the desired value. This result is important as almost the entire design of the control system is based on it. These results indicate that the system should be constructed with the transfer function, H(ω), as large as possible. Assuming that the gain of the system G(ω) is a property of the system and cannot be changed, the part that can be changed is H(ω) which must therefore be set as high as possible, which will hopefully be enough for locking. There is however one more relation that can be identified by looking at Eq. (18) that is very important and which will in fact limit the design to a much greater extent compared to the requirements of the transfer function.
The open loop transfer function H(ω)G(ω) must not approach −1 as this will make the term in front of the reference signal Xref infinite (or very large) in which case the system will start to oscillate and will as a result not be controllable. The open loop transfer function Hopen = H(ω)G(ω) can generally be written as
Hopen(ω) = A(ω)ejΦ(ω), (20)
where A(ω) is the frequency dependent amplitude of the open loop transfer function, Φ(ω) is the phase shift between the input and the output signals and j is the complex number.
The open loop transfer function will approach −1 when the phase shift Φ(ω) is close to
−180◦ while the amplitude A(ω) is close to unity gain. Because of this a safety buffer called phase margin of at least 30◦ should be kept at the point where the gain is unity (0 dB), which is important when designing the control loop in order to be sure of stability.
Furthermore a gain margin of at least 3 dB should be kept when the inevitable happens that the phase angle passes −180◦.
2.4 Bode plots
The Bode plot is a plotting technique that is very often used in electronics. The amplitude of a transfer function is converted to decibel and plotted versus the frequency. The scale of the x-axis in the plot is logarithmic with a base of ten. The gain, G(ω), is expressed in decibel according to the following relation
GdB(ω) = 20 log10(|G(ω)|), (21)
where ω is the angular frequency. The gain plot is often accompanied by a plot of the phase shift. That is, the difference in phase between the input and the output signal.
This plotting technique has quite a few advantages. First of all it visualises changes of the curve for small values on the x-axis as much as those for greater values, which is good in order to see the properties of the function. Besides that, another good thing about the technique is that two different signals multiplied by each other become additive in the logarithmic scale and thus may be added together in a new plot. The same goes for the plot of the phase shift, the phase shifts are also additive by nature. This is useful when designing circuits which will be seen later on in this thesis.
2.5 PID controller
PID stands for Proportional Integral Derivative and a PID-controller (or parts of it) may be used to control a system through negative feedback. The proportional part of the PID
is linear. When the signal is close to the reference value, the error signal is small and the effect of the term is decreased. The proportional part is used in most control systems but it does have one major weakness, it cannot handle drift for which other components are needed [7]. The integral part of the PID slows the system down generally, it makes the system react more slowly and makes it less sensitive to disturbances. It corrects for drifts in the system but the added slowness may make the system response too slow, in which case the corrections are made too late when the system changes. This is where the last part of the PID comes in, the derivative (or differential) part. This part adds speed to the system but it comes at the cost of stability. The differential component does indeed add some speed and thus an increased responsiveness which, as the name suggests, is due to it acting harder when the system is changing fast. The risk with such a component is that it is very sensitive to disturbances in the form of sudden extreme changes. Because of the above mentioned reasons it is often desirable to combine at least P and I components depending on the desired properties for the system, some times all three of them.
A schematic for a PID controller can be seen in Figure 4, which is used in this chapter for simple derivations of the parts that may be used.
Figure 4: Block diagram for a PID controller, the dashed rectangle containing the PID blocks represents the controller and the plant is the controlled system. To the right in the figure is a feedback loop and in the upper right corner the reference signal is added to the output and fed back to the controller [7].
For the PID controller model in Figure 4, signals in time domain are denoted by small letters and the corresponding capital version denotes the signal after transformation with the Laplace transform. In this section, e(t) is the error signal, u(t) is the correction signal, y(t) is the output signal and r(t) is the reference signal, all in time domain. The expressions within the boxes are used to indicate the effect each part of the controller will
have in time domain while the corresponding expressions in complex frequency domain are written outside of the boxes.
In time domain, each block acts on the same input signal as follows:
up(t) = Kpe(t) ui(t) = Ki
Z t 0
e(t0)dt0
ud(t) = Kdde(t) dt
Note that Kp, Ki and Kd here are some arbitrary constants. The Laplace transform is used on each of the relations in order to determine the effect in complex frequency domain, as can be seen below.
Up(s) = L {Kpe(t)} (s) = Kp· E(s) (22) Ui(s) = L
Ki
Z t 0
e(t0)dt0
(s) = 1
sKi· E(s) (23)
Ud(s) = L
Kdde(t) dt
(s) = s · Kd· E(s) − e(0) (24) In Eq. (24) the term e(0) may be taken as zero as the signal at time t = 0 is quite arbitrary and insignificant. Adding together the results from Eq. (22), Eq. (23) and Eq. (24) the result become that of Eq. (25) below.
U (s) = Up(s) + Ui(s) + Ud(s) =
Kp+ Ki
s + s · Kd
| {z }
H(s)
E(s) (25)
The output Y (s) can also be written as
Y (s) = U (s)G(s) (26)
and inserting the result from Eq. (25) into Eq. (26) gives the output
Y (s) = G(s)H(s)E(s). (27)
The rest of the loop is not considered interesting here as it was derived earlier in section 2.3.2 and for that reason it is left undone in this section. The transfer function H(s) is the interesting part here. If s → jω then the PID transfer function H(s) is transformed from
H(s) = Kp+ Ki
s + s · Kd in complex frequency domain into
H(ω) = Kp+Ki
jω + jω · Kd (28)
in frequency domain [6].
In order to understand the usefulness of these results, the concept of corner frequen- cies has to be introduced. The corner frequency appears where the imaginary and the real parts in a transfer function are equal. The PI corner is then the transfer region between
the region where the proportional term dominates and the region dominated by the inte- gral term. Likewise the PD corner is the region between the proportional and derivative dominated regions. An illustrating sketch of how it may look like in a Bode plot (with log log scale) can be seen in Figure 5 where both the PI corner and the PD corner can be seen.
Figure 5: Bode plot, PID schematic with the gain of the transfer function versus angular frequency [6].
2.5.1 Building blocks
A control loop may consist of many different components that are treated as building blocks. Many of the possible blocks, and versions of the blocks presented in this section can be found in [8] with more thorough descriptions and explanations. Before introducing the different blocks, the corner frequency introduced at the end of the previous section needs to be specified. For the case of a resistance and a capacitor in parallel, i.e. with a resulting term jωRC + 1, the corner frequencies can be written as
fc= 1
2πRC, (29)
where R and C are the resistances and capacitances accompanying the complex number j.
The first building block presented in this section is a non-inverting integrator with a low frequency gain limit that introduces a phase lag. It amplifies the signal for frequencies below a specified corner frequency and reaches unity gain for frequencies sufficiently high above the corner frequency. The circuit for the block and the corresponding gain and phase plots are shown in Figure 6. The gain is mathematically described as
G = 1 R0
R1
jωR1C1+ 1
+ 1, (30)
where G is the gain, R0 and R1 are some resistances, C1 is a capacitance and ω is the angular frequency equal to 2πf where f is the frequency. An example of Bode plots with gain and phase shift corresponding to Eq. (30) is shown in Figure 6(b) with the fraction R1over R0set to 10 and the corner frequency fctaken as 10 kHz (gives an amplification of 11, close to 21 dB). For derivation of the formula in Eq. (30), see section A.1 in appendix.
R0
R1 C1
(a)
100 101 102 103 104 105 106 107 108 0
10 20
Frequency [Hz]
Gain [dB]
100 101 102 103 104 105 106 107 108
−60
−30 0
Frequency [Hz]
Phase shift [deg]
(b)
Figure 6: Non-inverting integrator with low frequency gain limit, circuit layout (a) and the corresponding gain and phase plots (b) for R1/R0= 10 and fc= 10 kHz.
The second block has a switch, when the switch is open it is an ordinary inverter and when it is closed the block act as a differentiator. The circuit and corresponding plots are illustrated in Figure 7.
C0
R0
R1
(a)
100 101 102 103 104 105 106 107 108
0 20 40 60 80
Frequency [Hz]
Gain [dB]
Differentiation off Differentiation on
100 101 102 103 104 105 106 107 108
−180
−150
−120
−90
Frequency [Hz]
Phase shift [deg]
Differentiation off Differentiation on
(b)
Figure 7: Switchable inverting differentiator, circuit layout (a) and the corresponding gain and phase plots (b) for R1/R0= 1. The solid blue curve in (b) correspond to the case when the switch is open and the red dashed curve correspond to the case when the switch is closed with differentiation active and fc= 10 kHz.
The mathematical formula derived from the circuit in Figure 7(a) gives a gain G that follow the relation
G = −R1 R0
[jωR0C0+ 1] (31)
when the switch is closed resulting in a differentiator. When the switch is open the gain is
G = −R1
R0
(32) and the block acts as an ordinary inverter. This block is a suitable stage for adding variable gain in the form of a potentiometer with variable resistance in place of R1 as changing that
specific resistance only affect gain and neither corner frequencies nor phase shift, both in the case of an ordinary inverter and in the case of a differentiator. For derivation of Eq.
(31) and Eq. (32), see section A.2 in appendix.
The third block is both the most complicated and the most important block because it is the main integration block. The block has a switch offering two different integration modes. The circuit layout is shown in Figure 8(a). When the switch is closed the low frequency gain limited (LFGL) integration mode is active and the gain is
G = − 1 R0
R1
jωR1C1+ 1+ R2
jωR2C2+ 1
, (33)
where the components correspond to the circuit layout in Figure 8(a). When the switch is open, the circuit is in full integration mode and the gain becomes
G = − 1 R0
1 jωC1
+ R2
jωR2C2+ 1
. (34)
The plots of both the modes in Eq. (33) and Eq. (34) are shown in Figure 8(b). For derivations, see section A.3 in appendix.
R0
R1
C1
R2
C2
(a)
100 101 102 103 104 105 106 107 108
−80
−60
−40
−20 0 20 40
Frequency [Hz]
Gain [dB]
Limited Full
100 101 102 103 104 105 106 107 108
60 90 120 150 180 210
Frequency [Hz]
Phase shift [deg]
Limited Full
(b)
Figure 8: Switchable inverting integrator, circuit layout (a) and the corresponding gain and phase plots (b) with R1/R0= 1, R2/R0 = 1 and corner frequencies f1 = 100 Hz, f2 = 10 kHz and f3 = 1 MHz. The solid blue curve in (b) correspond to the case when the switch is closed (LFGL) and the red dashed curve correspond to the case when the switch is open and full integration is active.
The corner frequencies for this block are denoted f1, f2, f3, given by the relations f1 = 1
2πR1C1, f2 = 1
2πR2C1 and f3 = 1 2πR2C2,
where each corner number correspond to the number of the corner in Figure 8(b) counted from the left.
The last block of this chapter is an ordinary active inverting low pass filter, plots and circuit are shown in Figure 9. The gain of the block is
G = − 1 R0
R1
jωR1C1+ 1
, (35)
where the corresponding resistances and capacitors are shown in the circuit layout of Figure 9(a). The corner frequency is set to 10 kHz and is shown in Figure 9(b). For derivation of the formula in Eq. (35), see section A.4 in appendix.
R0
R1
C1
(a)
101 102 103 104 105 106 107
−60
−40
−20 0
Frequency [Hz]
Gain [dB]
101 102 103 104 105 106 107
90 120 150 180
Frequency [Hz]
Phase shift [deg]
(b)
Figure 9: Inverting low pass filter, circuit layout (a) and the corresponding gain and phase plots (b) for R1/R0= 1 and fc= 10 kHz.
3 Experimental setup
The setup had quite a lot of different parts and most of them are not of interest in this thesis. For that reason this section will only cover a brief and very simplified setup and a schematic of that setup can be seen in Figure 10.
Figure 10: Simplified experimental setup. EOM - electro-optic modulator, PBS - polarizing beam splitter, Cavity - external cavity, PD - photo detector, RF components - some radio frequency components, PZT - piezoelectric transducer.
The method for creating the error signal is called the Pound-Drever-Hall (PDH) technique, for more details on the subject see [9]. Starting at the comb, pulsed light is sent out from the laser. The light passes an external EOM and is modulated with a chosen modulation frequency. Two side bands are created and they are required for locking. The modulation frequency is chosen such that it lies between that of the cavity linewidth and the free spectral range of the cavity. The electric signal that is used for the modulation is also used as a reference signal, as can be seen in the schematic. The light passes a polarizing beam splitter and a quarter wave plate, the latter changes the polarization from linear to circular. After that the light is reflected back from the cavity, unless the modes are well matched, in which case the side bands are reflected back while the main mode is transmitted through the cavity. The light that is reflected back changes polarization from circular back to linear when passing the quarter wave plate and becomes perpendicular to the original beam. The light is reflected in the beam splitter and hits a grating where the light is dispersed and two photo detectors are used to detect two different wavelengths, as light with different wavelengths will reflect with different angles. The signals from the detectors goes to a box that is labelled RF (radio frequency) components in the figure where the detector signals are separately multiplied with the reference signal and the result is sent to the controllers in the form of two separate error signals. As can be seen in the schematic, the current controller has an error signal of its own while the PZT and EOM controllers are connected in parallel sharing the other error signal. The control signals (or correction signals) are fed back from the controllers to the actuators and the actuators respond with adjustments depending on those signals.
3.1 The laser
The optical frequency comb used in the experimental setup is an Er3+ fiber laser that is pumped by a continuous wave diode laser. The laser comb is a broadband light source
that emits at wavelengths between 1.5 and 1.6 µm with a power of approximately 20 mW.
The comb consists of quite a few parts as can be seen in Figure 11, in which a schematic of the internal parts of the comb is displayed.
Figure 11: Schematic for the internal parts of the comb [10]. WDM - wavelength division multiplexer, WPs - one half-wave plate and one quarter-wave plate, PBS - polarizing beam splitter, PZT - piezoelectric transducer, EOM - electro-optic modulator.
In this work, the interesting parts of the laser was the pump, the fiber, the PZT mounted on a stepper motor and the EOM. The stepper motor is just a way to manually make big changes to the displacement of the PZT and its mirror, it is indicated by an arrow at the PZT. The other parts of the laser are just mentioned briefly. The wave plates (WPs) and the cubic polarizing beam splitters (PBS) ensure that the light takes the desired path and has correct polarization for mode locking. The isolator makes sure no light can go in the wrong direction, the wavelength division multiplexer (WDM) makes sure the light that comes from the diode laser is coupled well for pumping the Er3+ fiber. The left one of the two PBS in the middle of the figure will let some parts of the light go to the output.
The wedge may be used for changing the offset frequency. A simple description of the path of the light in the laser is that the photons that are generated in the Er3+ fiber pass the fiber and the free space components, hit the mirror of the PZT and are reflected, go through the EOM and enter the fiber again. The interesting parts for this thesis are the pumping mechanism (the diode laser), the PZT and the EOM, since they are the three actuators that control the offset frequency and the repetition rate. They are described more in detail in section 3.1.1.
3.1.1 Laser actuators
There are three actuators in the frequency comb used in the setup targeted in this thesis.
These three act together but can be controlled separately and target different parameters of the comb, required for successful and stable locking. The actuators are used to control and change the properties of the comb with high precision. The first actuator is the pump diode laser current, the second is a piezoelectric transducer (PZT) and the third is an electro-optic modulator (EOM). These three actuators are crucial for the ability to change the frequencies and thus for stabilizing the comb to the cavity.
3.1.2 Current driver
The optical frequency comb is a pumped fiber laser where the pumping component is a diode laser. The diode laser in the comb is a solid state laser made of a semiconducting material and the intensity of the diode laser can be modulated by changing the input cur- rent. The current actuator affects both the optical offset frequency f0 and the repetition rate frep, the second more than the first. The current is however more or less the only way to control the offset frequency of the comb, so the offset is corrected by the current actuator and the repetition rate is then corrected for by other means, i.e. with the piezo- electric transducer and the electro-optic modulator. The current driver has a bandwidth of approximately 200 kHz.
3.1.3 Piezoelectric transducer
A piezoelectric transducer (PZT) is a solid state device that either changes size depending on the voltage applied to it, or produces a voltage when some pressure is applied. For ap- plications related to optical combs where it is desirable to be able to change the repetition rate, the PZT becomes very useful. This is because one of the most intuitive and powerful ways to change the repetition rate is to change the laser cavity length, which changes the boundaries of the cavity and as a consequence the repetition rate. The PZT has only a small effect on the offset frequency and provides powerful means to change the repetition rate. The PZT is limited in response time and does not work at high frequencies. It is limited to somewhere around 10 kHz and this means another actuator (EOM) is needed for correcting at frequencies higher than that, see section 3.1.4.
3.1.4 Electro-optic modulator
There is an electro optic modulator (EOM) in the laser, a device that changes the index of refraction in a medium through which the laser light passes and by doing that the phase of the light can be modulated. The medium itself is made of a solid state material which has a variable index of refraction depending on the voltage that is applied to it. The index of refraction can be changed very fast and the bandwidth of the EOM is around 900 kHz [11].
4 Experimental procedure
This section is dedicated to the experimental procedure related to this work and is divided into the following subsections:
1. Modelling - was done in Matlab and based on theoretical models.
2. Building - the results from the modelling were used to build the circuit.
3. Testing - done together with the optical system.
4.1 Modelling
Before the circuit could be built the design had to be determined. This was done by modelling the gain and phase shift of the open loop in Matlab where the properties of the laser and the cavity were taken into account. Some basic building blocks for the circuit were derived and the resulting formulas were plotted and their behaviour verified.
Each of these building blocks consisted of an operational amplifier with resistors and often capacitors and were represented as separate variables, as sub-parts of the total open loop gain, see section 2.5.1 in the theory section for more details of the building blocks. In the same way the optical system was represented by a block of its own consisting of a model for the laser and the cavity. The transfer function H of the controller was calculated by multiplying the desired building blocks. The transfer function G of the optical system was estimated using experimental data and in order to determine the open loop transfer function of the entire control loop with the optical system included, the transfer function H was multiplied with the transfer function G (see section 2.3.2). Note that the model for the cavity, the current driver and the PZT with related experiments was provided by the spectroscopy group and none of the experiments related to the models are part of this thesis. The results were however used as tools for designing the controllers.
The transfer function of the cavity with the PDH error signal was assumed to have the shape of a low pass filter where the properties of the cavity determine the cut off frequency [9]. The formula that was used to model the cavity is
Gcav(f ) = K 1 1 + jff
c
, (36)
where Gcav is the transfer function of the cavity, f is the frequency, K is the max gain and fcis the cut-off frequency of the model. The constant K was found to be K ≈ 1 (V/MHz), a value determined by measuring the peak to peak value of the error signal in Figure 12(b) and dividing it by the cavity linewidth 2Γc (see Eq. (8)), assuming that the slope of the line for the error signal was linear. The measured amplitude was approximately 70 mV and half the linewidth was determined as Γc≈ 70 kHz, resulting from a cavity length of 45 cm and a finesse F ≈ 2300. The cut-off frequency of the cavity was taken as fc = Γc in the model.
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−1
−0.5 0 0.5 1
Frequency detuning [νpdh]
Normalized error signal
(a)
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Frequency detuning [νpdh]
Normalized error signal
(b)
Figure 12: Illustration of the error signal with normalized amplitude, error signal with side bands (a) and a zoom on the main mode (b). Frequency detuning in terms of modulation frequency νpdh.
The cavity was the same for both the modelled controllers and the transfer function used in this thesis can be seen in Figure 13.
100 101 102 103 104 105 106
0.2 0.4 0.6 0.81.0
Frequency [Hz]
Amplitude [V/MHz]
100 101 102 103 104 105 106
−90
−60
−30 0
Frequency [Hz]
Phase shift [deg]
Figure 13: The transfer function of the cavity. The gain is shown in the upper plot and the corresponding phase shift is shown in the lower plot.
The transfer function of the current driver of the laser can be seen in Figure 14(a), where the measured data is plotted together with a fit model that is based on the data. The transfer function for the laser PZT can be seen in Figure 15(a). The resulting system transfer functions G for the current and the PZT multiplied by the cavity can be seen in Figure 14(b) and Figure 15(b) respectively. For more details and derivations, see section 2.3.2 where the transfer functions are explained.
The first building block in the controller circuit is the amplifying stage which can be seen in Figure 6, section 2.5.1. This stage is used as the first building block in the circuit and is the block where the major part of the amplification of the signal is performed. After the amplifying stage a block containing an integrator (Figure 8, section 2.5.1) was added and in order to compensate for the inverting effect of that block a third block containing just a simple inverter (Figure 7 with switch open) was added in between the amplifying stage and the integrator. In order to achieve variable gain the resistor R2 was replaced by a potentiometer with a maximum value equal to that of R1, see Figure 7(a) and Eq.
(32). By doing this the maximum gain of the block is 0 dB and thus no amplification is performed in this stage. The resulting system is considered the simplest possible system for the task and was therefore seen as a good starting point for modelling the system.
100 101 102 103 104 105
100 101 102
Frequency [Hz]
Amplitude [MHz/V] Experimental
Theoretical model
100 101 102 103 104 105
−150
−120
−90
−60
−30 0
Frequency [Hz]
Phase shift [deg] Experimental
Theoretical model
(a)
100 101 102 103 104 105 106
−60
−40
−20 0 20 40
Frequency [Hz]
Gain [dB]
100 101 102 103 104 105 106
−240
−180
−120
−60 0
Frequency [Hz]
Phase shift [deg]
(b)
Figure 14: The transfer function for the current input (a) where the solid blue curve is the model based on the experimental data that is indicated with blue asterisks. The transfer function of the current input combined with the transfer function of the cavity is plotted in (b). In each figure, the gain is plotted in the upper plot and the corresponding phase shift is plotted in the lower plot.
100 101 102 103 104 105
101 102
Frequency [Hz]
Amplitude [MHz/V]
Experimental (low freq) Experimental (high freq) Theoretical model
100 101 102 103 104 105
−150
−120
−90
−60
−30 0
Frequency [Hz]
Phase shift [deg]
Experimental (low freq) Experimental (high freq) Theoretical model
(a)
100 101 102 103 104 105 106
−40
−20 0 20 40
Frequency [Hz]
Gain [dB]
100 101 102 103 104 105 106
−240
−180
−120
−60 0
Frequency [Hz]
Phase shift [deg]
(b)
Figure 15: The transfer function for the laser PZT (a) where the solid blue curve is the model based on the experimental data that is indicated with red asterisks for lower frequencies and blue asterisks for higher frequencies. The transfer function of the PZT combined with the transfer function of the cavity is plotted in (b). In each figure, the gain is plotted in the upper plot and the corresponding phase shift is plotted in the lower plot.
An inverter with unity gain just changes sign, which equals a phase shift of −180◦ but does not affect the plots in any other way, so it was not interesting for the design at that point, though of course a quite necessary part of the circuit. The integrator on the other hand was very interesting for the design for two reasons mainly. The first reason was its effect on the gain for low frequencies as it drastically boosted the gain, which was crucial for a good lock with the control loop. The second reason and an important one as well was the effect on the phase shift of the circuit. As can be seen in Figure 8(b), it does lift
up the phase at a certain region at higher frequencies and was used to control, to some extent, where the total (open loop) phase was to pass −180◦. The boost of the phase from the integrator was first placed in a manner such that the unavoidable event of the phase passing −180◦ would happen for as high frequencies as possible, which more or less meant placing the top of the phase bump in Figure 8(b) at the point where the transfer function of the optical system passed −180◦ and then tweaking for the best effect. After that, the cut off frequencies of the integrating stage and the amplifying stage were placed at a frequency where they did not push down the phase too much. The gain was increased to a level such that the unity gain of the open loop appeared at a lower frequency than the −180◦ passing of the phase, thus the phase and gain margins (discussed at the end of section 2.3.2) were kept. When the plot of the open loop gain and corresponding phase shift for the entire system were optimized the building process could be started which meant building the circuit in the lab.
4.2 Building the controllers
The circuits for the control loops were built on ordinary protoboards (breadboards) nor- mally seen in school laboratories. The protoboards were mainly chosen for their customiz- ability as mentioned earlier. Those boards are generally used for temporary test builds and not for permanent high performance instalments due to their limitations. They do pose quite some challenges besides the obvious robustness-over-time issue, that must be considered when building more advanced circuits and especially when used for signals at high frequencies.
The first circuit that was built was the current controller, later the PZT controller was also built, after a working version of the current controller had been achieved. As mentioned in section 4.1 the first version of the current control circuit consisted of only three building blocks, the first amplifying stage, an ordinary inverter and an integrating stage. It did have variable input offset and a voltage limiter through a couple of diodes, the latter because of the properties of the current actuator on the laser that required the voltage to be limited between 0 V and 1 V. The circuit was gradually tested using a signal generator sending a sine wave through the circuit and measuring the amplitudes at the input and the output with an oscilloscope, while also measuring the phase shift between the two measuring points thus providing means to plot both the gain and the phase shift.
In order to make the signal as good as possible, to optimize the performance and minimize the noise and other factors that may cause errors, the following points were considered to be extra important throughout the building process.
• Amplification as early as possible in the circuit in order to avoid amplifying possible noise from operational amplifiers earlier in the circuit.
• Make wires as short as possible and twist them where possible.
• Use as few operational amplifiers as possible as each amplifier adds some noise.
• Avoid using small capacitors with capacitances below approximately 100 pF as the contact lines in the protoboard start acting like small capacitors at high frequencies and would change the actual overall capacitance, deviating it from the designed values.
• Keep the circuit length as short and compact as possible.
