Filter cavity design for filtering
photons at 1MHz separation
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
in
PHYSICS
Author : S.L.D. ten Haaf
Student ID : s1606034
Supervisor : Dirk Bouwmeester
Vitaly Fedoseev
2ndcorrector : Martin van Exter
Filter cavity design for filtering
photons at 1MHz separation
S.L.D. ten Haaf
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
January 22, 2021
Abstract
In order to detect single Stokes photons in the Raman interaction between a laser and a membrane in a high-Q optical cavity, high transmission of the Stokes photons and high suppression (>150dB) of the driving laser is
required. The frequency separation between the Stokes photons and the driving laser is 1 MHz. To achieve this goal a filter is developed consisting of eight mirrors forming four aligned, coupled filter cavities. The set-up is hoped to achieve above 70% transmission on resonance and 170dB suppression of the driving laser. Potential alignment problems are foreseen and the implementation of the filter in an experimental set-up is
Contents
1 Introduction 7
2 Multi Mirror Filters 11
2.1 The Transfer Matrix method 11
2.1.1 The Fresnel Coefficients 12
2.1.2 The Transfer Matrix 12
2.1.3 Absorption and Scattering loss 14
2.2 Three and Four Mirror Filters 15
2.2.1 Flat Top Transmission Peaks 18
3 Two Cavity Filters 21
3.1 Analytical Behaviour 21
3.1.1 The Cavity Spacing 22
3.1.2 The Central Peak 24
3.2 Numerical Calculations 26
3.2.1 Mirror Structure 26
3.2.2 Effect of Implementing Loss 29
3.2.3 Two Uncoupled Cavities 31
4 Multi Cavity Filters 33
4.1 Three and Four aligned cavities 33
4.1.1 Offset Parameter 34
4.1.2 Ideal Transmission Parameters 36
4.1.3 Loss in Cavity Spacings 38
4.1.4 Peak Transmission and 1 MHz Suppression 38
5 Final Design and Implementation 41
6 CONTENTS
5.1.1 Stablity 44
5.2 Aligning the Cavities 46
5.3 Maintaining Alignment 55
6 Conclusions 61
A Analytical Transmission Through Two Coupled Cavities 65
6
Chapter
1
Introduction
In cavity optomechanics the interaction between light of frequency ωland matter inside an optical cavity can lead to elastic scattering of light with frequency ω = ωl, but also inelastic scattering in a Stokes/Anti-Stokes process called Raman scattering if the mechanical modes are optomechan-ically coupled to the light. This Raman interaction leads to energy being transferred to/removed from the matters’ mechanical modesΩm.
The interaction between light with frequency ω0 coupled to e.g. a mem-brane in an optical cavity with mechanical modeΩmwill produce Stokes photons at ω = ωl −Ωm, transferring energy to the mechanical mode, and anti-Stokes photons at ω = ωl+Ωm, removing energy from the me-chanical mode[1]. The process is illustrated in Figure 1.1. The aim of this investigation is to design and build an optical filter able to isolate these ’up-converted’ or ’down-converted’ photons from the strong pump light at ωl.
The light-matter interaction inside a cavity can be tuned to suppress the Stokes process, by pumping the cavity at a frequency separated byΩm from the cavity mode ωcav. This leads to cooling of the mechanical mode and eventually drives the mechanical mode to its ground state[2][3]. At the ground state no further energy can be removed from the mechani-cal mode, meaning no anti-Stokes photons can be created. When the me-chanical mode transitions from the ground state to the single phonon state in interaction with light only a single Stokes photon will be emitted at
ω = ωl−Ωm. Such a membrane at the ground state coupled to light in an optical cavity can be used to study quantum optomechanical methods such as stimulated Raman adiabatic passage (STIRAP) [4] and Raman ra-tio thermometry [5]. In each case the detecra-tion of the mechanical mode
8 Introduction
Figure 1.1:Stokes photon scattering in the Raman interaction. Pumping an opti-cal cavity with cavity mode ωcav containing a membrane with mechanical mode
Ωmwith a laser ωldetuned from the cavity by±Ωmwill produce Stokes or
Anti-Stokes photons with frequency ωl±Ωm[1].
transition through detection of the Stokes photons is crucial.
In the system of interest to this investigation a 1064nm laser is used to drive a high-Q optical cavity containing a membrane with mechani-cal modes Ωm in the range 1−1.5Mhz. Once the membrane is cooled to its ground state, on transition to the one phonon state a single down-converted photon will be emitted. To detect this transition a single-photon detector needs to be placed in the path of the light reflected from the cavity.
The probability of producing Stokes photons is(g0
κ )
2[1], where g
0is the optomechanical single-photon coupling strength and κ is the linewidth of the optical cavity. In the system g0 ≈1 Hz and the linewidth of the cavity
κ ≈ 55kHz. The emitted photons need to be detected in reflection from
the cavity and these photons must be distinguished from photons coming from different mechanical modes, which can be as close as 70kHz from the mode of interest. To successfully detect the emitted Stokes photons an op-tical narrowband filter is thus required that provides 150dB suppression of the pumping light at ωl and (ideally) 100% transmission of the Stokes photons at ωl−Ωm. The efficiency of the available single-photon detector is>90%.
8
9
Optical Filters
A well understood optical filter for the purpose of distinguishing nearby frequencies is the Fabry Perot interferometer, consisting of two parallel reflective surfaces separated by a spacing d. The transmission of incident light depends on the spacing d, the frequency of light ω and the reflectance of the surfaces R. For high reflectance narrow transmission peaks with 100% transmission occur periodically. The transmission drops off from these peak frequencies ω0as [6]:
T(ω) = 1
1+ (ωo−ω
1 2κ
)2 =L(ω) (1.1)
known as the Lorentzian transmission function L(ω), where κ is the
full width at half maximum (FWHM), also known as the linewidth. The FWHM of a peak at ω0together define the quality factor of an optical filter cavity Q:
Q= ω0
κ (1.2)
The quality factor in turn relates to the time τ it takes for a fraction of 1e of the light to leave the cavity after building up inside the cavity:
τ = Q ω0
(1.3) To achieve 150dB suppression at 1MHz from ω0 with a simple Fabry Perot etalon one would need mirrors with a reflectance of less than 0.001ppm. The high end of commercially available high reflectivity mirrors have a re-flectance on the order of 10ppm. An improved, more complex design is thus required.
An option is to cascade multiple Fabry Perot filters. When light is pre-vented from travelling back and forth between the cavities so that consec-utive filters act independently after the other, the Lorentzian functions are multiplied and the transmission peak will drop off from ω0as:
Ln(ω) = 1 1+ (ωo−ω 1 2κ )2 n (1.4)
where n is the number of cascaded cavities and the κ the linewidth of a single cavity.
10 Introduction
Recent research by Galinskiy et al. with similar filtration requirements re-lied on four cascaded Fabry Perot cavities width a linewidth of 30kHz. The set-up used three optical circulators between the cavities to prevent light from travelling back and forth between the cavities. These extra op-tical parts introduce unwanted loss, reducing the maximum transmission of the peak frequency. The reported achieved optimal peak transmission was 30%[7].
The below thesis presents an investigation into multi-mirror filters with the aim of designing an optical filter that can provide 150dB suppression of the pumping light and high transmission of the detuned Stokes pho-tons. The discussion will be aimed at a filter designed for a wavelength
λ0 of 1064nm, corresponding to the frequency ω0 of 281.76 THz. Due to the set-up being required to be implemented on an optical table, the max-imum length of the filter is restricted to 2 metres. The modes of interest lie in the range 1−1.5 MHz and the investigation will focus on achieving the 150dB suppression at 1 MHz separation.
In chapter 2 the transfer matrix method that will be used for evaluating the properties of optical filter set-ups is introduced. The behaviour of three and four mirror systems is described analytically to provide a basis for un-derstanding the behaviour of two and more aligned cavities. In chapter 3 the properties of two aligned cavities are investigated and the numerical implementation of the transfer matrix method is outlined. These calcula-tions are extended to three and four aligned cavities in chapter 4. Lastly in chapter 5 the final chosen design is detailed and the implementation of such a filter in an experimental set-up is discussed.
10
Chapter
2
Multi Mirror Filters
To achieve the goal of 150dB suppression of light separated by 1MHz from
ω0using an optical filter, a set-up with more than two mirrors will be re-quired. As a starting point for the investigation the transmission response of three and four mirror systems will be described, expanding on the sim-ple Fabry Perot filter. As will be shown these set-ups provide a subtly different response over the Lorentzian transmission function that could be beneficial to the goal of filtering single photons. The calculations in this chapter will be done analytically, in later chapters computational methods will be employed. For both the analytical calculations and the numeri-cal numeri-calculations a method numeri-called the transfer matrix method is used. This method is explained in section 2.1.
2.1
The Transfer Matrix method
Any system consisting of aligned flat mirrors or of stacks of flat materials making up a mirror can be modelled as a multilayer system with N layers with refractive index ni and thickness di. To solve the Maxwell equations for such a multilayer system subjected to a uniform electric field the so-called transfer matrix method can be employed. This method uses a set of linear matrix equations that depend on the frequency of propagating light and the thickness and refractive index of each layer in the multilayer. Using these equations a so-called transfer matrix is obtained, which cou-ples light incident on one side of the multilayer to the light on the other side. From the transfer matrix the reflection and transmission coefficients can be derived that provide information about the fraction of reflected and transmitted light.
12 Multi Mirror Filters
2.1.1
The Fresnel Coefficients
When an electric field E(ω)is incident on the boundary between two
ma-terials with refractive indices n1 and n2, the amplitude of reflected light rE(ω) and transmitted light tE(ω) are determined by the Fresnel
coeffi-cients r and t, given by the Fresnel equations. In general, the reflection and transmission coefficients are different for S-polarized and P-polarized light and coefficients rp, rs, ts and tp need to be calculated separately. For normal incident light, however, rp = rs = r and tp = ts = t. The calcu-lations in this thesis concern only normal incidence, for which the Fresnel equations are [8]: t= 2n1 n1+n2 (2.1a) r = n1−n2 n1+n2 (2.1b)
In an ideal system without any loss, all incident light is either reflected or transmitted. In this case the following equation holds:
|t|2+ |r|2 =1 (2.2)
The absorption coefficients are related to intensity coefficients T and R:
T = |t|2, R= |r|2 (2.3)
In a non-ideal system light will be lost due to e.g. scattering or absorption, in which case:
|a|2=1− |t|2− |r|26= 0 (2.4) where a is taken to be the loss coefficient accounting for loss in the system. Ways to account for losses will be described in subsection 2.1.3.
2.1.2
The Transfer Matrix
The method starts off with setting up a system of N layers with varying re-fractive indexes niand thicknesses Li, as shown in Figure 2.1, with an elec-tric field incident from the left. Layer 0 is defined as the starting medium on the left, layer N+1 as the final medium to the right of the system. The starting and final medium are taken to be vacuum with n0 =nN+1=1, as the optical filter will be designed for use in free space. This is not generally required and any starting and finishing medium could be used.
12
2.1 The Transfer Matrix method 13
As light propagates through the multilayer it will be partially reflected and transmitted at the boundaries between layers, as discussed in subsec-tion 2.1.1, and will undergo a phase change while propagating through a layer. For ease of calculation the electric field Eninside layer n can be split into a forward propagating part E+n and backward propagating part E−n, writing the electric field in each layer as En = E+n +En−.
Figure 2.1:Schematic for a multilayer with N layers. Lines represent a boundary between two layers.
To relate the fields on the left of the multilayer to the fields on the right, the relation between fields on either side of a boundary and the relation between fields on either side of a layer must be known. To differentiate between the fields on the left and right within the same layer the notation En0±will be used to indicate the fields at left side of layer n and E0n± for the fields on the right side.
On propagation through a single layer the fields pick up a phase factor, such that the fields on the left side of layer n are related to the fields on the right by Equation 2.5: E0n+ E0n− ! =PnE + n En− (2.5a) Pn =e −iφn 0 0 eiφn (2.5b) where Pn is the propagation matrix and the phase φn is given by φn = 2πLnnc/ f .
The fields on the left and right side of a boundary between two layers are related through the Fresnel equations, leading to Equation 2.6:
E+ n−1 E−n−1 = In−1,n E 0+ n En0− ! (2.6a) In−1,n = 1 tn−1,n 1 rn−1,n rn−1,n 1 (2.6b)
14 Multi Mirror Filters
Multiplying the propagation and interface matrices for the entire mul-tilayer, the entire transfer matrix M can be calculated as:
E+ 0 E0− =ME + N+1 E−N+1 (2.7a) M = M00 M01 M10 M11 = I01P1I12P2· · ·PNIN(N+1) (2.7b)
Setting the field incident from the right to zero (E−N+1 =0), the overall reflection and transmission intensity coefficients are derived from Equa-tion 2.7a and given by [9]:
T = |E + N+1 E+0 | 2 = | 1 M00 |2 (2.8a) R = |E − 0 E0+| 2= |M10 M00 |2 (2.8b)
2.1.3
Absorption and Scattering loss
Implementing the transfer matrix method for a multilayer with only real refractive indices will give reflection and transmission coefficients for which Equation 2.2 holds: all light is either reflected or transmitted. In a non-ideal situation light will be lost in various ways, for example by being ab-sorbed by the material through which it propagates or by scattering from interfaces. Both effects can be accounted for through the introduction of a complex refractive index[9]:
n=n+iκ (2.9)
where n is the normal refractive index and κ known as the extinction co-efficient. That this gives rise to loss as a wave propagates can be seen through the complex wave vector k= 2πnλ :
E0ei(kz−ωt) = E0ei(
2π(n+iκ)
λ −ωt) =e−
2πκz
λ E0ei(kz−ωt) (2.10) The amplitude of light with wavelength λ propagating a distance d through a material with a complex refractive index n decays exponentially as e−2πκdλ.
If the extinction coefficient of a material for a certain wavelength is known, absorption in this layer can be accounted for by setting the com-plex part of the refractive index of the layer to the corresponding value. 14
2.2 Three and Four Mirror Filters 15
On the other hand, if the exact value for the extinction coefficient on prop-agation through a material is not known, an estimate of the expected per-centage loss can be used to tune the complex part of the refractive index until the loss upon propagation matches the expected loss.
To account for scattering loss methods have been implemented that es-timate the roughness of a surface and use this to introduce decay factors that act similarly to the factor above [10]. For the purpose of this investiga-tion however it is sufficient to estimate the expected loss due to scattering and implement it in the same way as the absorption loss. The effect on the calculated transmission will be the same, regardless of the mechanism through which the loss arises.
2.2
Three and Four Mirror Filters
Using the transfer matrix method described in subsection 2.1.2 the trans-mission through systems of three and four mirrors can be derived analyt-ically. By treating the n mirrors as boundaries between vacuum layers, with Fresnel coefficients ri and ti, Van der Stadt et al. characterized the behaviour of such set-ups with general separations and reflection coeffi-cients (in the abscence of loss)[11]. Due to the linear nature of the transfer matrix method, replacing the mirrors with boundaries of fixed ri and ti is valid over a frequency range where the reflection and transmission coeffi-cients of the mirror are constant.
For various set-ups with three and four mirrors Van der Stadt et al. pro-vide solutions that give unity intensity transmission (T = 1) and discuss the transmission behaviour of these set-ups in the context of filter cavities. Of special interest to this investigation are the cases sketched in Figure 2.2, for which the transmission properties will be repeated below. In both cases a solution for single-peak unity transmission exists with desirable flat-top properties and steep suppression, that will be further detailed in subsec-tion 2.2.1.
Three mirrors
Generally a three mirror system can be characterized by reflection coeffi-cients r1, r2 and r3 and spacings L1 and L2 between the mirrors. In the case of interest r1 = r3 6= r2 and L1 = L2, as sketched in Figure 2.2a. The intensity transmission coefficient T for this case was derived by Van der
16 Multi Mirror Filters
(a)Three mirror filter set-up. (b)Four mirror filter set-up.
Figure 2.2: Sketches of the three and four-mirror set-ups of interest, with the mirrors modelled as boundaries between vacuum with reflection coefficients ri.
Both set-ups have outer mirrors r1and inner mirror(s) r2 and equal separation L
that introduces a phase change φ=2πLcf. Stadt to be: T = (1−r 2 1)2(1−r22) (1−r21)2(1−r2 2) + [r2(1+r12) +2r1cos(2φ)]2 (2.11) By requiring T = 1, the condition for unity transmission is (Equa-tion 2.12):
r2(1+r21) +2r1cos 2φ =0 (2.12) Since cos(2φ) ≥ −1, the condition on R2 = |r2|2for a given R1 = |r1|2for obtaining unity transmission:
R2≤ 4R1
(1+R1)2 = R ideal(3)
2 (2.13)
When the inequality holds two periodic solutions exist to 2.12, which gives rise to a periodic pattern of two close peaks that move closer together as R2 approaches Rideal2 (3). At equality, when R2 = Rideal2 (3), only a single solution exists giving rise to a single flat top peak. At R2 > Rideal
(3)
2 no
solution for T =1 exists. The three cases are sketched in Figure 2.3a.
Four mirrors
Generally a four mirror system can be characterized by reflection coeffi-cients r1, r2, r3 and r4 and spacings L1, L2 and L3. For the four equally spaced mirror set-up sketched in Figure 2.2b, with outer mirrors r1 and inner mirrors r2, the intensity transmission coefficient is derived to be:
T = (1−r 2 1)2(1−r22)2 (1−r2 1)2(1−r22)2+2r21(1+cos 2φ)(2 cos 2φ+B−1)2 (2.14a) 16
2.2 Three and Four Mirror Filters 17
B = (1+r
2
1+r1r2)r2
r1 (2.14b)
Again looking for solutions where T = 1, the conditions for unity trans-mission are:
cos 2φ = −1 (2.15a)
cos 2φ = 1−B
2 (2.15b)
The first condition gives rise a central peak that always exists, regardless of the reflectivity of the mirrors used. The second condition gives two side-peaks that only exist if 1−2B > −1, leading to the following condi-tion on R2 = |r2|2 for a given R1 = |r1|2 for obtaining single-peak unity transmission (Equation 2.16): R2 ≥ (−R1+ q R21+14R1+1−1)2 4R1 =Rideal2 (4) (2.16)
For R2 < Rideal2 (4) the central peak is surrounded by two sidepeaks. At R2= Rideal
(4)
2 the side-peaks merge to form a single flat-top peak. At R2 > Rideal2 (4) a narrow single peak forms. The three different cases are sketched in Figure 2.3b.
(a)3 Mirror Filter (b)4 Mirror Filter
Figure 2.3: Plots for various values of R2 for R1 = 0.9, where for a) the ideal
value for flat top transmission Rideal2 (3)is given by Equation 2.13 and in b) Rideal2 (4) is given by Equation 2.16.
18 Multi Mirror Filters
2.2.1
Flat Top Transmission Peaks
Both the three mirror and four mirror set-up have conditions on R2 in relation to R1 where the split transmission peaks merge to form a single flat-top peak. The advantage of this will be discussed below. The shape of these peaks, when R2 =Rideal2 , has been derived by Van der Stadt et al. and found experimentally by Hogeveen et al.[12] to not be L(ω)(Equation 1.1)
or Ln(ω)(Equation 1.4), but rather follow:
Tn(ω) = 1
1+ (ω0−ω
2κ )2n
(2.17) around the peak transmission ω0, where n indicates the number of cavities formed between the mirrors. When applying n Lorentzian filters one after the other, without interference between the cavities, the expected trans-mission shape would be Ln(ω). The three and four mirror systems
pro-vide a subtly different type of response. These transmission shapes have an advantage in having a larger range around ω0 where T = 1, a steeper drop-off and higher suppression, resulting in a more square transmission filter . A comparison to Ln(ω) is plotted in Figure 2.4 for both the three
and four mirror set-up.
(a)3-Mirror filter transmission peak (b)4-Mirror filter transmission peak
Figure 2.4: Shapes of the transmission response of three and four mirror set-ups where R2 = Rideal2 and R1 = 0.999. Equation 2.17 is fitted in yellow, Equation 1.4
is fitted in red. For the three mirror set-up the linewidth is 63.4kHz. For the four mirror set-up the linewidth is 44.9 kHz.
In the context of this investigation two advantages can be seen. First of all the suppression at 1 MHz is higher compared to Ln(ω) with the same
linewidth. In the above example for 3 mirrors the transmission at 1 MHz 18
2.2 Three and Four Mirror Filters 19
is 1.01·10−6compared to 6.05·10−6for L2(ω). For the 4 mirrors
transmis-sion at 1 MHz is 1.27·10−10 compared to 8.11·10−9for L3ω.
Secondly the Stokes photons that need to be transmitted by the filter have a Lorentzian linewidth themselves on the order of a couple of kHz. Ideally the filter provides unity transmission over the entire linewidth of the photons or beyond this linewidth and so the flat unity peaks can be very advantageous compared to the sharper Lorentzian peaks.
Despite offering steeper suppression, the mirror combinations required to achieve 150dB suppression at 1 MHz using a three or four mirror set-up would still need unattainably high quality (inner) mirrors. The theoreti-cally required mirrors for 150dB suppression and ideal flat peaks are listed in Table 2.1, keeping in mind the total available length of 2 metres:
R1(ppm) R2(ppm) L (cm) 3 mirror set-up 10 2.5·10−5 100 4 mirror set-up 240 7.2·10−3 65
Table 2.1: Required mirrors R1and R2to create an optical filter achieving 150dB
suppresion at 1 MHz from peak transmission.
Even 10ppm is already at the high end of available mirrors and any loss in such mirrors forming a cavity will be greatly amplified, as will be shown in subsection 3.2.2.
Chapter
3
Two Cavity Filters
In concluding the previous chapter a comparison is made between the multi mirror set-ups and uncoupled cascaded Fabry Perot filters and the advantages are discussed of the former over the latter. To set up uncou-pled cascaded Fabry Perot filters additional elements are required that block light from travelling backwards between each consecutive cavity. Without these elements a set-up of two equal aligned cavities with a vari-able distance between them, as sketched in Figure 3.1, is in essence a spe-cific case of a four mirror system that can be treated in the same way as the four mirror set-up in the previous chapter. The question arises how this affects the resulting transmission response.
While being a specific case of a four-mirror set-up, the two aligned ’coupled’ cavity set-up is discussed here separately as its properties are an important basis for the final filter design to be discussed in chapter 5. In section 3.1 the analytical transmission is derived and the properties of the set-up as a filter cavity are discussed. In section 3.2 a numerical imple-mentation of the transfer matrix method is introduced to be able to model the individual mirrors in more detail and to investigate the effect of losses in the system as discussed in subsection 2.1.3.
3.1
Analytical Behaviour
Starting from the same point noted by Van der Stadt in deriving Equa-tion 2.14, but now setting r1 = r2 = r3 = r4(= r1) and φ1 = φ2 = φ 6= φ3=α, the general expression for transmission through two equal aligned
22 Two Cavity Filters
Figure 3.1: Set-up of four mirrors forming two equal cavities of length L1, with
variable distance L2 between the cavities. The associated phase factors with the
spacings are φ=2πL1cf and α=2πL2cf.
mirror cavities with variable spacing between the cavities is derived to be:
T = (1−r
2 1)4
(1−r21)4+8r2
1(1+cos(2φ))[r21cos(φ−α) +cos(φ+α)]2
(3.1) Setting α = φin the above equation and setting r1 = r2 in Equation 2.14 yields the same equation, as it should. Again two conditions arise for solving T =1:
cos(2φ) = −1 (3.2a)
r12cos(φ−α) = −cos(φ+α) (3.2b)
As for the four mirror case treated in Equation 2.2, a central peak is always present when cos(2φ) = −1, so that L1 controls the location of the central peaks. The shape of this peak shows similarities to the central peak in the three mirror set-up and is affected by the spacing L2, further explored in subsection 3.1.2. The sidepeaks from the second condition are dependant on the difference between φ and α and therefore on the differ-ence between distances L1 and L2, which will be further detailed in sub-section 3.1.1. The derivation of Equation 3.1 and the conclusions drawn in the following sections are discussed in more detail in Appendix A.
3.1.1
The Cavity Spacing
The location of the main peaks of interest associated with Equation 3.2a de-pend only on the spacings L1. The second solution (Equation 3.2b) on the other hand is slightly more complex, and gives rise to side-peaks whose location depends on the difference between L1and L2. For ease of descrip-tion, the notation δo will be used to indicate the relative offset between L1 and L2modulo the wavelength of the desired resonant frequency λ0.
δo = |L1−L2|%λ0 (3.3)
22
3.1 Analytical Behaviour 23
The side-peaks around λ0occur when the cavity formed by the inner two mirrors is resonant with the propagating frequency. Two boundary cases can be identified. If δo =0 or δo = λ2 the side-peaks coincide with the main peak at λ0. If δo = λ4 or δo = 3λ4 the nearest side-peaks are at the maximal distance from the main peak. Previous research on two aligned microcav-ities confirm this behaviour[13]. To illustrate this, the transmissions of the set-up for a variety of cases have been plotted in Figure 3.2, together with the transmission spectrum of the inner cavity. To clarify: throughout this thesis reference will be made to offsets of λ
4 and
λ
2 by which the offsets of 3λ
4 and 0 are implied as the behaviour is periodic with a period of λ2.
(a) L2<L1and λ4 offset (b) L2=L1and λ4 offset (c) L2>L1and λ4 offset
(d) L2<L1and λ2 offset (e) L2=L1and λ2 offset (f) L2>L1and λ2 offset
Figure 3.2: Plots showing the effect of the offset δo on the transmission through
two aligned cavities set up as shown in Figure 3.1. Various configurations of L1
and L2 with an offset δo of either λ2 or λ4 are plotted in black. The dashed lines
show the transmission spectrum of the cavity formed by the inner two mirrors, indicating that the side peaks occur where the inner cavity is resonant with the propagating frequency. For an offset of λ
2 the side-peaks coincide with the main
resonant peak ω0. For an offset of λ4 the nearest side-peaks to ω0 occur at ω0± 1
2FSRinner
When δo = λ2 the side-peaks coincide with the main-peak and a triple peak occurs in the same way as for the four equally spaced mirror set-up when R2 < Rideal
(4)
2 . Changing δo such that the side-peaks occur away from the central peak, the central peak at ω0behaves like the main peak in the three-mirror set-up. This is further detailed in subsection 3.1.2.
24 Two Cavity Filters
In all the intermediate cases, when δo is varied between λ40 and λ20, the nearest side-peaks shift from ω0±12FSRinner to ω0. This affects the shape of the main peak at ω0, as will become clear in subsection 3.1.2, but also in-fluences the transmission at ω0+1 MHz. The transmission at 1 MHz from
ω0 is plotted in Figure 3.3 for varying δo between 0 and λ0and highlights the importance of controlling the spacing between the two cavities when considering the set-up as an optical filter.
Figure 3.3: Effect of the offset δo on the transmission at 1MHz from the main
peak at ω0 in a set-up with four equal mirrors with reflectivity R forming two
equal cavities of length 50cm. Data was collected for R = 0.99, R = 0.995 and R=0.999. As R→1 the ideal offset for the purpose of suppression of the 1 MHz frequency approaches λ0
4 while the worst case offset approaches
λ0
2.
3.1.2
The Central Peak
When the resonance frequency of the cavity formed by the inner two mir-rors is tuned away from the frequency of the central peak ω0, the reflectiv-ity of the inner cavreflectiv-ity is reasonably constant over the width of the central peak (Figure 3.2). In this case the central peak at ω0 behaves similarly to the three mirror set-up transmission as plotted in Figure 2.3a
Indicating the reflectivity of the inner cavity around the central peak as Rinner, Figure 3.4 showcases how Rinner affects the shape of the central peak in relation to Rideal2 (3) from Equation 2.13. It is apparent that the two 24
3.1 Analytical Behaviour 25
inner mirrors can be treated as a single mirror R2 such that R2 = Rinner, creating a three mirror set-up with the same transmission response around the central peak.
In order to get the same ideal flat-top transmission curve for the two aligned cavities as for the three mirror set-up, the reflectance of the inner cavity Rinner should thus be equal to Rideal
(3)
2 in Equation 2.13. At λ4 off-set the reflectivity of the inner cavity is maximal (Rinnermax ) around ω0. The formula for the maximal reflectivity of a single cavity with mirrors R1 is known[14]:
Rmaxinner = 4R1
(1+R1)2 (3.4)
so that we find Rmaxinner = Rideal2 (3). Coincidentally the non-general case of using four equal mirrors provides the ideal Rmaxinner for the desired flat-top transmission. This will not turn out to hold for set-ups with more than two aligned cavities, as will be discussed in subsection 4.1.2.
(a) Rinner=Rideal (b) Rinner <Rideal (c) Rinner<Rideal
Figure 3.4: Plots comparing the main transmission peak of two aligned cavities to the transmission through a three mirror set-up as described in Figure 2.2. Solid red: the ideal value Tideal(= 1−Rideal2 ) as given by Equation 2.13. Dashed gold: the transmission response Tinner of the cavity formed by the inner two mirrors.
Solid black: transmission through two aligned cavities with R1 = 0.99 . Dashed
black: Transmission through three mirrors with R1= 0.99 and R2 = Rinner, with
Rinner the reflectivity of the cavity formed by the inner two mirrors of the two
aligned cavities set-up. The transmission curve of the three mirror system has been shifted to the left for ease of comparison.
Since in the ideal case the transmission is comparable to the transmis-sion through the three mirror set-up, the parameters required to achieve 150dB suppression at 1 MHz from ω0 are comparable as well. While no longer needing an exceedingly high central mirror, four 10ppm mirrors would be required. This value matches the predicted value for the re-quired outer mirrors in a three mirror set-up presented in Table 2.1.
26 Two Cavity Filters
3.2
Numerical Calculations
It is clear from section 3.1 that two aligned cavities can be used to create a filter with the same flat top transmission response as was seen for the three and four mirror set-ups in section 2.2. While still not suitable for achieving 150dB suppression at 1 MHz from ω0, the set-up provides an interesting basis to extend to three and four aligned cavities that poten-tially can achieve the right amount of suppression while maintaining the flat top transmission.
Since the analytical equations for set-ups with five and more mirrors become quite cumbersome, set-ups with more than two aligned cavities need to be treated numerically. Setting up the layers and related param-eters numerically, the Fresnel intensity coefficients R and T can be calcu-lated for any frequency by calculating and multiplying the propagation and interface matrices (Equation 2.5 and Equation 2.6) to calculate the transfer matrix. Once a program is set up to carry out this calculation for a two layer system and an arbitrary frequency it can be easily extended to any number of layers and a range of frequencies.
No longer limited to a low number of layers for ease of calculation, the internal structure of each mirror can be considered as well so that the effect of loss at any point inside the mirrors can be investigated. In subsec-tion 3.2.1 the structure of the mirrors and the numerical implementasubsec-tion of the transfer matrix method is outlined. In subsection 3.2.2 the effect of the implementation of loss on the transmission is described for two coupled cavities.
3.2.1
Mirror Structure
Rather than treating each mirror as a boundary with chosen Fresnel coef-ficients r and t as done in the analytical evaluations, each mirror can be broken down into the actual physical layers that would constitute a real mirror. The thickness, refractive index and extinction coefficient of each layer in each mirror can then be tuned to control the reflectivity and loss of each mirror, which will arise through the transfer matrix method.
A common method of creating high reflectivity mirrors is through the use of a structure called a Bragg reflector[15][16]. By stacking two ma-terials with high refractive index and low refractive index, with thickness 26
3.2 Numerical Calculations 27
equal toλB
4n, light with a wavelength close to λB(known as the Bragg wave-length) will interfere destructively inside the layers so that a very high re-flectivity can be reached. The rere-flectivity of a Bragg reflector with high index layer nH, low index layer nL and N repeated pairs of layers will achieve a maximum reflectivity at λBof[17]:
R= " n2NH −n2NL n2NH +n2NL #2 (3.5) The reflectivity over a large frequency range varies and is plotted in Fig-ure 3.6.
Such a stack of layers, called a Bragg stack, is created by depositing layers one by one on a substrate. This substrate material should be chosen so that it has minimal loss for the propagating wavelength. Furthermore an anti-reflective coating is required on the opposite side to minimize the amount of incoming light reflected off the substrate. If the substrate has refractive index nS, the ideal anti-Reflective coating has refractive index nAR =
√
nS and thickess 4nλARB .
A simple schematic showing the mirror structure to be simulated is shown in Figure 3.5.
Figure 3.5: Simple schematic of the layers to be included in simulation of high reflectivity mirrors.
By precisely tuning the refractive indexes of the high and low layer and increasing or decreasing the number of layers, any desired reflectiv-ity for the simulated mirrors can be achieved. In practice many different materials are used to create high reflectivity Bragg mirrors, each suited for a specific wavelength range. The numerical calculation is not limited to physical materials, but to keep the indexes reasonable the mirrors are modelled after a SiO2-TiO2Bragg reflector. A value of nL =1.4496 is used
28 Two Cavity Filters
for the low index layer, corresponding to the refractive index of silicon dioxide at 1064nm. The index of the high index layer will be varied to get the desired mirror reflectivity, but kept near nH = 2.4789 corresponding to the refractive index of titanium dioxide at 1064nm. If a higher index is required to create a higher reflectivity mirror, the number of layers is increased instead. As substrate a layer of 5mm with a refractive index of 1.55 is included, corresponding to the refractive index of Silicon. The anti-reflective coating layer was set based on this substrate layer as described above.
Examples of the required parameters for simulating Bragg stacks with certain chosen reflectivities and an example of the reflection response of such a stack is shown in Figure 3.6. The numerically calculated reflection corresponds closely to the expected analytical response.
Figure 3.6: Table showing the required Bragg stack parameters for reaching cer-tain reflectivities and an example of the reflectivity of a simulated Bragg stack over a 150-450 THz range, using the parameters for R=0.9999.
By setting up multiple individual mirrors as described above, sepa-rated by layers of vacuum, the transfer matrix for a system of multiple aligned mirrors can be calculated numerically to obtain the reflection and transmission coefficients. As an example a sketch of the two coupled cav-ity set-up is shown in Figure 3.7 with all relevant layers included. The numerically calculated transmission through this set-up was compared extensively to the analytical transmission to confirm the predictions in section 3.1. All numerical calculations in this thesis were carried out in python using a script implementing the transfer matrix method, written specifically for this thesis.
28
3.2 Numerical Calculations 29
Figure 3.7: Sketch of all the layers to include in a numerical calculation of the transmission through two aligned cavities.
Implementation of losses
Loss can easily be included in any layer in the numerical calculation through the use of complex refractive indices as described in subsection 2.1.3. While data exists on the extinction coefficients of materials used in Bragg mirrors, it is not always accurately known at the wavelength of interest to this in-vestigation. Furthermore other losses beside absorption will need to be accounted for as well, so the preferred method is to look at the expected total loss associated with the modelled structures and implement this loss through tuning of the complex parts of the refractive indices.
In an experimental system a certain amount of absorption loss is ex-pected to be present in the Bragg stacks and the substrates and a certain amount of light will be scattered from the surfaces as they are not be per-fectly smooth. Furthermore other optical parts such as lenses might be required for alignment, which have an associated loss as well.
For each of these parts the total loss is to be estimated and implemented as absorption loss through setting the complex refractive indices of the relevant layers. For lenses the total loss, depending on the quality of the anti-reflective coating, is expected to be 0.1%-0.3% and the loss in the sub-strates around 0.2%-0.5%. As will be shown both count as ’loss in the cavity spacing’ and need not be accounted for as accurately. The total loss associated with the Bragg stacks is a much more sensitive factor, sensitive to even 1ppm changes. Mirrors with a total loss as low as 3ppm have been reported for our desired frequency[18].
The effect of the inclusion of these losses will be discussed below in subsection 3.2.2 for the two coupled cavity set-up.
3.2.2
Effect of Implementing Loss
With the mirrors structured and set up as sketched in Figure 3.7, the effect of loss in the Bragg stacks, substrates, cavity spacings and mirror spacings
30 Two Cavity Filters
is investigated. Loss is implemented as described in subsection 2.1.3. Two distinguished effects are observed. Loss in the Bragg stacks and the mir-ror spacings will heavily affect the main peak at ω0, plotted in Figure 3.8b. Loss in the substrates or the cavity spacing mostly affects the side-peaks, plotted in Figure 3.8a for the set-up with δ0 = λ2 to clearly show the sup-pression of the side-peaks.
(a)Loss in substrate/cavity spacing (b)Loss in mirror stack/mirror spacing
Figure 3.8:Effect of loss in two aligned cavities with 10ppm mirrors (R = 0.9999). a) The effect of loss in the cavity spacing and substrates. The side-peaks are heav-ily suppressed, while the main peak transmission is still 70% even at 50% loss in the cavity spacing. b) The effect of introducing loss in the Bragg stacks and mirror spacings. The peak transmission is much more sensitive to this loss and at 10ppm loss the peak transmission drops to 67%.
Mirrors with a reflectivity of 10ppm were used to demonstrate the ef-fect of loss more clearly. It is important to note that a lower reflectivity will result in less transmission loss. Higher reflectivity mirrors cause light to make more bounces within each cavity and so any loss due to absorption or scattering is increased.
The expected maximum transmission of a single cavity, formed by mirrors with reflectivity R and loss A, is given by[19]:
Tmax = (1−R−A)
2
(1−R)2 (3.6)
The peak transmissions obtained through numerical calculation in Fig-ure 3.8b match the expected maximum transmission through two uncou-pled cavities(Tmax)2.
Looking at Figure 3.8a, the side-peaks seem to disappear when going to 15% loss in the cavities. This is observed regardless of the mirror reflec-30
3.2 Numerical Calculations 31
tivities. It would therefore appear that controlling δo is no longer needed when the loss inside the cavity spacing is high enough. It turns out how-ever that even at 15% loss the suppressed side-peaks can significantly af-fect the transmission at 1 MHz. This is shown in the discussion on the stability of the final design in subsection 5.1.1.
3.2.3
Two Uncoupled Cavities
In the introduction and in subsection 2.2.1 the transmission through two aligned cavities was said to be L2(ω) if light is prevented from travelling
between the cavities. In an experimental set-up this is realised through the use of an optical isolator or optical circulator, which allows the propaga-tion of forward travelling light, but blocks or redirects backwards travel-ling light. In the numerical calculations this can be replicated by setting the amplitude of backwards propagating light inside the cavity spacing to zero. A comparison can then be made with the same set-up where light is not blocked. This comparison is plotted in Figure 3.9 for two identical set-ups with 500ppm mirrors and no included losses.
Figure 3.9:Comparison between the transmission response of two coupled (blue) and two uncoupled (red) aligned cavities. Both cavities consist of mirrors with R=0.9995.
The linewidth of the transmission through the uncoupled cavities plot-ted above is 36.08kHz, whereas the linewidth for the coupled cavities is 39.6kHz. To get a sense of the ’flatness’ of the transmission through the coupled cavities, at±quarter the linewidth distance from the centre of the peak the transmission is still 92%. For the uncoupled cavities the transmis-sion at this point has dropped to 77%.
Chapter
4
Multi Cavity Filters
The previous section characterised the filter properties of two equal length cavities separated by a variable distance. With the right separation be-tween cavities such a set-up provides flat-top and steep drop-off trans-mission peaks, similar to a three mirror set-up, but can not comfortably achieve the required 150dB suppression at 1 MHz within the available mir-ror range.
To further improve the design the properties seen in two aligned cavi-ties are explored for extended set-ups with three and four aligned cavicavi-ties through the numerical method introduced in section 3.2.
4.1
Three and Four aligned cavities
For two aligned cavities two factors were important for obtaining a filter with a desirable flat-top transmission peak. The relative offset between the mirror spacings and the cavity spacing determined the location of the sipeaks around the main peak. The shape of the main peak was de-termined by the reflectivity of the cavity formed by the inner two mirrors and related to the shape of the main peaks in the three mirror set-up.
In the section below these effects are investigated for three and four aligned cavity set-ups through numerical implementation of the transfer matrix method. The implementation of this method is done in the same manner as described in section 3.2 for the two cavity set-up. For clarity the three and four mirror set-ups are sketched in Figure 4.1.
34 Multi Cavity Filters
Figure 4.1:Sketches of the three and four aligned cavities set-ups.
4.1.1
Offset Parameter
For two aligned cavities an important parameter proved to be δo, indicat-ing the relative offset between the mirror spacindicat-ings and cavity spacindicat-ings. With three and four aligned cavities there are now five and seven spac-ings to control, but the general behaviour turns out very similar to the two aligned cavities. To start off the mirrors used to create the cavities will be set to all have the same reflectivity R.
As with the two aligned cavities, the mirror spacings have to be equal or no transmission peak will be obtained. The frequency of the main peaks are determined solely by this spacing. For the cavity spacings the relevant parameter is again δo. If for all spacings δo = λ2 the side-peaks all coincide with the main peak. For three and four cavities this results in four and six side-peaks surrounding the main peak respectively. If for all the spacings
δo = λ4, the nearest side-peaks are at maximum separation from the main peak.
It should be noted that the cavity spacings do not necessarily have to be equal, but for maximal interference with the main peak all cavities require
δo = λ2 and for least interference all cavities require δo = λ4. The two ex-tremes are plotted in Figure 4.2.
The main peaks in the two cavity set-up related to the main peaks in a three mirror set-up through the reflectivity of the cavity formed by the inner two mirrors. Through the same logic, the main peaks of the three and four aligned cavities can be expected to behave like the main peaks in the four and five mirror set-up’s, when δois near λ4. This comparison is discussed in subsection 4.1.2.
34
4.1 Three and Four aligned cavities 35
(a)Three aligned cavities. (b)Four aligned cavities.
Figure 4.2: Numerical transmission through three and four aligned cavities with equal mirrors, for cases δo = λ4 (blue) and δo = λ2 (red).
In the two cavity set-up with four mirrors with equal reflectivities R1, the maximum reflectivity of the inner cavity Rinner happens to be equal to Rideal2 (3), so that at δo = λ4 the ideal flat top transmission peak is obtained. Focussing on the main peaks in Figure 4.2 reveals that this is not the case for three and four cavities with equal reflectivity mirrors. Close ups of the transmission peaks are plotted in Figure 4.3, showing that the peaks do not quite correspond to the ideal flat top peaks. A possible explanation is that the reflectivities of the cavities formed between the main cavities (Rinner) do not quite correspond to Rideal2 for each case.
(a)Three aligned cavities. (b)Four aligned cavities.
Figure 4.3: Shape of main transmission peak for three and four aligned cavities, when all mirrors have equal reflectivity. Increasing or decreasing the reflectivity narrows/broadens these peaks, but does not get rid of the rippled tops.
36 Multi Cavity Filters
4.1.2
Ideal Transmission Parameters
For two aligned cavities the inner cavity could virtually be replaced by a single mirror with the same reflectivity, to find the ideal reflectivity of the inner mirrors with given outer mirror reflectivities R1. In this case the ideal reflectivity for the inner mirrors turned out to be R1as well (subsec-tion 3.1.2). Since the ideal reflectivity of the inner mirrors in a four mir-ror set-up is known as well, a similar prediction can be made about three aligned cavities in relation to a corresponding four mirror set-up.
Figure 4.4: Diagram showing the virtual replacement of the inner cavities by a single mirror to treat the two and three aligned cavities as a three and four mirror system when δo = λ40.
Setting up six mirrors, with two outer mirrors R1 and four inner mir-rors R3(sketched in Figure 4.4), the transmission peak when δo = λ4 can be compared to a four mirror set-up with outer mirrors R1 and inner mir-rors R2 = Rmaxinner. A prediction can thus be made for which value for R3 will yield the ideal flat-top response of the four mirror set-up with R2 =Rideal2 (4) by requiring: Rmaxinner = 4R3 (1+R3)2 = Rideal2 (4) (4.1) Solving for R3 gives a prediction for the required inner mirrors to obtain flat top peak transmission in three aligned cavities:
R3 =
4−2Rideal2 (4)−4 q
1−Rideal2 (4)
2Rideal2 (4) (4.2)
Unlike in the two aligned cavities case, plugging in R2ideal(4) in terms of R1 does not result in R3 =R1, as was expected from Figure 4.3.
36
4.1 Three and Four aligned cavities 37
(a)Three aligned cavities (b)Four aligned cavities
Figure 4.5: Transmission response of three and four aligned cavities for various effective reflectivities of the inner cavities. In a) the reflectivity parameters were chosen so that Rinnerwas equal to (purple), larger than (blue) or smaller than(red)
Rideal2 (4). The required parameters for flat-top transmission matched the predic-tion in Equapredic-tion 4.2. In b) the parameters were chosen by trial and error as no analytical expression has been derived for transmission through five mirrors, but importantly again a combination of reflectivities could be found that results in a steep, flat top (purple).
In Figure 4.5a the reflectivities of the inner mirrors R3are varied so that Rinner is bigger than, smaller than or equal to R
ideal(4)
2 . The ideal value for R3predicted by Equation 4.2 indeed results in a flat top transmission peak and lower and higher values result in the same behaviour seen for the four mirror set-up in Equation 2.2.
It is expected that a comparison can also be made between four aligned cavities and a five mirror set-up to predict the ideal mirror reflectivities for four aligned cavities. As an analytical prediction has not been derived for a five mirror up, the mirror reflectivities for the four aligned cavity set-up were varied by trial and error and the resulting transmission curves are plotted in Figure 4.5b. Set-ups with three different reflectivity mirrors (R1, R2 and R3) were created, with mirrors in the order R1−R2−R2− R3−R3−R2−R2−R1, to keep the system symmetric. Again an ideal combination of mirrors was found for given outer mirrors R1that results in obtaining a steep, flat top transmission peak. The requirement R3 > R2 > R1 is apparent (but not conclusive), which was expected from the R2> R1requirement for the three and four mirror set-ups.
38 Multi Cavity Filters
4.1.3
Loss in Cavity Spacings
For two aligned cavities any loss in the spacing between the cavities heav-ily affected the height of the side-peaks while leaving the main peaks rel-atively unaffected. In the same way Figure 3.8a was created the effect of loss in the cavity spacing is plotted in Figure 4.6 for three and four aligned cavities. Again the effect was the same whether the loss was located in the cavity spacing or in the substrates and the side-peaks disappear at 15% loss.
(a)Three aligned cavities (b)Four aligned cavities
Figure 4.6:Effect of loss in the spacing between cavities in three and four aligned cavities with δo = λ2. Cavities were set up with equal mirrors of R = 0.999. At
15% loss in the cavity spacings the main peak transmission is 71% for the three cavity set-up and 61% for the four cavity set-up.
Any loss in the Bragg stacks again highly suppresses the peak mission, influenced by the reflectivities of the mirrors. The maximal trans-mission for three and four cavities was found to match Tmax3 and Tmax4 , with Tmax the maximal transmission through a single cavity with lossy mirrors as given by Equation 3.6.
4.1.4
Peak Transmission and 1 MHz Suppression
In the previous sections it was shown that three or four aligned cavities obey the same behaviour that was shown for two aligned cavities and that they can be used to obtain flat-top transmission peaks with steep suppres-sion. The most important question is now whether these set-ups can be used to satisfy the filtration requirements discussed in the introduction.
The important factors to consider are the transmission at ω0 and at
ω0+1MHz. While using higher reflectivity mirrors will be beneficial to 38
4.1 Three and Four aligned cavities 39
the suppression of light at ω0+1MHz, the transmission at ω0 will be re-duced due to increases loss as discussed above. A trade-off will therefore be reached between better suppression and lower peak transmission and both must be taken into account carefully.
For three and four aligned cavities the transmission at ω0and at ω0+ 1Mhz was calculated numerically for a range of mirror reflectivities. For ease of comparison the set-ups with identical mirrors are used rather than using the ideal parameters discussed in subsection 4.1.2. As the expected loss in each mirror stack is not known, the peak transmission is plotted for 3 ppm and 6 ppm Bragg stack loss. The substrate/cavity spacing loss was kept constant at 1%.
(a)Transmission at ω0vs mirror
reflectiv-ity
(b)Transmission at ω0+1MHz vs mirror
reflectivity
Figure 4.7: a) Comparison between the transmission at ω0 for three and four
aligned cavities, using equal mirrors with 3ppm and 6ppm loss. b) Comparison between the transmission at ω0+1MHz. The golden dashed line indicates the
desired 150dB suppression.
It is clear that both a three cavity set-up or a four-cavity set-up could be used to achieve 150dB suppression of the ω0+1MHz signal within the range of available mirrors. In the next chapter a final design is chosen based on the properties presented in Figure 4.7.
Chapter
5
Final Design and Implementation
Having gone through the calculations for a range of possible set-ups, a decision can now be made on what the final optical set-up could be and to start looking at how to realise this set-up in a laboratory. A very im-portant factor that up till now has been ignored is that in any filter cavity there must be a way for all the distances between the mirrors to be con-trolled so that the right frequency is transmitted. Furthermore a method must be developed to move the mirrors to these correct positions and keep them there. In section 5.1 the specifications of the final chosen design are outlined. In section 5.2 and section 5.3 the way in which the design can be realised is discussed.
5.1
Optical Cavity Specifications
The final design specifications were chosen through comparison of the properties of the three cavity set-up and the four cavity set-up plotted in Figure 4.7. For three cavities the minimally required 110ppm mirrors would mean a maximum transmission of the peak frequency of less than 85% for 3ppm mirror loss and less than 75% for 6ppm mirror loss. For four cavities only 360ppm mirrors are required, giving a maximum transmis-sion of above 90% at 3ppm mirror loss and 85% at 6ppm mirror loss. On this basis the decision was made to use four aligned cavities to ensure a higher attainable peak transmission.
While 360ppm mirrors are sufficient, it was decided to use 250ppm mirrors in the final design to potentially provide 170dB suppression at
42 Final Design and Implementation
3ppm mirror loss). As a technical sidenote, for easier alignment mirrors will be used with a curvature of 30cm, which does not affect the expected transmission behaviour when light is aligned through the centre of each mirror. Furthermore a pair of lenses will be placed inside each cavity to allow for easier alignment of the cavities, which also does not affect the expected transmission behaviour except for the loss introduced by the lenses. This loss is estimated at an additional 1% loss in each cavity spac-ing and accounted for in the calculations.
The expected properties of the final optical filter with eight 250ppm mirrors and 2% loss in each cavity spacing are displayed in Figure 5.1. The properties are plotted for a Bragg stack loss of 1, 3 and 6ppm as the actual loss in the Bragg mirrors is unknown and will have to be measured for each mirror to be used in the experimental set-up.
Figure 5.1:Summary of the expected properties of the final cavity design and an example of the transmission curve around ω0 when the loss in the Bragg stack
is 3ppm. At 1ppm and 6ppm Bragg stack loss the overall height of the peak is higher/lower, but the shape does not change significantly.
The data in Figure 4.7 on which the final design is based was calcu-lated for three and four aligned cavities with equal mirrors. On the other hand Figure 4.5 showed that slightly tuning the reflectivities of the inner mirrors will result in improved flat-top transmission peaks. In settling on 42
5.1 Optical Cavity Specifications 43
a final design the preference went to having eight identical mirrors that can be produced in bulk and which allow for the set-up to be built as four identical cavities, simplifying construction.
The plots in Figure 5.2 show two options of set-ups with slightly dif-ferent mirror reflectivities, chosen to give the ideal flat peaks. Since no analytical expression was derived for a five mirror set-up, the presented reflectivities were again derived through trial and error.
Figure 5.2:Possible flat top transmission peaks when using slightly tuned mirrors compared to the final design using eight identical 250ppm mirrors with 3ppm loss (dashed blue). For the purple curve R1 = 400ppm, R2 = 280ppm, R3 = 195ppm
and transmission at 1MHz is 5.4·10−17. For the red curve R1 = 250ppm, R2 =
170ppm, R3 =110ppm and transmission at 1MHz is 8.5·10−19.
Similar to the two cavity comparisons in subsection 3.2.3, a comparison can be made between four ’coupled’ cavities and four ’uncoupled’ cavities where light is blocked from travelling backwards between the cavities. The comparison is plotted in Figure 5.3, again by setting the backwards propagating light to zero in the numerical calculation in each cavity spac-ing. An important factor not included in this comparison is the loss that would be introduced by optical elements realising such a process, which can be quite large depending on the elements used and the accuracy of alignment.
As discussed the linewidth of a single Stokes photon is on the order of a few kHz. Taking the Lorentzian linewidth of the photon to be 5kHz, the expected total transmission percentage of such a photon in the final design is 72.5%. For four ’uncoupled’ cavities this percentage reduces to 62.3%,
44 Final Design and Implementation
assuming no extra loss is associated with a device that prevents the back-wards propagating light and that 100% of this light is removed. Factoring in an estimated additional 1% loss in each cavity spacing the transmission percentage drops below 60%. In each case the frequency of the photon was taken to be perfectly on resonance with the filter cavities, any slight shift up to 10kHz will greatly favour the coupled cavities. Furthermore, the higher the linewidth of the photon to detect, the larger the advantage of the coupled cavities over the uncoupled cavities. As a side-note, the transmission percentage for a 5kHz photon through the tuned set-up in Figure 5.2 (purple) is 76.5%.
(a)3 MHz range around peak (b)80 kHz range around peak
Figure 5.3:Comparison between the transmission through the final chosen set-up with four coupled cavities (blue) and the transmission through the same cavities when light is blocked from going backwards between cavities (red), including 3ppm Bragg stack loss. a) Indication of the improved suppression at 1 MHz. b) Indication of the wider and steeper transmission peak of the coupled cavities. The linewidth of the coupled cavities is 22.8kHz and transmission at 1 MHz is 2.3·10−17. For the uncoupled cavities this is 12.3kHz and 1.4·10−15respectively.
5.1.1
Stablity
An important factor that ties in with aligning the cavity and maintaining alignment, is the sensitivity of the system to small changes in the posi-tions of the mirrors once they are in the correct position. The transmission peaks discussed this far were creating using cavities set to be perfectly on resonance with the desired frequency ω0. As discussed the transmission response is sensitive to both the spacing between mirrors and the spacing between the cavities and both need to be accounted for. The spacing be-tween the mirrors is important for maintaining high transmission at ω0, 44
5.1 Optical Cavity Specifications 45
while the spacing between the cavities is important for maintaining high suppression of the pump frequency at ω0+1MHz. The stability of both spacings with regards to those goals are shown in Figure 5.4.
An important distinction is that the system is much more sensitive to the mirror spacings compared to the cavity spacings. As was shown ana-lytically for the two aligned cavities in Figure 3.3, there is a 200nm range in which the cavity spacing results in good suppression at ω0+1MHz. It should be noted that this effect persists even at 15% loss in the cavity spacings, so that the necessity to control the cavity spacings can not be re-moved by suppressing the side-peaks through introducing losses without significantly decreasing the peak transmission at ω0.
Moving a single mirror to decrease or increase the mirror spacing by only 20 picometers is enough to reduce the peak transmission by 50%. As such the mirror spacings must be controlled much more accurately than the cavity spacings.
(a)Stability of mirror spacing. (b)Stability of cavity spacing.
Figure 5.4:a) highlights the stability of the mirror spacings, showing that a single mirror offset by only 20pm reduces peak transmision by 50%. b) highlights the importance of controlling the spacing between the cavities in order to maintain > 150dB suppression of the ω0+1Mhz signal. Data is plotted for 0%, 15% and
50% loss between the cavities to indicate that, unless the loss between the cavities is extremely high, the cavity spacing needs to be accounted for.
46 Final Design and Implementation
5.2
Aligning the Cavities
With all the mirrors in place a final task is tuning the distances between the mirrors so that the cavities are all resonant with ω0and the cavity spacings are all offset by δo = λ4. For this purpose the position of at least seven of the eight mirrors must be accurately controlled, for example through the use of piezoelectric motors.
With an optical circulator placed between the cavities, an additional advantage arises besides not having to worry about the distances between cavities: all the cavities can be locked one by one based on the reflection signal from each cavity. In deciding on a final design using these extra op-tical parts was purposely avoided as they will negatively affect the 88% peak transmission that is hoped to be achieved. Therefore a different method must be found that allows the system to be aligned based only on the reflection signal of the entire eight mirror structure. This locking problem is illustrated in Figure 5.5.
Figure 5.5:Sketch showing the signals to be used for locking if additional optical components redirect the reflected signal of each cavity (top) and the situation in the current set-up (bottom) where only the single reflection signal of the entire set-up can be used. The green ray represents the forward propagating light. The red ray represents the backwards propagating light.
In the set-up with additional optical circulators, each individual cav-ity can be locked and kept on resonance through ’dither locking’ (further discussed in section 5.3), as was demonstrated by Galinskiy et al. [7]. The aligned cavities are pumped with the desired frequency ω0. Initially all light will be reflected as cavity 1 is not resonant with the pumping fre-quency. Moving mirror two will cause a dip in the reflection signal from 46
5.2 Aligning the Cavities 47
cavity 1 when the cavity becomes resonant with ω0, allowing the locking of cavity 1 through dither locking. The light is then transmitted through cavity 1 and reaches cavity 2 so that the second cavity can be locked in the same manner. When cavity 2 is locked light is transmitted through to cavity 3, and so on.
In an ideal situation without any losses, moving any of the mirrors in the set-up without extra optical parts can not result in any change in the reflection signal when the cavities are not locked. When the four cavities are all at resonance with different frequencies, pumping the cavities with
ω0 will again initially result in all light being reflected. Moving mirror 2 to attempt to lock cavity 1 will now not result in a reflection dip as it did for the set-up with optical circulators, since any light transmitted through the first cavity will be reflected from cavity 2 and passes back through cav-ity 1. It would therefore seem that no useful information can be obtained about the position of the mirrors through observing the reflection signal. Furthermore moving any of the mirrors 2-7 will affect both the bordering cavity and the bordering cavity spacing, adding further difficulty.
The solution to this problem appears to come from the fact that the set-up at hand is not expected to be lossless: there is a certain amount of loss expected to arise in the Bragg stack, the substrates and the lenses between the cavities. These losses were already accounted for in calculating the fi-nal set-up’s transmission response in section 5.1. Upon investigation the loss in the Bragg stacks causes two separate effects that can be observed in the reflection signal, which could be used for locking all four cavities as well as controlling the spacings between the cavities.
When cavity 1 is resonant with ω0, the field builds up inside the cavity and the effect of any loss in the Bragg stacks will be amplified. When scan-ning mirror 2 a reflection dip should thus be observable on resonance: not because light is transmitted but rather because more light is lost inside the first cavity. The four aligned cavities were set-up for the numerical calcula-tions with a chosen added offset to each spacing in the range from -250nm to 250nm. The first cavity was given an offset of -100nm. The effect of ’moving’ mirror 2 was investigated numerically by increasing/decreasing the distance between mirror 1 and mirror 2, and decreasing/increasing the distance between mirror 2 and mirror 3 by the same amount. Fig-ure 5.6 shows the reflection signals when mirror 2 is scanned from -150nm to 150nm around its current position, for set-ups with 1ppm, 3ppm and 6ppm loss in the Bragg stack.
