Continuous phase stabilization and active interferometer control using two modes

2.2 MB·English

by  Gregor Jotzu

Continuous phase stabilization and active interferometer control using
two modes
Gregor Jotzu, Tim J. Bartley∗, H. B. Coldenstrodt-Ronge, Brian J. Smith, and I. A. Walmsley
University of Oxford, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom
∗Corresponding author: [email protected]
CompiledJanuary11,2011
Wepresentacomputer-basedactiveinterferometerstabilizationmethodthatcanbesettoanarbitraryphase
differenceanddoesnotrelyonmodulationoftheinterferingbeams.Theschemeutilizestwoorthogonalmodes
propagating through the interferometer with a constant phase difference between them to extract a common
1 phase and generate a linear feedback signal. Switching times of 50ms over a range of 0 to 6π radians at
1 632.8nmareexperimentallydemonstrated.Thephasecanbestabilizeduptoseveraldaystowithin±3◦. (cid:13)c
0 2011 OpticalSocietyofAmerica
2 OCIS codes: 120.3180(Interferometry);120.5050(Phasemeasurement);260.3160(Interference)

n
a
J The ability to continuously adjust and stabilize the spatial, temporal, frequency or polarization modes de-

0 opticalphasedifferencebetweentwoarmsofaninterfer- pending upon the nature of the experiment and noise
1 ometer is of great importance to a wide range of appli- involved. Indeed, this mode multiplexing approach can
cationsincludinghomodynedetectionforquantumstate be seen as the key physical principle behind previous
]
s tomography[1],phase-shiftkeyinginopticaltelecommu- schemestocontinuousphasestabilization,whichutilized
c nications [2], phase-shifting interferometry [3], ultrafast frequency [9], and spatial modes [10]. Both modes expe-
i
t pump-probe spectroscopy [4–6], interference-based op- rienceacommonphaseshiftφ,andonemodeundergoes
p
tical lattices [7] and near-field scanning microscopy [8]. an additional, but constant phase offset δ. Monitoring
o
. Previousapproachestypicallyreliedonfringe-lockmeth- both modes at the output of the interferometer, with
s
c ods, in which a useful error signal can only be produced knowledgeoftheoffsetphasedifferenceδ,enablesanac-
i near integer multiples of π/2. These methods are prone curate linear estimation of the phase common to both,
s
y tofringeskipping,wherenoisecausesthephaselockcir- andthusstablefeedbackcontroloveracontinuousrange
h cuit to hop to a neighboring interference fringe, a half of φ.
p
wavelength away. In many applications the ability to
[
continuously adjust the phase difference to an arbitrary

1 value is required, which cannot be achieved with fringe-
v lock methods. Continuous phase locking has been previ-
6
ously accomplished in two ways. An arbitrary phase can
0
be locked by modulation of one of the interfering beams
9
1 and detection at the fundamental and second harmonic
. of the modulation frequency [9]. This method, which
1
0 produces a sinusoidal error signal, is still susceptible to
1 fringe skipping. Furthermore, the phase modulation re-
1 quiresspecializedsignalanalysisandisundesirablewhen
v: signal acquisition rates faster than the modulation fre-
i quency are necessary. A recent technique that utilizes
X
tilting of the beam in one arm of the interferometer and
r Fig.1.SetupofourMach-Zehnderinterferometer(MZI)
spatially resolved measurements at the output to cre-
a
with two polarization modes. Both horizontal (H) and
ate a linear error signal was introduced [10]. However,
vertical(V)modesaresubjecttoacontrollablecommon
this scheme requires precise alignment and stabilization
phaseφ,whileonemode,V,picksupanadditionalphase
ofphotodetectorpositioning,andsuffersfromchromatic
δinthelowerbranchoftheMZI.Theinterferencesignals
aberrations of the glass wedge used for the tilt.
S and S of each mode are measured independently,
InthisLetter,wepresentageneralapproachtophase 1 2
from which the phase φ is calculated by software.
controlcapableofstabilizationtoanarbitraryphaseset-
tingbyproducingalinearerrorsignalinthephase.The
basicprincipleliesinutilizingtwodistinctopticalmodes As a concrete example, consider two polarization
passing through the interferometer with non-identical modesasdepictedinFig.1.Areferencelaserpolarizedat
optical path length differences resulting in two phase 45◦ withrespecttothehorizontalislaunchedintoabal-
offsets. The modes could consist of different transverse- anced Mach-Zehnder interferometer (MZI), constructed
with non-polarizing 50:50 beam splitters (BS). Both po-
1
larizations traverse the upper path containing a control- byscanningthephaseφ,andcollectingthedetectorvolt-
lable phase φ. A waveplate in the lower path of the MZI ages.Aparametricplotof(S (φ),S (φ))asthephaseφ
2 1
produces a constant relative phase shift δ between the is scanned sweeps out a rotated and displaced ellipse, as
horizontalandverticalpolarizationmodes.Thepolariza- showninFig.2(a).FittingthisellipsetoEq.(1)directly
tionmodesfromoneoutputportoftheMZIaresplitbya yieldsthecalibrationparameterstobeusedinthephase
polarizingbeamsplitter(PBS),anddirectedtophotodi- stabilizationprotocol.Toobtainmoreaccuratevaluesof
ode detectors. This scheme is easily generalized to other the feedback parameters, we repeated this process mul-
modes, e.g. two transversely displaced co-propagating tiple times. As long as δ differs sufficiently from 0 and
laser beams. π, i.e. the ellipse does not become a line, successful sta-
The output currents from the photodiodes are con- bilization can be accomplished. Roughly speaking, the
verted into voltages across resistors and monitored by a phase φ can be stabilized to within
computerusingafastanalog-to-digitalconverter(ADC). √
∆P 1+ 2
The expected signal voltages have the form ∆φ≥ , (4)
P¯ 2V|sin(δ/2)|
S1 =A1cos(φ)+B1, S2 =A2cos(φ+δ)+B2. (1) where P¯ and ∆P are the average and standard devi-
ation in the reference laser power, V is the fringe visi-
Here A and B , which depend on the input in-
1(2) 1(2) bility for the two modes (assumed equal), and δ is the
tensity, interference visibility, detector gain, background
phaseoffsetbetweenthetwomodes.Thiscanbederived
light and electronic noise levels, are calibrated prior to
from assuming the signal voltages are given by S =
stabilization. These equations can be combined to esti- 1
gP(1+V cos(φ))/2 and S = gP(1+V cos(φ+δ))/2,
mate the common phase value 2
wheregisthedetectorgain,assumedtobeequalforboth
(cid:26) (cid:20) (cid:21)(cid:27) detectors. This result puts quantitative bounds on the
cos(δ) A (S −B )
φ=U tan−1 − 1 2 2 , (2) intensity fluctuations, fringe visibility, and offset phase
sin(δ) A (S −B )sin(δ)
2 1 1 that can be tolerated to yield phase stabilization within
∆φ.
where the U is the phase unwrapping function. This es-
Once calibrated, the interferometer can be stabilized
timate works accurately as long as the phase does not
at a desired setting φ , by continuous monitoring of
change by more than π, corresponding to a half wave- 0
the phase φ and feedback to the control PZT stage on
lengthpathdifference,withinasamplingperiod.Ifφ is
0
a ms timescale. The feedback voltage applied to the
the desired phase setting at which the interferometer is
PZT, derived from Eq. (3), was V = −f(cid:104)φ−φ (cid:105),
to be locked, the estimated phase in Eq. (2) can be used f 0
where the average was taken over previous measure-
to implement a linear error signal to drive the feedback
ments. The best stability was achieved with a feedback
control of φ
gainoff =3.65mV/degree.Theoptimalfeedbackfunc-
err =−f(φ−φ ), (3)
0
tion, that is, the relationship between error signal and
where f is a constant feedback gain parameter. the feedback control voltage, will generally depend on
To experimentally test this phase stabilization tech- the details of the experiment [11].
nique we used a Helium-Neon (HeNe) reference laser The performance of our active feedback stabilization
(JDS Uniphase 1125P, 5 mW average power) polarized scheme was limited primarily by power fluctuations of
at45◦ withrespecttothehorizontal,asshowninFig.1. the reference laser and the feedback response time (i.e.
ThephaseφintheupperarmoftheMZIissetbyacom- the time delay between estimating the phase and the
puter controlled piezo-electric translation (PZT) stage implementation of the feedback signal voltage). Power
(ThorlabsNF5DP20/M)delayline.Aquarterwaveplate fluctuations can be eliminated by monitoring the input
(QWP) with vertically aligned slow axis sets the con- laser power and normalizing the output data, or use of
stant relative phase δ ≈90◦ in the lower MZI arm. Note a more stable laser system. The time lag was mainly
that a wide range of δ can be tolerated, so that a multi- due to the PC processing time and PZT response time,
order wave plate or arbitrary birefringent medium could which varied between 4 ms and 7 ms. This implies high-
potentially be used. A PBS at one output of the MZI frequency noise could not be compensated. This issue
splits the horizontal and vertical polarization modes, couldbeimprovedusingfasterAD/DACelectronicsand
which are subsequently focused onto fast photodiodes lower-levelprogrammingtoimplementthephaseestima-
(Thorlabs DET10A/M). The voltages produced by the tionandfeedback.TheresultsshowninFig.2(b)and(c)
photodiodes are monitored using a fast 16-bit analog- were achieved with calibration parameters A = 0.260,
1
to-digital / digital-to-analog converter (AD/DAC) (Na- A = 1.083, B = 1.859, B = 5.511, and δ = 93.5◦. At
2 1 2
tional Instruments USB-6221 BNC) connected to a per- what would be an unstable phase setting for fringe lock
sonal computer (PC) at a rate of 10 kHz. A software methods, φ = 15◦, we are able to stabilize to within
0
program calculates the error signal in Eq. (3), which is a standard deviation of ∆φ ≈ 3◦. This corresponds to
sent to the PZT stage using the AD/DAC. lengthvariationsofapproximately5nmbetweenthetwo
Toimplementthestabilizationschemetheparameters 3.5m long arms of the interferometer. Given we have
A , B and δ must first be calibrated. This is done power fluctuations ∆P/P¯ of 0.6%, Eq. 4 yields phase
1(2) 1(2)
2
Fig. 2. (a) Parametric plot of (S (φ),S (φ)) when scanning the phase φ yields the calibration ellipse. By fitting the
1 2
data, the values of the constants in Eq. (2) are determined. (b) After allowing the phase to fluctuate naturally for
10 minutes, the stabilizer is switched on to lock the phase at 15◦. The root-mean-squared error in the locked phase
is 3◦, which can be maintained for over 24 hours. (c) The phase of the stabilizer can be arbitrarily set to any value
within the range of the computer controlled delay stage. Here, we show phase locking to various phases from 0 to
6π radians.
fluctuations of 2.9◦, which is in good agreement. theapplicabilityofthestabilizationschemetoalloptical
This stability can be maintained for over 24 hours. degrees of freedom.
Furthermore, the desired phase setting φ can be This research was supported by the EU Integrated
0
changed in real time as shown in Fig. 2 (c). Changing Project Q-ESSENCE through the European Commu-
the phase setting φ by more than 6π could be achieved nity’s Seventh Framework Programme FP7/2007-2013
0
with switching time near 50ms, which is primarily lim- under grant agreement no. 248095. The authors would
ited by the feedback response time. The current scheme like to thank Gaia Donati for assistance with the data
thus allows one to set and stabilize an interferometer to analysis.
any optical path length difference with a precision of a
few nanometers and a range limited only by the move- References
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tweenthetwointerferometerarmsasdemonstratedhere.
Frequency modes could be used in optical fiber based
interferometers where the dispersion is known. Improve-
mentstothefeedbackelectronicswilllikelyleadtofaster
switching times and increased stability against higher-
frequency noise.To increase theprecision of this scheme
requires increased stability or continuous monitoring of
the reference laser, as eluded to in Eq. (4). Further-
more, implementing a more sophisticated feedback sig-
nal, e.g. using a PID control algorithm [11], should lead
toimprovedperformance.Thetechniquesdevelopedhere
should be of use for a wide range of applications due to
3