arXiv:0906.4563v1 [quant-ph] 24 Jun 2009
On the Application of a Monolithic
Array for Detecting Intensity-Correlated
Photons Emitted by Different Source
Types
D. L. Boiko
Centre Suisse d’Electronique et de Microtechnique SA, 2002, Neuchâtel, Switzerland
dmitri.boiko@csem.ch
N. J. Gunther
Performance Dynamics, 4061 East Castro Valley Blvd., Castro Valley, California, USA
njgunther@perfdynamics.com
N. Brauer, M. Sergio, C. Niclass
Ecole Polytechnique Fédérale de Lausanne, Quantum Architecture Group, 1015, Lausanne,
Switzerland
nilsbenedict.brauer@epfl.ch, maximilian.sergio@epfl.ch, cristiano.niclass@epfl.ch
G. B. Beretta
HP Laboratories, 1501 Page Mill Road, Palo Alto, California, USA
giordano.beretta@hp.com
E. Charbon
Technische Universiteit Delft, Mekelweg 4, 2628 CD Delft, The Netherlands
e.charbon@tudelft.nl
Abstract: It is not widely appreciated that many subtleties are involved
in the accurate measurement of intensity-correlated photons; even for
the original experiments of Hanbury Brown and Twiss (HBT). Using a
monolithic 4×4 array of single-photon avalanche diodes (SPADs), together
with an off-chip algorithm for processing streaming data, we investigate the
difficulties of measuring second-order photon correlations g(2) (x′ ,t ′ , x,t) in
a wide variety of light fields that exhibit dramatically different correlation
statistics: a multimode He-Ne laser, an incoherent intensity-modulated
lamp-light source and a thermal light source. Our off-chip algorithm treats
multiple photon-arrivals at pixel-array pairs, in any observation interval,
with photon fluxes limited by detector saturation, in such a way that a
correctly normalized g(2) function is guaranteed. The impact of detector
background correlations between SPAD pixels and afterpulsing effects
on second-order coherence measurements is discussed. These results
demonstrate that our monolithic SPAD array enables access to effects that
are otherwise impossible to measure with stand-alone detectors.
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Table 1. Values of first and second order correlation functions for incoherent, coherent and
thermal light states. Single-mode states are considered, θ is the angular width of the source,
λ is the wavelength, τc is the coherence length. Integration effects due to limited detector
response times and resolution of coincidence counter are not indicated.
Function
Incoherent
Coherent
Thermal
g(1) (x, τ )
0
1
g(2) (x, τ )
1
1
2
sinc πθ λx exp −π 2ττ 2
2
1 + sinc2 πθ λx exp −π ττ 2
0
2
sinc2 πθ λx exp −π ττ 2 ,
[ g(1) (0) = 1 ]
Entangled
c
[ g(2) (0) = 1 ]
1.
c
[ g(2) (0) = 2 ]
c
Introduction
The first-order correlation function, g(1) , is widely used in optical applications because a simple
experimental arrangement makes it easily accessible, e.g. Young’s double-slit interferometer
[1]. In Young’s experiment, two points on the wavefront
separated by a distance x12 produce a
√
fringe pattern with resultant intensity I1 + I2 + 2 I1 I2 |g(1) (x12 , τ )| cos(∆ϕ12 ) at a screen location where the phase difference is ∆ϕ12 and the propagation time-difference is τ . The fringe visi1 +I2
V
bility V measures the magnitude of the first-order correlation function |g(1) (x12 , τ )| = 2I√
I I
1 2
between field points at the double pinhole. When the light-state distribution, |g(1) (x12 , τ )|, is
known then the phase difference is all that is required to define relationship between the field
points. But in the case of unknown field distributions, measuring this quantity alone is ambiguous because it cannot distinguish between light states like: entangled photons, incoherent light,
or coherent and thermal light (See Table 1). The ability to resolve such potential ambiguities
is highly significant, for example, for the current debate over the existence of microcavitypolariton Bose-Einstein condensation (BEC).[2]–[10]
Table 1 shows that to distinguish coherent [11] and thermal (chaotic) light states [12]
or entangled photons and incoherent light state, the second-order correlation function
I2 (t+τ )i
g(2) (x12 , τ )= hIhI1 (t)
(t)ihI (t)i , associated with Hanbury Brown and Twiss (HBT) correlated inten1
2
sity fluctuations [13], must also be measured. Here, I1,2 (t) is the light intensity at a point ± 21 x12
and time t. However, recalling some of the early controversies surrounding photon-correlation
measurements [14, 15, 16, 17], reminds us that second-order correlation measurements are technically more difficult to accomplish in the optical range and requires dedicated single-photon
detectors as well as a coincidence-counting apparatus. An additional difficulty arises due to the
impossibility of taking single, simultaneous image of the g(2) (x,t, x,t) distribution. Instead, a
pair of detectors and a beam splitter must be used to sample the distribution over a range of
detector-pair positions. More recent correlation measurements, such as those for BEC cavity
polaritons, often require the ability to resolve the spatial dependence of g(2) in a single run.
Taken together, all these requirements demand an integrated, monolithic, photon detector (as
flexible as a camera), that is capable of imaging intensity noise-correlations.
Elsewhere [18], we have presented such a device and applied it in a table-top implementation
of a stellar HBT interferometer [19], which in their original paper was used to measure the
Fig. 1. Micrograph of the 4×4 SPAD array (a), schematics of the SPAD pixel structure (b)
and electronic readout circuit (c). For a typical operation conditions VOP = −21V , which
by 4 V exceeds the breakdown voltage, VDD = 3.3V and VBIAS = 0V .
apparent angular diameter of the star Sirius from the current-noise correlations of two detectors
separated along a variable baseline. As seen from the Table 1, row for a thermal light source,
the angular width of Sirius star θ = 0.0063′′ assumes that detector separation baseline x12 can
be as large as a 106 × λ , yet enabling to observe second-order correlations.
However for a table-top implementation, for which the angular width of a typical source is
θ ∼ 1000′′ (e.g., a 200 µ m spot seen from a distance of 2 cm), the detectors have to be placed
at a distance x12 no more than a few tens of wavelength λ in order to observe correlations. It
was thus not possible to measure g(2) in the optical spectral range with stand alone detectors
located in one plane. A conventional table-top implementation necessitates the introduction of
a beam splitter together with two detectors at equivalent planes related by reflection, as in the
original HBT implementation [13]. At the same time for such delicate experiments demanding a clear-cut answer whether the intensity correlations exist or not, as in the polariton BEC
experiments, a strong sensitivity to variations in the relative detector position might render the
measurement procedure impractical and the results biased by an error in the relative detector position of the few tens of wavelength scale. Therefore, we proposed [18] the use of single photon
avalanche detectors (SPADs) integrated into a monolithic array with a simple data-treatment
algorithm for realizing table-top (laboratory) measurements of an HBT interferometer in the
original ”stellar”-like configuration.
Here, we report on the peculiarities of our SPAD detector-array for probing local g(2) correlations. The structure of this paper is as follows. In Sec. 2, we describe our detector system, which
comprises a 4×4 array of Si SPADs implemented in CMOS technology (Fig. 1) and a simple
algorithm to treat the multiphoton time-of-arrival distributions from different SPAD pairs, implemented off-chip. The device also incorporates on-chip high-bandwidth I/O circuitry which
facilitates the external data processing. We discuss the data treatment procedure to correct for
spurious correlations in measurements (Sec. 3) and we demonstrate the operation of our device
for cases of very different light statistics (Sec. 4): measuring the beat note of a multimode coherent source, measuring the depth of intensity oscillations in a superposition of incoherent and
thermal source, and measuring the instrument response function of a table top interferometer
as a function of the source angular-width.
2.
Structure of g(2) imager
Our g(2) imager is based on a single photon avalanche diode (SPAD) array [20], whose photomicrograph is shown in Fig. 1 (a). The array pitches are 30 and 43 µ m in the horizontal and
vertical directions, respectively. The chip is fabricated in a 0.35 µ m CMOS technology. Each
pixel of 4×4 SPAD array is independently accessible and has a structure as indicated in Fig.
1 (b). The SPADs p+ -n-well junction is reverse biased above breakdown by a voltage known
as excess bias voltage. In this mode of operation, called Geiger mode, single photons may be
detected and counted using auxiliary electronics. The anode is enclosed in the p+ -diffusion obtained from the standard implant used to define a PMOS channel. To prevent premature edge
breakdown in the periphery of the multiplication region, a 3.5 µ m guard ring of lightly doped
p-type implant is designed to surround the anode.
The p+ anode of each SPAD pixel is kept at a negative voltage VOP of 21 V, which by
4 V exceeds the breakdown voltage (Fig. 1 (c)). When an avalanche is triggered, e.g. as a
result of photoabsorption in the depleted n-well region or thermally induced carrier release
from a trap defect, the avalanche is subsequently quenched via a resistive current path in the
PMOS transistor structure with the transistor gate kept at a ground potential ( VBIAS = 0). This
ballast resistivity brings the bias voltage across the p+ -n junction below the breakdown voltage
and is used to read out the photodetection events. After avalanche quenching followed by a
recharge of the depletion region, the detection cycle is completed and can be initiated again by
arriving photons or thermally excited carriers. Each SPAD pixel comprises a high-bandwidth
inverter so as the Geiger pulses are converted into digital signals and transmitted for off-chip
data processing. This approach considerably improves the signal-to-noise ratio, however as
discussed in section 3, the data should be corrected for spurious correlations due to a possible
crosstalk between SPAD pixel outputs.
During the time required to complete a detection cycle, the detector is insensitive to arriving
photons. In our structures, the dead time is 12-15 ns, so as a theoretical limit on the count rate
set by detector saturation (pile-up) effects is about 30 MHz.
Tunneling of minority carriers through the shallow p-doped guard ring and/or trap-assisted
carrier release may trigger an avalanche and thus create spurious Geiger pulses. The lowest
detectable photon flux is set by the dark count rate (DCR) of SPADs, which at room temperature
conditions, is in the 5–10 Hz range. Such low DCR is achieved by implementing n-wells of
small diameter d=3.5 µ m. The lowest detectable photon flux density in our experiments is thus
10−8 photons/s/cm2 and the dynamic response range is 64 dB.
At 4V excess bias voltage, the measured photon detection probability (PDP) of such 0.35µ mCMOS SPAD pixels is 40 % at a wavelength 450 nm and exceeds 25 % in the wavelength range
at 400 - 550 nm. PDP is proportional to excess bias. The spectral sensitivity curve and pixel
statistics on dark count rate can be found in [20]. For our experiments, the chip was mounted on
a standard ceramic case and was wire-bounded with 20 µ m-thick aluminum wires of 6-7 mm
length spaced by 100 µ m. The bonding wires can be seen on the peripheries of the chip in Fig.
Fig. 1 (a). For what follows it is important to know that the bonding wires of individual detector
data lines are ordered with the SPAD pixel number i indicated on the microphotographic
image.
Since all 16 detectors in the array have separate parallel outputs, 16
=
120
HBT
interfer2
ometer measurements are possible simultaneously between various detector pairs at temporal
resolution set by the SPAD timing jitter characteristics ( 80ps). Measurements of pairwise second order correlation are based on implementing expression [21]:
g(2) (xi j , τ ) =
+
Tr(ρ̂ â+
i â j â j âi )
+
Tr(ρ â+
i âi )Tr(ρ â j â j )
(1)
with a simple algorithm. Here, integers i, j (i 6= j) enumerate detector pixels Di and D j in
Fig. 1, âi, j is the photon annihilation operator at two detectors, ρ̂ is the density operator for
the field and detectors, and trace is taken over the detector and field states [22]. At this stage,
the pairwise intensity noise correlations g(2) (xi j , τ ) are computed from multiphoton arrivals
at different detector pairs of the array, using a programmable four-channel 6-GHz bandwidth
digital oscilloscope (Wavemaster 8600A, LeCroy) and implementing a simple expression
N/2
(2)
g̃i j (τ ) = g̃(2) (xi j , τ ) =
(m)
NM ∑M
m=0 ∑n=−N/2 Xi
N/2
(m)
∑M
m=0 ∑n=−N/2 Xi
(m)
(n) ∧ X j (n + l)
N/2
(m′ )
(n) ∑M
m′ =0 ∑n′ =−N/2 X j
(n′ + l)
(2)
where Xi and X j are discrete random variables whose values 0 (no event) or 1 (photon detection) correspond to the binary data stream emanating from any pair of detectors Di and D j ,
respectively. The spatial lag xi j is set by the separation of the detector pair within the SPAD
array. Time-lag increments τ = lT are set by multiples of temporal resolution T (equivalent to
the bin width in conventional methods), where NT is the width of the measurement window
and M is the overall number of measurements series. Note that both addition and the bitwise
AND operator (∧ in the numerator of Eq. 2) could be implemented easily using on-chip digital
electronic circuits.
As can be seen from Eq. 2 , the concept of our g(2) imager [18] is drastically different to
conventional approaches based on measuring start-stop times and computing a timing histogram of delayed photon arrivals at two detectors, followed by renormalization of the histogram according to a statistical hypothesis for the field (e.g., a hypothesis like g(2) (τ = 0) = 1
or g(2) (τ = ±NT /2) = 1). Our g(2) -imager algorithm (Eq (2)) does not suffer from missing detection events in case of subsequent photon arrivals at the same detector and operates correctly
with multiphoton arrivals. Here, we will show that differently from the conventional approach,
it operates correctly with modulated intensity signals at any count rates within the dynamical
response range of SPADs and any width NT of the temporal window of interest.
An inherent limitation of our current prototype detector-array is the off-chip data treatment
implemented with a general purpose digital oscilloscope that enables user-defined Matlab data
processing subroutines. As a result of the relatively slow Matlab interpreter, a data trace of 5
µ s length (N=5000 points at T=1ns resolution) requires 75 ms of computations to calculate
the second-order correlation function in 200 points (in the interval −100ns < τ < 100ns). To
achieve a low standard variation of the processed g̃(2) data, about M=106 such data traces needs
to be acquired and treated. The undesirable latency of this numerical processing step could be
reducesd dramatically by integrating our robust algorithm into the chip.
3.
Multiphoton correlation measurements
The performance of our g(2) imaging device (Fig. 1) was tested by measuring the distributions
g(2) (xi j , τ ) when the 4×4 SPAD array detector is uniformly illuminated with an incandescent
light bulb (incoherent broadband light source). Imperfections in g(2) imaging with the SPAD
array chip might be caused by several reasons including the concept of monolithic implemen(2)
tation itself. Fig. 2 shows two time-resolved correlation patterns g̃i j (τ ) (left panels) and corre(2)
lation maxima images g̃i j (0) (right panels) acquired with respect to detectors D8 (top panels
, j = 8) and D9 (bottom panels, j = 9) respectively. In the figure, the pairwise correlations
(2)
(2)
g̃i8 (τ ) (g̃i9 (τ )) are ordered according to the detector index i in Fig.1. For an ideal g(2) -imager,
one would expect to measure no correlations (g(2) (xi j , τ ) = 1) and a uniform correlation maxima map for all pairwise detector combinations. However, Fig. 2 reveals a deviation from the
expected distribution.
(2)
Measured second-order auto-correlation function of a detector g̃ii (τ ) shows a large coincidence peak g(2) (0) ≫ 1 at zero time lag [green scale curves in (a)]. In fact, measurements
with one and the same detector at τ = 0 do not represent statistical correlations between field
states projected on two independent detectors in Eq. 1 [21, 22]. Instead, the statistical algorithm in Eq. 2 is applied to one and the same field state projection. The excess correlations at
(a)
g(2)
150
1.2
D0 & D8
D4 & D8
100
log(g(2))
g(2)
100 1.2
g(2)
D8 & D8
D12 & D8
10
1.0
50
0.8
1.0
1
0.8
0
g(2)
1.2
D1 & D8
D5 & D8
D9 & D8
D13 & D8
D2 & D8
D6 & D8
D10 & D8
D14 & D8
D3 & D8
D7 & D8
D11 & D8
D15 & D8
1.0
0.8
g(2)
1.2
1.0
0.8
g(2)
1.2
1.0
0.8
–20 –10
0
Time [ns]
10 20
–20 –10
0
Time [ns]
10 20
(b)
–20 –10
0
Time [ns]
10 20
–20 –10
0
Time [ns]
10 20
(c)
D0
D4
D8
D12
D0
D4
D8
D12
D1
D5
D9
D13
D1
D5
D9
D13
D2
D6
D10
D14
D2
D6
D10
D14
D3
D7
D11
D15
D3
D7
D11
D15
Fig. 2. Second-order correlation functions g(2) (xi j , τ ) of SPAD array pixels (a) and the
maps of correlation maxima g(2) (xi j , 0) [(b) and (c)] measured for incoherent light using
detector D8 [(a) and (b)] or D9 (c) as a reference. The time resolution is 1 ns. The panels
corresponding to individual SPAD pixels are arranged in the same order in which they
appear in the imager pattern and corresponds to SPAD pixel position in Fig.1. Green scale
curves in (a) show autocorrelation curves g(2) (0, τ ) in linear (olive curve, left axis) and
logarithmic (green curve, right axis) scales
zero time-lag can be understood by noting that the bitwise AND-ing of the binary data-streams
(m)
(m)
(m)
in Eq.(2) yields hXi (n) ∧ Xi (n)i = hXi (n)i = µ T for the average detection rate µ and
NT )
= 1/ µ T . In Fig 2, the correlation curves
time window NT . The result is then g̃ii (0) = N(µ(µNT
)2
are acquired at resolution time T =1 ns and the average rate of detections 6 MHz, yielding an
(2)
estimate g̃ j j (0) = 166, in perfect agreement with the acquired data in Fig.2 (olive curve, left
axis).
(2)
In the vicinity of the self-correlation peak, the dead-time bands of g̃ii (τ ) = 0 are due to a
recharge of the detector depletion region after each Geiger pulse. Measurements in our SPAD
array indicate the dead time τD of 12-15 ns (Fig.2 (a), green curve, right axis). At larger time
(2)
lags |τ | > τD , the self-correlation function g̃ii (τ ) again shows an excess of delayed coincidences, exponentially decaying to g(2) = 1. [Note a logarithmic scale of the right axis in (a).]
This time, i.e., the excess of delayed coincidences, is due to the thermal release of minority
carriers from traps in the depletion region, that were captured during the preceding avalanche
current pulse. A casuality relationship between afterpulses and their generating photodetection
events explains the measured increase of delayed coincidences for the incoherent light source
[23, 24]:
pA (|τ |)
(2)
(3)
g̃ii (τ ) = 1 +
,
|τ | > τD ,
(1 + ε )2 µ
pA (t)dt is the probability to detect an afterpulse in the time interval (t,t + dt) and ε =
Rwhere
∞
p
(t)dt
is the overall probability of afterpulsing. We stress that the impact of afterpulsing
τD A
effects on auto-correlation function (3) varies with the incident photon flux and is particularly
pronounced at small count rates µ . Measuring auto-correlation curve (3) for a larger time lag
(not shown in the figures) shows the characteristic decay time of afterpulsing probability 40 ns
with an integral effect ε = 7% at 4 V excess above threshold. The afterpulsing probability ε
is determined by the number of trap defects in the active (n-well) region material, the number
of carriers involved in an avalanche and the recharge time. Its impact on (3) can be drastically
reduced by increasing the deadtime of the detector, e.g. by using a smaller recharge current for
the depletion region. We have also noticed a slight decrease of afterpulsing probability with
excess of the biasing voltage above the avalanche breakdown. In any case, the auto-correlation
measurements, as in (3), do not allow the second-order correlation statistics of a source to be
accessed but can be used to measure the count rate, dead-time, and afterpulsing probability of
SPAD detectors.
The results of pairwise measurements in Fig.2 deviate slightly from the expected cross(2)
correlation function gi j (τ ) = 1 as well. For i 6= j, the axis scale is limited to the range
(2)
(2)
˙
0.8 < gi j < 1.3. In each of the correlation maps in Figs. (b) and (d), one can distinguish discrepancies of two kinds: (i) There are a couple of SPAD pixel-pairs exhibiting a bunching of
Geiger pulses at zero time-lag (red curves), while (ii) the rest of detector pairs exhibit anticorrelations at τ = 0 (black curves).
Thus in Fig.2 (a), which was acquired using SPAD pixel D8 as a reference, detectors D7 and
D9 exhibit correlation maxima g(2) (0) ∼ 1.3 measured at temporal resolution T =1ns. At higher
temporal resolution and shorter integration time (T =100ps), its height increases while the natural FWHM measures 320 ps in width (not shown in the figures). Measuring these spurious
correlations at various excess voltages above the avalanche threshold and incident light intensities, we have not seen any impact. Such behavior attests for the electrical cross-talk origin of
spurious correlations that occur at the SPAD output data lines and have no impact on the detection process. In particular, the detectors D7 and D9 have data line bonding pads on the CMOS
chip adjacent to the wire of SPAD pixel D8. (The bonding wires of SPAD data outputs seen at
the chip edges in Fig.1 are ordered with the SPAD pixel index i.) These wire lines are subjected
to high currents when the logical state changes between 0 (0 V) and 1 (3.3 V). The measured
excess of correlations can thus be attributed to electrical cross-talk between the SPADS pixels
with neighboring bonding wires. This is confirmed in measurements of pairwise correlations
with a reference detector D9 (Fig. 2 (c)) showing the excess of correlations for detector pairs
D8-D9 and D9-D10. The patterns in Fig.2 indicate that there is no direct optical crosstalk [25]
between the adjacent SPADs within the array, which would appear in Figs.2 (b) and (d) as a
cross pattern centered at the reference pixel.
All other SPAD pixel pairs in Fig.2 with larger index difference (|i − j| ≥ 2) have non-nearest
(2)
neighbor bonding wires and show small anti-correlations gi j (0) ∼ 0.85. The anti-bunching dip
of Geiger pulses is of 2 ns width followed by strongly damped oscillations of 4 ns period
(they can be seen in Fig.3 (b)). This feature corresponds to reflections in the 90 cm length
RF cables transmitting photodetection events data to the processing timing electronics (oscilloscope Wavemaster 8600A, LeCroy). In particular, we attribute this to the impedance mismatch
between CMOS output inverters of the SPAD array chip and the 50 Ohm RF cable. Measurements of spurious anti-bunching dips at various illumination power and excess above threshold
conditions have not revealed any variations in the dip height or width, attesting the non-detector
(2),bg
origin of measured g̃i j (τ ) bias.
Spurious correlations and afterpulsing effects have to be taken into account in the photon
coincidence measurements utilizing our SPAD arrays. Starting from the results of Ref. [23] and
(2),bg
spurious background correlations g̃i j (τ ) between Xi and X j data lines, one can conclude that
for incoherent and coherent light sources as well as for thermal, entangled or single photon
sources with coherence time smaller than the detector dead time τD , the measured correlation
function (2) is
(2)
gi j (τ ) − 1
(2),bg
(2)
i 6= j
(4)
+ (g̃i j (τ ) − 1),
g̃i j (τ ) = 1 +
(1 + ε )2
(2)
where gi j is the correlation function for photons that would be measured by ideal detectors.
The second term takes a reduction of the photon bunching peak (anti-bunching dip) due to the
spurious afterpulses of one detector uncorrelated with photon detection events and afterpulses
of another detector. Note that spurious uncorrelated dark counts have the same effect (see also
Eq.5), however for illumination intensity levels used in the experiments discussed here, their
impact is negligible. The third term is the correction due to background correlation caused by
data transmission lines from the SPAD chip to the timing electronics.
We shall stress that in agreement with Eq.(4), there were no variations observed in the impact
of afterpulsing effects on the cross-correlation function with increasing (decreasing) incident
light power. This feature allows the detector’s pairwise correlation measurements (2) to be
used as an efficient tool for accessing the field statistics g(2) (xi j , τ ). Unlike conventional detection methods based on start-stop timing histograms of delayed single photon arrivals, which
are not capable to treat multiple photon arrivals at one detector during the start-stop interval measurements, our approach implements properly normalized multiphoton distribution. As
such it is robust against missing detection events, the impact of Poisson-like distribution decay
∝ exp(−µτ ) (see Ref [18] for details) and intensity modulation. As opposed to standard methods, Eq. 2 permits arbitrary count rates and temporal window of interest and does not require a
statistical hypothesis to normalize g(2) .
4.
Experimental results and discussion
The imager was tested by measuring statistical properties of several light states including multimode coherent state, near field of an extended intensity-modulated thermal light source and
the far field of stationary thermal light source.
4.1. Multimode coherent state
As a model system for a multimode coherent state we use the emission of a He-Ne laser
operating on a fundamental Gaussian mode of the cavity at 633 nm wavelength transition.
The cavity length of L=21 cm assumes a large separation of the longitudinal modes c/2L=710
MHz, such that a small optical gain and narrow linewidth allows only two or three longitudinal
modes to reach the lasing threshold (in function of detuning from the gain line center). No
special provision for stabilization of the cavity length or lasing modes has been made. The
relative phases of modes vary rapidly in time such that no intensity modulation at the intermodal
frequency can be observed in the emitted output radiation.
The output beam of the laser impinging the g(2) imaging detector was attenuated to reduce
the count rate at detectors down to 2 MHz. The temporal resolution of the timing electronics
was set to T =100 ps so as to observe dynamics of lasing modes making a cavity roundtrip in
1.4 ns.
(2)
Fig.3 (a) shows the correlation function g̃5,9 (τ ) [Eq.(2)] measured by two detectors in the
middle of array (detectors D5 and D9) at 30 µ m baseline. The background component due
(2),bg
to a spurious crosstalk at the same detector pair g̃5,9 (τ ), which was measured with an incoherent broadband light source, is shown in (b). The corresponding correlation function of
the multimode laser beam g(2) (x5,9 , τ ) calculated from Eq.(4) after corrections for background
correlations and afterpulsing effects in the detectors is plotted in (c). It can be seen that this
procedure is an efficient tool for removing spurious (anti-) correlations seen as a dip in both the
(2)
(2),bg
measured correlations g̃i j (τ ) and reference background g̃i j (τ ).
According to the Table 1, for each of the modes of the laser, one measures g(2) (τ ) = 1.
However, as evidenced by the measured second order correlation function in Fig.3 (c), when
these modes are simultaneously excited in the lasing spectrum, one can see the effective beat
signal of intensity and phase noise correlations in the modes [30]. The numerical fit with a
cosine function (green curve) reveals the noise beat period of 1.4 ns, that is the time of a
roundtrip in the laser cavity. The amplitude of harmonic oscillations 0.2 indicates that the noise
of different modes is just partially correlated. We stress that no such periodic signal is observed
in the intensity of the output laser beam.
4.2. Intensity-modulated extended thermal light source
As a model system for an extended quasi-monochromatic thermal (chaotic) light source, the
green line of mercury (546 nm) from the emission spectrum of a low-pressure Hg-Ar discharge
lamp (CAL2000, Ocean Optics) is used. The lamp is driven with an original power supply
incorporating a 30-KHz AC source producing 1.8 KV pulses to initiate the discharge and 120V
in a steady state operation conditions. The green line of mercury is filtered with a 10 nm band
pass filter (FL543.5- 10, Thorlabs).
Such a quasimonochromatic chaotic light source exhibits the first order correlations |g(1) | > 0
in the Young’s interference experiment (Fig.4 (a)). It is characterized by Gaussian distribution
of photons in the Glauber P-representation (Plank distribution in n-representation) so as one
should expect measuring second order correlations g(2) (x, τ ) = 1 + q1 |g(1) (x, τ )|2 with q being
the number of photon modes (of equal intensities) reaching the detectors [11]. Here we report
on measurements in the near field of the source such that the number of modes is large. (The
angular width of the source θ ∼1 in Table 1). Therefore, for a detector pair with the baseline
of the order of λ and more, that is practically in all cases, the second order correlation function
should be close to one, indicating no second order coherence.
To measure the second order correlations in the near field of such chaotic light source, a 3
mm size domain of the lamp discharge region was imaged onto the SPAD array at magnification
(a)
(b)
(c)
Fig. 3. Multimode coherent state emitted by a He-Ne laser at 632.8 nm wavelength. (a): Ac(2)
quired second order coherence function g̃5,9 (τ ) [Eq.(2)]. (b): Spurious correlations back(2),bg
ground g5,9
(τ ) measured with the help of an incandescent light bulb. (c): Second order
correlation function of the field g(2) (x5,9 , τ ) corrected for spurious correlations [Eq.(4)].
The laser cavity length is L=21 cm with the cavity roundtrip time of 1.4 ns (the period of
oscillation seen in (c)). The detectors D5 and D9 separated by x5,9 =30 µ m baseline are
used at a temporal resolution of timing electronics T =100ps.
µ
Fig. 4. Extended quasi-monochromatic thermal light source (546nm line of mercury). (a):
Young’s interference fringes indicate phase correlation |g(1) | > 0. (b) and (c): Measured
(2)
second order correlation function g5,9 in the near field of the source obtained after corrections (4) at temporal resolution 100ps (b) and 100ns (c). The oscillations are due to the AC
power supply of the Hg-Ar discharge lamp.
0.44. The image created by a 150 mm focal distance lens was overilluminating the entire SPAD
array such that we were able to test correlations at two pairs of detectors, one in the middle
of the array (SPAD pixels D5 and D9) and one pair with detectors at the opposite array corners (pixels D0 and D15). These detector pairs with baselines of 30 µ m and 158 µ m measured
correlations g(2) (x, τ ) at the points of the lamp discharge that are separated by 68 and 360 µ m
distance, respectively. We were expecting to see no correlations (g(2) ≡ 1), since the correlation
measurements were done between different emission points of the extended source. However,
in both cases, we have observed excess of coincidence counts. Fig.4 details the correlation func(2)
tion g5,9 (τ ) measured at the detector pair in the array center. The correlations measured with
SPAD pixels at the array corners exhibit the same correlation curve with excess of coincidences
(2)
g0,15 (0) = 2.7 (not shown in the figure).
The difference between the usual statistical data analysis and our method, can be understood as follows. The distribution g(2) (τ ) is assumed to be an even function about the origin
with boundary condition g(2) (τ → ±∞) ∼ 1. However, when applied to a given measurement
window of interest that is at τ = ±NT /2, this assumption can be invalid. Consider the high
resolution data in Fig. 4 (b) sampled at T =100ps. Those data, which may be regarded as a
uniformly distributed histogram of two-photon arrival times, can be renormalized to meet the
above boundary condition, viz., g(2) (τ ) = 1 at the edges of the sample window. But because of
the renormalization, g(2) will not have the required value of 2.7.
By limiting the temporal resolution to T =100 ns and increasing the width of the measurement
window to NT =50 µ s, the origin of the excessive correlations can be made visible [Fig. 4 (c)].
The measured second order correlation function just reveals the fact of long-period intensity
oscillations of the lamp at the double frequency of the AC power supply, at about 60 KHz.
Because of these oscillations and the large angular width of the source, the photon bunching at
τ = 0 due to Planck’s statics [13] is not visible. The change in the width of the measurement
window and the temporal resolution (in Figs. (b) and (c) ) has no impact on the measured
excess value g(2) (0) = 2.7, attesting for correct normalization of the correlation function when
multi-photon arrivals are treated with the algorithm (2)-(4). Interestingly, after correction for
the afterpulsing effects at the detectors [ε in Eq.(4)], the minima of g(2) (τ ) function are close
to zero, in agreement with fluorescence lifetime considerations.
Modulation depth of the correlation function can be tailored in a mixed state. Fig. 5 details
second-order correlation measurements for a superposition of incoherent light (as in Fig.2)
and a quasimonochromatic chaotic light beam (as in Fig.4) at various probabilities to detect
photons in chaotic light state. The incandescent lamp is used as a source of incoherent emission,
which is superimposed on mercury green line fluorescence from the intensity-modulated HgAr lamp. The probability P(Hg) that a detected photon has been emitted by mercury atoms, is
given by the intensity ratio IHg /I estimated from the count rates at the detectors. For example,
P(Hg) = 0.38 when IHg = 6.8 kHz, I = 17.8kHz.
One can easily show that for a superposition of mutually uncorrelated photonic states, characterized by partial intensities I (α ) and correlations g(2),α (xi j , τ ), the second order coherence
hI (t) I (t+τ )i
(α )
j
function g(2) (xi j , τ )= hIi (t)ihI
for the overall photon fluxes Ii, j = ∑α Ii, j impinging the dei
j (t)i
tectors assumes the expansion :
(2),α
gi j (τ ) = 1 + ∑(gi j
(2)
α
(α )
(τ ) − 1)
hIi
(α )
ihI j i
hIi ihI j i
(5)
This expression can be simplified in the case of equal count rates of the detectors, yielding
the partial weights hI (α ) i2 /hIi2 for state contributions of g(2),α (τ ) − 1 to the overall excess
(or lack) of coincidences of the correlation function. For example, for the overall rate hIi =
(b)
(a)
τ
τ µ
Fig. 5. Intensity correlations measured for the superposition of incoherent light (from
incandescent lamp) and intensity-modulated quasimonochromatic thermal light (green
546nm line of mercury from Hg-Ar lamp CAL2000) at various relative intensities. Temporal resolution is T =100ns. (a): Second order correlation function g(2) (τ ). (b): Correlation
function maximum g(2) (0) plotted as squared relative intensity of thermal light incident on
detector pair.
(1 + ε )hI (α ) i of photodetection events caused by a process of intensity hI (α ) i and statistics
(2),α
gi j (τ ) accompanied by spurious (incoherent) afterpulses occurring with probability ε , Eq.(5)
predicts a reduction of the bunching peak (or antibunching dip). The second order coherence
(2),α
function of the entire process is then 1 + (gi j (τ ) − 1)/(1 + ε )2 , in agreement with Eq.4.
Along the same lines, the effect of spurious dark counts of detectors can be taken into account
at low count rates hIi, j i.
For the experimental results reported in Fig.5, the incoherent light constituent does not contribute to the excess of correlation (the correlation function g(2),incoh (τ ) ≡ 1). The average intensities at the detectors (SPAD pixels D5 and D9) are equal so as the contribution from the chaotic
(2),Hg
light source of Fig.4 is reduced as 1 + (g5,9 (τ ) − 1)hIHg i2 /hIi2 [Fig.5 (a)]. The maximum
(2)
values of the second order correlations g5,9 (0) − 1 linearly grows with the squared probability
of detecting a thermal light photons hIHg i2 /hIi2 [Fig.5 (b)].
4.3. Quasi-monochromatic thermal light source
So far we have not yet observed the HBT photon bunching in the emission from a quasimonochromatic chaotic light source. To make it visible, we replace the power source of the
lamp and conduct measurements in the far field. In particular, we drive the Hg-Ar lamp bulb (the
same as used in previous experiments) from a DC voltage source. This lamp bulb (CAL-2000B, Ocean Optics) having a cold cathode and U-folded discharge and designed for operation
in the AC regime, also operated well with a DC power supply (160V @ 15 mA). To start the
discharge, we used the original AC power supply of the lamp, which was connected in parallel
with a DC source via a filter (2H inductance) such that the AC supply was gradually turned off,
whereas the DC source was gradually turned on.
As in the experiment of Fig.4, the light emitted by the lamp was transmitted through a 10nmwidth bandpass filter, which keeps only the emission at the green line of mercury (546 nm).
The inherent difficulty is related to the fact that the number q of uncorrelated photon modes
contributing to the intensity interference at SPAD array detectors g(2) (x, τ ) = 1 + q1 |g(1) (x, τ )|2
has to be kept small. (The requirement q ∼ 1 can be also seen from Eq.(5) for a superposition
of q modes of equal intensities.) For a light beam of transversal size w seen from detectors at a
R 2 2k
∼
distance L in the ray cone of angle Θ = x12 /L, the number of photon modes is q = d(2rd
π )2
(b)
(a)
(c)
(d)
(e)
Fig. 6. (a) Experimental setup of the table-top stellar HBT interferometer. (b) Correlations
measured at various detector separation xi j and distance L to the fiber end. The Fresnel pawx
rameter FN = λ Li j is indicated in the panels. (c) Measured (points) and modeled (curves)
second-order correlations in function of detector separation for the model Eq.6 (dashed blue
curve) and Eq. 8 (solid red curve). The inset shows the corresponding near field distributions at the fiber end (indicated with same type ). (d) shows the corresponding modelled
g(2)m ax (r) patterns (top line of panels) and far field patterns (bottom line of panels). (e)
Imaged second-order correlation maxima along the row of the array. Its position in the
g(2) plane is indicated in (d) with green solid lines. It is assumed that g(2) = 2 along the
diagonal. The green line of Hg (546nm) is used. Temporal resolution T = 1 ns.
1
2
12 2
= 16
(π wx
λ L ) . For a SPAD array of a few 100 × 100 µ m size, which defines a
characteristic baseline x12 of detectors, the requirement q ∼ 1 imposes the angular width of
such extended source to be θ = w/L ∼ 10−2 or less (see also Table 1).
To compromise the brightness of the source and its angular width at the detectors, the light
from the lamp bulb was injected into a multimode fiber of 1 m length and core diameter w ∼ 200
µ m [Fig.6(a) ]. The other end of the fiber was used to illuminate the SPAD array in the far field
zone of the fiber end, at a variable distance L (1 - 3 cm) from the fiber such that the whole
4×4 SPAD array is over illuminated. Such extended thermal light source is of the small angular
width w/L=10−2 rad and exhibits first-order correlations g(1) , when the SPAD array is replaced
by Young double-pinhole interferometer as in Fig. 4 (a), which was taken in the same setup
configuration but with the Hg-Ar lamp driven from the AC power supply.
In intensity interference measurements with a quasi monochromatic thermal light source, the
coherence width and Doppler-broadened spectral line have the same impact as in the case of
the Young double slit interferometer. The classical expression for second-order spatio-temporal
correlations for a non-polarized single-mode chaotic light√source are determined by the firstorder correlations [11, 21, 26] with coherence time τc =2 2π ln 2/∆ω due to the inhomogeneous broadening ∆ω (FWHM) of the emission line
1
2
64 (kwΘ)
g(2) (xi j , τ ) = 1 +
1 (1)
g (xi j , τ )
2
2
πw
τ2
1
xi j exp −π 2 ,
= 1 + sinc2
2
λL
τc
(6)
where the second term in the right hand side takes into account the decorrelation effects due to
unpolarized light (the coefficient 1/2), the zero-delay degree of spatial coherence and the Gaussian profile of the delayed first-order correlation function for an inhomogeneously-broadened
line.
Fig.6 (b) shows several traces g(2) (τ ) measured at different distances to the fiber end at
wx
several detector pairs (the Fresnel like parameter FN = λ Li j is indicated for each trace). At
small FN < 0.5 (large distance to detectors and small baseline), the detector counts show a
correlation peak. At FN = 1, it disappears and then at FN ∼ 1.2, it reappears but with a smaller
amplitude of correlation excess. At first sight, such behavior corresponds well to the expected
oscillations of the sinc function in Eq.(6). However measuring correlations at various distances
from the fiber end with several detector pairs and correcting the data for afterpulsing effect
(Fig.6 (c), points), we find large discrepancies with expected dependence (Fig.6 (c), dashed
(2)
blue curve). The difference is outside the instrumental error and the fast growth of gmax at
FN ∼ 1.2 cannot be compensated for with a numerical fit to the model in Eq.(6).
To interpret the experimental results in Fig.6, one shall admit that g(2), being a function of
two variables, can be tailored not only in the time domain, as exemplified in previous sections,
but it can also be tailored via spatial modulation of the near field distribution. This fact is often
omitted in experiments on photon bunching, leading to wrong data interpretation. In the same
way as the time domain variations of intensity (intensity noise) renders invalid the exponential
term of the famous expression (6), the sinc term in that expression is obsolete if the source has
a nonuniform intensity distribution in the near field.
(2)
(1) (1)
(1) (1)
(1)
Using Glauber’ expression g12 = 1 + G12 G21 /G11 G22 [11] with Gi j = hEi E ∗j i for
an extended source with near field intensity distribution I(x, y) and detector pair with
baseline xi j centered at x = 0, after integration over the wavelet contributions E1,2 ∝
1
2
λ L |E0 (x, y)| exp(−ikx12 x/L)dxdy from mutually incoherent elements dxdy of the extended
source, one finds
R
2
τ2
1 I0 (x, y)exp −i 2λπLx xi j dxdy
(2)
R
(7)
g (xi j , τ ) = 1 +
exp −π 2 ,
I0 (x, y)dxdy
2
τc
which is nothing else than the Van Cittert-Zernike theorem applied for visibility of the second order correlations. Here, as in (6), the coefficient 1/2 takes the decorrelation effects due
to unpolarized light into account. We shall stress that only the 1-D Fourier transform of the
near field intensity in the direction of the detector baseline is involved. For a uniform intensity
distribution, integration over the source area from −w/2 to w/2 yields expression (6).
Field distribution in multimode fiber highly depends on excitation conditions and uniformity
of the fiber core material [27, 28, 29]. Due to cylindrical symmetry of the fiber, the intensity distribution has an intensity bump or a dip at the core center. To simplify analysis, we consider only
the main radial harmonic of frequency Ω, assuming the approximation I(r) ∝ 1 + γ cos(Ωr):
γ
γ
2
1
1
π
FN
+
π
FN
+
π
FN
−
sinc
sinc
Ωw
+
sinc
Ωw
2
2
2
2
τ2
1
exp −π 2
g(2) (xi j , τ ) = 1+
1
2
τc
1 + γ sinc( 2 Ωw)
(8)
where γ is the intensity modulation depth and we have used Fresnel parameter FN = wxi j /λ L
to shorten the expression. The solid red curve in Fig.6 (c) shows the result of the numerical fit
with the model (8), reporting γ = −0.3 and spatial frequency Ω = 2.8 π /w. The corresponding
near field (shown in the inset) has a small intensity dip at the center. Note that the difference to
(2)
uniform intensity distribution is remarkable only in the correlation maxima distribution gmax (r)
(Fig.6 (d), top line panels) while it is not visible at all in the far field distribution impinging the
detectors (Fig.6 (d), bottom line panels).
Being limited by the number of acquisition channels, we were able to record simultaneous
correlations between four independent detectors form one row of the SPAD array. In Fig.6
(e), g(2) (xi j , 0) measured along the array row is plotted as a pairwise correlation map g(2) (i, j)
for measured (left panel) and calculated with the models (8) and (6) correlation maxima (two
right panels). In this image map, the spatial oscillations of the coherence factor are clearly
visible. The measured excess of correlations for the corner pixels on the secondary diagonal
corresponds better to the model (8), confirming previous conclusion on impact of the spatial
modulation of the light source.
5.
Conclusion
We have presented a g( 2)-imager built with conventional CMOS technology, which is capable of measuring second-order spatio-temporal correlated photons. We have discussed several
important aspects related to temporal and spatial modulation of the source intensity when conducting HBT correlation measurements with such imager. Such detectors will find various applications. This approach allows the functions g(2n) of other even orders to be implemented as
well.
The application in Sec. 4.3 used a green line emission of mercury in a Hg-Ar discharge
lamp as a reliable source of HBT photons pairs. The lamp was emitting around hundred µ W
of thermal light at the green-line transition, only small fraction of which can be coupled into
a multimode fiber. Yet the photon flux µ emanating from the fiber was quite high, about 1011
photons/sec. If not decorrelations due to the angular width of such source, one would need
detectors and coincidence electronics with integration time T ∼10ps to detect coincidences at
the highest rate Rc ∼ 2µ 2 T of about 1011 photon pairs/sec in the near field of the fiber.
In practice, because of the low collection efficiency of detectors in the far field of the fiber
end, the count and coincidence rates are very low, about µ =100 kHz and Rc =10 Hz respectively, for integration time T =1 ns. Using detectors of large diameter d will not overcome the
problem, yielding a reduction of measured excess of correlations (g(2) (τ ) − 1) by a factor of
∼ w2 d 2 /λ 2 L2 [see Sec.4.3 ], like long integration time T results in a decoherence factor of
τc2 /(τc2 + T 2 ) ∼ τc2 /T 2 [31].
Although the alternative of building a quasi-thermal source by using a He-Ne laser beam impinging a rotating sintered disk offers higher intensity, larger coherence time, and hence much
faster acquisition, we scrupulously avoided this type of source so as not to introduce the undesirable possibility of spurious correlations such that to observe a clear impact of detector
imperfections, non-stationarity and non-uniformity of the field on measured second-order correlations.
It is interesting to stress the impossibility of detecting a normalized correlation function
g(2) (τ ) for entangled biphotons (Table 1) [12]. A state of the art entangled photon source based
on spontaneous parametric down conversion in periodically-poled nonlinear crystal shows
conversion efficiency of 10−7 and is capable of producing bi-photons at rates of 108 photon
pairs/sec per miliwatt of pump power. The principal difference to a thermal light source illuminating detectors is that all generated bi-photon pairs produce coincidence detections, provided
perfect collection efficiency and no decoherence due to the angular width of the source. Technical difficulties arise from low overall power and a very short coherence time of bi-photons
(typically, τc ∼0.1 ps) so as to measure the peak value g(2) (0) = 1 for entangled photons (see
Table 1) one will need detectors with the response time T ∼0.1 ps, which do not exist yet. On
practice, one just measures non-normalized second order correlations G(2) (τ ) with the time lag
τ defined by the optical path difference to two detectors.
Future work will include the development of larger arrays of SPADs, the integration of onchip data processing based on equation (1), and the extension to other g(2) -imaging applications.
Acknowledgements
We would like to acknowledge Christian Depeursinge fabricated the double pinhole for interferometeric measurements. This research was supported, in part, by a grant of the Swiss National
Science Foundation.