SUPPLEMENTARY INFORMATION
DOI: 10.1038/NPHOTON.2014.40
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Supplementary Information
S1. Theory of TPQI in a lossy directional coupler
Following Barnett, et al. [
24], we start with the probability of detecting one photon in each
output of a lossy, symmetric beam splitter in a Hong-
Ou-Mandel measurement,
(1)
where
and
label the two outputs of the splitter,
and
are the complex transmission and reflection
coefficients that characterize it,
and
is a quantity between zero and one that describes the overlap of
the photons at the splitter. In this notation,
and
are the fractions of power transmitted and
reflected at the splitter, respectively,
and
and
are the phases of the transmitted and
reflected waves with respect to the incident wave. For indistinguishable photons arriving
simultaneously,
, while delaying the arrival of one by much more than its coherence time
gives
. Because the beam splitter is lossy, these quantities obey the inequality
, but not
the corresponding e
quality.
We can apply this equation directly to the case of a lossy directional coupler (as distinct from a
beam splitter) if we can calculate
and
for a given coupling length. To do so, we first represent the
field amplitudes in a pair of coupled waveguides using a two-component vector, where each component
represents the field amplitude in one of the two waveguides. Using this simplified notation, we can
write the even and odd supermodes of the coupled waveguides (see Fig. S1) as follows:
(2)
(3)
Here,
represents the mode of a single, isolated waveguide. The transverse spatial arguments are
unimportant and have been suppressed. The labels
and
refer to the symmetric and antisymmetric
modes, respectively, and
and
are the real and imaginary parts of the effective index of each mode.
The wavevector,
, is determined by the free-space wavelength,
, and
is the direction
parallel to the axis of both waveguides.
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DOI: 10.1038/NPHOTON
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Figure S1: Even (left) and odd (right) supermodes of the DLSPPW directional coupler. The vertical
component of the electric field is plotted
as a function
of position, and the direction of propagation is
into the page.
If both supermodes are excit
ed equally, the field amplitude vector
is
(4)
where
is the difference between the real parts of the effective indices of the two
supermodes, and
is the difference between their imaginary parts. While
measures the
strength of the coupling between the two waveguides, the physical significan
ce of
is not yet
intuitively clear.
Note that the amplitude vector is simply
at
, so all of the energy
starts in the first waveguide. The magnitude of the starting field,
, is important for
normalization
.
For an equal
splitting r atio, we choose
such that
. This choice ensures that the
field
amplitudes in the two waveguides (i.e. the top and botto
m components of the vector in (4)
) are equal in
magnitude.
The result is
(5)
where we have omitted the irrelevant overall phase factor,
. Dividing by
for normalization, we
end up with the complex transmission and reflection coefficients,
and
, which are the top and bottom
components of this vector
, respectively:
(6)
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(7)
Note that in the lossless case, we have
,
, and
, as expected.
Moreover, the phase difference between the transmitted and reflected waves is
for the lossless coupler
—
and even for a lossy coupler with
—
which is necessary for perfect TPQI.
In contrast, a lossy coupler with
will impart a different relative phase to the two waves, resulting
in the reduced visibility of TPQI.
To make these last observations quantitative, we substitute (6) and (7) into (1). Writing
for the magnitude of
and
and
for the
phase difference between them, with
, we have
(8)
The theoretical
maximum
visibility of TPQI is then:
(9)
This expression confirms that a lossy directional coupler with
, and therefore
, cannot give
perfect visibility of TPQI.
Fortunately for our experiment,
in our DLSPPW couplers is small relative to
, so this effect
is small. Using a commercial finite-difference frequency-domain mode solver, we calculate
,
, and
, giving a maximum visibility of
0.999.
S2. Frequency response of plasmonic directional coupler and inverted Gaussian fit
Because the bandwidth of the down-converted beams (5 nm, set by the filters) is much larger
than the bandwidth of the pump laser (< 1 nm), the pairs of photons we create are entangled by
frequency, as was also the case in the original paper by Hong, Ou, and Mandel [4]. There the authors
show that such an input (eq. 3) gives precisely the result that we observe when the filters that
determine the spectra of the down-converted photons are Gaussian, as is the case in the present
measurement.
Unlike the original HOM experiment, however, this experiment uses a plasmonic directional
coupler in lieu of a beam splitter. In principle, the splitting ratio of our coupler and the phase difference
between the transmitted and reflected waves both depend on frequency, and this dependence could
play a role in the theory of our measurement. If that were so, the frequency correlations between
photons might thereby become important. Our calculations suggest that this is not the case, however.
From the calculations described in Section S1, we find that the difference between the real parts of the
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refractive indices of the symmetric and antisymmetric supermodes of our coupler changes by less than
+/- 1% over the range 811
-817 nm. That is, the quantity
in eq. 4 above deviates from
by less
than 1% over this range, causing the complex transmission and reflection coefficients, as well as the
phase difference between the transmitted and reflected beams, to change by less than 1% as well.
The
direct effect on the visibility of the in
terference is negligible (less than one part in 1000), so we conclude
that any indirect effect based on the frequency correlations of the input photons is probably also small.
In addition,
the shape of the HOM dip depend
s on the biphoton spectral density
—
and hence on
frequency correlations between the photons
—
in the general case. In our case, however, the fact that
our 5 nm filters are identical, approximately Gaussian, and centered at the degenerate wavelength (814
nm) allows us to fit a simple inverted Ga
ussian function to the data.
In the original
HOM paper
[4]
(specifically eq. 11 and the a
nalysis leading up to it),
the authors show that an inverted Gaussian fit is
appropriate
.
S3. Estimated losses in plasmonic waveguides
To determine the propagation length of the plasmonic mode in our DLSPPWs, we fabricated
10,
20, and 30 μm long waveguides and measured transmission through them using an 800 nm diode laser.
The results are shown in Figure S2.
The exponential fit gives a 1/
e decay length of 6.8 μm, while the
overall transmission through the 10 μm DLSPPW was 3.2%, roughly a factor of ten lower than the 30
-
35% transmission we observed through dielectric waveguides without plasmonic components. From
these two numbers
we estimat
ed that the coupling efficiency between the plasmonic and d
ielectric
waveguides was approximately
0.66 per transition.
Figure S2: Measurements of transmission through DLSPPWs of different lengths.
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S4. Raw TPQI data and normalization
The raw TPQI data from Figure 3 of the main text are shown in Figure 1. Over the course of
these measurements the in-coupling optics drifted slightly out of alignment, reducing the number of
photon pairs counted at later times (longer delay settings) compared to earlier times. To correct for this
artifact and to allow direct comparison between the two measureme
nts , we fit a line to the data points
far from the TPQI dip in each case and divided each data set by its respective background line.
Figure S3: Raw TPQI data for the dielectric (left) and plasmonic (right) 50-50 couplers.
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