Table Of ContentObservation of ultra-broadband, beamlike parametric downconversion
Kevin A. O’Donnell and Alfred B. U’Ren
Divisi´on de F´ısica Aplicada, Centro de Investigaci´on Cient´ıfica y de Educaci´on Superior de Ensenada
Apartado Postal 2732, Ensenada, Baja California, 22800 M´exico
(Dated: February 1, 2008)
Wereportspontaneousparametricdownconversionhavinganunusuallywidespectralbandwidth.
A collinear type-1 phase-matching configuration is employed with degeneracy near the zero group
velocitydispersionfrequency. Withaspectralwidthof1080nmanddegeneratewavelength1885nm,
thesourcealsoemitsahighfluxof3.4×1011s−1W−1 photonpairsconstrainedtoaconeofonly≈2◦
half-angle. Arigoroustheoreticalapproachisdevelopedthatconfirmstheexperimentalobservations.
The source properties are consistent with an ultra-short photon-pair correlation time and, for a
narrowband pump,extremely high-dimensional spectral entanglement.
7
0 PACSnumbers: 42.50.Ar,03.67.-a
0
2
The process of spontaneous parametric downconver- pump was mapped into a yet wider range of downcon-
n
sion (SPDC), in which pump photons split into pairs verted frequencies. A different approach was taken in
a
of signal and idler photons, can exhibit a wide range Ref. 8, which reported a bandwidth of 185nm at center
J
of emission characteristics. In particular, the spectral wavelength812nm. In this casethe pump lightwasnar-
6
1 propertiesvarywidelyinpracticeandareintimatelytied rowband but tightly-focused; the bandwidth arose from
to the phase-matching configuration in the second-order phase matching the wide angular range of pump wave
1 nonlinear crystal employed. The purpose of this Letter vectors.
v is to demonstrate that, when certain conditions related In contrast, here we develop a broadband SPDC
5 to phase matching are met, even a narrowband pump source with a narrowband, weakly-focused pump. In a
0
can produce downconversionwith an exceptionally large periodically-poled crystal, energy conservation and per-
1
bandwidth. These same conditions also lead to a high fect phase matching imply that[9]
1
0 flux of downconverted photons emitted into a narrow,
ω −ω −ω =0 and (1)
7 beam-like angular distribution. p s i
0 The significance of broadband downconversion has ∆k≡kp(ωp)−ks(ωs)−ki(ωi)−kg =0, (2)
/
h been discussed previously. A large bandwidth is con- where the subscripts p, s, andi denote quantities associ-
p sistent with a short correlation time between the signal ated with, respectively, pump, signal, and idler photons.
t- andidlerphotonsofagivenpair[1]. Thistimerepresents Here, ∆k is the wave vector mismatch, ωj (j=p,s,ori)
n the resolution limit in metrology applications, such as is the frequency, k (ω ) is the wave vector within the
a quantum optical coherence tomography[2], that rely on crystal, and k is thje pjoling wave vector.
u g
signal/idler arrival-time differences. Also, for a narrow-
q To produce wideband phase matching, consider the
: bandpump, awide downconversionbandwidth createsa case when all vectors of Eq. (2) are collinear and are re-
v two-photon state having many Schmidt mode pairs or,
placedbycorrespondingscalars. Weassumeamonochro-
i
X equivalently, high-dimensional spectral entanglement[3]. matic pump with fixed ω , but allow ω and ω to vary
p s i
Many modes implies a large mutual information, which
r fromthedegeneratefrequencyωd ≡ωp/2asωs=ωd+δωs
a is a measure of the information that can, in principle, and ω =ω +δω . Expanding ∆k into a power series in
i d i
be shared by two parties through spectral photon pair
δω and δω , it follows that
s i
correlations[4]. Further, Schmidt modes can be occu-
∞
piedindependently,sothathavingmanyavailablemodes
∆k = [(−1)n+1k(n)(ω )−k(n)(ω )]δω n/n!, (3)
means that individual photon pair behavior can be pre- i d s d s
X
served at high light levels. Such behavior has been n=1
demonstrated with a two-photon process that is linear where the superscripts (n) denote nth frequency deriva-
instead of quadratic in its power-dependence[5]. tives,k hasbeenchosentoprovideperfectphasematch-
g
There have been previous experimental demonstra- ingatdegeneracy,andithasbeennotedthatEq.(1)im-
tions of broadband SPDC, where bandwidths of tens of plies that δωi = −δωs. In what follows, we broaden the
nm are not uncommon. For example, in Ref. 5 a band- bandwidthoverwhich|∆k|issmallbyremovingtermsof
width (defined throughout as full width at half maxi- Eq. (3) of increasing order in δωs. First, for a type 1 in-
mum) of 31nm was reported for periodically-poled bulk teraction,ks(ωs)andki(ωi)areassociatedwiththesame
KTP, while Ref. 6 obtained a bandwidth of ≈ 40nm crystal index, so all terms having odd n in Eq. (3) van-
with a KTP waveguide, with both results being in the ish. Then, the n=2 term of Eq. (3) may be eliminated
(2)
near infrared. Elsewhere, a downconversion bandwidth by choosing ω so that k (ω ) = 0 for the particular
d s,i d
of ≈ 115nm was reported near a mean wavelength of crystal,which is the conditionofzerogroupvelocity dis-
840nm[7]. There, the frequency spread of a broadband persion. The series does not then begin until the fourth
2
order in δω , implying a wide emitting bandwidth. It is sistentwithbroadbandphasematching. Thusthecrystal
s
notable that these conditions can only occur at degener- wasinsteadoperatedatroomtemperatureandthe spec-
acy;otherwise,the nthtermofEq.(3)containsks(n) and trumwaswidth-optimizedbyvaryingλp andΛg. Indeed,
k(n) evaluated at different frequencies, and term cancel- λp and Λg as will be quoted in the experimental results
laitions for n=1,2, and 3 are not possible. differ slightly from λˆp and Λˆg calculated earlier. These
differences may be attributed to optimization of phase
We employ periodically-poled lithium niobate, with
matching throughout the limiting aperture of Fig. 1.
pump, signal, and idler waves polarized along the z-
With λ =942.5nm and Λ =27.4µm, Fig. 2(a) shows
axis and propagation along the x-axis of the crystal. p g
measured spectra for several values of the limiting aper-
Using a Sellmeier equation for the extraordinary re-
ture angular radius ∆θ. For small ∆θ, the spectrum
fractive index[10], it is straightforward to calculate the
rapidly rises with increasing ∆θ, while the curves for
temperature-dependent ωd for which ks(2,i)(ωd) = 0. At ∆θ = 1.5◦, 2.0◦, and 2.5◦ are of similar height. Most
◦
20 C, this condition corresponds to a pump wavelength
significantly, the spectra are remarkably broad. For
λˆ =2πc/ω =958.7nm, which in turn implies a poling
p p the three larger ∆θ of Fig. 2(a) the frequency band-
period Λˆg=2π/kg=28.1µm from Eq. (2) at degeneracy. width is 0.53ω¯, where ω¯ is the average frequency; in
Thus, cancellation of all terms of Eq. (3) for n < 4 is terms of wavelength, this corresponds to an unprece-
consistent with reasonable experimental parameters. dented 1080nm bandwidth. The spectra are normalized
In experiments, the pump was a narrowband, in units of photon rate per unit frequency per watt of
◦
continuous-wave, Ti:sapphire laser having ≈1 W pump power. Integration of the spectrum for ∆θ=2.5
power[11] with a birefringent filter added to tune over implies a photon pair rate of 3.4 × 1011s−1W−1, or a
910−980nm. As indicated in Fig. 1, lenses focused the SPDC power of 3.6 × 10−8 times the incident power.
output beam to a wide 110µm waist inside the L=1cm These values (and, for that matter, the spectra) are in-
long crystal, which had poling periods Λ in steps be- ternal quantities within the nonlinear crystal, which are
g
tween 25.5µm and 28.7µm at 8 positions along its en- inferred by applying calculated losses to external mea-
tranceface. The lightexiting the crystalwasincidenton surements.
a Si Brewster plate that absorbed the pump but trans- In other data, we have found that for shorter λ the
p
mitted the downconverted flux. This flux then passed SPDC power falls; for λ ≤ 939nm the spectra fall to
p
through a limiting aperture and was focused by a CaF ≤10%ofthelevelsofFig.2(a). Forλ >942.5nmthereis
2 p
lens into a spectrometer, which employed a highly ef- still significant SPDC power,but the spectra sag for fre-
ficient Littrow grating and a sensitive cooled InSb de- quenciesnear the center ofFig.2(a). Thus, while results
tector. The detector signal was processed by a lock-in for other λ are not shown here, the results of Fig. 2(a)
p
amplifier. with λ =942.5nm may be considered near-optimal.
p
It was verified that if either the pump or detected po- To calculate the SPDC photon number spectrum
◦
larization was rotated by 90 , the detector signal fell
to low levels as expected. Even at room temperature,
there wasnoevidence ofphotorefractivecrystaldamage. λs(µm) λs(µm)
3.0 2.4 2.0 1.6 1.4 3.0 2.4 2.0 1.6 1.4
It was also found that temperature-tuning the crystal
(a) (b)
could flood the spectrometer with blackbody radiation 7
but only weakly affected the SPDC spectrum, as is con-
6
Hz5 u.)
a.
W4 ) (s
ω
TS s3 R(1
0
12
C
1
L1
WPL2 LN BP L3 SP 0.6 0.8 1.0 1.2 1.4 0.6 0.8 1.0 1.2 1.4
ωs (1015 Hz) ωs (1015 Hz)
FIG. 2: (a) Experimental downconversion spectra for a
FIG. 1: Broadband downconversion experiment. The 1 cm lithium niobate crystal with poling period 27.4µm,
Ti:sapphire laser (TS) beam ischopped(C) and focused into pumpedatwavelength942.5nm. Casesshowcollection angle
alithiumniobatecrystalLNbylensesL1andL2. Ahalf-wave ∆θ=0.5◦ (squares), ∆θ=1.0◦ (inverted triangles), ∆θ=1.5◦
plateWProtatesthepumppolarization. ASiBrewsterplate (circles), ∆θ =2.0◦ (triangles), ∆θ =2.5◦ (diamonds). (b)
(BP, with plate normal directed out of figure plane) isolates Corresponding theoretical spectra for 942.5nm, convolved
the downconversion, and the emission half-angle subtended overthespectrometerslit integration width. Horizontallines
bythelimiting apertureafter BPdefines∆θ. TheCaF2 lens denote these ωs-dependent widths; vertical lines denote the
L3 focuses thelight into thespectrometer SP. degenerate frequency.
3
◦
R (ω ), we develop a theoretical approach based on Fig. 2(a) is excellent. Theoretical spectra for ∆θ>2.5
1 s
Ref. 12. It follows that are not shown, but are nearly identical to those for
∆θ=2.0◦ and 2.5◦; thus essentially all flux is restricted
z0 ◦
ǫ z to a cone of ≈ 2 half-angle. This compact, beam-
R (ω )= 0 dx′ dy′ dz′hIˆ(r′, 0)i (4)
1 s 2δt~ω Z Z Z c like angular emission is a novel property that, in future
s
z0−cδt work, could be essential in obtaining high free-space or
fiber-coupled collection efficiencies. There is also near-
where Iˆ(r,t)≡Eˆ(−)(r,t)Eˆ(+)(r,t) is the intensity oper- symmetry about the degenerate frequency in Fig. 2(b)
ator, h·i indicates an expectation value with respect to for ∆θ=2.0◦ and2.5◦. For a monochromaticpump, this
the two-photon state, z is the distance along the prop- symmetry is a consequence of photon energy conserva-
0
agationaxis from the crystalto the detector, r′ refers to tionifallemittedfluxiscollected. Forthemostpart,the
coordinates outside the crystal, and cδt is the length of symmetry is present in the analogous data of Fig. 2(a),
the volume detected in time δt. In the following θs,i and although there is a slight decrease at ωs≈0.8×1015Hz
φs,i denote, respectively, the polar and azimuthal inter- that could arise from light falling outside the collection
nal propagation angles of the signal/idler waves. For a aperture. Wenotethatrelatedcalculationsnotpresented
Gaussian pump beam with waist w at the crystal mid- here predict the signal/idler correlation time to be only
0
point, R1(ωs) may be written as 4.9fs on the optic axis for our experimental parameters
(λ =942.5nm and Λ =27.4µm).
p g
R (ω )∝ ωsks2ks(1) πdθ sinθ 2πdφ d3k (ω /n2) In conclusion, we have produced downconverted light
1 s n2s s s s i i i having an unprecedented bandwidth of 1080nm at a
R0 R0 R
central wavelength of 1885nm. The wideband phase-
×η (θ ,φ )|g(ω ,θ ,φ ;ω ,θ ,φ )|2δ(ω −ω −ω )(5)
r s s s s s i i i p s i
matching conditions occur for a type 1 process with
where the ω -dependence of k is now implicit, d3k = collineardownconversionatthe degenerate frequency ωd
s,i s,i i
okni2lkyi(1w)istihniθnidthωeidliθmiditφini,gaanpderηtur(rθes(,tφhse)pishyosficuanliatpheretiguhret wpahierns sk−s(12,i)W(ω−d1))≈is0a.ssAocliaartgeedflwuixthotfhpisholatorgnes b(3a.n4d×wi1d0t1h1,
function η(θ ,φ ) is mapped to the internal η (θ ,φ ) andtheangle-dependenceofthephasemismatchrestricts
s s r s s
◦
by refraction at the crystal exit face). The function g the light to an easily-collected cone of half-angle ≈2 .
represents the joint amplitude[13] Theexperimentaldatapresentedexhibitexcellentagree-
ment in direct comparisonwith theoretical photon num-
sin(βL) ber spectra. Even though our broadband conditions oc-
g(ωs,θs,φs;ωi,θi,φi)=exp[−(k⊥w0/2)2] (βL) (6) cur only at a particular pump frequency ωp for a given
crystal, in principle it is possible to employ metamate-
where β =k⊥2/(4kp)−∆k/2,and k⊥ and∆k now follow rials having dispersion designed to produce such effects
from at any ω desired. The generation of photon pair states
p
having ultra-short correlation time and an exceptionally
k⊥2 =ks2sin2θs+ki2sin2θi+ high-dimensional spectral entanglement may, in the fu-
2k k sinθ sinθ cos(φ −φ ) ture, be of key importance for many applications.
s i s i s i
and ∆k =k −k cosθ −k cosθ −k . (7)
p s s i i g
In numerical integration of Eq. 5 we ignore the Acknowledgments
(θ ,φ )-dependence of k itself, which is a good ap-
s,i s,i s,i
proximation for propagation near the pump direction. We are grateful for discussions with R. S. Cudney, H.
The theoretical spectra are shown in Fig. 2(b) and, with F. Alonso, S. Gu¨nther, and A. Hemmerich. This work
parameters as in the experiments, the agreement with was supported by Conacyt grants 46370-Fand 49570-F.
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