Negative frequency resonant radiation

0.31 MB·English

by  E. Rubino

Negative frequency resonant radiation.
E. Rubino,1,∗ J. McLenaghan,2,∗ S. C. Kehr,2 F. Belgiorno,3, D. Townsend,4
S. Rohr,2 C.E. Kuklewicz,4 U. Leonhardt,2,∗ F. K¨onig,2,∗ and D. Faccio1,4,∗
1 Dipartimento di Scienza e Alta Tecnologia, Universita` dellInsubria, Via Valleggio 11, IT-22100 Como, Italy
2 School of Physics and Astronomy, SUPA, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK
3 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo 32,20133 Milano, Italy
4 School of Engineering and Physical Sciences, SUPA,
Heriot-Watt University, Edinburgh EH14 4AS, UK ∗
(Dated: January 16, 2012)
Optical solitons or soliton-like states shed light to blue-shifted frequencies through a resonant
2
emission process. Wepredictamechanism bywhichasecond propagatingmodeisgenerated. This
1
mode, called negative resonant radiation originates from the coupling of the soliton mode to the
0
2 negativefrequencybranchofthedispersionrelation. Measurementsinbothbulkmediaandphotonic
crystal fibresconfirm ourpredictions.
n
a
J
Introduction - Resonantradiation(RR),oftenalsore- and are associated with positive frequencies only [11].

2
ferred to as dispersive-wave or Cherenkov radiation is a A process such as that highlighted here, that mixes
1
nonlinear optical process by which a soliton propagat- positive and negative frequencies will therefore change

] ing in an optical fibre in the presence of higher-order the number of photons, leading to amplification or even
s dispersion sheds light through a resonant-like process to particle creation from the quantum vacuum [12, 13].
c
i a shifted frequency [1–5]. This process and the precise In this work we show how alongside the usual resonant
t
p frequencyofthe RRis determinedbyawave-vectorcon- radiation spectral peak observed in many experiments,
o servation relation, a second, further blue-shifted peak is also predicted.
.
s This new peak may be explained as the result of
c k(ωRR)−kωIN−(ωRR−ωIN)/v−KNL =0 (1) the excitation of radiation that lies on the negative
i
s frequency branch of the dispersion relation. We first
y where ωIN and ωRR are the soliton and RR frequencies,
explain why this radiation should be observed and then
h v is the soliton velocity and KNL = ωINn2I/c is a
provide experimental evidence of what we call “negative
p nonlinear correction term that may be small or even
[ negligible atlow intensities, I [5]. A verysimilar process frequency resonant radiation” in both bulk media and

1 occurs also in bulk media. The stationary 1D fibre photonic crystal fibres.
Theory - In order to show how the negative frequency
v soliton is now replaced by the stationary 3-dimensional
RR arises, we consider without any loss of generality a
9 X-wave [6]. X-waves may form spontaneously in Kerr
8 basicdispersionrelationthatcontainshigherorderterms
media at high enough powers in much the same way
6 such as in fused silica glass and shown in Fig. 1(a). The
that solitons form spontaneously in a fibre [7, 8]. A
2 dashed curves indicate the negative frequency branches.
. blue-shiftedpeakwillalsobeobservedthatwillformone
1 ofthe twoX-wavetails: the wholeX-wave,including the The phase-matching relation (1) (with kNL = 0) is a
0 straight line that intersects the dispersion relation in a
RR is therefore described by Eq. (1) [9], which indeed
2
number of points that define the allowed modes. The
1 reflects the non-dispersive nature of the wave-packet
point indicated with IN is simply the input mode, or
: considered, i.e. the soliton in 1D and the X-wave in 3D.
v soliton mode, and B indicates a backward propagating
These frequency conversion processes may be under-
i
X stoodin terms ofenergytransfer betweenspecific modes modethatisnotexcitedintheexperimentsandisthere-
fore neglected. There is then a third positive frequency
r identified by Eq. (1) and the dispersion curve [4, 5, 9].
a mode, RR, that indicates the resonant radiation mode.
However, to date only the positive frequency branch of
However, there is also a negative frequency mode, N’,
the dispersion has been considered when this actually
equally predicted by Eq. (1) yet always neglected. The
also has a branch at negative frequencies. This branch
object of this work is precisely this negative frequency
is usually neglected or even considered meaningless
branch mode. All these various modes are easier to
when, in reality, it may host mode conversion to a new
visualise in the comoving reference frame coordinates
frequency. The fact that a mode on the negative branch
as shown in Fig. 1(b). These curves are obtained from
of the dispersion relation may be excited has a number
the original dispersion relation by transforming to
of important implications, beyond the simple curiosity
the comoving coordinate system ω′ = γ(ω − vk) and
of the effect in itself. Indeed, light always oscillates
with both positive and negative frequencies, but the k′ = γ(k−ωv/c2), with γ = 1/p1−v2/c2 and v = vg
is the soliton group velocity. Transforming also Eq. (1)
negative-frequency part is directly related to its positive
to the comoving frame using the same relations, gives
counterpart and seems redundant [10]. On the other
′ ′
ω = ω . In other words, momentum conservation in
hand, light particles, photons, have positive energies IN
2
in the spectrum with a higher frequency than the RR
(a)
mode. In analogy with the usual positive frequency
RR, we callthis new mode “negativefrequency resonant
RR
IN radiation”.
B
We note that the fact that the negative RR mode is a
solution to Eq. (1) does not, alone, imply that it will
k
’ actually be excited. The negative RR mode lies on a
N
dispersion curve that is separate from the positive mode
branchofthe INandRRmodesanditisnotguaranteed
’ that energy may be transferred between the two. A
(b)
similarity can be sought for example with higher-order
N’ ! ’ RR spatial modes in a waveguide that also lie on separate
IN dispersion curves. In the presence of an adiabatic
B IN
(spatial) variation of the waveguide there will be no
’ energy exchange between the various dispersion curves
k
" ’ and higher order modes will not be excited. Conversely,
IN N
any sudden changes in the waveguide geometry will lead
to an energy exchange between the modes. Similarly,
in our situation any adiabatic or smooth (temporal)
FIG.1: Typicaldispersionrelation,e.g. forfusedsilicaglass, perturbation of the medium will not excite the negative
in the laboratory reference frame (a) and in the reference RR mode: a non-adiabatic variation within the input
frame comoving at the soliton velocity (b). Dashed curves pumppulse,e.g. asteepshockfront,isrequired. Indeed,
indicate the (laboratory frame) negative frequency branches preliminary numerical simulations albeit in simplified
of the dispersion relation.
setting (see e.g. [14]) do indicate that the actual
intensity of the negative mode depends critically on the
steepness of the refractive index variation induced by
the laboratory reference frame corresponds to energy the nonlinear Kerr effect.
conservation in the comoving reference frame. The al- Finally, we note that in the comoving frame both the
lowedmodesarethereforenowfoundbysimplytracinga RR mode and the negative RR mode propagate with
horizontal line through the input soliton mode (that by negative group velocities (as can be deduced from the
definition,haszerogroupvelocityinthecomovingframe slopeofthe dispersioncurveatthese frequencies),i.e. in
and thus lies at a local minimum) and as before, looking the backward direction. The phase velocities of the two
for the intersections with the dispersion relation. The modes are however opposite to each other. Conversely,
main point here is that the dispersioncurves tell us that in the lab. frame both the RR and the N modes have
it should be possible, starting from two positive modes, positive phase and group velocities, i.e. they both
IN and RR, to excite a third negative, N’, mode. When propagate in the forward direction.
trying to assign a physical meaning to the negative Experiments - We performed two sets of experiments
frequency mode we should recall that in reality any in order to capture the formation of the negative RR
electromagnetic field is a real-valued quantity that can mode: (i) in a bulk medium and (ii) in a few mm long
be written as a sum of a complex term with its complex photonic crystal fibre.
coniugate: E ∼ cosωt = exp(+iωt) + exp(−iωt). In the first experiment, we chose a 2 cm long bulk
However, considering only the modes obtained from the calcium flouride, CaF2 sample as host material. Light
intersections with ω′ = +ω′ , amounts to considering pulses of 60 fs duration and 800 nm carrier wavelength
IN
only the first complex term and neglecting the com- are provided by an amplified Ti:sapphire laser system of
plex coniugate. In order to recover the full field we 1 kHz repetition rate. We reshape the pulses into Bessel
obviously need to also sum the modes obtained from beamswith aconeangle(inthe medium) ofθ =0.6deg,
the intersections with ω′ = −ω′ : the sum of N’ and using a conical lens of fused silica with 2 deg base angle.
IN
N in Fig. 1(b) therefore will give a real-valued field The Bessel pulse in the sample moves with uniform
with a positive frequency in the laboratory reference speed v = vg/cosθ where vg denotes the group velocity
frame. Nevertheless, as explained above, the origin of of a Gaussian pulse of carrier wavelength 800 nm. The
this mode lies in the coupling of one or more modes on spectrumattheoutputfromthesampleiscollectedwith
the positive frequency branch of the dispersion relation alensandafibre-basedspectrometer. Afilterwithaflat
to a mode that lies on the negative frequency branch. response in the visible-near-UV region is placed before
We alsonote thatthe negative mode has a truly distinct the spectrometer in order to reduce the input pump
frequency from all the other modes in Fig. 1 and, if it intensity without affecting the shape of the spectrum
is generated, it should appear as a clearly distinct peak between 300 nm and 720 nm. The input pulse energy is
3
characterised by two distinct peaks in the spectrum
(a) 16 !J
that do not shift with increasing energy. The first peak
is located around 620 nm, the second is much weaker
0.3 0.5 0.7 and is located around 341 nm wavelength. Examples
x0.2 (!m)
) of these spectra (15 and 16 µJ) are shaded in red in
s
t
ni the figure. At higher input energies the pulse starts to
b. u 50 !J develop complex dynamics, typical of the filamentation
ar regime during which the pulse breaks up and creates a
(
y broad-band and highly structured spectrum known as
t 29 !J
densi N RR 0 wfohriteex-laimghptlesuopnerctohnetinspueucmtru[1m7].mWeaesfuorceuds ofourr aatnteinntpiount
ral 16 !J energy of 16 µJ: the spectrum is not substantially
t
ec 15 !J modified if we account for the filter response, as shown
p
S in the inset to Fig. 2(a). Three clear peaks are indicated
13 !J with λ0, λRR and λN and we indentify these with the
IN, RR and negative RR modes, respectively. Indeed,
12.8 !J these correspond exactly to the positions for the RR
12 !J and negative RR modes given the IN mode and the
0.3 0.4 0.5 0.6 0.7 0.8 dispersion relation for CaF2 [16], as shown in Fig. 2(b).
(!m) We note that attempts to generate similar features
(b)
40 in other glasses or media, e.g. BK7, fused silica and
Hz) $#’0 RR 0 water, failed. Spectral broadening through self-phase
T modulation and the steepness of Kerr-induced shock
d 0 20x fronts are both strongly limited by dispersion. Our
a
’ #(r N "#’0 eshxopwereidmtehnatst ienvefnusaetdtshielichaigahnedstwinapteurt(ednaetragineso,tsspheocwtrna)l
-40 broadening exhibited a sharp cut-off around ∼ 450 nm,
0.3 0.4 0.5 0.6 0.7 0.8
(!m) whereasthenegativeRRpeakwas,inallcases,predicted
to appear at shorter wavelengths. On the other hand
FIG. 2: Experimental results for negative RR generation in CaF2 (as other fluoride glasses) is quite unique as it
bulkCaF2. (a)showsthemeasuredspectraforincreasingin- exhibits significantly lower dispersion [16], in particular
put energies (indicated next to each curve). The spectra are in the UV spectral region and thus allows the formation
vertically displaced to increase visibility. The inset shows a of steeper shock fronts and broader continua. In this
samplespectrum(16µJinputenergy)corrected forthefilter
specific case it allows a relatively efficient excitation of
response. (b) shows the CaF2 dispersive relation in the co- the negative RR peak in the UV.
moving frequency versus lab. frame wavelength coordinates.
Inasecondexperiment,wesent7fslightpulses,centred
PositionsofthepredictedRRandnegativeRRspectralpeaks
are indicated. The inset is a 20x enlargement of the curve around 800 nm, with a 77 MHz repetition rate into a
around the λRR wavelength. fusedsilicaphotonic-crystalfibre. Photoniccrystalfibres
have the advantage of enhanced nonlinear effects due
to tight mode confinement combined with a remarkable
varied from 10 µJ up to 50 µJ at which point the input flexibility in tailoring the waveguide dispersion that
pulse is in a strongly nonlinear regime and develops can therefore strongly modify the corresponding bulk
a complex and structured spectrum. Generation of medium dispersion and thus allow observation and
negative RR modes is observed at intermediate energies control of a variety of novel effects (see e.g. [5] for an
∼ 15 µJ. Examples of the resulting spectra for varying extensive review). We selected fibres where the spec-
input energies are shown in Fig. 2(a). The spectra are trum of the incident light lies in a region of anomalous
vertically displaced in order to render them visible. At group-velocity dispersion such that it can propagate as
lower energies (12-14 µJ) the output spectrum shows a a soliton-like pulse. The 7 fs input pulses are coupled
distinct single peak that shifts to shorter wavelengths into the fiber using a 90 deg off-axis parabolic mirror.
with increasing input energy. This process has been We estimate the coupling efficiency to be 20%. In a pre-
described in detail [15] in similar conditions and is a liminary experiment we confirmed that most of the RR
direct manifestation of the formation of a steep shock emergedusingonlyafew millimetersoffiber. Therefore,
front on the trailing edge of the pump pulse. As energy short pieces of fiber of approximately 4-5 mm are used.
is increased the shock front steepens and the spectral Figure 3 shows the output spectrum after 5 mm of
peak shifts towards shorter wavelengths. Between 15 fibre for three different input energies (246, 324 and 366
and ∼ 20 µJ input energy a different regime sets in, pJ). The ultraviolet part of the spectrum, Fig. 3(a) was
4
)
s
unit (a) N (b) RR Fibre λRR λN pred. λN meas.
b.
r NL-1.6 615a 542 nm 233.4 nm 233.1 nm
a
(
ty NL-1.6 615b 542 nm 233.3 nm 232.1 nm
si
n NL-1.5 590 516 nm 228.7 nm 227.0 nm
e
d
t. NL-1.5 670a 478 nm 221.4 nm 218.1 nm
c
pe 220 225 230 235 450 500 550 NL-1.5 670b 480 nm 221.8 nm 218.9 nm
S (nm) (nm)
120 TABLE I: Predictions of λRR and predictions and measured
) (c) RR 0 valuesfor λN for thethreefibres used in theexperiment.
z
H
T 25x
d 0
a
r
(
’! one of the two orthogonal polarization-maintaining
-120 N axes of the fibre. The table lists the wavelength of
the negative RR emission predicted from the relative
200 400 600 800
(nm) dispersion relations compared to the actually measured
wavelengths: as can be seen, very good agreement is
FIG.3: ExperimentalresultsfornegativeRRgenerationina obtained in a variety of settings.
photoniccrystalfibre. (a)and(b)showthemeasuredspectra Conclusion - Frequency conversion through a resonant
in the visible and UV regions for three different input ener- transferofenergyfromaninputlaserpulsetoatypically
gies. (c) shows the fibre dispersive relation: positions of the blue shifted peak is a well-studied process in nonlinear
predicted RR and negative RR spectral peaks are indicated.
optics andhas attractedsubstantialattentionin the last
The inset is a 25x enlargement of the curve around the λRR
few years due to the high conversionefficiencies that are
wavelength.
attainable with short pulses [3, 19] and more recently
even to predicted mode-squeezing properties [20]. Here
wehaveshownhowthesameprocessgeneratesasecond,
measured with a monochromator and a photomultiplier so far un-noticed peak that corresponds to resonant
tube and shows a clear peak that we identify with the transfer of energy to the negative frequency branch of
negative RR mode. The part of the output spectrum the dispersion relation. The energy transfer is favoured
that lies in the visible range, shown in Fig. 3(b), was in the presence of steep shock fronts or more generically,
measured with a compact CCD spectrometer. The peak by a non-adiabatic variation within the pump pulse.
observed here corresponds to the RR: the frequency of Experiments were performed in both bulk media and
this mode shifts to shorter wavelengths with increased waveguides with optimised dispersion landscapes so as
input pulse energy due to the nonlinear modification to allow the process to occur with a relatively high
of the refractive index from the pulse [5]. Figure 1(c) efficiency. These experiments also conceptually demon-
shows the predicted RR and negative RR frequencies stratethe classicallimit ofanimportantquantumeffect,
based on the dispersion relation for the photonic crystal i.e. the particle creation by the mixing of positive and
fibre. The measured peaks at λRR and λN are, similarly negative frequencies [12]. This implies that the process
to the bulk measurements, the main spectral features in could, if sufficiently optimised, generate a squeezed
the whole spectrum and both correspond very precisely vacuum state that couples wavelengths around λRR to
to the predictions. We note that the negative RR peak wavelengths at the negative RR peak, λN. At a classical
does not shift noticeably with input energy because level, the mixing of positive and negative frequencies
the nonlinear refractive index change from the pulse is leads to simultaneous wavelength conversion to the UV
negligible compared to the dispersive index changes in region and amplification (at the UV wavelengths) of
the UV. an input seed pulse that co-propagates with the input
Experiments were repeated for a series of photonic pump pulse. These ideas will be developed in future
crystalfibresasshowninTableI:NL-1.6615andNL-1.5 work.
590 (used for the data shown in Fig. 3) consist of a We thank Simon Horsely, Thomas Philbin, Scott
solid silica core surrounded by a hexagonal pattern of Robertson, Philip Russell and Sahar Sahebdivan for
air holes [18] and the fibre indicated with NL-1.5 670 discussions. ER wishes to acknowledge Fondazione
consists of a solid silica core surrounded by a cobweb of Cariplo,UnivercomoandBancadelMontediLombardia
silica strands. The size and spacing of these holes and for financial support. Our work was supported by
the thickness (∼ 1 µm) of the strands determine the Heriot-Watt University, the University of St Andrews
dispersionprofileofthe fibre. The letters“a”and“b”in and the Royal Society.
the fibre name indicate measurements performed along
5
∗ E.R. and J.M. contributed equally to this work. [9] D. Faccio et al., Opt. Express,15, 13077 (2007).
[10] M. Born and E. Wolf, Principles of Optics (Cambridge
University Press, Cambridge, 1999).
[11] L.MandelandE.Wolf, OpticalCoherence andquantum
optics (Cambridge UniversityPress, Cambridge, 1995).
[12] N.D.BirrellandP.C.W.Davies,Quantumfieldsincurved
∗ Electronicaddress: D.Faccio: [email protected]; F. Konig: fewk@spsatc-ean(dCraemwsb.raicd.guekUniversityPress, Cambridge, 1982).
[1] N.Akhmedievet al., Phys. Rev.A,51, 2602 (1995). [13] R. Brout et al., Phys. Rep., 260, 329 (1995).
[2] P.K. A. Waiet al., ” Opt.Lett. 11, 464 (1986). [14] E. Rubino,et al., New J. Phys.,13, 085005 (2011).
[3] L. Tartara et al., Appl.Phys. B, 77, 307 (2003). [15] D. Faccio et al., Phys. Rev.A., 78, 033825 (2008).
[4] D.V. Skryabinet al., Phys. Rev.E 72, 016619 (2005). [16] I.H. Malitson, Appl.Opt.,103 1103 (1963).
[5] J. Dudley et al., Rev.Mod. Phys., 78, 1135 (2006). [17] A. Couairon et al., Phys.Rep., 441, 47 (2007).
[18] J.C.Knightetal.,IEEEPhoton.Technol.Lett.,12,807
[6] M. Zamboni-Rached, E. Recami, and H. E. Hernandez-
(2000).
Figueroa Ed. Localized Waves (Wiley-Blackwell, 2008).
[7] M. Kolesik et al., Phys.Rev. Lett.92, 253901 (2004). [19] G. Chang et al., Optics Express, 19, 6635 (2011).
[8] D.Faccio et al., Phys.Rev.Lett., 96, 193901 (2006). [20] T.X. Tran et al., Phys. Rev.A,84, 013824 (2011).