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Optics Communications 251 (2005) 405–414
www.elsevier.com/locate/optcom
Broadband amplification in non-linear crystalsusing controlled angular dispersion of signal beam
Luis Cardoso *, Goncalo Figueira
L2I/Grupo de Lasers e Plasmas, Instituto Superior Tecnico, Av Rovisco Pais, 1049-001 Lisboa, Portugal
Received 8 November 2004; received in revised form 2 March 2005; accepted 4 March 2005
Abstract
Recent advances in high intensity laser technology, such as the optical parametric chirped pulse amplification tech-
nique, allow the amplification of broadband pulses to high energies, by parametric interaction in a non-linear crystal. In
this article, angular dispersion of the signal beam inside the non-linear media is added to a common setup to improve
the non-collinear phase matching range. A comparative study of the performance of BBO, LBO and KDP using this
geometry is undertaken. Computer simulations show that both BBO and LBO are excellent broadband performers.
Although the maximum bandwidth of KDP does not broaden, we show how to avoid its severe narrowing below
the degeneracy wavelength.
� 2005 Elsevier B.V. All rights reserved.
PACS: 42.65.Yj
Keywords: Broadband; OPCPA; Femtosecond pulses
1. Introduction
The need for shorter laser pulses drives thesearch for broadband gain materials, usually also
with the requirement to withstand high power or
energy shots. Broadband non-linear crystals such
as b-barium borate (BBO) and lithium triborate
0030-4018/$ - see front matter � 2005 Elsevier B.V. All rights reserv
doi:10.1016/j.optcom.2005.03.009
* Corresponding author. Tel.: +351 218419379; fax: +351
218464455.
E-mail address: luis.cardoso@ist.utl.pt (L. Cardoso).
(LBO) have already proved their fitness to such de-
mands since they are used in optical parametric
amplifiers (OPAs). These can provide very highgains from IR to UV, if properly pumped and
tuned. By tuning one means to use temperature,
collinear, non-collinear or quasi phase matching
techniques to optimise the gain around the signal
wavelength. A non-collinear geometry is often
chosen because it makes it simpler to separate
the signal from the idler, and usually allows a lar-
ger bandwidth. In particular, depending on the
ed.
406 L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414
crystal and the pump wavelength used, one can
find a very broad and reasonably flat gain curve
for a certain range of signal wavelengths. Out of
that range, it has already been shown that the
bandwidth can be increased, but at the expenseof a custom tailored, chirped pump beam [1].
The use of angular dispersion to bring each com-
ponent of a broadband pulse closer to its phase
matching condition has already been shown for
femtosecond laser second harmonic generation
[2–6] and, more generally, for sum frequency gener-
ation [7–9]. Recently, controlled angular dispersion
was applied to a difference frequency generationprocess, such as optical parametric chirped pulse
amplification (OPCPA) [10,11], in both coplanar
[12] and non-coplanar setups [13]. In those works,
the simulations were developed for a BBO crystal,
showing remarkably promising results in terms of
attainable bandwidth.
In order to complement the results obtained for
BBO in the non-coplanar setup, in this paper, weexplore the use of signal angular dispersion with
LBO and KDP and compare the results for all
these three crystals, which are among the most
used in OPAs for their exceptional optical charac-
teristics (BBO and LBO) and large size growth
ability (KDP).
In Section 2, we describe the model used, the
setup needed and its grounds. Section 3 presentsthe results and the configurations at which they
were obtained and discusses the benefits and limi-
tations of this technique. Finally, in Section 4 we
present the conclusions.
2. Model
We built a simulation code [13] to assess the
possible bandwidth increase in OPCPA processes
using this new technique. It was designed to calcu-
late phase matching in 3D space, for realistic use
of the degrees of freedom available in the labora-
tory. Assuming a slowly varying signal envelope
and flat top spatial and temporal pump profiles,
with no depletion, the gain G and phase u of theamplified signal in an OPA can be estimated using
the analytical solution of the coupled wave equa-
tions defined in [10]:
G ¼ 1þ ðcLÞ2 sinhBB
� �2
; ð1Þ
u ¼ tan�1 B sinA coshB� A cosA sinhBB cosA coshB� A sinA sinhB
; ð2Þ
where
A ¼ DkL=2; ð3Þ
B ¼ ½ðcLÞ2 � ðDkL=2Þ2�1=2; ð4Þ
c ¼ 4pdeffðIp=2e0npnsnickskiÞ1=2; ð5Þ
DkL ¼ ðkp � ks � kiÞL ð6Þwith the restriction xp = xs + xi. Here, e0 and c are
the permittivity and speed of light in vacuum, c is
the gain coefficient, DkL the phase mismatch, L
the amplifier length and deff the effective non-linear
coefficient. I, n, x, k and k stand for the intensity,
refractive index, frequency, wavelength and wave
vector, respectively, with the appropriate sub-scripts for pump (p), signal (s) and idler (i).
For calculation of the refractive indexes and re-
lated quantities, we adopt the same Sellmeier equa-
tions as in the SNLO software [14].
As mentioned before, non-collinear phase
matching helps increase the bandwidth. Still, fixed
phase matching hPM and non-collinear hNC angles
providing perfect phase matching at a certainwavelength give only partial phase matching for
the neighbouring wavelengths. Following the
treatment for non-collinear geometry used in
[11], we find that the wavelength dependent ideal
angle between a signal component and the pump
beam for perfect phase matching is given by
hNC;idealðksÞ ¼ cos�1k2p þ k2s � k2i
2kpks
!: ð7Þ
Fig. 1 shows the plot of this function at some con-
stant phase matching angles for the three studied
crystals.These graphs can be interpreted as follows: in a
traditional setup with a fixed hNC, the flatness of
the curve for each phase matching angle hPM gives
an indication of how close to perfect phase match-
ing is a given range of wavelengths; the broader the
flat region, the broader the quasi-phase-matched
Fig. 1. Dependence of the ideal non-collinear angle hNC,ideal with the signal wavelength ks for some phase matching angles (hPM for
BBO and KDP, /PM for LBO), using a 532 nm pump for (a) BBO, (b) LBO and (c) KDP crystals. The dots mark stationary points.
L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414 407
bandwidth. In fact, this is the principle followed for
choosing the set of parameters in the traditional
non-collinear geometry. For instance, for
hPM = 23.87� in BBO there is an almost flat region
between 750 and 900 nm, typical of Ti:sapphire la-
sers, allowing broadband amplification.
We can also interpret the graphs in anotherfashion: for a given hPM, provided we can some-
how introduce an angular dependence to the
signal wave vector that follows exactly the corre-
sponding hNC,ideal (ks), we would have unlimited
phase matching, i.e., DkL = 0 for all wavelengths.
Naturally, this effect is accomplished over a lim-
ited wavelength range, since it is virtually impossi-
ble to reproduce these curves with typicaldispersive devices. The idea of introducing this
dependence in the form of an intentional linear
angular dispersion b0 to operate in a range
[k0 � Dk1,k0 + Dk2] where hNC,ideal (ks) exhibits analmost linear behaviour comes straightforward,
and compensation to first order is accomplished
by making
b0 ¼ ðdhNC;ideal=dksÞks¼k0: ð8Þ
This concept has been applied to BBO [12], and
can be extended to other crystals, namely KDP,
as we show in this article. In previous work [13],
we showed that improved broadband amplifica-
tion in BBO could be achieved by using a different
approach, consisting of describing the behaviour
of hNC,ideal (ks) around stationary points by meansof an ad hoc function (instead of a polynomial
approximation), which has a straightforward geo-
metrical correspondence. Here, we apply that pro-
cedure to LBO as well given the similarity of the
curves for hNC,ideal.
We start our analysis by evaluating the behaviour
of hNC,ideal for BBO and LBO around the stationary
points, marked with dots in Fig. 1(a) and (b), obey-ing dhNC,ideal/dks = 0 and d2hNC;ideal=dk
2s > 0. Let us
define k0 as the signal wavelength at some such sta-
tionary point, and hNC,0 ” hNC,ideal (k0), for a given
hPM. Using the fact that around those points the
function shows a hyperbolic-like behaviour (degen-
erating into two crossing lines for the lowermost
point), we introduce the ad hoc function
n2ðksÞ ¼ b20ðks � k0Þ2 þ h2NC;0 ð9Þ
Fig. 2. Setup geometry and definition of the angles used in the
simulation for BBO and LBO. The idler becomes dispersed in
both planes.
Fig. 3. Definition of the angles used in the simulation for KDP.
The idler is dispersed in a single plane.
408 L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414
such that
dndks
����k¼k0
¼ 0; ð10aÞ
d2n
dk2s
�����k¼k0
¼ b20
hNC;0
: ð10bÞ
Additionally, n(ks) tends asymptotically to
n(ks) � b0(ks � k0) for (ks � k0)� hNC,0/b0, corre-sponding to the long wavelength regions where
the linear approximation described above – Eq.(8) – is normally applied. In order to find the angu-
lar dispersion to be applied, and since we want to
match n to hNC,ideal, we take Eq. (10b) with the
proper substitution and obtain the second order
correspondence
b0 ¼ hNC;0
d2hNC;ideal
dk2s
!1=2
ks¼k0
: ð11Þ
For the degenerate case, when hNC,0 � 0, we can
revert to Eq. (8), this time evaluated at ks = k0 ±dks with dks such that hNC,ideal is locally linear.
Normally, it would be unlikely that the same
value for b0 would serve both as a second-orderapproximation at k0 and a first order asymptotic
approximation for large ks, unless hNC,ideal (ks)were actually locally described by a perfect hyper-
bolic curve. Therefore, some numerical adjustment
of b0 and hPM may be required in order to obtain
the broadest bandwidths. Note that, given the
shape of the curves, and especially when large
bandwidths are available, k0 is not necessarily theamplified signal central wavelength, which is gen-
erally shifted towards the longer wavelengths.
In what concerns the geometrical correspon-
dence, we can notice that Eq. (9) also corresponds
to the squared norm of a vector whose orthogonal
components are b0(ks � k0) and hNC,0. From this,
the idea of using a phase matching geometry with
the non-collinearity (hNC,0) in one plane and theangular dispersion (b0) in an orthogonal one arises
naturally. In Fig. 2, we present the setup consid-
ered and illustrate the role of the different angles
involved. The non-collinear plane yz is defined
by the crystal�s optical axis (OA) and the pump
beam (kp), and it also contains the central wave-
length signal wave vector ks0, such thatOA�kp = hPM and kp
�ks0 = hNC,0. The angular dis-
persion takes place in the xz plane, containing ks0and orthogonal to the first plane. We have repre-
sented two wave vectors ks1 and ks2, above and be-
low the yz plane, respectively, to illustrate the
angular spread. Considering a first order approxi-
mation for the angular dispersion, we have
~ks0^~ks1 � hdð~ks1Þ ffi b0 � ðks1 � k0Þ; ð12Þwhere hd is the dispersion angle and ks1 is the
wavelength corresponding to the wave vector ks1.
Consider now the right triangle defined by theintersection of kp, ks0 and ks1 with a plane given
by z = constant (shown in the right edge of the
box depicted in Fig. 2). From it, and using Eq.
(12), one can readily recover Eq. (9), provided all
the angles involved are small (tanh � h). Although
not shown for the sake of clarity, one can also rea-
lise that the corresponding idler beams will be dis-
persed in both planes for this geometry.
L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414 409
We turn now to the case of KDP. Contrarily to
the above, Fig. 1(c) does not show stationary points
with d2hNC;ideal=dk2s > 0, so this scheme is unusable.
For a fixed hPM and hNC one is limited to obtaining
a first order approximation around the dhNC,ideal/dks = 0 points present only at wavelengths above
the degeneracy. In order to operate at a wavelength
k0 below that point, we may use the already known
first-order polynomial approximation
hNC;idealðksÞ ffi hNC;idealðk0Þ þ b0 � ðks � k0Þ ð13Þ
Fig. 4. BBO: 8.7 mm long crystal, pumped with kp = 532 nm, Ip = 1
simulation. (a) Spectral gain profile (nm) vs. k0 (nm). (b) Similar to (a)
matching near k0, using Eq. (11). (c) Similar to (b), but with signal ang
At degeneracy, signalled with the dashed line, we find the broadest ba
are two possible optimised configurations for each k0. That could be
together back to back at the dashed line. Note that this reverses absc
corresponding to approximating the curves by
their local tangent at the target wavelength, with
b0 given by Eq. (8). Geometrically, this represents
a linear angular dispersion in the same plane as
that of the non-collinearity, as depicted in Fig. 3.One final note for both setups: the fundamental
issue in these geometries is the relative orientation
of hNC and the angular dispersion plane; since kp is
fixed and the signal is an ordinary ray, the relative
orientation of hNC and hPM in the crystal holds no
constraint.
GW cm�2. Some pixelation results from the step used in the
, but with signal angular dispersion calculated for perfect phase
ular dispersion manually optimised for the broadest bandwidth.
ndwidth, detailed in Fig. 8. At slightly lower wavelengths there
presented in two separate graphics, but we opted to join them
issa counting beyond that line.
410 L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414
3. Results
To test this setup, some simulations were run
for BBO, LBO and KDP crystals, pumped by a
532 nm, 1 GW cm�2 beam, typical figures of acommercial Nd:YAG laser. These simulations
are targeted at a gain of 1000, and the length of
each crystal is set accordingly. We then optimise
the bandwidth by introducing a small phase mis-
match; however, this must not affect the gain curve
by more than ±10% to minimise signal degrada-
tion. In this optimised set of parameters hNC is
never set below 0.3� to allow easy signal–idler sep-aration. The model assumes a linear source of
angular dispersion. To test our setup more realisti-
Fig. 5. Similar to Fig. 4 but for a 20.1 mm long LBO crystal. Unlik
configuration near the degeneracy point.
cally we simulate instead the first diffraction order
output of a grating with a normally incident beam.
Dispersion rate is controlled by either changing
groove density or telescoping.
Figs. 4–6 show the performances for each of thethree crystals, with and without angular disper-
sion, when varying k0.To summarise these results, the next figure com-
pares the bandwidth when no angular dispersion
and optimised angular dispersion are used on each
crystal.
As we can see in Fig. 8, for BBO the optimisa-
tion allowed a maximum bandwidth exceeding700 nm while still leaving gain and phase very
smooth.
e BBO (c), LBO (c) does not reveal an interesting alternative
Fig. 6. KDP: 56.7 mm long crystal, pumped with kp = 532 nm, Ip = 1 GW cm�2. (a) Spectral gain profile (nm) vs. k0 (nm). (b) Similar
to Fig. 4(c) but with signal angular dispersion added in the non-collinear plane to maximise the bandwidth below the degeneracy point.
L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414 411
LBO performs almost as well as BBO, but the
steeper slopes on the short wavelength side of
Fig. 1(b) deny it a broader bandwidth. KDP is
completely different in the sense that it does not
have stationary points with positive second deriv-
ative. This means that a simpler geometry mustbe used instead, which gives only first order
Fig. 7. Full width half maximum gain bandwidth with undispersed (do
(a) BBO, (b) LBO and (c) KDP. Abscissas for the dotted lines are as u
central wavelengths for the gain bandwidth, not k0 because of the gr
approximation. The result is a maximum band-
width no better than KDP already had, but
decreasing much less below the degeneracy point.
The main setup parameters for each crystal are
plotted in Fig. 9. To avoid cluttering only two con-
figurations are shown in each graph. Notice thatFig. 7 is intended to show the best and worst band-
tted curve) and optimally dispersed (solid curve) signal beam for
sual in the literature, but for the solid lines they are the effective
eater gain asymmetry around it.
Fig. 8. Calculated gain (solid) and phase (dotted) for BBO at
k0 = 1060 nm with optimised angular dispersion
b0 = 101.2 lrad/nm – lineout at the dashed line in Fig. 4(c).
The FWHM bandwidth is 724 nm, centred at 1170 nm.
Fig. 9. Wavelength-dependence of the phase matching angle (left), no
BBO, LBO and KDP (top, middle and bottom row of figures, respectiv
(with angular dispersion) are plotted; the configurations with the flatte
KDP the dispersionless configuration is shown dotted. For all the thr
allow extended bandwidth. Note alternative configurations in BBO n
412 L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414
width cases while Fig. 9 serves primarily to charac-
terise the new configurations, hence the dotted
curves do not always correspond to the same set-
up. However, solid lines do always mean an opti-
mised angular dispersion configuration.From the previous graphics, it is evident the
major bandwidth increase when angular dispersion
is properly used, reaching one order of magnitude
for some configurations. It is necessary, however,
to control two more quantities: k0 and b0. It is
out of the scope of this article to detail quantita-
tively each variable�s error sensitivity, as that
would require the definition of a criterion for mis-behaviour, difficult to implement automatically in
n-collinear angle (centre), and angular dispersion rate (right) for
ely). For BBO and LBO only the newly proposed configurations
st gain and phase for wavelengths near k0 are shown dotted. For
ee crystals the solid lines represent the optimised parameters to
ear degeneracy.
Fig. 10. (a) Normalised electric field of a transform limited 7 fs
gaussian pulse at 1053 nm and the simulated amplified pulse
after a gain and phase modulation as in Fig. 8. (b) Contrast
remains better than 106.
L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414 413
the simulation code, and to apply it for each crys-
tal while scanning the signal wavelength. Wefound in general that k0 has a tolerance in the
nanometre range, b0 accepts deviations of a few
percent and that hNC,0 is just about as sensitive
as hPM, that is in the order of the hundredths of de-
gree for BBO and LBO and thousandths of degree
for KDP.1 So, although each variable demands ex-
tra care, they are not critically sensitive as they are
well within common alignment accuracy.To illustrate the potential of this scheme, we
simulated an amplifier stage with a BBO crystal
set for the maximum bandwidth configuration,
like in Fig. 8. Inputting transform-limited pulses
centred at 1053 nm, a common wavelength in high
power lasers, these will experience minimal broad-
ening even when phase terms down to the third or-
der are allowed at the output:
� a 7 fs (FWHM) Gaussian pulse enlarges to
<10 fs with contrast ratio >106 (Fig. 10);
� a 5 fs Gaussian pulse enlarges to <6.5 fs with
contrast ratio >104.
In what concerns the comparative performance
of the three crystals, BBO features the broadest
1 See web available pictures for graphs, based on Figs. 4–6,
showing the effects of misadjusted variables (See Appendix A).
bandwidth, while LBO withstands the highest
pumping power and fluence. A preamplifier of
the first combined with a final stage of the second,
which can already be grown up to 20 · 20 mm2
cross-section, could then support at least a 5 J,10 fs pulse, which is in the petawatt range. KDP,
with its narrower bandwidth, is in general unsuited
for amplification of such short pulses. However,
being grown to large crystal sizes, it allows the cre-
ation of much more energetic pulses, thus, entering
in the petawatt range once again. Looking at Fig.
7(c), we notice that this technique now allows
amplifying beams below the degeneracy point setby the pump wavelength, where hitherto band-
width falls down to hardly usable values. Signal
wavelength choice can now be less limited as most
laboratories, using their well established power la-
ser chains to pump their OP(CP)A experiments,
have a fixed pump wavelength.
In addition to the amplification of ultra-broad-
band pulses, another possible use of this setup is asa tuneless amplifier for tunable laser sources with
longer pulse durations.
4. Conclusion
This paper explores further the application of a
recently proposed scheme, allowing the use of aBBO crystal as an ultra broadband amplifier, to
LBO and KDP. We described the setup and the
conditions for which extra bandwidth can be
achieved in these three non-linear crystals. A basic
error sensitivity assessment was made to show that
this setup is feasible with commonly available
commercial opto-mechanics. For BBO and LBO,
the bandwidth increase is notorious for signalwavelengths above 900 nm. The peak bandwidth
is obtained at the degeneracy wavelength, reaching
more than 600 and 700 nm in LBO and BBO,
respectively. KDP does not extend its bandwidth
but below the degeneracy point, where it used to
become very narrow band. Simulations presented
in this work use the second harmonic of a
Nd:YAG laser as the pump source, but equallygood results are expected for most valid pumping
wavelengths; successfully tested were Nd:YAG
third harmonic and iodine�s 1315 nm second and
414 L. Cardoso, G. Figueira / Optics Communications 251 (2005) 405–414
third harmonics. Other negative crystal types can
also benefit from this scheme.
Acknowledgement
Work supported by FCT under project POCTI/
FAT/41586/2001 and grant SFRH/BD/10305/2002.
Appendix A. Supplementary data
Supplementary data associated with this articlecan be found in the online version at doi:10.1016/
j.optcom.2005.03.009.
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