Broadband amplification in non-linear crystals using controlled angular dispersion of signal beam

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: [email protected] (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.

mailto:[email protected]

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.


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.


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.


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.


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


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.


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


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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Broadband amplification in non-linear crystals using controlled angular dispersion of signal beam