optimum operation of femtosecond parametric oscillation of a noncollinear phase match in ktp

Optimum operation of femtosecond parametricoscillation of a noncollinear phase match in KTP

Wei Quan Zhang

For group-velocity (GV) matching, tunable femtosecond parametric oscillation must use noncollinearphase matching (PM). The tuning curves of femtosecond parametric oscillation are described; tuning canbe continuous from the visible to the mid infrared. We demonstrate that GV matching and PM can besatisfied in type II PM for KTP. The effective nonlinear coefficient, the walk-off angles, the acceptanceangles (a and �s), the acceptance spectral width, and the duration of the output idler pulse are calculated.Consequently, optimum femtosecond parametric oscillation of noncollinear phase matching is obtained inKTP. © 2005 Optical Society of America

OCIS codes: 170.5280, 310.2790, 310.3840.

1. Introduction

KTP is a widely used crystal for optical parametricgeneration. Rines et al.1 implemented 3-�m opticalparametric oscillation (OPO) with type II noncriticalphase matching in KTP. Tuning the pump wave-length, yielded a cw signal range of 1.05–1.37 �m andan idler range of 2.2–3.1 �m.2 Another implementa-tion of noncritically phase-matched KTP OPO pro-vided a pulsed signal range of 1.03–1.2 �m and anidler range of 2.18–3.03 �m.3 Zhang studied tuning ofoptical parametric generation in KTP. Although theirtuning ranges are quite large, the ranges of the signaland the idler wavelengths are not contiguous (fortype II PM).4

To obtain chirped-pulse amplification techniquesfor efficient energy exchange it is necessary thatphase-matching (PM) and group-velocity (GV) match-ing be achieved simultaneously. A decrease in signaland idler GVs in the direction of the pump to syn-chronize all pulses along the pump’s wave vector canbe obtained by adoption of noncollinear PM geome-tries. Noncollinear matching was studied in Refs. (5)–(9) for uniaxial and biaxial crystals. Zhang10,11

studied femtosecond optical parametric generation ofnoncollinear PM for KTP crystal. That research

showed that simultaneous GV matching and PM canbe verified as type II noncollinear PM in KTP.

In this paper we first describe the tuning curve ofnoncollinear PM. There are two sets of matching an-gle (� and �s) for the same signal or idler wavelength.By choosing the appropriate PM angle we can connectthe idler wavelength ranges for continuous tuningfrom 0.457 to 1.55 �m ��p � 0.395 �m� or from 0.757to 3.17 �m ��p � 0.6 �m�. Second, we calculate theeffective nonlinear efficient, the walk-off angle, theacceptance angles, and the acceptance spectralwidths. Finally, the optimum PM angles and relativeparameters for continuous-tuning OPO are found.

2. Phase-Matching Angles

In principal axis coordinates, the direction cosines ofthe pump wave vector are

kpx � sin � cos �, kpz � cos �,

kpy � sin � sin �, (1)

where � is the angle between the wave vector andmain axis z and � is the angle between the projectionof the wave vector in the x�y plane and the x axis. x,y, and z are the major axes of the crystal �nx � ny

� nz�. Assume that the pump, the signal, and theidler wave vectors all lie in same meridian plane; thedirection cosines of the signal and the idler wavevectors are, respectively:

kxs � sin(� �s)cos �, kzs � cos(� �s),kxi � sin(� �i)cos �, kzi � cos(� �i),

(2)

W. Q. Zhang ([email protected]) is with the Department ofPhysics, Zhejiang Science University, Hangzhou 310033, China.

Received 27 February 2004; revised manuscript received 10 Oc-tober 2004; accepted 6 November 2004.

0003-6935/05/122431-07$15.00/0© 2005 Optical Society of America

20 April 2005 � Vol. 44, No. 12 � APPLIED OPTICS 2431

or

kxs � sin(� �s)cos �, kzs � cos(� �s),kxi � sin(� �i)cos �, kzi � cos(� �i).

(3)

With reference to Fig. 1, Eqs. (2) correspond to coun-terclockwise �s angles with respect to pump wavevector Kp and Eqs. (3) correspond to clockwise �s an-gles. The momentum-conservation relation is

Kp � Ks Ki, (4)

where Kj � 2�nj��j �j � p, s, i�. When an ultrashortpulse propagates through a crystal, the GV is

GV � vg � c�[n �(dn�d�)]�j. (5)

We consider two conditions of GV matching:

cos �s�[n �(dn�d�)]�s � 1�[n �(dn�d�)]�p, (6)

cos �i�[n �(dn�d�)]�i � 1�[n �(dn�d�)]�p. (7)

In Eq. (6) the component of GVs parallel to Kp isequal to GVp; in Eq. (7) the component of GVi parallelto Kp is equal to GVp. The calculations of K and GVwere made in Ref. 11.

Figures 2 and 3 show the tuning curves �� 28.6°. Figures 2(a) and 3(a) are FFS PM and satisfyEq. (6) and Figs. 2(b) and 3(b) are FSF PM and satisfy

Eq. (7), where FFS represents a fast signal light anda slow idler light; FSF, vice versa. Figure 2 shows thedependence of angle �s on signal wavelength �s for�p � 395, 527, 600 nm. Figure 3 shows the depen-dence of pump angle a on �s.

We can see that there are two tuning curves forthe same pump wave and that the tunable femto-second pulse can be obtained continually from thevisible to the mid infrared. The tuning range of FFSPM is larger than that of FSF PM. For FFS PM thesignal wavelengths are 0.53–2.9 �m and the idlerwavelengths are 0.457–1.55 �m at �p � 395 nm. For�p � 527 nm the signal wavelengths are 0.7–2.9 �mand the idler wavelengths are 0.644–2.13 �m. For�p � 600 nm the signal wavelengths are 0.74–2.9 �mand the idler wavelengths are 0.757–3.17 �m. Toachieve such a continuously and broadly tunablerange with angle tuning of KTP pumped by a laser ofa fixed wavelength we must rotate angle �s from 0 to10° and angle � from 20° to 85° (assuming that �

Fig. 1. Wave vectors of the three interacting noncollinear phase-matched waves: (a) counterclockwise angle �s with respect to Kp, (b)the clockwise angle �s.

Fig. 2. Tuning curves: Ks to Kp angle �s as a function of signalwavelength �s at three pump wavelengths �p for (a) FFS PM andGVs, par � GVp matching and (b) for FSF PM and GVi, par � GVp

matching.

2432 APPLIED OPTICS � Vol. 44, No. 12 � 20 April 2005

� 28.6°). For this purpose, more than one KTP crystalis usually required.

3. Calculation of Parameters for OPO

Methods for calculating the PM parameters, includ-ing the effective nonlinear coefficient, the walk-offangle, the acceptance angles, and the acceptancespectral widths of a biaxial crystal such as KTP havebeen discussed by Zhang.4 The effective nonlinearcoefficient is

deff � �idijk�j�k, (8)

where �i � kiG��1 ni2�n2� and G � �� ki

2��1 ni

2�n2�2�1�2 �i � x, y, z�. Figures 4(a) and 4(b) showthe results that correspond to FFS PM, GVs, par� GVp and FSF PM, GVi, par � GVp, respectively. deff

are 2.33–6.78 pm�V. Comparing Figs. 4(a) and 4(b),one can find that the effective nonlinear coefficient ofFSF PM is larger than that of FFS PM for same �p.

The walk-off angle between the signal and thepump Poynting vectors is4

�PS � cos1(SpxSsx SpySsy SpzSsz), (9)

where Sij �i � s, p and j � x, y, z) are the directioncosines of the pump and the signal Poynting vectors,respectively. Sij � nr�1�nij

2 1�nr2�kij��1�nij

2

1�n2��n and 1�nr2 � 1�n2 n2 ��kij��1�nij

2

1�n2��2�. Figures 5(a) and 5(b) show the dependenceof the walk-off angle on signal wavelength �s for FFSand FSF PM, respectively. For FFS PM the walk-offangle is approximately 0.2°–6°. For FSF PM thewalk-off angle is approximately 2.12°–12.8°. Obvi-

Fig. 3. Tuning curves: angle � between Kp and the crystal’s z axisas a function of signal wavelength �s: (a) For FFS PM andGVs, par � GVp matching, (a1) is the counterclockwise �s and (a2) isthe clockwise �s. (b) For FSF PM and GVi, par � GVp matching,thicker curves correspond to counterclockwise �s; thinner curves, toclockwise �s.

Fig. 4. Effective nonlinear coefficient deff as a function of signalwavelength �s: (a) For FFS PM and GVs, par � GVp matching, (a1)is the counterclockwise �s and (a2) is the clockwise �s. (b) For FSFPM and GVi, par � GVp matching, thicker curves correspond tocounterclockwise �s; thinner curves, to clockwise �s.


ously the walk-off angle of FFS PM is smaller thanthat of FSF PM.

The acceptance angles for efficient nonlinear con-version are usually defined as angular ranges �� and��s, such that wave-number mismatch �K satisfies

|(�K�2)| ��L, (10)

where L is the crystal’s length. For its calculation, seeRef. 4. With the aforementioned two sets of PM andGV matching conditions, ��L and ��sL are plotted asfunctions of signal wavelength �s in Figs. 6 and 7,respectively, for �p � 395, 527, 600 nm. The crystallength is 8 mm. Figure 6 shows that ��L is 0.2 to58.7 rad��m at FFS PM. From Fig. 7, ��sL is0.27–15.2 rad��m for FFS PM. For FSF PM, ��sL is0.53–21.3 rad��m.

If the wavelength of the pump wave is shifted by��p, the wavelengths of the signal and the idler waveswill be shifted by ��s and ��i, respectively. Assumingthat �vs � �vi, ��p, can be calculated from4

K � |d(�K)�d(��p)|�p0��p. (11)

We can obtain the acceptance spectral widths ��p�L.Figure 8 shows acceptance spectral widths ��p�Lplotted as functions of signal wavelengths. ��p�L is0.084–0.21 nm-nm and does not change much with

Fig. 5. Walk-off angles as a function of signal wavelength �s: (a)For FFS PM and GVs, par � GVp matching and (b) for FSF PM andGVi, par � GVp matching, thicker curves correspond to counterclock-wise �s; thinner curves, to clockwise �s.

Fig. 6. Acceptance angles �� multiplied by crystal length L as afunction of signal wavelength �s: (a) For FFS PM and GVs, par

� GVp matching, (a1) is the counterclockwise �s and (a2) is theclockwise �s. (b) For FSF PM and GVi, par � GVp matching, thickercurves correspond to counterclockwise �s; thinner curves, to clock-wise �s.


signal wavelength �s. For �p � 600 nm the acceptancespectral width is larger.

Not only is a shorter pulse duration required fortime resolution but it is also required for sufficientlyhigh peak power. In Ref. 11, by the Fourier methodthe coupled-wave equations of femtosecond opticalparametric generation of noncollinear PM weresolved for biaxial crystal. The equations include thelowest and second-order GV dispersion and the GVmismatch. Assume that the incident pump pulse andsignal pulse are, respectively,

Ap(�, 0) � Ap(0)�[exp(��T0) exp(��T0)],As(�, 0) � As(0)�[exp(��T0) exp(��T0)].

(12)

The thickness of the crystal is 8 mm. The peak of the

electric-field intensity of the pump pulse is 1.94� 106 V�m, and T0 � 30 fs.

Figure 9 shows the durations of output idler pulseversus wavelength �i with a pump’s central wave-lengths at �p � 395, 527, 600 nm. The durations are45–80 fs. Because GV matching conditions are satis-fied, the duration is smaller than that of collinearPM. When �p � 600 nm the duration of the outputidler pulse is the smallest for FSF PM.

4. Optimum Operation of Femtosecond OPO

The properties of the femtosecond OPO are summa-rized in Table 1 for the two sets of noncollinear PMand GV matching conditions and the three pumpwavelengths.

From Table 1 we can see that, when �p � 600 nmand the PM type is FFS or FSF, the tuning range ofthe idler light is the largest, the effective nonlinearcoefficient deff is the largest (for FSF PM), the walk-offangle is the smallest (for FFS PM), acceptance angle��L is the largest (for FFS PM), and the duration of

Fig. 7. Acceptance angles ��s multiplied by crystal length L as afunction of signal wavelength �s: (a) For FFS PM and GVs, par

� GVp matching, (a1) is the counterclockwise �s and (a2) is theclockwise �s. (b) For FSF PM and GVi, par � GVp matching, thickercurves correspond to counterclockwise �s; thinner curves, to clock-wise �s.

Fig. 8. Acceptance spectral widths ��p multiplied by crystallength L as a function of the signal wavelength �s: (a) for FFS PMand GVs, par � GVp matching and (b) for FSF PM and GVi, par

� GVp matching. Thicker curves correspond to counterclockwise �s;thinner curves, clockwise �s.


the output idler pulse is the smallest for FSF PM. Itis an optimum operation for femtosecond opticalparametric generation of noncollinear PM in biaxialcrystal KTP.

5. Conclusions

In summary, we have studied femtosecond opticalparametric oscillation of noncollinear phase match-ing. The signal and idler waves can be tuned contin-uously from the visible to the mid infrared. Theeffective nonlinear coefficient, the walk-off angle, ac-ceptances ��L and ��sL, and acceptance spectralwidth ��p�L were calculated for the two sets of PMand GV matching types and three pump wave-lengths. As GV matching is satisfied at noncollinearPM, the duration of output idler pulse is smaller.Comparing these results yields the optimum opera-tion for femtosecond tunable parametric generationof noncollinear phase matching in KTP.

Fig. 9. Durations of output idler pulses as functions of idler wave-lengths. (a) Continuous curves, to FFS PM and GVs, par � GVp;dashed curves, FSF PM and GVi, par � GVp for (a) �p � 395 nm and(b) �p � 600, 527 nm.

Tab

le1.

Pro

per

ties

of

Fem

tose

cond

OP

Oo

fN

onc

olli

near

Pha

seM

atch

ing

for

the

Tw

oS

ets

of

PM

and

GV

Mat

chin

gan

dT

hree

Pum

pW

avel

eng

thsa

Pro

pert

yP

Mty

pe

�p

�nm

�

395

527

600

FF

SF

SF

FF

SF

SF

FF

SF

SF

Tun

ing

rang

e,�

s��i�

�m

0.53

(1.5

5)–2

.9(0

.457

)0.

59(1

.2)–

1.59

(0.5

25)

0.7

(2.1

3)–2

.9(0

.644

)1.

1(1

.01)

–2.8

(0.6

49)

0.74

(3.1

7)–2

.9(0

.757

)0.

9(1

.8)–

2.9

(0.7

57)

def

f�p

m�V

�2.

33–4

.83

3.38

–6.8

72.

63–4

.57

3.15

–6.6

2.57

–5.0

03.

88–6

.78

Wal

k-of

fan

gle

(deg

)0.

2–5.

474.

13–1

2.8

1.6–

64–

10.8

1.07

–4.3

2.12

–10.

1�

�L

�rad

-�m

�4–

20.7

10–3

40.

2–25

.718

–53.

30.

13–5

8.7

0–46

.7�

�sL

�rad

-�m

�0.

36–1

5.2

2–3.

60.

27–1

5.2

4.93

–16.

30.

27–1

4.4

0.53

–21.

3�

�pL

�nm

-nm

�0.

085–

0.09

0.08

4–0.

090.

153–

0.15

70.

147–

0.15

70.

098–

0.20

30.

196–

0.20

6D

urat

ion

ofid

ler

wav

e(f

s)46

.1–8

058

.3–7

8.3

45.1

–68.

347

.2–7

247

.4–6

8.3

44.4

–57.

7

aP

rope

rtie

sin

clud

etu

nabl

era

nges

,eff

ecti

veno

nlin

ear

coef

ficie

nts,

wal

k-of

fang

les,

acce

ptan

cean

gles

��

and

��

s,an

dac

cept

ance

spec

tral

wid

ths

��

pm

ulti

plie

dby

crys

tall

engt

hL

,and

dura

tion

sof

outp

utid

ler

puls

es.


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optimum operation of femtosecond parametric oscillation of a noncollinear phase match in ktp

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