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Optics Communications 281 (2008) 2271–2278

Analysis of broadband third harmonic generation withnon-collinear angular dispersion in KDP crystals

Kun Li a,b, Bin Zhang b,*

a School of Optoelectronic Information, University of Electronic Science and Technology of China, Sichuan Chengdu 610054, Chinab College of Electronic Information, Sichuan University, Sichuan Chengdu 610064, China

Received 3 July 2007; received in revised form 7 December 2007; accepted 8 December 2007

Abstract

A new scheme with non-collinear sum frequency generation (SFG) process of broadband third harmonic generation (THG) with twoKDP crystals (one doubler and one mixer) using angular dispersion was proposed. The principle of broadband harmonic generation withnon-collinear angular dispersion was presented. The comparison between the schemes of non-collinear and collinear SFG process wasperformed. The effects of the angular dispersion on the conversion efficiency, the pulse shape and the spectrum of the third harmonic fieldwere discussed. The results show that, if proper angular dispersion is added to the fundamental and the second harmonic field outside themixing KDP crystal, respectively, the broadband third harmonic conversion efficiency can be improved significantly. However, the dif-ficulty of this scheme arises due to the requirement of two gratings with different angular dispersion outside the mixer. With the newscheme, it can be simpler that only one grating is needed to realize the broadband phase matching in the non-collinear SFG process.Although the fundamental and the second harmonic field inside the mixing crystal are non-collinear except the center wavelength, a blaz-ing grating with proper angular dispersion for the fundamental field (twice as that for second harmonic field) can yield the well compen-sation for phase mismatching in the SFG process. Consequently, the conversion efficiency and the characteristics of the third harmonicfield for the broadband third harmonic generation can be improved significantly.� 2007 Elsevier B.V. All rights reserved.

Keywords: Third harmonic generation; Broadband laser; Angular dispersion; Linear chirp pulse; Non-collinear

1. Introduction

Broadband ultraviolet laser plays an important role inthe research of interaction of laser light and materialsand inertial confinement fusion (ICF) [1–5], etc. Usually,it can be obtained by third harmonic generation (THG)of the infrared laser. However, the broadband third har-monic conversion efficiency decreases sharply with theincreasing of the input bandwidth [5,6] due to the phasemismatching of the non-center wavelength components.

Commonly compensatory approaches include multi-crystal designs [7], chirp compensation [8,9], angular spec-

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.12.014

* Corresponding author. Tel.: +86 028 85464312; fax: +86 02885463871.

E-mail addresses: anryco@sohu.com (K. Li), zhangbinff@sohu.com(B. Zhang).

tral dispersion (ASD) [10–12], quasi-phase-matched tech-nique [13], etc. However, for the harmonic generation ofthe wide bandwidth laser (e.g., time–frequency modulatedlaser with bandwidth larger than 1 nm or femtosecondlaser with bandwidth of several tens of nm), ASD is oneof the most effective methods.

Using prisms or gratings to provide ASD in broadbandharmonic generation was first presented in 1970s [14], andbecame more important in ICF for the purpose of beamsmoothing [1,2] in 1990s. By introducing ASD, the differentwavelength components are incident into the frequencyconversion crystal at different angles. Consequently, thephase matching condition can be satisfied in the wholebandwidth range when proper angular dispersion is added.Usually, the angular dispersion provided by a prism is notlarge enough. Although multi-prisms in series [10] canprovide large angular dispersion, it can also enlarge the dis-

2272 K. Li, B. Zhang / Optics Communications 281 (2008) 2271–2278

persion and the reflective, as well as the absorptive loss.However, gratings are applicable for wide bandwidth har-monic generation due to the large angular dispersion andhigh diffractive efficiency [11,12].

Up to now, the research of harmonic generation oflarge broadband laser with ASD has mostly concentratedin frequency doubling of femtosecond pulse [15–19].Experiment of third harmonic generation of frequency-modulated laser with the bandwidth of about 1 nm byusing ASD has been carried out [5]. In this experiment,one doubling and one mixing KDP crystal were usedand one grating was set to provide angular dispersion inthe front of the ICF system before the fundamental fieldwas amplified. However, this scheme was reviewed to beless effective to larger bandwidth. Another numericalresearch showed that only partial compensation of thephase matching could be achieved in the type I frequencymixing process of 800 nm and 400 nm in BBO (b-bariumborate) and KDP [20]. There were also some SFG meth-ods for ultra broadband laser, but in most cases the inputtwo fields are manually separated into two beam paths[20–23].

In this paper, an improved scheme with collinear SFGprocess by setting gratings directly in front of the doublerand the mixer was mentioned. Furthermore, a new schemewith non-collinear SFG process of broadband THG withgrating angular dispersion was proposed and analyzed the-oretically. Comparisons of the non-collinear SFG processwith the collinear SFG process were made in order to showthe improvement and the merits of the former. The effectsof the angular dispersion on the conversion efficiency, thepulse shape and the spectrum of the third harmonic fieldwere also discussed.

For KDP crystal is the optimal and commercial choicefor the frequency conversion in ICF which requires largeaperture to ensure enough energy and moderate intensity[24] which is irreplaceable in ICF so far, KDP crystalwas taken as the example in our discussion. The more gen-erality implications of our method were mentioned in theconclusion part of this article.

2. Theory of the broadband THG with angular dispersion

2.1. The coupled-wave equations of broadband THG

Broadband harmonic conversion can be described bythe theory of wave mixing in dispersive birefringent nonlin-ear crystals [6].

The coupled-wave equations of type I phase matchingsecond harmonic generation (SHG) in the doubling crystalcan be written as

oA1o

oz¼ � 1

vg 1o

oA1o

ot� i

2b1o

o2A1o

ot2þ i

2k1o

o2

ox2þ o2

oy2

� �A1o

þ ix1o

n1ocd2A2eA

�1o expðiDk2zÞ � a1o

2A1o ð1aÞ

oA2e

oz¼ � 1

vg 2e

oA2e

ot� i

2b2e

o2A2e

ot2þ i

2k2e

o2

ox2þ o2

oy2

� �A2e

� q2eðhÞoA2e

oyþ ix2e

n2ecd2A2

1o expð�iDk2zÞ � a2e

2A2e

ð1bÞ

The coupled-wave equations of type II phase matchingsum frequency generation (SFG) in the mixing crystal areas follows:

oA1e

oz¼ � 1

vg 1e

oA1e

ot� i

2b1e

o2A1e

ot2þ i

2k1e

o2

ox2þ o

2

oy2

� �A1e

� q1eðhÞoA1e

oyþ ix1e

n1ecd3A�2oA3e expðiDk3zÞ � a1e

2A1e

ð2aÞoA2o

oz¼ � 1

vg 2o

oA2o

ot� i

2b2o

o2A2o

ot2þ i

2k2o

o2

ox2þ o2

oy2

� �A2o

þ ix2o

n2ocd3A�1eA3e expðiDk3zÞ � a2o

2A2o ð2bÞ

oA3e

oz¼ � 1

vg 3e

oA3e

ot� i

2b3e

o2A3e

ot2þ i

2k3e

o2

ox2þ o2

oy2

� �A3e

� q3eðhÞoA3e

oyþ ix3e

n3ecd3A1eA2o expð�iDk3zÞ � a3e

2A3e

ð2cÞ

In Eqs. (1) and (2): Am is the optical field; the subscriptm = 1o, 1e, 2o, 2e, 3e denote the fundamental ordinaryfield, the fundamental extraordinary field, second harmonic(SH) ordinary field, SH extraordinary field and the thirdharmonic (TH) extraordinary field, respectively; nm is therefractive index; d2, d3 is the efficient nonlinear couplingcoefficient (for type I SHG d2 = �vsinhsin2U, and for typeII SFG d3 = vsin2hcos2U, where v is the nonlinear suscep-tibility, v = 0.78 pm/v (for KDP crystal), h is the anglebetween the crystal dielectric axes z and the beam vectork, h = h0 + Dh, h0 is the phase matching angle, Dh is thetilting angle, U is the azimuth angle); qm is the transversespatial walk-off coefficient of the extraordinary field; vg_m

is the group-velocity; Dk2 and Dk3 is the wave-vector-mis-matching of SHG and SFG, respectively; c is the lightvelocity in vacuum; aj is the absorbing coefficient.

Third harmonic field can be calculated from Eqs. (1) and(2) by applying numerical techniques such as split-stepFourier transformation and fourth-order Runge–Kuttamethod [6,25]. This numerical solution is base on the pre-mise that the amplitudes of the fields are slowly varying,and has adequate calculation precision if Dz (step size inthe propagation direct) and Dt (step size of the temporalcoordinate) is properly chosen.

2.2. The principle of broadband harmonic generation with

angular dispersion

For the sake of simplicity but without loss of practicalinterest, suppose that the input fundamental field is linearly

Δk2 (cm-1)

0-2

n(m

rad)

.1

.2

.3

.4

K. Li, B. Zhang / Optics Communications 281 (2008) 2271–2278 2273

frequency-chirped Gaussian pulse with super-Gaussianspatial intensity distribution, i.e.

Eðx; y; tÞ ¼ E0 exp � xw0

� �n

þ yw0

� �n� �� 1þ i

C2

� �tt0

� �2( )

ð3ÞIn Eq. (3), E(x, y, t) is the complex amplitude of the

input fundamental field; E0 is the maximum amplitude;w0 is the half width of the beam; t0 is the 1/e half widthof the pulse; C denotes the chirp parameter.

The instantaneous frequency of the input fundamentalfield is

xðtÞ ¼ x0 þ Ct=t20 ð4Þ

and the half width of the frequency spectrum becomes

Dx ¼ ð1þ C2Þ1=2=t0 ð5Þ

The intensity of the field can be obtained by

Iðx; y; tÞ ¼ 1

2cne j Eðx; y; tÞj2 ð6Þ

where n is the refractive index, e is the dielectric constant.Fig. 1 shows the different wavelength components of the

broadband fundamental field incident into the frequencyconversion crystal at different angles after dispersed bythe grating, where k0 is the central wavelength.

The refractive index of uniaxial negative crystal can bedetermined by the equation:

1

n2eðk; hÞ

¼ sin2ðhÞn2

eðkÞþ cos2ðhÞ

n2oðkÞ

ð7Þ

where the refractive index ne(k) and no(k) of KDP crystalare calculated from the Sellmeier equations [26], i.e.

n2oðkÞ � 1 ¼ 1:260476þ 12:99707

1� ð20=kÞ

2

þ 1:011279 � 10�2

k2 � 0:11376462ð8Þ

n2eðkÞ � 1 ¼ 1:133831þ 3:227935

1� ð20=kÞ2þ 8:653247 � 10�3

k2 � 0:1108752ð9Þ

As the directions of the wave-vector of the fundamentalfield and the second harmonic field are the same for type ISHG, the wave-vector-mismatching Dk2 can be written as

0

0

gratingcrystal

e

z

Δλ/2-λ

0 Δλ/2+λ

λ

Fig. 1. Scheme of broadband harmonic generation with ASD.

Dk2 ¼2pk2e

n2eðk2e; hÞ � 2� 2pk1o

n1oðk1oÞ ð10Þ

Combining with Eqs. (7)–(10), the variation of the wave-vector-mismatching Dk2 of type I SHG with the fundamen-tal wavelength k1 and the detuned angle Dh2in inside theKDP crystal can be calculated and shown in Fig. 2.

2.2.1. Collinear SFGFor type II SFG, under the precondition that the funda-

mental field and the SH field inside the mixing crystal arecollinear, the wave-vectors of the two incident fields andthe TH field are in the same direction, so the wave-vec-tor-mismatching Dk3 can be expressed as

Dk3 ¼2pk3e

n3eðk; hÞ �2pk2o

n2oðkÞ �2pke

neðk; hÞ ð11Þ

By using Eqs. (7,8,9) and (11), the variation of the wave-vector-mismatching Dk3 of type II SFG with fundamentalwavelength k1 and the detuned angle Dh3in inside theKDP crystal were calculated and shown in Fig. 3.

It can be seen from Figs. 2 and 3 that, phase matchingcondition can be satisfied in the whole bandwidth rangeof the fundamental field if every wavelength componentof the fundamental field incidents into the KDP crystal atproper angle. Within the bandwidth of 20 nm, the variationof the detuned angle with the wavelength of the fundamen-tal field is almost linear, i.e., the angular dispersion(D = dDh/dk) is nearly a constant for type I SHG and typeII SFG in KDP crystal, when the phase matching conditionis satisfied. The exact angular dispersion for the fundamen-tal field inside the doubling crystal (D2in, D2in = dDh2/dk1o)

02

λ1 (nm)1044 1047 1050 1053 1056 1059 1062

Δθ2 i

-.3

-.2

-.1

0.0

Fig. 2. Contour plot of the wave-vector-mismatching Dk2 vs. thefundamental wavelength k1 and the detuned angle Dh2in inside thedoubling KDP crystal. The variation in the wavelength of the fundamentallaser pulse can be compensated by adjusting the detuned angle of thedoubling KDP crystal. In this calculation, the cutting angle of the axis inthe KDP crystal is 41�110 so that phase matching can be achieved at afundamental wavelength of 1053 nm.

Δk3(cm-1)

-100

-50

0

50

100

λ1 (nm)1044 1047 1050 1053 1056 1059 1062

Δθ3i

n(m

rad )

-10.0

-5.0

0.0

5.0

10.0

Fig. 3. Contour plot of the wave-vector-mismatching Dk3 vs. thefundamental wavelength k1 and the detuned angle Dh3in inside the mixingKDP crystal. The variation in the wavelength of the fundamental laserpulse can be compensated by adjusting the detuned angle to the mixingKDP crystal. The SH wavelength is half of the fundamental wavelength,and the incident angle of the SH pulse inside the crystal is the same as thatof the corresponding fundament field. In this calculation, the cutting angleof the axis in the KDP crystal is 59� so that collinear phase matching canbe achieved at a fundamental wavelength of 1053 nm.

1(t)

KDP

2(t)= 1(t)/2

air

λ

λλ

Fig. 4. Refractions of the non-center wavelength components of thefrequency chirped-fundamental field and the SH field between theinterface of the air and the mixing KDP crystal.

k1

k2

2

31 m

c

air

KDPair k3

θθΔ

θΔθ θΔ

Fig. 5. Refraction of the non-center wavelength components of thefundamental field and the SH field between the interface of the air and themixing KDP crystal.

2274 K. Li, B. Zhang / Optics Communications 281 (2008) 2271–2278

and the fundamental inside the mixing crystal (D3in,D3in = dDh3in/dk1e = 0.5dDh3in/dk2o) is 33 lrad/nm and�1130 lrad/nm, respectively.

In this scheme, the center wavelengths of the fundamen-tal and the SH field are both vertical to the surface of thecrystal. However, considering the refractive effect betweenthe interface of the air and the crystal, the non-center wave-length components of the fundamental field (k1(t)) and thecorresponding SH field (k2(t) = k1(t)/2) are required to beincident at different angles (as shown in Fig. 4) to ensurethat they are collinear inside the mixing KDP crystal.

According to the Snell’s law, i.e.

sinðDhairÞ ¼ n sinðDhinÞ ð12Þwhere the incident angle of the fundamental field to theKDP surface is Dhair (for the center wavelength, Dhair = 0);Dhin is the refractive angle inside the KDP crystal, i.e., theinside detuned angle; n is the refractive index.

By using Eq. (12), the angular dispersion outside thecrystal for the fundamental field of SHG, the fundamentaland the SH field of the collinear SFG have been optimizedto be 50 lrad/nm, �1660 lrad/nm, �3420 lrad/nm,respectively.

2.2.2. Non-collinear SFG

As mentioned above, in order to ensure that the funda-mental extraordinary field and the SH ordinary fieldare collinear inside the mixing KDP crystal, the angulardispersion of them outside the crystal should be different.Consequently, two gratings with different angular disper-

sion are required, leading the complication of the experi-mental scheme of the collinear SFG.

Fortunately, a simpler scheme using one blazing gratingfor non-collinear SFG is available to achieve high conver-sion efficiency.

In the scheme of non-collinear SFG, the blazing gratingprovides high diffractive efficiency to the fundamental fieldat the first-order diffraction at the blaze angle, and so doesto the SH field at the second-order diffraction at the sameincident angle as the fundamental field. At the meantime,the angular dispersion of SH field is twice as that of thefundamental field.

Thus, after providing dispersion by the blazing grating,every wavelength component of the fundamental field(ke(t)) and the corresponding wavelength component ofthe SH field (k2o(t) = ke(t)/2) incident into the mixingKDP crystal at the same angle Dhair (as shown in Fig. 5).The center wavelength components of the two fields areincident vertically into the surface of the KDP crystal.

In Fig. 5, hair is the incident angle of the fundamentaland the SH field; hm is the phase matching angle; Dh1,Dh2, Dh3 are the detuned angle of the fundamental, theSH field and the TH field inside the mixing crystal,respectively.

It can be seen that the fundamental field and the SH fieldare non-collinear inside the mixing KDP crystal. However,

K. Li, B. Zhang / Optics Communications 281 (2008) 2271–2278 2275

broadband phase matching can be achieved if proper angu-lar dispersion is added.

As the wave-vector of the fundamental field, the SH fieldand the TH field is ~k1; ~k2 and ~k3, respectively, for the non-collinear type II SFG, the wave-vector-mismatching Dk3 isthe difference between the sum-vector~ksum ð~ksum ¼ ~k1 þ ~k2Þand the wave-vector of the TH field ~k3 at the sum-vector’sdirection, i.e.

Dk3 ¼ k3 � ksum ð13Þwhere

k1 ¼j ~k1 j¼2pk1

n1eðk1; h1Þ ð14Þ

k2 ¼j ~k2 j¼2pk2

n2oðk2Þ ð15Þ

ksum ¼j~ksum j¼j ~k1 þ ~k2 j¼ ½k2

1 þ k22 þ 2k1k2 cosðDh1 � Dh2Þ�1=2 ð16Þ

Dh3 ¼ Dh1 � arccosk2

1 þ k2sum � k2

2

2k1ksum

� �ð17Þ

k3 ¼j ~k3 j¼2pk3

n3eðk3; h3Þ ð18Þ

In Eqs. (14) and (18), hi (i = 1, 3) is the angle of the funda-mental field and the TH field between the direct of thewave-vector and the crystal’s optical axis. hi = hm + Dhi.

Using Eqs. (7)–(9) and (12)–(18), the variation of thewave-vector-mismatching Dk3 of the three wave interactionin the non-collinear SFG process with the fundamentalwavelength k1 and the outside detuned angle Dhair can becarried out, as shown in Fig. 6.

Δk3 (cm-1)

-100

-500

050

100

λ1 (nm)1044 1047 1050 1053 1056 1059 1062

Δθai

r (m

rad)

-20

-10

0

10

20

Fig. 6. Contour plot of the wave-vector-mismatching Dk3 vs. thefundamental wavelength k1 and the detuned angle Dhair outside themixing KDP crystal. The variation in the wavelength of the fundamentallaser pulse can be compensated by detuning the incident angle to themixing KDP crystal. The SH wavelength is half of the fundamentalwavelength, and the incident angle of the SH pulse outside the crystal isthe same as that of the corresponding fundament field. In this calculation,the cutting angle of the axis in the KDP crystal is 59�.

It can be seen from Fig. 6 that phase matching conditioncan also be achieved for non-collinear broadband SFGwhen the angular dispersion is about �2450 lrad/nm forthe fundamental field (and twice as that for the SH field)outside the mixing KDP crystal.

3. Results and analysis

As is well known that, multiple factors [6,20,27], such asthe parameters of crystals, the pulse shape, the spatialphase and the spectrum distribution of the input fields,affected the conversion efficiency. In order to focus onexplanation of our method, numerical examples are givento illustrate the validity of the non-collinear method inimproving the conversion efficiency, the pulse shape andthe spectrum distribution of the TH field in the followingdiscussion.

The optimized parameters for KDP crystals are shownin Table 1. The parameters of the fundamental field:k = 1.053 lm, n = 12, t0 = 0.1 ns. C is set to be 2000 asan example to simulate the case with large bandwidth(the corresponding bandwidth is 14 nm).

Fig. 7 shows the pulse shapes and the spectrum distribu-tions of the input fundamental field (the solid line), the SHfield of the SHG process (the dotted line) and the third har-monic field of the SFG process (the dash line) withoutASD.

The TH conversion efficiency without ASD is only 4%for the case of the input fundamental field with the band-width of 14 nm. Moreover, Fig. 7 shows that the shapeof the pulse and the spectrum of the TH field are narrowedand distorted severely. One of the main restriction of theconversion efficiency and the bandwidth of TH field isthe acceptance bandwidth of the conversion crystal. Theacceptance bandwidth of KDP is about 2 nm and 0.2 nmfor SHG and SFG, respectively [27].

3.1. Collinear SFG

The variations of the conversion efficiency of the SHfield of the SHG process and the TH field of the SFG pro-cess with the bandwidth of the input fundamental fieldwithout ASD and with ASD for the scheme of collinearSFG are compiled in Fig. 8.

In Fig. 8, the solid line and the dotted line denote theconversion efficiency of the TH field of SFG process andthe SH field of SHG process without ASD, respectively.The short dash line denotes the TH field of SFG whenASD is added only to the SFG process. The dash–dot line

Table 1The calculation parameters for KDP crystals

Length (cm) Dh (lrad) U (�) hm

Doubler 1.1 220 45 41�110

Mixer 0.9 0 0 59�

bandwidth (nm)0 2 4 6 8 10 12 14

conv

ersi

on e

ffici

ency

0.0

.2

.4

.6

.8

1.0

3ω, D2=0, D3=02ω, D2=03ω, D2=0, D3= -1130μrad/nm2ω, D2=30μrad/nm3ω, D2=30μrad/nm, D3= -1130μrad/nm

Fig. 8. Variation of the conversion efficiency of the SH and the TH fieldwith the bandwidth of the input fundamental field for the scheme ofcollinear SFG.

λj-λj0 (nm)-30 -20 -10 0 10 20 30

norm

aliz

ed in

tens

ity

0.0

.2

.4

.6

.8

1.0

1.21ω2ω3ω

t (ps)-.3 -.2 -.1 0.0 .1 .2 .3

inte

nsity

(GW

/cm

2 )

0

1

2

3

4

5

61ω2ω3ω

a

b

Fig. 7. The pulse shapes (a) and the spectrum distributions (b) of the inputfundamental field (the solid line), the SH field of the SHG process (thedotted line) and the TH field of the SFG process (the dash line) withoutASD.

2276 K. Li, B. Zhang / Optics Communications 281 (2008) 2271–2278

represents the SH field of SHG process with ASD. Thelong dash line represents the third harmonic field of SFGprocess with ASD adding to both the SHG and the THGprocesses.

Fig. 8 implies that, for the case without ASD, the band-width of the input fundamental field has little effect on theconversion efficiency of the SH field (e.g., only decreasesfrom 60% to 57%), while it has significant effect on the con-version efficiency of the TH field. The conversion efficiencyof the TH field is about 80% for the case of the narrow-band. However, it decreases to 40% for the bandwidth of1 nm and even less to 4% for the bandwidth of 14 nm.The conversion efficiency of the TH field decreases sharplywith the increasing of the bandwidth of the fundamentalfield due to phase mismatching of the non-center wave-length components of the fundamental and the SH field.When proper angular dispersion is added only to theSFG process, the conversion efficiency of the TH fieldincreases to 50% for the bandwidth of 14 nm. However,when proper angular dispersions are added to both theSHG and the SFG processes, the conversion efficiency ofthe TH field can be increased greatly to be 75%, and theconversion efficiency maintains high in the range of thebandwidth of 0–10 nm.

For collinear broadband harmonic generation, thewave-vector-mismatching:

Dk ¼ oDkok

� �Dkþ oDk

oh

� �DhþO Dkð Þ2 þO Dhð Þ2 þ � � �

ð19Þ

By introducing angular dispersion Din ¼ oDkok =

oDkoh and ignor-

ing the second and higher order item of the Eq. (19),Dk ¼ oDk

ok

� �Dkþ oDk

oh

� �Din � Dkð Þ � 0. The waves with differ-

ent frequency components can propagate with differentincident angles into the crystal so that the phase matchingcondition is satisfied for the entire frequency componentsof the incident waves.

3.2. Non-collinear SFG

The variation of the conversion efficiency of the SH fieldof the SHG process and the TH field of the SFG processwith the bandwidth of the input fundamental field withoutASD and with non-collinear ASD are compared in Fig. 9.

It can be shown from the comparison between Figs. 9and 8 that the scheme of non-collinear SFG can alsoimprove the conversion efficiency of the broadband THGsignificantly as the collinear scheme. However, the non-col-linear scheme is much simpler than the collinear schemebecause of using one blazing grating.

For the non-collinear THG of broadband laser (e.g., thebandwidth of 14 nm), when ASD is only added to the SFGprocess and the angular dispersion of the fundamental fieldoutside the mixing crystal (D3) is �2.450mrad/nm, THGconversion efficiency of 55% can be achieved, which ismuch higher than that without ASD (about 4%). More-over, when ASD is added to both the SHG and the SFGprocesses, and the angular dispersion of the fundamentalfield outside the doubling crystal (D2) and outside the mix-ing crystal (D3) are chosen to be 40 lrad/nm and

bandwidth (nm)0 2 4 6 8 10 12 14

conv

ersi

on e

ffici

ency

0.0

.2

.4

.6

.8

1.0

3ω, D2=0, D3=02ω, D2=03ω, D2=0, D3= -2450μrad/nm2ω, D2=40μrad/nm

3ω, D2=40μrad/nm, D3= -2450μrad/nm

Fig. 9. Variation of the conversion efficiency of the SH field (2x) and theTH field (3x) with the bandwidth of the input fundamental field.

conversion efficiency

.35

.45

.40

.50

.50.45.40.35 .30

.55

.55 .60.65.70.75

.70.65

.30

.60

.50

D2 (μrad/nm)0 10 20 30 40 50 60 70 80

D3 (

μ rad

/nm

)

-2700

-2600

-2500

-2400

-2300

-2200

Fig. 11. Contour plot of the TH conversion efficiency vs. the outsideangular dispersion D2 and D3 for the scheme of non-collinear SFG.

K. Li, B. Zhang / Optics Communications 281 (2008) 2271–2278 2277

�2.450mrad/nm, respectively, the maximum TH conver-sion efficiency of 78% can be achieved.

The corresponding pulse shapes and spectrum distribu-tions of the input fundamental field (the solid line), theSH field of the SHG process (the dotted line) and the third

t (ps)-.3 -.2 -.1 0.0 .1 .2 .3

inte

nsity

(GW

/cm

2 )

0

1

2

3

4

5

61ω2ω3ω

bandwidth (nm)-30 -20 -10 0 10 20 30

norm

aliz

ed in

tens

ity

0.0

.2

.4

.6

.8

1.0

1.21ω2ω3ω

a

b

Fig. 10. The pulse shapes (a) and the spectrum distributions (b) of theinput fundamental field (the solid line), the SH field of the SHG process(the dotted line) and the TH field of the SFG process (the dash line) withASD for the scheme of non-collinear SFG.

harmonic field of the SFG process (the dash line) with ASDfor the scheme of non-collinear SFG are compiled inFig. 10a and b.

It can be seen from Fig. 10 that, if proper angular dis-persions are added to the SHG and the non-collinearSFG process, the conversion efficiency and the peak inten-sity of the TH field increase significantly. The pulse shapeof the TH field becomes smoother and more uniform com-pared with Fig. 7 (without ASD). In addition, the spectralwidth of the SH field of the SHG process and the TH fieldof the SFG process are both increased. Especially, the spec-tral width of the TH field increases to be about 1/3 of theinput fundamental field. This further demonstrates that,compared with the scheme of collinear SFG, the schemeof non-collinear SFG is equivalently effective in improvingboth the conversion efficiency and the characteristics of theTH field.

In order to show the effect of the angular dispersion onthe conversion efficiency, Fig. 11 gives the variation of theTH conversion efficiency with the angular dispersion D2

(outside the doubling crystal for the fundamental field ofthe SHG process) and D3 (outside the mixing crystal forthe fundamental field of the SFG process) using the non-collinear scheme. It can be seen from Fig. 11 that, theTH conversion efficiency keeps above 60% for D2 in therange from 10 to 70 lrad/nm and D3 in the range from�2350 to �2550 lrad/nm.

4. Conclusion

In this paper, a new scheme with non-collinear SFGprocess of broadband THG with grating angular disper-sion was proposed and analyzed. Comparisons of thenon-collinear SFG process with the collinear SFG processwere made.

2278 K. Li, B. Zhang / Optics Communications 281 (2008) 2271–2278

For the third harmonic generation with two KDP crys-tals (one doubler and one mixer) of the large bandwidthlaser, phase mismatching caused by the bandwidth can bewell compensated if proper angular dispersions are addedin the SHG and the SFG processes.

If the outside angular dispersion of input fundamentalfield of the SHG process, the fundamental and the SH fieldof the SFG process is chosen to be 50 lrad/nm,�1660 lrad/nm and �3420 lrad/nm outside the crystals,respectively, the fundamental and the SH field is collinearinside the mixing KDP crystal, and the third harmonic con-version efficiency can be improved greatly to above 75%. Inthis scheme, two gratings are needed to the SFG process.

A simpler scheme with one blazing grating to the SFGprocess is found to be also effective. Every wavelength com-ponent of the fundamental and the corresponding wave-length component of the SH field are incident into themixing crystal at same angles. They are non-collinear insidethe mixing crystal except the center wavelength after pro-viding dispersion by a blazing grating with proper angulardispersion (e.g., �2450 lrad/nm for the fundamental fieldand twice as that for the SH field outside the crystal). Phasemismatching of the nonlinear SFG process can also be wellcompensated, and similar results can be obtained com-pared with the scheme of collinear SFG.

In addition, this method of using one blazing grating togenerate appropriate angular dispersion to the fundamen-tal and the SH field is not limited to the KDP crystal only.It is also effective to the certain kind of frequency conver-sion crystals with nearly linear angular dispersions, in otherwords, the detuned angle has nearly proportional relation-ship with the input wavelength when phase-matched, suchas BBO crystal etc. [28].

Acknowledgements

This work was supported by the Youth Science Founda-tion of Sichuan Province (No. 05ZQ02613) and the Pro-gram for New Century Excellent Talents in University(NCET-05-0784).

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