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Journal of Nonlinear Optical Physics & MaterialsVol. 26, No. 1 (2017) 1750014 (10 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S021886351750014X

Bandwidth analysis of non-collinear fourth and fifth harmonicgeneration in nonlinear uniaxial crystals

Dongsheng Song, Yuanlin Zheng, Xiaohui Zhao,Zengyan Cai and Xianfeng Chen∗

State Key Laboratory of Advanced Optical Communication Systems and Networks,School of Physics and Astronomy, Shanghai Jiao Tong University,

800 Dongchuan Road, Shanghai 200240, P. R. China

Key Laboratory for Laser Plasma (Ministry of Education),Collaborative Innovation Center of IFSA (CICIFSA),

Shanghai Jiao Tong University,800 Dongchuan Road, Shanghai 200240, P. R. China

∗xfchen@sjtu.edu.cn

Received 23 February 2014

The optimal angle bandwidth and wavelength bandwidth of fourth-harmonic generation(FHG) and fifth-harmonic generation (FIFHG) of the 1064 nm laser are analyzed basedon the numerical calculation results of non-collinear type-I and type-II phase match-ing processes for general nonlinear uniaxial crystals with 1 cm length. The non-collinearphase matching angles and effective nonlinear coefficients of FHG and FIFHG are cal-culated. The optimal angle bandwidth and wavelength bandwidth are obtained. The

results are beneficial to broadband and efficient non-collinear phase matching FHG andFIFHG experiments and studies.

Keywords: FHG; FIFHG; non-collinear phase matching; angle bandwidth; wavelengthbandwidth.

1. Introduction

The broadband high efficiency fourth harmonic generation (FHG) of lasers cannotonly be used as the optical probe of physical diagnostics,1 but also has the probabil-ity of directly interacting with the nuclear target in ICF instead of third harmonicgeneration (THG) lasers widely used at present.2 Shorter wavelength lasers increasethe energy absorption efficiency, reduce the instability of lasers and plasmas, andsuppress stimulated Brillouin scattering (SBS) and stimulated Raman scattering(SRS).3 However, FHG can only support narrow bandwidth lasers limited by thestrong dispersion properties of materials in the UV region, which makes it difficultto combine it with the present beam smoothing methods such as smoothing by

∗Corresponding author.

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D. Song et al.

spectral dispersion (SSD),4 so it is necessary to study the basic theory and developkey technologies of producing efficient FHG for physical applications.

Second harmonic generation (SHG) and THG have been widely studied to getultrashort and ultra-intense lasers, and many effective technology methods havebeen used for the phase matching of broadband pulses, such as achromatic phasematching,5 chirp matched harmonic generation,6 multi-crystals design,7 quasiphase matching,8 retracing point phase matching,9 etc. However, the non-collinearphase matching technology is relatively a new study direction for its free geometryconfiguration,10 and it can be used in the prefocusing scheme to get higher damagethreshold to avoid high-power laser damage.11 It is the primary purpose of thispaper to get optimal bandwidths for a cone or parallel laser beam in the prefocus-ing scheme with a narrowband frequency propagating in a thin nonlinear crystal(1 cm), to find a proper material for the broadband experiment and study.

In this paper, non-collinear phase matching theory is analyzed firstly, which pro-vides the numerical calculation foundation for non-collinear FHG and FIFHG phasematching angle (NCPMA), effective nonlinear coefficients (ENC), two kinds of anglebandwidths (AB) and wavelength bandwidth (WB) of a 1064nm fundamental laser,using KDP as an example in Sec. 2. In Sec. 3, the optimal angle bandwidth andwavelength bandwidth and the corresponding phase matching angle and effectivenonlinear coefficients are numerically calculated in the same procession as FHG ofKDP for general nonlinear crystals such as DKDP, ADP, BBO, CLBO, and KBBF.

2. Non-Collinear Phase Matching Theory and Calculationin KDP Crystal

FHG and FIFHG non-collinear phase matching theory is similar to SHG or THG,which can all be described by the three-wave coupled equation.12 There are twomethods to get FHG 226 nm (4ω) laser, one is sum-frequency generation (SFG)of a fundamental wave of 1064nm (ω) and its third harmonic wave 355nm (3ω)(ω + 3ω → 4ω), the other is SHG of a 532nm (2ω) laser (2ω + 2ω → 4ω). Bothmethods are calculated in type-I (o + o → e) and type-II (e + o → e) in this paper.Meanwhile, the two methods for the FIFTH 213 nm (5ω) laser are SFG of a 1064nm(ω) laser and a 226 nm laser (ω+4ω → 5ω), or SFG of a 532nm laser and a 355 nmlaser (2ω + 3ω → 5ω) with two types same as the FHG. The NCPMA, ENC, ABand WB are calculated for the KDP crystal with 1 cm.

The calculation of angle bandwidth is based on two schemes. In scheme one, thetwo inject waves are both cone beams, and in scheme two the first beam is parallel,and the second beam is cone. These two schemes are both useful for the prefocusingscheme used to avoid high power laser focusing damage.11

2.1. Non-collinear phase matching angle

The geometry configuration of non-collinear phase matching has been analyzed inRef. 13. The following equations are used to calculate the type-I phase matching

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Bandwidth analysis of non-collinear fourth and fifth harmonic generation

angles for a given exit angle θ3,

(n1ω1

c

)2

+(

n3(θ3)ω3

c

)2

−(n2ω2

c

)2

= 2n1ω1

c

n3(θ3)ω3

ccos(θ3 − θ1), (2.1)

(n2ω2

c

)2

+(

n3(θ3)ω3

c

)2

−(n1ω1

c

)2

= 2n2ω2

c

n3(θ3)ω3

ccos(θ2 − θ3), (2.2)

where θ1, θ2 and θ3 are, respectively, the angle between z axis and three waves.For type-II, the equations are almost the same, except that n1 is an extraordinarylight and should be replaced by n1(θ1). For KDP crystal, the phase matching anglesfor FHG by non-collinear mixing the fundamental wave at 1064nm and the thirdharmonic wave at 355 nm are numerically calculated by the above equations. Twotypes of phase matching angle relationship curves are shown in Fig. 1.

In Fig. 1(a) for type-I, the region of θ3 satisfying the non-collinear phase match-ing condition is from 59.95◦ to 120.05◦, and there are two pairs of θ1 and θ2 foreach given θ3. The blue pair of curves is for θ1 < θ3 < θ2 and the red pair isfor θ1 > θ3 > θ2. Points A and B, θ1 = θ3 = θ2, represent the collinear phasematching angles for type-I. Similarly, the region of θ3 is from 72.03◦ to 107.97◦

for type-II in Fig. 1(b). The collinear phase matching angle is marked as point E,but meanwhile, for this θ3 at point E, there is another θ1 and θ2 fitting the non-collinear triangle. However, at points C and D, there is only one non-collinear phasematching situation, while point C is for θ2 and point D is for θ1. Similar to type-I,the blue pair is for θ1 < θ3 < θ2 and the red pair is for θ1 > θ3 > θ2. The twopairs are center symmetry with each other about the point of (90◦, 90◦) in bothfigures, because of the axis symmetry of the uniaxial crystal. The phase matchingangle region of type-I is bigger than type-II, the same situation also happens inother crystals calculated in Sec. 3. This provides a convenient condition for theexperiment.

(a) (b)

Fig. 1. FHG NCPMA. The blue curve is for θ1 < θ3 < θ2, the red curve is for θ1 > θ3 > θ2. (a)Type-I. Point A and B represent the collinear point. (b) Type-II. Point E represents the collinearpoint, Point C and D represent the smallest value of θ2 and θ1 fitting non-collinear condition.

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(a) (b)

Fig. 2. (a) Type-I FHG ENC of KDP. (b) Type-II FHG ENC of KDP.

2.2. Effective nonlinear coefficients

The ENC for the nonlinear crystals is distinguished by the point group theybelong to. The expression for different point group may be different.14 The non-collinear ENC is more complex than the collinear situation. For KDP belong-ing to 4̄2m point group, the expression is deff =−d36 sin θ3 sin 2ϕ for type-I anddeff = d14 sin θ1 cos θ3 cos 2ϕ+d36 cos θ1 sin θ3 cos 2ϕ for type-II. ENC for FHG type-I deff and FHG type-II d2

eff — the square index is eliminating the negative valueappearing in type-II — is calculated and shown in Figs. 2(a) and 2(b) respectivelybased on the numerical calculation results of the FHG phase matching angles forKDP crystal. The blue curve represents θ1 < θ3 < θ2 and the red curve representsθ1 > θ3 > θ2. For type-I, the ENC of θ1 < θ3 < θ2 matching type is bigger whenθ1 < 90◦, so it should be chosen to get higher conversion efficiency. It should choosethe θ1 > θ3 > θ2 matching type for type-II for the same reason. Overall, the ENCof type-I is much larger than that of type-II.

2.3. Angle bandwidth

According to the relationship of the conversion efficiency and phase mismatchingfactor on the conditions of slowly varying amplitude approximation and small sig-nal approximation, one can easily define the bandwidth of angle or wavelength.Meanwhile, the phase matching factor can be expressed as

∆k = ∆k0 +∂(∆k)

∂θ

∣∣∣∣θ=θm

∆θ +∂(∆k)

∂λ

∣∣∣∣λ=λ0

∆λ + · · · , (2.3)

where θm is the phase matching angle, and λ0 is the center wavelength of the beam.Two kinds of schemes mentioned in Sec. 2.1 are calculated for each set of θ1,

θ2, θ3. In scheme one the two inject beams are cone beams, where the edge-ray canbe considered as the center beam rotates by a small angle. Because every ray of�k1 with a small departure angle from the center ray must has a corresponding rayof �k2 with the same departure angle, the geometry relationship between the twodeparture rays is the same as the center rays (Fig. 3(a)). While in scheme two thefirst fundamental beam is a parallel light, and the second fundamental beam is a

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Bandwidth analysis of non-collinear fourth and fifth harmonic generation

k1 k

2

k3

∆k∆θ1 k

1

k2

k3

∆k

(a) (b)

Fig. 3. Two kinds of angle bandwidth scheme. (a) �k1 and �k2 are both cone beams and (b) �k1 iscone and �k2 is parallel.

cone beam, in this situation, the changing of θ2 causes the phase mismatching. Theschematic diagrams for the angle bandwidth of the two institutions are shown inFig. 3, separately.

Figure 4 shows the numerical calculation results of the angle bandwidth versusθ3 for KDP crystal. For type-I (the dash-dotted curves), the blue curve showsthat the angle bandwidth has the maximum value of ∆θ1 = 1.813◦ at the non-critical angle of θ3 = 90◦ for the cone inject beams. The red one shows that thefirst fundamental beam is parallel and the second fundamental beam is cone. Itshows that the maximum value is ∆θ2 = 0.9935◦ at the minimal value of θ1 whenθ3 = 62.2◦. Meanwhile, for type-II (the solid curves), the angle of the maximumvalue is a little larger than that from type-I for cone beams, and the maximum anglebandwidth is 1.9897◦ when θ3 = 91.02◦, which is 0.9776◦ for the other matchingsituation when θ3 = 73.77◦. The situation is almost the same for other crystalsmentioned below.

Fig. 4. Non-collinear FHG angle bandwidth of scheme one (blue curves) and scheme two (redcurves) with type-I (dash-dotted curves) and type-II (solid curves) for KDP.

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(a) (b)

Fig. 5. FHG Wavelength bandwidth for KDP. (a) Wavelength bandwidth region. (b) The maxi-mum wavelength bandwidths for type-I (blue curve) and type-II (red curve).

2.4. Wavelength bandwidth

With the frequency relationship between the three beams, ∆ω2 = 2∆ω1, ∆ω3 =3∆ω1, we have ∆λ2 = − λ2

22πc∆ω2 = − λ2

18πc2∆ω1 = − 1

2λ21

2πc∆ω1 = ∆λ1/2 and ∆λ3 =∆λ1/3. Figure 5 shows the numerical calculation results based on the definitionof wavelength bandwidth in Eq. (2.3). Figure 5(a) evidences that the wavelengthbandwidth for type-I has the maximum value around the uncritical angle, shownas the yellow line, and the range of half wavelength bandwidth for all the possiblephase matching situation is within about 0.012 nm for the KDP crystal, shown asthe lines in the region between the yellow line and blue line. Figure 5(b) shows themaximum wavelength bandwidths for type-I (the blue curve) and type-II (the redcurve), they are more or less equal to each other, which evidences that the phasematching situation has almost no influence on the WB. The situation is almost thesame for other crystals mentioned in Sec. 3.

3. Calculation in Other Crystals and Discussion

We calculate the NCPMA, ENC, optimal AB and WB for FHG and FIFHG ofother usual uniaxial crystals in the same method as for the KDP crystal in Sec. 2.These crystals are DKDP, ADP, BBO, BeSO4·4H2O, DADP, ADA, DADA, CLBO,KABO, BABF, KBBF, LB4. They are selected for two reasons, one is that theirtransparencies include the FHG and FIFHG of a 1064nm laser,15,16 the other is thatthe non-collinear phase matching condition can be satisfied. The results are shownin Tables 1 and 2. Not all crystals calculated are mentioned in the table but theones with the best performance in one or two items are mentioned. Table 1 showstype-I non-collinear phase matching results, and Table 2 shows type-II non-collinearphase matching results. Each table contains the angle bandwidth for the cone beams(ABCC), the angle bandwidth for the first beam is parallel and the second beam iscone (ABPC), wavelength bandwidth (WB), the corresponding non-collinear phasematching angle (NCPMA) and effective nonlinear coefficient (ENC).

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Bandwidth analysis of non-collinear fourth and fifth harmonic generation

Table

1.

Calc

ula

tion

resu

lts

for

type-

I.

AB

CC

NC

PM

AE

NC

AB

PC

NC

PM

AE

NC

WB

NC

PM

AE

NC

Cry

stal

Sch

eme

(◦)

(◦/◦ )

a(p

m/V

)(◦

)(◦

/◦ )

(pm

/V

)(n

m)

(◦/◦ )

(pm

/V

)

KD

P4(1

+3)b

1.8

128

90/77.3

70.3

90.9

935

62.2

1/57.5

90.3

450

0.4

142

89.7

9/77.1

60.3

900

4(2

+2)

1.8

122

90/86.6

90.3

90.8

801

77.0

3/76.2

50.3

800

0.1

48

90/86.6

90.3

900

DA

DP

4(1

+3)

1.7

876

90/77.2

90.4

30.9

883

62.3

5/47.6

80.3

809

0.4

006

89.7

9/77.0

80.4

300

4(2

+2)c

1.7

888

90/87.2

70.4

30.8

761

79.4

5/78.7

90.4

227

0.1

3938

90/87.2

70.4

300

5(1

+4)d

1.4

685

90/86.1

30.4

30.8

851

83.8

9/82.0

30.4

275

0.2

011

90.0

5/86.1

90.4

300

BB

O4(1

+3)

1.0

978

90/58.3

30.1

60.9

585

47.8

7/32.6

41.6

61

0.2

759

89.1

5/57.4

90.1

941

4(2

+2)

1.0

978

90/74.4

10.1

60.8

400

50.0

1/45.2

11.6

01

0.0

9055

90/74.4

10.1

600

5(1

+4)

0.8

996

90/57.3

40.1

60.9

025

60.5

0/40.7

11.2

72

0.1

091

90.8

1/58.1

70.1

275

5(2

+3)

0.8

994

90/79.3

70.1

60.7

028

71.8

6/67.0

30.8

681

0.0

6383

90.1

7/79.5

40.1

532

BeS

O4·4

H2O

4(1

+3)

1.9

172

90/78.8

80.2

31.0

061

64.4

5/60.4

90.2

075

0.3

856

89.8

1/78.6

90.2

300

4(2

+2)

1.9

172

90/87.9

80.2

30.8

915

81.7

6/81.3

10.2

276

0.1

396

90/87.9

80.2

300

CLB

O4(1

+3)

1.6

42

90/72.0

80.7

41.1

124

53.7

8/47.1

20.7

422

0.3

716

89.8

4/71.9

20.7

400

4(2

+2)

1.6

421

90/82.3

60.7

40.8

88

62.3

4/60.5

00.6

554

0.1

3349

90/82.3

60.7

400

5(1

+4)

1.3

86

90/76.9

80.7

40.8

992

70.5

9/64.2

00.6

979

0.1

56

90.3

1/77.2

80.7

400

KB

BF

4(1

+3)

1.4

472

90/61.8

10

1.1

11

36.0

5/26.8

30.3

962

0.6

866

88.7

4/60.5

60.0

108

4(2

+2)

1.4

473

90/75.0

00

0.9

239

37.9

1/34.8

50.3

866

0.2

399

90/75.0

00

5(1

+4)

1.2

691

90/59.0

80

1.0

281

43.9

1/30.6

50.3

530

0.3

702

91.3

4/60.4

30.0

115

5(2

+3)e

1.2

695

90/74.2

50

0.8

186

49.5

7/44.8

40.0

778

0.2

275

90.4

8/74.7

40.0

010

Note

:a(θ

3/θ 1

).b4(1

+3):

ω+

3ω→

4ω.

c4(2

+2):

2ω

+2ω

→4ω.

d5(1

+4):

ω+

4ω→

5ω.

e5(2

+3):

2ω

+3ω

→5ω.

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D. Song et al.

Table

2.

Calc

ula

tion

resu

lts

for

type-

II.

AB

CC

NC

PM

AE

NC

AB

PC

NC

PM

AE

NC

WB

NC

PM

AE

NC

Cry

stal

Sch

eme

(◦)

(◦/◦ )

a(p

m/V

)(◦

)(◦

/◦ )

(pm

/V

)(n

m)

(◦/◦ )

(pm

/V

)

KD

P4(1

+3)b

1.9

898

91.0

9/84.0

60.0

330

0.9

776

73.7

7/70.7

50.2

264

0.3

857

87.9

6/80.9

30.0

751

DA

DP

4(1

+3)

2.0

04

90.8

5/85.9

10.0

243

0.9

687

78.2

2/75.9

20.1

875

0.3

95

89.8

9/84.8

00.0

398

BB

O4(1

+3)

1.1

957

93.8

8/66.8

30.0

6124

0.9

36

52.0

2/36.9

51.1

31

0.2

77

90.8

6/63.4

20.0

154

4(2

+2)c

1.4

382

90.8

4/87.9

90.0

012

0.8

113

81.1

6/79.6

70.0

634

0.0

874

89.8

0/86.8

70.0

004384

5(1

+4)d

0.9

271

86.7

9/58.2

80.0

677

0.7

895

64.0

6/44.9

20.7

124

0.1

074

90.7

7/62.4

80.0

1428

BeS

O4·4

H2O

4(1

+3)

2.1

49

90.1

9/89.0

90.0

029

0.9

865

87.2

9/86.8

00.2

368

0.3

698

89.6

8/88.5

70.0

070

CLB

O4(1

+3)

1.8

289

92.1

0/79.3

70.1

098

0.9

921

62.3

9/56.5

80.6

474

0.3

856

89.6

2/76.7

30.1

746

KB

BF

4(1

+3)

1.6

267

94.3

1/70.5

90.0

1224

1.0

870

40.5

6/30.9

00.3

194

0.7

532

97.5

0/74.2

50.0

1736

4(2

+2)

2.0

014

92.4

4/84.8

70.0

019

0.9

001

56.7

6/53.8

60.1

584

0.2

448

90.7

4/83.0

60.0

00765

5(1

+4)

1.3

327

85.2

9/57.8

60.0

214

1.0

058

47.6

8/34.3

10.2

725

0.3

876

94.7

7/68.0

30.0

1524

5(2

+3)e

1.5

968

91.7

7/85.2

20.0

0126

0.7

932

70.2

6/67.1

80.0

6419

0.2

302

90.5

6/83.9

10.0

005081

Note

:a(θ

3/θ 1

).b4(1

+3):

ω+

3ω→

4ω.

c4(2

+2):

2ω

+2ω

→4ω.

d5(1

+4):

ω+

4ω→

5ω.

e5(2

+3):

2ω

+3ω

→5ω.

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In Table 1 for type-I, the results evidence that for ABCC, BeSO4 · 4H2O has themaximum value, but CLBO has the maximum ENC value. For ABPC, CLBO hasthe maximum value, while BBO has the maximum ENC value. For WB, KBBF hasthe maximum value, but CLBO has the maximum ENC value. Overall, for everykind of bandwidth, CLBO has bigger ENC value than most of others, KDP andDADP have the best aggregate performance.

In Table 2 for type-II, the results evidence that for ABCC, BeSO4 · 4H2O hasthe maximum value, but CLBO has the maximum ENC value. For ABPC, KBBFhas the maximum value, while BBO has the maximum ENC value. For WB, KBBFhas the maximum value, but CLBO has the maximum ENC value. Overall, forevery kind of bandwidth, CLBO has bigger ENC value than most of others, KDPand DADP have the best aggregate performance.

The optimal non-collinear phase matching angle bandwidth is much bigger thanthat of collinear situation, while the wavelength bandwidth is more or less the same.The reason is that, for the collinear angle bandwidth, the phase matching happensat the intersection point of the �k1 +�k2 curve and �k3 curve in the wave vector space,while the phase matching for the optimal bandwidth happens at the tangent pointfor the non-collinear situation. Therefore, the phase mismatch factor changes moreslowly for the same angle changing in the latter. But for the wavelength bandwidth,there is no such tangent point as the angle wavelength. Although BBO and KBBFhave bigger angle bandwidths or wavelength bandwidths, they are difficult to growto a proper size for most applications.17,18 CLBO has easy deliquescence in atmo-spheric environment.19 Overall, KDP crystal has the best aggregate performance.

4. Conclusion

We have analyzed the theory of non-collinear phase matching for FHG and FIFHG.Two kinds of angle bandwidth configurations and one kind of wavelength bandwidthmodel were investigated. The phase matching angles for type-I and type-II FHGand FIFHG were numerically calculated in detail, and the maximum angle band-width and wavelength bandwidth were obtained for all the possible uniaxial crystalssatisfying the conditions as we know so far. The results are of use to help the exper-imentalists choose the proper crystal and suitable configuration for non-collinearFHG or FIFHG at 1064nm.

Acknowledgments

This work was supported in part by the National Natural Science Foundation ofChina Under Grant Nos. 61235009 and 11604206.

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