Abstract

We present a theoretical model to deal with second harmonic generation (SHG) of ultrashort pulses in refractive-index-linear-modulating (RILM) nonlinear crystals. New coupled-wave equations were derived based on plane wave propagation. Under the undepleted-pump approximation, the analytic solutions for SHG of ultrashort pulses were obtained. The second harmonic (SH) pulse width and the efficiency were optimized in terms of the RILM parameter and the chirp coefficient of fundamental pulses. The results show that modulation of the refractive index (RI) in nonlinear crystals leads to a spectral localization of SHG that compensates the group velocity mismatch (GVM) and therefore compresses SH pulses. The modulation complicates the SHG process, which determines the pulse width, and the conversion efficiency varies with length and bandwidth. Especially, an overshooting compression phenomenon was found, which arose mainly from internal interference of the carrier frequency component in a SH pulse.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. J. J. Huang, W. Gao, T. Shen, B. L. Mao, Y. M. Andreev, A. V. Shaiduko, G. V. Lanskii, U. Chatterjee, and V. V. Atuchin, “Influence of composition ratio variation on optical frequency conversion in mixed crystals. II. Random variation of composition ratio,” J. Opt. Soc. Am. B 24, 3081–3090 (2007).
    [CrossRef]
  20. J. J. Huang, B. L. Mao, H. Dai, T. Shen, and L. Y. Zhang, “Research on static frequency doubling in refractive-index modulated crystals,” Acta Opt. Sin. 30, 2634–2638 (2010).
    [CrossRef]

2012

2011

J. J. Huang, L. Y. Zhang, and T. Shen, “Properties of periodic multicrystal configurations in walk-off-compensating second harmonic generation of ultrashort pulses,” Chin. Phys. B 20, 044206 (2011).
[CrossRef]

2010

J. J. Huang, B. L. Mao, H. Dai, T. Shen, and L. Y. Zhang, “Research on static frequency doubling in refractive-index modulated crystals,” Acta Opt. Sin. 30, 2634–2638 (2010).
[CrossRef]

2008

J. J. Huang, Y. Q. Chang, T. Shen, and Y. Q. Yang, “Proposal high quality walk-off compensated sum frequency generation of ultra-short pulses,” Opt. Commun. 281, 5244–5248 (2008).
[CrossRef]

2007

2006

2005

2004

P. Baum, S. Lochbrunner, and E. Riedle, “Tunable sub-10-fs ultraviolet pulses generated by achromatic frequency doubling,” Opt. Lett. 29, 1686–1688 (2004).
[CrossRef]

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, “Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology,” J. Cryst. Growth 264, 565–577 (2004).
[CrossRef]

2003

2002

2000

1998

1997

1995

R. A. Hass, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113, 523–529 (1995).
[CrossRef]

1990

J. D. Bierlein, D. B. Laubacher, and J. B. Brown, “Balanced phase matching in segmented KTIOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

J. Khurjin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism for efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Alford, W. J.

Andreev, Y. M.

Arbore, M. A.

Armstrong, D. J.

Ashihara, S.

Atuchin, V. V.

Baum, P.

Bhargava, R. N.

J. Khurjin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism for efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Bierlein, J. D.

J. D. Bierlein, D. B. Laubacher, and J. B. Brown, “Balanced phase matching in segmented KTIOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Brown, J. B.

J. D. Bierlein, D. B. Laubacher, and J. B. Brown, “Balanced phase matching in segmented KTIOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Brown, M.

Carlson, D.

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, “Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology,” J. Cryst. Growth 264, 565–577 (2004).
[CrossRef]

Chang, Y. Q.

J. J. Huang, Y. Q. Chang, T. Shen, and Y. Q. Yang, “Proposal high quality walk-off compensated sum frequency generation of ultra-short pulses,” Opt. Commun. 281, 5244–5248 (2008).
[CrossRef]

Charbonneau-Lefort, M.

Chatterjee, U.

Colak, S.

J. Khurjin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism for efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Dai, H.

J. J. Huang, B. L. Mao, H. Dai, T. Shen, and L. Y. Zhang, “Research on static frequency doubling in refractive-index modulated crystals,” Acta Opt. Sin. 30, 2634–2638 (2010).
[CrossRef]

Fejer, M. M.

Fermann, M.

Fujioka, N.

Galvanauskas, A.

Gao, W.

Gavrilin, N.

Gehr, R. J.

Harter, D.

Hass, R. A.

R. A. Hass, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113, 523–529 (1995).
[CrossRef]

Huang, J. J.

J. J. Huang, L. Y. Zhang, and T. Shen, “Properties of periodic multicrystal configurations in walk-off-compensating second harmonic generation of ultrashort pulses,” Chin. Phys. B 20, 044206 (2011).
[CrossRef]

J. J. Huang, B. L. Mao, H. Dai, T. Shen, and L. Y. Zhang, “Research on static frequency doubling in refractive-index modulated crystals,” Acta Opt. Sin. 30, 2634–2638 (2010).
[CrossRef]

J. J. Huang, Y. Q. Chang, T. Shen, and Y. Q. Yang, “Proposal high quality walk-off compensated sum frequency generation of ultra-short pulses,” Opt. Commun. 281, 5244–5248 (2008).
[CrossRef]

J. J. Huang, G. J. Ji, T. Shen, Y. M. Andreev, A. V. Shaiduko, and U. Chatterjee, “Influence of composition ratio variation on optical frequency conversion in mixed crystals. I. Gradual variation of composition ratio,” J. Opt. Soc. Am. B 24, 2443–2453 (2007).
[CrossRef]

J. J. Huang, W. Gao, T. Shen, B. L. Mao, Y. M. Andreev, A. V. Shaiduko, G. V. Lanskii, U. Chatterjee, and V. V. Atuchin, “Influence of composition ratio variation on optical frequency conversion in mixed crystals. II. Random variation of composition ratio,” J. Opt. Soc. Am. B 24, 3081–3090 (2007).
[CrossRef]

Imeshev, G.

Ji, G. J.

Khurjin, J.

J. Khurjin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism for efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Kimmel, M. W.

Kuroda, K.

Lanskii, G. V.

Laubacher, D. B.

J. D. Bierlein, D. B. Laubacher, and J. B. Brown, “Balanced phase matching in segmented KTIOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Lochbrunner, S.

Mao, B. L.

Marco, O.

Motakef, S.

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, “Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology,” J. Cryst. Growth 264, 565–577 (2004).
[CrossRef]

Ono, H.

Raciukaitis, G.

Regelskis, K.

Riedle, E.

Schober, A. M.

Shaiduko, A. V.

Shen, T.

J. J. Huang, L. Y. Zhang, and T. Shen, “Properties of periodic multicrystal configurations in walk-off-compensating second harmonic generation of ultrashort pulses,” Chin. Phys. B 20, 044206 (2011).
[CrossRef]

J. J. Huang, B. L. Mao, H. Dai, T. Shen, and L. Y. Zhang, “Research on static frequency doubling in refractive-index modulated crystals,” Acta Opt. Sin. 30, 2634–2638 (2010).
[CrossRef]

J. J. Huang, Y. Q. Chang, T. Shen, and Y. Q. Yang, “Proposal high quality walk-off compensated sum frequency generation of ultra-short pulses,” Opt. Commun. 281, 5244–5248 (2008).
[CrossRef]

J. J. Huang, G. J. Ji, T. Shen, Y. M. Andreev, A. V. Shaiduko, and U. Chatterjee, “Influence of composition ratio variation on optical frequency conversion in mixed crystals. I. Gradual variation of composition ratio,” J. Opt. Soc. Am. B 24, 2443–2453 (2007).
[CrossRef]

J. J. Huang, W. Gao, T. Shen, B. L. Mao, Y. M. Andreev, A. V. Shaiduko, G. V. Lanskii, U. Chatterjee, and V. V. Atuchin, “Influence of composition ratio variation on optical frequency conversion in mixed crystals. II. Random variation of composition ratio,” J. Opt. Soc. Am. B 24, 3081–3090 (2007).
[CrossRef]

Shimura, T.

Smith, A. V.

Stolzenberger, R.

J. Khurjin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism for efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Wang, C. A.

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, “Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology,” J. Cryst. Growth 264, 565–577 (2004).
[CrossRef]

Wargo, M. J.

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, “Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology,” J. Cryst. Growth 264, 565–577 (2004).
[CrossRef]

Wiegel, M.

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, “Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology,” J. Cryst. Growth 264, 565–577 (2004).
[CrossRef]

Yang, Y. Q.

J. J. Huang, Y. Q. Chang, T. Shen, and Y. Q. Yang, “Proposal high quality walk-off compensated sum frequency generation of ultra-short pulses,” Opt. Commun. 281, 5244–5248 (2008).
[CrossRef]

Zeludevicius, J.

Zhang, L. Y.

J. J. Huang, L. Y. Zhang, and T. Shen, “Properties of periodic multicrystal configurations in walk-off-compensating second harmonic generation of ultrashort pulses,” Chin. Phys. B 20, 044206 (2011).
[CrossRef]

J. J. Huang, B. L. Mao, H. Dai, T. Shen, and L. Y. Zhang, “Research on static frequency doubling in refractive-index modulated crystals,” Acta Opt. Sin. 30, 2634–2638 (2010).
[CrossRef]

Acta Opt. Sin.

J. J. Huang, B. L. Mao, H. Dai, T. Shen, and L. Y. Zhang, “Research on static frequency doubling in refractive-index modulated crystals,” Acta Opt. Sin. 30, 2634–2638 (2010).
[CrossRef]

Appl. Phys. Lett.

J. D. Bierlein, D. B. Laubacher, and J. B. Brown, “Balanced phase matching in segmented KTIOPO4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

J. Khurjin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism for efficient blue second-harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57, 2540–2542 (1990).
[CrossRef]

Chin. Phys. B

J. J. Huang, L. Y. Zhang, and T. Shen, “Properties of periodic multicrystal configurations in walk-off-compensating second harmonic generation of ultrashort pulses,” Chin. Phys. B 20, 044206 (2011).
[CrossRef]

J. Cryst. Growth

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, “Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology,” J. Cryst. Growth 264, 565–577 (2004).
[CrossRef]

J. Opt. Soc. Am. B

G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17, 304–318 (2000).
[CrossRef]

A. V. Smith, D. J. Armstrong, and W. J. Alford, “Increased acceptance bandwidths in optical frequency conversion by use of multiple walk-off-compensating nonlinear crystals,” J. Opt. Soc. Am. B 15, 122–141 (1998).
[CrossRef]

S. Ashihara, T. Shimura, and K. Kuroda, “Group-velocity matched second-harmonic generation in tilted quasi-phase-matched gratings,” J. Opt. Soc. Am. B 20, 853–856 (2003).
[CrossRef]

A. M. Schober, M. Charbonneau-Lefort, and M. M. Fejer, “Broadband quasi-phase-matched second-harmonic generation of ultrashort optical pulses with spectral angular dispersion,” J. Opt. Soc. Am. B 22, 1699–1713 (2005).
[CrossRef]

J. J. Huang, G. J. Ji, T. Shen, Y. M. Andreev, A. V. Shaiduko, and U. Chatterjee, “Influence of composition ratio variation on optical frequency conversion in mixed crystals. I. Gradual variation of composition ratio,” J. Opt. Soc. Am. B 24, 2443–2453 (2007).
[CrossRef]

J. J. Huang, W. Gao, T. Shen, B. L. Mao, Y. M. Andreev, A. V. Shaiduko, G. V. Lanskii, U. Chatterjee, and V. V. Atuchin, “Influence of composition ratio variation on optical frequency conversion in mixed crystals. II. Random variation of composition ratio,” J. Opt. Soc. Am. B 24, 3081–3090 (2007).
[CrossRef]

Opt. Commun.

J. J. Huang, Y. Q. Chang, T. Shen, and Y. Q. Yang, “Proposal high quality walk-off compensated sum frequency generation of ultra-short pulses,” Opt. Commun. 281, 5244–5248 (2008).
[CrossRef]

R. A. Hass, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113, 523–529 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

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Figures (6)

Fig. 1.
Fig. 1.

Time-domain pictures of SHG at different times t0, t1, t2, t3 in a RILM nonlinear crystal with a RI modulation depth Δn. Two different spectral components of a FH pulse, ω (solid curve) and ω (dashed curve) are converted to 2ω (solid curve) and 2ω (dashed curve), respectively.

Fig. 2.
Fig. 2.

Dependence of the SH pulse width, peak intensity [Iout(0)], and SHG efficiency (η) on the modulation parameter C2, where G=1 is taken.

Fig. 3.
Fig. 3.

SH pulse width, peak intensity [Iout(0)], and SHG efficiency (η) versus normalized prechirping coefficient C1 with fixed C2=1 and assumed condition G=1.

Fig. 4.
Fig. 4.

SH peak intensity and SHG conversion efficiency as functions of C2 at the optimal case: C1=C2 with G=1.

Fig. 5.
Fig. 5.

(a) Optimal chirp compensated contour plot of the SH pulse width and (b) variation with μ for several PM bandwidths. Four estimated overshooting compression loci [Eq. (30)] with n=1, 2, 3, 4 are shown in (a).

Fig. 6.
Fig. 6.

Optimal chirp compensated contour plot of the conversion efficiency (a) with four estimated loci as in Fig. 5(a), and efficiency variation with μ for several PM bandwidths (b) where G=1 is assumed.

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

2E(r,t)μ0ε02E(r,t)t2=μ02P(r,t)t2,
Ej(z,t)=12Aj(z,t)exp[i(kj0zωj0t)],
[2z2+μ0ε0ε^j(z,ω)ω2]E^j(z,ω)=μ0ωj02P^jNL(z,ω),
nj(z,ω)=nj(ω)+[βj+βj(ωωj0)]z,
[kj(ω)2kj02]A^j(z,ω)+2ikj0A^j(z,ω)z=μ0ωj02P^jNL(z,ω)exp(ikj0z),
kj(ω)kj0+ωj0cβjz+(1vj+zβjc)(ωωj0)+αj2(ωωj0)2.
(1v1+zcβ1)A1t+A1zik0β1zA1=iσA1*A2exp(iΔkz),
(1v2+zcβ2)A2t+A2z2ik0β2zA2=iσA12exp(iΔkz),
A1zik0β1zA1+(δv+zcΔβ)A1t=iσA1*A2eiΔkz,
A2z2ik0β2zA2=iσA12eiΔkz,
a1ςπix1ςa1+(μ+δTς)a1τ=iBa1*a2eiπyς,
a2ς2πix2ςa2=iBa12eiπyς,
μ=PW·L/Ltw,PW2ln2.
b1ς+(μ+δTς)b1τ=iBb1*b2exp[iπ(yς+xς2)],
b2ς=iBb12exp[iπ(yς+xς2)],
b1(ς,τ)=exp{[τς(μ+ςδT/2)]2/2}.
bout(τ)=1/21/2iBexp[(τμς)2+iπ(yς+xς2)]dς=eτ21/21/2iBexp[μς(2τμς)+iπς(y+xς)]dς,
bout(τ)=iG(C21iC2)1/2exp(τ21iC2),
C2=μ2/(πx)=PW2Ltw2k01ΔβμΓ
Iout(τ)|bout(τ)|2=G2|C2|1+C22exp(2τ21+C22),
Δτ2=2ln2(1+C22),
η=n2n1|bout(τ)|2dτ|b1(0,τ)|2dτ=G22|C2|.
bout(τ)i2G[erf(μ2+τ)+erf(μ2τ)],
I20=G2[erf(μ/2)]2G2,
b1(ς,τ)=11iC1exp[(τςμ)22(1iC1)],
bout(τ)=iG(1iC1)1/2[C21i(C1+C2)]1/2exp[τ21i(C1+C2)].
Iout(τ)=G2|C2|(1+C12)1/2[1+(C1+C2)2]1/2exp[2τ21+(C1+C2)2].
Δτ2={2ln2[1+(C1+C2)2]}1/2,
η=G2|C2|2(1+C12).
bout(τ)=11iC11/21/2iBexp[(τμς)21iC1+iπxς2]dς=G2(μμiΓ)1/2eτ2[erf(iτΓ/21iΓ/μ)erf(iτ+Γ/21iΓ/μ)],
Iout(τ)=G24(1+Γ2μ2)1/2e2τ2|erf(iτ+Γ/21iΓ/μ)erf(iτΓ/21iΓ/μ)|2.
D(x,y)|2x|1{[S(yx|2x|)S(y+x|2x|)]2+[C(yx|2x|)C(y+x|2x|)]2},
Δxn=8n,n=±1,±2,.
Γn=(8n+x0)πμ1.
bout(τ)=iB1/21/2b12(ς,τ)exp[iπ(yς+xς2)]dς.
b^out(Ω2)=iBdΩ11/21/2b^1(ς,Ω1)b^1(ς,Ω2Ω1)dς×exp[iπ(yς+xς2)iδκ(Ω1,Ω2)ς],
b^1(ς,Ω1)=12πb1(ς,τ)exp[iκ1(Ω1)ςiΩ1τ]dτ,
b1(ς,τ)=b^1(ς,Ω1)exp[iΩ1τiκ1(Ω1)ς]dΩ1.
b^out(Ω2)=D^(Ω1,Ω2)b^12(Ω2)
D^(Ω1,Ω2)=iB1/21/2exp[iπxς2iδκ(Ω1,Ω2)ς]dς,
b^12(Ω2)=b^1(Ω1)b^1(Ω2Ω1)dΩ1.
D^(Ω2)=iBexp[iς(πxςδvLΩ2)]dς=B(iπΓμ)1/2exp(iC2T2Ω224).
n1(0,ω10)=n2(0,ω20=2ω10).
δn(z,ω)n1(z,ω)n2(z,2ω)=0
δnzδz+δnωδω=0.
δn0δnω|ω=ω10={ω[n1(0,ω)2n2(0,2ω)]}|ω=ω10.
Δn+δn0δω=0.
δn0=k01δv,
δω=Δnk0/δv,Δn=δωδvk01.
δΩΔT=PW2.
ΓPW·δω/δΩ.
|δT|=L2|Δβ|cT|Δn|LcT=|δωδv|Lck0T=|δωμ|ω10=|Γ|PWδΩω10|μ||μ|,

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