Efficient second-harmonic generation of a broadband radiation by control of the temperature distribution along a nonlinear crystal

doi:10.1364/OE.20.028544

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Efficient second-harmonic generation of a broadband radiation by control of the temperature distribution along a nonlinear crystal

K. Regelskis, J. Želudevičius, N. Gavrilin, and G. Račiukaitis »View Author Affiliations

Optics Express, Vol. 20, Issue 27, pp. 28544-28556 (2012)
http://dx.doi.org/10.1364/OE.20.028544

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– Abstract
1. Introduction
2. Numerical model and simulations
3. The experiment
4. Conclusions
– References and links

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Abstract

We demonstrate an efficient technique for the second harmonic generation (SHG) of the broadband radiation based on the temperature gradient along a nonlinear crystal. The characteristics of Type I non-critical phase-matched SHG of broadband radiation in the LiB₃O₅ (LBO) crystal with the temperature gradient imposed along the crystal were investigated both numerically and experimentally. The frequency doubling efficiency of the broadband pulsed fiber laser radiation as high as 68% has been demonstrated.

1. Introduction

The main limitation to efficient broadband frequency conversion originates from the chromatic dispersion of the nonlinear crystal. The crystal acceptance bandwidth, which in the time domain is directly related to the pulse group velocity mismatch, is of high importance for frequency conversion of broadband laser pulses. In order to improve the efficiency of the frequency conversion process, there is a definite advantage in using shorter crystals and higher irradiance. However, the increase of the laser pulse irradiance is limited by the damage thresholds of the crystal. The efficiency of frequency conversion in a nonlinear crystal is proportional to the square of the interaction length, but the wavelength acceptance bandwidth is inversely proportional to the same interaction length (we assume the regime of low conversion, i.e. without pump depletion), resulting in a trade-off between the conversion efficiency and bandwidth. Several methods to reduce or eliminate this trade-off have been demonstrated. One method suggested for improving the frequency conversion efficiency is to use multiple short crystals, each of them phase-matched for different spectral components of the input pulse [1

M. Brown, “Increased spectral bandwidths in nonlinear conversion processes by use of multicrystal designs,” Opt. Lett. 23(20), 1591–1593 (1998). [CrossRef] [PubMed]

]. According to results of numerical modeling, this approach can increase the conversion efficiency up to 50% but requires multiple crystals and accurate crystal angle alignment [2

W. J. Alford and A. V. Smith, “Frequency-doubling broadband light in multiple crystals,” J. Opt. Soc. Am. B 18(4), 515–523 (2001). [CrossRef]

]. Another approach is to use multiple crystals with an opposite group-velocity mismatch [3

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(1), 122–141 (1998). [CrossRef]

]. This method can increase the conversion efficiency N times, where N is equal to the number of the used group-velocity-mismatch (and walk-off) compensating crystals. Additional well known approach is to use angular dispersion so that each wavelength enters the nonlinear crystal at its appropriate phase-matching angle [4

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25(12), 2464–2468 (1989). [CrossRef]

, 5

B. A. Richman, S. E. Bisson, R. Trebino, E. Sidick, and A. Jacobson, “Efficient broadband second-harmonic generation by dispersive achromatic nonlinear conversion using only prisms,” Opt. Lett. 23(7), 497–499 (1998). [CrossRef] [PubMed]

]. In these designs combinations of prisms and lenses are typically used to produce the angular dispersion that closely matches those angle-tuning characteristics of the nonlinear crystal. All described approaches can provide some enhancement in the SHG efficiency of the broadband light pulses but all of them have significant drawbacks as they require numerous additional optical components and alignment precision. In addition to above mentioned methods, quasi-phase-matched SHG of broadband radiantion can be achieved by use of a chirped or aperiodic grating stucture [6

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30(7), 1596–1604 (1994). [CrossRef]

, 7

Y. L. Lee, Y. C. Noh, C. Jung, T. Yu, D. K. Ko, and J. Lee, “Broadening of the second-harmonic phase-matching bandwidth in a temperature-gradient-controlled periodically poled Ti:LiNbO₃ channel waveguide,” Opt. Express 11(22), 2813–2819 (2003). [CrossRef] [PubMed]

]. However, it is difficult to fabricate chirped grating with smooth period change. This problem can be solved with introducing a step-chirped grating, which provide more convenient route for fabrication and poling [8

A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” J. Lightwave Technol. 26(3), 343–349 (2008). [CrossRef]

, 9

A. Tehranchi and R. Kashyap, “Engineered gratings for flat broadening of second-harmonic phase-matching bandwidth in MgO-doped lithium niobate waveguides,” Opt. Express 16(23), 18970–18975 (2008). [CrossRef] [PubMed]

One promising concept for improving the frequency conversion efficiency by using a constant-temperature gradient imposed along the nonlinear crystal has been proposed by R. A. Haas [10

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113(4-6), 523–529 (1995). [CrossRef]

]. His theoretical analysis of Type I SHG in the quasi-continuous-wave undepleted-fundamental regime indicates that a constant temperature gradient significantly increases the tuning spectral bandwidth for fixed crystal orientation. Spectral distribution of the conversion efficiency is analogous to the light irradiance spatial distribution for a single-slit diffraction.

In Fig. 1 the phase matching temperature versus fundamental harmonic (FH) wavelength for non-critical phase-matched (NCPM) SHG in LBO crystal is depicted [11

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB₃O₅,” IEEE J. Quantum Electron. 30(12), 2950–2952 (1994). [CrossRef]

]. NCPM of LBO is featured by the absence of spatial walk-off, very wide acceptance angle and the maximized effective nonlinear optics coefficient [12

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, 1st ed. (Springer, 2005).

]. Type I NCPM takes place along the x-axis of the crystal (phase matching angles:

ϑ = 90^{o}, φ = 0^{o}

), and the temperature-controllable NCPM wavelength range is ~950…1800 nm. Furthermore, these spectral characteristics are very suitable for doubling broadband laser radiation near 1300 nm [13

X. Liu, L. J. Qian, and F. W. Wise, “Efficient generation of 50-fs red pulses by frequency doubling in LiB₃O₅,” Opt. Commun. 144(4-6), 265–268 (1997). [CrossRef]

]. The wavelengths for phase-matched SHG depend on the crystal temperature. With the temperature gradient imposed along the crystal, the phase-matching conditions are satisfied for different wavelength at different positions along the crystal and efficient SHG takes place at small distance till the wave-number mismatch is close to zero at corresponding wavelength. The perfect phase-matching of different spectral components of the broadband FH radiation in the whole wavelength range can be achieved in different parts of the crystal with a proper selection of the temperature distribution along the crystal. Thus, this approach is somewhat similar to the method involving multiple crystals but requires only a single long crystal and a modified crystal oven. Moreover, this method does not require any sophisticated alignment, it is insensitive to mechanical vibrations and can be easily (electronically) tuned for the required laser wavelength interval.

Fig. 1 Phase-matching temperature versus FH wavelength for NCPM SHG in LBO [11

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB₃O₅,” IEEE J. Quantum Electron. 30(12), 2950–2952 (1994). [CrossRef]

In this paper we both numerically and experimentally investigated the characteristics of Type I NCPM SHG of broadband radiation in the LBO crystal with the temperature gradient imposed along the crystal and compared with the characteristics of the conventional schemes of SHG. The final goal of our investigation was to develop practical and reliable SHG unit for the broadband frequency doubling of the short-pulse ytterbium-doped fiber laser radiation.

2. Numerical model and simulations

In this section we present the results of numerical simulations of the broadband laser pulse SHG when a temperature gradient is imposed along the nonlinear crystal, and we have compared this method with a conventional SHG scheme – with uniform crystal temperature.

The computer simulations were performed using a plane-wave model by solving the set of coupled-wave equations [14

A. Dubietis, G. Tamosauskas, and A. Varanavicius, “Femtosecond third-harmonic pulse generation by mixing of pulses with different duration,” Opt. Commun. 186(1-3), 211–217 (2000). [CrossRef]

, 15

T. Zhang and M. Yonemura, “Pulse shaping of ultrashort laser pulses with nonlinear optical crystals,” Jpn. J. Appl. Phys. 38(Part 1, No. 11), 6351–6358 (1999). [CrossRef]

] numerically:

\begin{array}{l} \frac{\partial A_{1}}{\partial z} = - ν_{13} \frac{\partial A_{1}}{\partial t} + i \frac{g_{1}}{2} \frac{\partial^{2} A_{1}}{\partial t^{2}} + \frac{h_{1}}{6} \frac{\partial^{3} A_{1}}{\partial t^{3}} - \frac{α_{1}}{2} A_{1} - i σ_{1} A_{3} A_{2}^{*} e^{- i Φ (z)}, \\ \frac{\partial A_{2}}{\partial z} = - ν_{23} \frac{\partial A_{2}}{\partial t} + i \frac{g_{2}}{2} \frac{\partial^{2} A_{2}}{\partial t^{2}} + \frac{h_{2}}{6} \frac{\partial^{3} A_{2}}{\partial t^{3}} - \frac{α_{2}}{2} A_{2} - i σ_{2} A_{3} A_{1}^{*} e^{- i Φ (z)}, \\ \frac{\partial A_{3}}{\partial z} = i \frac{g_{3}}{2} \frac{\partial^{2} A_{3}}{\partial t^{2}} + \frac{h_{3}}{6} \frac{\partial^{3} A_{3}}{\partial t^{3}} - \frac{α_{3}}{2} A_{3} - i σ_{3} A_{1} A_{2} e^{+ i Φ (z)}, \end{array}

(1)

where

z

and

t

are propagation and time coordinates, respectively, relative to a frame of reference which is moving with the pulse of the frequency

ω_{3}

at the group velocity

u_{3}

(the so-called retarded frame),

ν_{j 3} = u_{j}^{- 1} - u_{3}^{- 1}

is the group velocity mismatch,

u_{j}^{- 1} = \frac{d k_{j}}{d ω} |_{ω = ω_{j}}

is the inverse group velocity,

g_{j} = \frac{d^{2} k_{j}}{d ω^{2}} |_{ω = ω_{j}}

is the group velocity dispersion,

h_{j} = \frac{d^{3} k_{j}}{d ω^{3}} |_{ω = ω_{j}}

is the third-order dispersion,

k_{j}

is the wave number corresponding to the central frequency

ω_{j}

at which the pulse spectrum is centered,

α_{j}

is the absorption coefficient,

σ_{j} = \frac{ω_{j} d_{e f f}}{n_{j} c}

is the coupling coefficient and

d_{e f f}

is the effective second-order nonlinear optic coefficient. Subscripts

j = 1, 2

refer to FH waves and

j = 3

refers to the second harmonic (SH) wave. The irradiance (sometimes called “intensity”) of the

j -th

wave is related to the complex amplitude (

A_{j}

) via

I_{j} = \frac{c ε_{0} n_{j}}{2} {| A_{j} |}^{2}

. Because the refractive index

n_{j} (T (z))

is dependent on the crystal temperature, the phase mismatching due to temperature change along the crystal can be written as [10

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113(4-6), 523–529 (1995). [CrossRef]

, 16

M. Sabaeian, L. Mousave, and H. Nadgaran, “Investigation of thermally-induced phase mismatching in continuous-wave second harmonic generation: a theoretical model,” Opt. Express 18(18), 18732–18743 (2010). [CrossRef] [PubMed]

, 17

S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B 27(8), 1504–1512 (2010). [CrossRef]

Φ (z) = \int_{0}^{z} Δ k (T (z^{'})) d z^{'} = \int_{0}^{z} [k_{3} (T (z^{'})) - k_{1} (T (z^{'})) - k_{2} (T (z^{'}))] d z^{'},

(2)

where

k_{j} (T (z)) = ω_{j} n_{j} (T (z)) / c

is the wave number. Furthermore, our model includes temperature dependence of

u_{j}

g_{j}

and

h_{j}

through

k_{j} (T (z))

. In the case when a constant temperature gradient (

d T / d z = const .

) is imposed along the crystal or z-axis (frame of reference of laboratory), the linear dependence of the temperature within the crystal of length L is then given by

T (z) = T_{c} + (d T / d z) (z - L / 2),

(3)

where

T_{c}

is the central temperature at the midplane of the crystal. We assume that the temperature gradient does not alter the crystal structure and refractive indices depend only on the crystal temperature and that the temperature dependence of the effective second-order nonlinear optical coefficient of the crystal can be neglected. In addition, the distinctive feature of LBO crystal is high linear thermal expansion coefficients. Mean values of the linear thermal expansion coefficients for temperature range 298–423 K are equal to:

66.4 \times 10^{- 6} K^{- 1}

- 52.8 \times 10^{- 6} K^{- 1}

and

27.3 \times 10^{- 6} K^{- 1}

along x, y and z-axis of the crystal, respectively [12

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, 1st ed. (Springer, 2005).

]. Hence, apart from the temperature-induced changes in the refractive indices, temperature changes result in the appearance of strain-induced changes in the LBO crystal – rotations of the entry face of the crystal and of the crystal-optical coordinate system [18

S. G. Grechin, V. G. Dmitriev, V. A. D'yakov, and V. I. Pryalkin, “Anomalous temperature-independent birefringence in a biaxial optical LBO crystal,” Quantum Electron. 30(4), 285–286 (2000). [CrossRef]

]. Thermal-strain changes must be taken into account when phase-matching directions other than along the principal axes of the crystal are chosen. Moreover, a linear temperature variation within a crystal does not cause stress [10

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113(4-6), 523–529 (1995). [CrossRef]

] (also see ref. 13

X. Liu, L. J. Qian, and F. W. Wise, “Efficient generation of 50-fs red pulses by frequency doubling in LiB₃O₅,” Opt. Commun. 144(4-6), 265–268 (1997). [CrossRef]

and 14

A. Dubietis, G. Tamosauskas, and A. Varanavicius, “Femtosecond third-harmonic pulse generation by mixing of pulses with different duration,” Opt. Commun. 186(1-3), 211–217 (2000). [CrossRef]

inside [10

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113(4-6), 523–529 (1995). [CrossRef]

]). Because Type I NCPM takes place along the x-axis of the crystal, all the investigated in this paper effects rely only on the dependence of refractive indices on temperature [11

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB₃O₅,” IEEE J. Quantum Electron. 30(12), 2950–2952 (1994). [CrossRef]

The numerical integration of Eqs. (1) was carried out using a split-step technique in which propagation is handled by fast Fourier transform methods, whereas nonlinear interaction is handled by Runge–Kutta integration. This technique is well described in the papers [19

A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B 16(4), 609–619 (1999). [CrossRef]

, 20

G. Arisholm, “General numerical methods for simulating second-order nonlinear interactions in birefringent media,” J. Opt. Soc. Am. B 14(10), 2543–2549 (1997). [CrossRef]

]. The computer simulation program was written in C + + programming language. The accuracy of the numerical calculations without the presented temperature gradient along the crystal (temperature gradient was set to zero) was initially tested in comparison with the public domain free software SNLO [21

A. V. Smith, “How to select nonlinear crystals and model their performance using SNLO software,” Proc. SPIE 3928, 62–69 (2000). [CrossRef]

]. In order to verify the correctness of the numerical calculations when a temperature gradient is imposed along the crystal, the initial crystal was divided into a large number of short crystals, where each of them was at a constant temperature set in accordance with temperature variation of the initial crystal and the output of the each previous crystal was set as input of each next crystal in sequence. Further checks of the solutions were obtained from conservation of energy.

Type I NCPM (the waves propagate along x-axis of the crystal, phase matching angles:

ϑ = 90^{o}, φ = 0^{o}

) SHG of broadband laser pulses in the LBO crystal was considered in our numerical modeling. Case when the spectral bandwidth is much wider than the transform limit of the pulse duration is mainly analyzed. The aim is to explore SHG of broadband laser pulses when a constant temperature gradient (

d T / d z = const .

) is imposed along the crystal and to determine conditions when the said method has an advantage over conventional SHG without the temperature gradient (

d T / d z = 0

) along the crystal. The broadband laser pulses were assumed to be linearly chirped Gaussian pulses:

A_{j} (t) = a_{j 0} \exp (- (1 + i γ_{j}) 2 l n (2) t^{2} / τ_{j}^{2}),

(4)

where

a_{j 0}

is the pulse peak amplitude,

γ_{j}

is the chirp parameter and

τ_{j}

is the pulse width defined at the full width at half maximum (FWHM) of irradiance. The duration of a chirped pulse is consequently larger than the transform limit. The carrier frequency is varying on the time scale of the pulse. The spectral width of the pulse at the FWHM of irradiance is

Δ λ_{j} \approx λ_{j}^{2} 2 \ln 2 \sqrt{1 + γ_{j}^{2}} / (π τ_{j} c),

(5)

where

λ_{j} = 2 π c / ω_{j}

is the central wavelength at which the pulse spectrum is centered. The linearly chirped Gaussian pulses as broadband laser pulses were chosen for clarity and more comprehensible examination of the SHG characteristics. Similar results can be obtained with other broadband pulses of different temporal and spectral characteristics.

Initially, the SHG conversion efficiencies of broadband FH pulses (Eq. (4)) without the temperature gradient (

d T / d z = 0

) along the crystal were calculated. Temperature of the crystal was set to

T_{c} = 149.2 ° C

, which corresponds to a phase-matched FH wavelength at 1064 nm. Type I NCPM SHG was calculated using temperature-dependent Sellmeier equations for the LBO crystal [11

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB₃O₅,” IEEE J. Quantum Electron. 30(12), 2950–2952 (1994). [CrossRef]

], where the refractive indices

n_{1} = n_{z} (λ_{1}, T)

n_{2} = n_{z} (λ_{2}, T)

and

n_{3} = n_{y} (λ_{3}, T)

correspond to the waves that are polarized along

z

and

y

axes of the crystal. A value of

d_{e f f} = 0.85 pm / V

was chosen in accordance with [12

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, 1st ed. (Springer, 2005).

] and for simplicity we neglected losses (

α_{j} = 0

). The losses are insignificant only when the low-power radiation (several watts) is applied to the crystal, however at a higher average power (a hundred watts level) the heating induced due to optical absorption has a major impact on the SHG efficiency [16

, 22

Q. Liu, X. Yan, M. Gong, X. Fu, and D. Wang, “103 W high beam quality green laser with an extra- cavity second harmonic generation,” Opt. Express 16(19), 14335–14340 (2008). [CrossRef] [PubMed]

, 23

K. H. Hong, C. J. Lai, A. Siddiqui, and F. X. Kärtner, “130-W picosecond green laser based on a frequency-doubled hybrid cryogenic Yb:YAG amplifier,” Opt. Express 17(19), 16911–16919 (2009). [CrossRef] [PubMed]

Since both FH waves of Type I SHG are identical, subscripts (1 and 2) are omitted in further considerations and assuming that

I_{1} = I_{2}

, the irradiance of FH is

I = I_{1} + I_{2}

. The peak irradiances of the broadband FH pulses (Eq. (4)) that are required for achieving adequate conversion efficiency are presented in Fig. 2 . Each value of the SH conversion efficiency with the corresponding peak irradiance and spectral width of the FH pulses was calculated by optimizing the crystal length for the highest conversion efficiency. At the beginning of the crystal, the SH pulse grows smoothly and the FH pulses are depleted. Eventually, the SH conversion efficiency reaches the maximum and, for longer distances the energy from the SH pulse is returned to the FH wave. A structure is developing in the temporal and spectral distributions of both the FH and SH pulses passing the optimal length. An undistorted pulse is not again recovered for nonlinear crystal lengths longer than optimal, although the energy oscillate between the FH and SH wave as the crystal length is increased [24

R. Eckardt and J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20(10), 1178–1187 (1984). [CrossRef]

]. We observed that the SH conversion efficiency is independent of the pulse duration >1ps, while it has a strong dependence on the peak irradiance and spectral width. To maintain the conversion efficiency constant with the increasing spectral width, for instance

N

times, the peak irradiance of the FH pulse should be increased roughly

N^{2}

times (

I_{0} ~ Δ λ^{2}

). For example, to achieve 50% conversion efficiency at

Δ λ = 10 nm

the peak irradiance should be up to 3.35 GW/cm², while at

Δ λ = 20 nm

the peak irradiance should be increased four times, i.e. 13.4 GW/cm². In the case of femtosecond pulses this dependence is no longer correct. Numerical simulation shows that the conversion efficiency grows up as the crystal length is increased and for transform limited femtosecond pulses the conversion efficiency can exceed 90%. The SH spectrum is narrowed by temporal walk-off and the efficiency falls down proportionally to the ratio of the spectral width of the SH to the FH. However, the spectral wings of the FH pulse contribute to SHG by sum frequency mixing and the conversion efficiency continues growing up [2

W. J. Alford and A. V. Smith, “Frequency-doubling broadband light in multiple crystals,” J. Opt. Soc. Am. B 18(4), 515–523 (2001). [CrossRef]

]. The pulse duration of the SH increases with the spectrum narrowing. This applies only for transform-limited femtosecond pulses.

Fig. 2 Peak irradiance of the broadband FH pulses required to achieve the conversion efficiency of 30%, 50% and 70%. The temperature along the LBO crystal is uniform and equal to 149.2°C.

The results are slightly different when femtosecond pulses are chirped. In this case, the frequencies on opposite sides of the spectra do not overlap in time, there is no contribution from sum-frequency mixing and the conversion efficiency is limited by the crystal acceptance bandwidth.

As it was said earlier, in order to maintain the sufficiently high conversion efficiency with increasing the spectral width the irradiance should be increased according to the relation

I_{0} ~ Δ λ^{2}

. Increasing of the FH irradiance is limited by optical damage of the crystal. The damage may occur in the crystal bulk, at its surface, or in dielectric coatings. For example, the surface damage threshold irradiance of the LBO crystal is 25 GW/cm² at

τ = 100 ps

@1064 nm [12

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, 1st ed. (Springer, 2005).

, 25

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB₃O₅,” J. Opt. Soc. Am. B 6(4), 616–621 (1989). [CrossRef]

]. The damage threshold for antireflection coatings is close to 500 MW/cm². The optical damage actually depends on many factors: not only peak irradiance or wavelength, but also the pulse duration and the pulse repetition rate. So, with increasing the spectral width it becomes impossible to achieve the adequate conversion efficiency since the irradiance is limited by the optical damage of the crystal or antireflection coating.

Consequently, a significant increase in the spectral bandwidth of the SHG can be achieved by applying a temperature gradient along the crystal. This increase occurs together with a decrease in the conversion efficiency compared to the conversion efficiency for a uniform temperature crystal. However, this limitation may be overcome by increasing the crystal length. Computer simulation results of the SHG conversion efficiency when a constant temperature gradient (

d T / d z = const .

) is imposed along the crystal (temperature changes linearly along the crystal) are presented in Fig. 3 . In these calculations the central wavelength (

λ

) of the FH pulse is 1064 nm, the LBO crystal length (

L

) is 3 cm, the midplane temperature (

T_{c}

) is 149.2°C, which corresponds to a phase matched FH central wavelength, and the temperature gradient imposed along crystal is varied. The pulse duration was chosen to be equal

τ = 10 ps

for simulations, and identical results were obtained for all pulse durations that are >1 ps. Each plot in Fig. 3 represents the conversion efficiency dependence on

d T / d z

at five different values of the FH pulse peak irradiances: 0.5, 1, 2, 5 and 10 GW/cm². These results illustrate the significant SHG improvement of the broadband FH pulse. The high conversion efficiencies are achievable with much weaker peak irradiance of the FH pulse. For example, in order to achieve the 70% conversion efficiency at uniform crystal temperature when

Δ λ = 100 nm

, the required peak irradiance is 1.79 TW/cm² (see Fig. 2), which is more than two orders above the optical damage threshold of the LBO crystal. When a temperature gradient is imposed along the crystal, the 70% conversion efficiency can be achieved only with 10 GW/cm² (see Fig. 3(d)). In addition to the improvement of conversion efficiencies for very broadband pulses when

Δ λ = 100 nm

, the method also has advantage even when

Δ λ = 10 nm

(compare Fig. 2 and Fig. 3(a)). The conversion efficiency close to 90% can be achieved at moderate level of the FH pulse peak irradiance when

d T / d z \approx 10 ° C / cm

Fig. 3 SHG conversion efficiency of the broadband FH pulses dependence on

d T / d z

for various values of the peak irradiances (0.5, 1, 2, 5 and 10 GW/cm²) and spectral bandwidth: 10 nm (a), 20 nm (b), 50 nm (c) and 100 nm (d).

With the temperature gradient imposed along the crystal, the phase matching conditions are satisfied for different wavelength at different positions along the crystal. Proper selection of temperature distribution along the crystal allows for whole spectrum of broadband FH radiation to be converted to SH. Together with increasing the peak irradiance of the FH pulse, the temperature gradient along the crystal should be increased for the optimal conversion efficiency. The maximum of the conversion efficiencies moves toward a higher value of

d T / d z

when increasing the peak irradiance (see Fig. 3). This can be explained by a shorter interaction length needed for each small part of spectrum to achieve the same conversion efficiency. Thus it becomes possible to increase temperature gradient by such amount that conversion efficiency for each small part of spectrum stays the same but a range of phase matched wavelength increases. This lets to convert to SH broader part of FH spectrum and overall conversion efficiency increases. The spectral range (

δ λ

) can be obtained directly from dependence of phase-matched wavelength on temperature of the LBO crystal (Fig. 1). The dependence is approximately linear close to the 1064 nm wavelength and the spectral range can be expressed as linear equation:

δ λ \approx 0.808 L d T / d z = 0.808 Δ T

. Here

Δ T

is a temperature difference at the crystal endfaces. Contrary to the case when the crystal temperature is uniform, the spectral range (phase matching bandwidth) is not related to a crystal length and it is proportional to

Δ T

. It should be noted that phase matching bandwidth with the uniform crystal temperature is inversely proportional to

L

. So, there is no limitation on a crystal length, and high conversion efficiencies of the broadband FH pulses can be obtained when a temperature gradient is applied along the crystal.

With increasing

d T / d z

the conversion efficiency reaches its maximum and then it starts to decrease slowly. At this point the conversion efficiency weakly depends on temperature variation (Fig. 3). This feature may have a potential application in the field of high-average-power SHG [22

Q. Liu, X. Yan, M. Gong, X. Fu, and D. Wang, “103 W high beam quality green laser with an extra- cavity second harmonic generation,” Opt. Express 16(19), 14335–14340 (2008). [CrossRef] [PubMed]

, 23

]. In the case of the uniform crystal temperature, the conversion efficiency of the SHG significantly decreases when the temperature of the crystal is detuned. Thermally induced phase mismatching occurs due to optical absorption of the high-average-power FH and the SH waves by the crystal. Since the absorption coefficient of the SH wave is much larger than that of the FH wave, the positive longitudinal temperature gradient is induced within the LBO crystal [23

]. The temperature difference, at input and output surfaces of the crystal, becomes comparable to the temperature-dependent phase matching bandwidth. As a result, the back conversion due to the phase mismatch can degrade the SH conversion efficiency. However, the situation is quite different when a strong temperature gradient is artificially imposed along the crystal. The temperature variation induced by optical absorption of the FH and SH waves slightly modifies the strong temperature gradient and has no significant impact on the SH conversion efficiency. However, more detailed experimental analysis is required to confirm this statement.

Typical temporal and spectral characteristics of the FH and SH pulses when a constant temperature gradient is imposed along the LBO crystal are presented in Fig. 4 . The parameters of the FH pulse are:

τ = 10 ps

λ = 1064 nm

Δ λ = 20 nm

, peak irradiance

I_{0} = 5 {GW/cm}^{2}

. The crystal length (

L

) is 3 cm (Figs. 4(a) and 4(b)) and 6 cm (Figs. 4(c) and 4(d)),

d T / d z = 20 ° C / cm

T_{c} = 149.15 ° C

. The temporal and spectral profiles show a modulation structure which becomes more uniform as the crystal length and temperature difference between the crystal endfaces increase (compare Figs. 4(a) and 4(b) with Figs. 4(c) and 4(d)). The spectral fluence is given in units of (mJ/cm²/nm) in Fig. 4. Integration over all wavelengths gives energy fluence in units of (mJ/cm²) and integration of the irradiance over pulse temporal profile also gives energy fluence of equal value.

Fig. 4 Temporal and spectral characteristics of the FH and SH pulses when a constant temperature gradient is imposed along the LBO crystal. LBO crystal length is 3 cm (a, b) and 6 cm (c, d). The initial temporal and spectral profiles of the FH pulse are plotted as dotted-dashed curves.

When the FH pulse enters the crystal, the spectral components of the FH pulse are not phase-matched with the SH waves. The direction of energy transfer changes periodically according to the change in the phase relation between the interacting waves. The oscillations of the energy grow up as the waves continue to propagate through the crystal. The irradiance of the SH starts to increase rapidly close to the point where the perfect phase matching conditions are satisfied for corresponding wavelength. The most rapid increase of the SH wave irradiance is at the point where the perfect phase matching is achieved. Then the energy is transferred in a constant direction between the interacting waves. After passing this point the irradiance of the SH wave again starts to oscillate and a depth of the oscillation decreases with every next oscillation. Each spectral component of the FH and SH pulses meets a perfect phase matching conditions at different place along the crystal and experiences different characteristics of oscillations. Thus, the pulse temporal and spectral profiles are modulated. Modulations can be suppressed with further increasing the temperature difference between the crystal endfaces [10

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113(4-6), 523–529 (1995). [CrossRef]

]. To maintain the conversion efficiency constant with the increasing temperature difference, the length of the crystal also should be increased.

Because each spectral component of the FH and SH pulses experiences different characteristics of oscillations, for this reason a slight modulation also appears in the SHG conversion efficiency dependence on

d T / d z

(see Fig. 3). The modulation is clearly seen when the spectral bandwidth of the FH is 10 nm (Fig. 3(a)). With the increasing spectral width, modulation caused by different spectral components average out and modulation of conversion efficiency disappears (Fig. 3(d)).

The SH conversion efficiency when a temperature gradient is imposed along the crystal is not substantially sensitive to the input wavelength, crystal temperature disturbance, pulse irradiance, crystal length and the angle of incidence. Furthermore, this technique together with NCPM is advantageous because of its large angular acceptance and because it eliminates spatial walk-off between FH and SH radiation, which leads to the highest conversion efficiency. The technique may also be used to enhance the spectral bandwidth of other frequency conversion processes that require phase matching [10

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113(4-6), 523–529 (1995). [CrossRef]

]. The direction of the conversion process depends only on initial irradiancies of interacting waves and there is no requirement for the optimal crystal length to avoid back conversion. When

I_{1} > 0

I_{2} > 0

and

I_{3} = 0

the sum frequency generation takes place. On the other hand, in the same way a parametric amplification of the weak signal of irradiance

I_{1}

in the strong pump field of irradiance

I_{3}

can be realized.

The proposed technique with the LBO crystal could be an ideal choice for rugged industrial applications where high power and reliability are required. LBO is an excellent nonlinear crystal with a wide transparency range, a relatively large effective nonlinear optic coefficient, high damage threshold and good chemical and mechanical properties. LBO allows temperature-controlled Type I NCPM SHG in the wavelength range of ~950…1800 nm. This covers the whole gain spectrum of the ytterbium-doped fiber laser [26

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]

3. The experiment

Experimental investigation of influence of the temperature gradient along the nonlinear crystal for optical frequency doubling of optical pulses from the fiber laser system in the LBO crystal was carried out. The experimental fiber laser system consisted of the oscillator, preamplifier and the power amplifier (Fig. 5 .) The passively mode-locked fiber oscillator was based on ytterbium-doped gain medium tuned for lasing at 1064 nm center wavelength. It generated 8 ps duration pulses at the repetition rate of 58 MHz. The repetition rate was reduced down to 1.92 MHz using an acousto-optic modulator (AOM). These pulses were then amplified in the ytterbium-doped core-pumped polarization-maintaining single-mode fiber preamplifier up to 3.6 nJ pulse energy. In the following stage – the power amplifier – a double-clad Chirally-Coupled-Core (CCC) ytterbium-doped fiber with the core diameter of 33 µm was used. Pump radiation from the laser diode module was launched into inner cladding of active CCC fiber in the counter-propagating direction. At pump power of 27 W it was possible to amplify pulses up to 2.8 μJ energy with no significant Raman scattering or other detrimental nonlinear effects. Because of phase self-modulation, which took place mainly in single-mode optical fiber, the pulse spectrum was broadened up to ~12 nm and the pulse duration was increased up to 11 ps because of dispersion in the fiber.

Fig. 5 Experimental setup – fiber laser system. Insets: (a) autocorrelation of amplified FH pulse, (b) microscopic picture of CCC fiber cleaved endface.

Amplified pulses were used for generation of the SH in the LBO crystal both in conventional critical phase-matching setup and in noncritical phase-matching setup with/without the longitudinal temperature gradient. In the latter case, the 3 cm long LBO crystal cut normal to x plane was used. The nonlinear crystal was mounted in a custom-designed crystal oven with independent heaters at each end of the crystal (Fig. 6 ). The crystal oven was made with high precision – the air gap between the crystal sidewall and the inner wall of the oven was less than 10 microns. This high tolerance ensures uniform temperature distribution in transversal direction of the crystal and a constant axial temperature gradient in longitudinal direction of the crystal. Using a two-channel temperature controller (TK1, UAB Ekspla), the different temperature values were set for opposite crystal ends and in a such way the longitudinal temperature gradient was controlled. The fundamental laser beam was slightly focused inside the crystal (beam waist diameter was 130 µm at

1 / e^{2}

irradiance full width) in order to increase the peak irradiance and improve frequency doubling efficiency. In the experiment, the spectrum and the average power of FH and SH were measured. The conversion efficiency was estimated and compared with results achieved using the conventional SHG setup without the temperature gradient.

Fig. 6 Custom-designed crystal oven.

For the 3 cm long LBO crystal in conventional (without longitudinal temperature gradient) noncritical phase-matching configuration, the phase-matching bandwidth is limited to 1.3 nm (@1064 nm center wavelength) and is much narrower than the spectrum of FH pulses used in this experiment. This leads to poor conversion efficiency and pulse/spectrum distortions as it was discussed earlier. In this case (when crystal temperature was constant) conversion efficiency not higher than 17% was achieved. However, when the temperature gradient along the nonlinear crystal was present, almost the whole fundamental spectrum was converted to the SH with little distortions. This means that the phase-matching bandwidth can be essentially improved by implementation of the temperature gradient along the nonlinear crystal. This can be seen from Fig. 7 where normalized spectra of FH and SH are compared for different pulse energy of the FH.

Fig. 7 Comparison of the generated SH spectrum with FH spectrum at different FH pulse energy (0.73, 1.65 and 2.5 µJ), when the temperature gradient is applied along the crystal.

Δ T

is a temperature difference between crystal ends. Grey curves with filled regions represent FH spectra and thick red curves represent SH spectra.

Numerical calculations (presented in previous section) predicted that the optimal temperature gradient increased when first harmonic pulse energy was increased and this was observed experimentally. During measurements the temperature gradient was tuned for the optimal conversion efficiency for each pulse energy value. Experimental results also showed a significant conversion efficiency improvement when the longitudinal temperature gradient was present along the crystal. Conversion efficiency versus fundamental pulse energy for three investigated SHG configurations is depicted in Fig. 8 . By using the same 3 cm long crystal and setting temperature gradient conversion efficiency was improved significantly in comparison to the case when the crystal temperature was uniform (compare Fig. 8 squares and circles). In Fig. 8 the conversion efficiency of SHG configuration with the longitudinal temperature gradient is also compared with conventional critical phase matching configuration (phase matching angles:

ϑ = 90^{o}, φ = {11.6}^{o}

) using the 1 cm-long crystal and optimal focusing (according to [27

S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B 21(3), 591–598 (2004). [CrossRef]

]) achieving the beam waist diameter of 50 µm (at

1 / e^{2}

irradiance full width). Sharper focusing so as to improve conversion efficiency was hardly possible in our case as the peak irradiance in the crystal was already close to the reported damage threshold of 25 GW/cm² [25

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB₃O₅,” J. Opt. Soc. Am. B 6(4), 616–621 (1989). [CrossRef]

]. From the graph presented in Fig. 8 it can be seen that using SHG configuration with the longitudinal temperature gradient it is possible to greatly improve (up to 1.8 times) the conversion efficiency of broadband radiation in comparison with the critical phase matched SHG when crystal temperature is uniform. In our specific case, when the fundamental pulse spectrum width was ~12 nm and the pulse energy was 2.5 µJ the conversion efficiency as high as 68% was achieved using SHG configuration with the longitudinal temperature gradient.

Fig. 8 SHG conversion efficiency versus pulse energy in case of 3 cm long crystal with longitudinal temperature gradient (circles), 3 cm long crystal without longitudinal temperature gradient (squares) and 1 cm long crystal without the longitudinal temperature gradient (critical phase matching) (triangles).

The spatial beam quality of the fundamental and SH was characterized by the measurement of the

M^{2}

values. The

M^{2}

values of the fundamental beam in horizontal and vertical directions at the fiber laser output were 1.1 and 1.1, respectively. The SH beam generated with the optimal temperature gradient along the crystal and > 50% conversion efficiency was characterized. The SH beam quality (

M^{2}

) in vertical and horizontal directions was obtained to be equal to 1.1 and 1.2, respectively.

4. Conclusions

In summary, the characteristics of Type I NCPM SHG of broadband radiation in the LBO crystal with the temperature gradient imposed along the crystal were investigated numerically and experimentally. The results of numerical calculations and experimental investigations indicate that the conversion efficiency of the broadband pulse can be significantly increased when a constant temperature gradient is imposed along the nonlinear crystal. Advantage of this method is obvious when the frequency conversion of broadband phase modulated pulses (

Δ λ > 10 nm

) is considered. Using the 3 cm long LBO crystal we were able to generate the SH of the ~12 nm broad FH radiation at the conversion efficiency of 68%. This value was 1.8 times higher than the conversion efficiency achieved using the conventional method when the crystal temperature was uniform. The presented method can be attractive for implementation in practical applications because the enhancement in the conversion efficiency is achievable with very little additional components. In fact, by using this technique, a single frequency conversion module can be made suitable for a wide range of different practical applications involving all wavelengths over whole ytterbium-doped fiber laser gain spectrum.

Acknowledgments

Postdoctoral fellowship is being funded by European Union Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania”, Project No 74 (SKAIDULA).

References and links

1.	M. Brown, “Increased spectral bandwidths in nonlinear conversion processes by use of multicrystal designs,” Opt. Lett. 23(20), 1591–1593 (1998). [CrossRef] [PubMed]
2.	W. J. Alford and A. V. Smith, “Frequency-doubling broadband light in multiple crystals,” J. Opt. Soc. Am. B 18(4), 515–523 (2001). [CrossRef]
3.	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(1), 122–141 (1998). [CrossRef]
4.	O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25(12), 2464–2468 (1989). [CrossRef]
5.	B. A. Richman, S. E. Bisson, R. Trebino, E. Sidick, and A. Jacobson, “Efficient broadband second-harmonic generation by dispersive achromatic nonlinear conversion using only prisms,” Opt. Lett. 23(7), 497–499 (1998). [CrossRef] [PubMed]
6.	K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30(7), 1596–1604 (1994). [CrossRef]
7.	Y. L. Lee, Y. C. Noh, C. Jung, T. Yu, D. K. Ko, and J. Lee, “Broadening of the second-harmonic phase-matching bandwidth in a temperature-gradient-controlled periodically poled Ti:LiNbO₃ channel waveguide,” Opt. Express 11(22), 2813–2819 (2003). [CrossRef] [PubMed]
8.	A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” J. Lightwave Technol. 26(3), 343–349 (2008). [CrossRef]
9.	A. Tehranchi and R. Kashyap, “Engineered gratings for flat broadening of second-harmonic phase-matching bandwidth in MgO-doped lithium niobate waveguides,” Opt. Express 16(23), 18970–18975 (2008). [CrossRef] [PubMed]
10.	R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113(4-6), 523–529 (1995). [CrossRef]
11.	K. Kato, “Temperature-tuned 90° phase-matching properties of LiB₃O₅,” IEEE J. Quantum Electron. 30(12), 2950–2952 (1994). [CrossRef]
12.	D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, 1st ed. (Springer, 2005).
13.	X. Liu, L. J. Qian, and F. W. Wise, “Efficient generation of 50-fs red pulses by frequency doubling in LiB₃O₅,” Opt. Commun. 144(4-6), 265–268 (1997). [CrossRef]
14.	A. Dubietis, G. Tamosauskas, and A. Varanavicius, “Femtosecond third-harmonic pulse generation by mixing of pulses with different duration,” Opt. Commun. 186(1-3), 211–217 (2000). [CrossRef]
15.	T. Zhang and M. Yonemura, “Pulse shaping of ultrashort laser pulses with nonlinear optical crystals,” Jpn. J. Appl. Phys. 38(Part 1, No. 11), 6351–6358 (1999). [CrossRef]
16.	M. Sabaeian, L. Mousave, and H. Nadgaran, “Investigation of thermally-induced phase mismatching in continuous-wave second harmonic generation: a theoretical model,” Opt. Express 18(18), 18732–18743 (2010). [CrossRef] [PubMed]
17.	S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B 27(8), 1504–1512 (2010). [CrossRef]
18.	S. G. Grechin, V. G. Dmitriev, V. A. D'yakov, and V. I. Pryalkin, “Anomalous temperature-independent birefringence in a biaxial optical LBO crystal,” Quantum Electron. 30(4), 285–286 (2000). [CrossRef]
19.	A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B 16(4), 609–619 (1999). [CrossRef]
20.	G. Arisholm, “General numerical methods for simulating second-order nonlinear interactions in birefringent media,” J. Opt. Soc. Am. B 14(10), 2543–2549 (1997). [CrossRef]
21.	A. V. Smith, “How to select nonlinear crystals and model their performance using SNLO software,” Proc. SPIE 3928, 62–69 (2000). [CrossRef]
22.	Q. Liu, X. Yan, M. Gong, X. Fu, and D. Wang, “103 W high beam quality green laser with an extra- cavity second harmonic generation,” Opt. Express 16(19), 14335–14340 (2008). [CrossRef] [PubMed]
23.	K. H. Hong, C. J. Lai, A. Siddiqui, and F. X. Kärtner, “130-W picosecond green laser based on a frequency-doubled hybrid cryogenic Yb:YAG amplifier,” Opt. Express 17(19), 16911–16919 (2009). [CrossRef] [PubMed]
24.	R. Eckardt and J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20(10), 1178–1187 (1984). [CrossRef]
25.	C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB₃O₅,” J. Opt. Soc. Am. B 6(4), 616–621 (1989). [CrossRef]
26.	R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33(7), 1049–1056 (1997). [CrossRef]
27.	S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B 21(3), 591–598 (2004). [CrossRef]

OCIS Codes
(140.3510) Lasers and laser optics : Lasers, fiber
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing
(140.3515) Lasers and laser optics : Lasers, frequency doubled

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: October 10, 2012
Revised Manuscript: November 24, 2012
Manuscript Accepted: November 26, 2012
Published: December 10, 2012

Citation
K. Regelskis, J. Želudevičius, N. Gavrilin, and G. Račiukaitis, "Efficient second-harmonic generation of a broadband radiation by control of the temperature distribution along a nonlinear crystal," Opt. Express 20, 28544-28556 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28544

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References

M. Brown, “Increased spectral bandwidths in nonlinear conversion processes by use of multicrystal designs,” Opt. Lett.23(20), 1591–1593 (1998). [CrossRef] [PubMed]
W. J. Alford and A. V. Smith, “Frequency-doubling broadband light in multiple crystals,” J. Opt. Soc. Am. B18(4), 515–523 (2001). [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. B15(1), 122–141 (1998). [CrossRef]
O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron.25(12), 2464–2468 (1989). [CrossRef]
B. A. Richman, S. E. Bisson, R. Trebino, E. Sidick, and A. Jacobson, “Efficient broadband second-harmonic generation by dispersive achromatic nonlinear conversion using only prisms,” Opt. Lett.23(7), 497–499 (1998). [CrossRef] [PubMed]
K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron.30(7), 1596–1604 (1994). [CrossRef]
Y. L. Lee, Y. C. Noh, C. Jung, T. Yu, D. K. Ko, and J. Lee, “Broadening of the second-harmonic phase-matching bandwidth in a temperature-gradient-controlled periodically poled Ti:LiNbO3 channel waveguide,” Opt. Express11(22), 2813–2819 (2003). [CrossRef] [PubMed]
A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” J. Lightwave Technol.26(3), 343–349 (2008). [CrossRef]
A. Tehranchi and R. Kashyap, “Engineered gratings for flat broadening of second-harmonic phase-matching bandwidth in MgO-doped lithium niobate waveguides,” Opt. Express16(23), 18970–18975 (2008). [CrossRef] [PubMed]
R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun.113(4-6), 523–529 (1995). [CrossRef]
K. Kato, “Temperature-tuned 90° phase-matching properties of LiB3O5,” IEEE J. Quantum Electron.30(12), 2950–2952 (1994). [CrossRef]
D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, 1st ed. (Springer, 2005).
X. Liu, L. J. Qian, and F. W. Wise, “Efficient generation of 50-fs red pulses by frequency doubling in LiB3O5,” Opt. Commun.144(4-6), 265–268 (1997). [CrossRef]
A. Dubietis, G. Tamosauskas, and A. Varanavicius, “Femtosecond third-harmonic pulse generation by mixing of pulses with different duration,” Opt. Commun.186(1-3), 211–217 (2000). [CrossRef]
T. Zhang and M. Yonemura, “Pulse shaping of ultrashort laser pulses with nonlinear optical crystals,” Jpn. J. Appl. Phys.38(Part 1, No. 11), 6351–6358 (1999). [CrossRef]
M. Sabaeian, L. Mousave, and H. Nadgaran, “Investigation of thermally-induced phase mismatching in continuous-wave second harmonic generation: a theoretical model,” Opt. Express18(18), 18732–18743 (2010). [CrossRef] [PubMed]
S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B27(8), 1504–1512 (2010). [CrossRef]
S. G. Grechin, V. G. Dmitriev, V. A. D'yakov, and V. I. Pryalkin, “Anomalous temperature-independent birefringence in a biaxial optical LBO crystal,” Quantum Electron.30(4), 285–286 (2000). [CrossRef]
A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B16(4), 609–619 (1999). [CrossRef]
G. Arisholm, “General numerical methods for simulating second-order nonlinear interactions in birefringent media,” J. Opt. Soc. Am. B14(10), 2543–2549 (1997). [CrossRef]
A. V. Smith, “How to select nonlinear crystals and model their performance using SNLO software,” Proc. SPIE3928, 62–69 (2000). [CrossRef]
Q. Liu, X. Yan, M. Gong, X. Fu, and D. Wang, “103 W high beam quality green laser with an extra- cavity second harmonic generation,” Opt. Express16(19), 14335–14340 (2008). [CrossRef] [PubMed]
K. H. Hong, C. J. Lai, A. Siddiqui, and F. X. Kärtner, “130-W picosecond green laser based on a frequency-doubled hybrid cryogenic Yb:YAG amplifier,” Opt. Express17(19), 16911–16919 (2009). [CrossRef] [PubMed]
R. Eckardt and J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation,” IEEE J. Quantum Electron.20(10), 1178–1187 (1984). [CrossRef]
C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB3O5,” J. Opt. Soc. Am. B6(4), 616–621 (1989). [CrossRef]
R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron.33(7), 1049–1056 (1997). [CrossRef]
S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B21(3), 591–598 (2004). [CrossRef]

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