Abstract

Angular dispersion of the signal beam inside the nonlinear media is introduced to improve the non-collinear phase-matching range. Simulations run for BBO crystals predict that bandwidth increase is possible for most of the application spectral range and that it can surpass one order of magnitude in some particular configurations.

©2004 Optical Society of America

1. Introduction

In the quest for increasing optical power, shorter laser pulses are under continuous development, requiring broader bandwidth amplifier media. Moreover, as tunable optical parametric oscillators (OPOs) become more widespread, attention is getting dispersed from the traditional wavelengths, set by renowned amplifier media such as dyes, doped glasses, Ti:sapphire and other crystals. On the other hand, broadband nonlinear crystals, such as β-Barium Borate (BBO) and Lithium Triborate (LBO), used in optical parametric amplifiers (OPAs) can provide very high gains from IR to UV, if properly pumped and tuned. By tuning one means to use temperature, collinear, non-collinear or quasi phase matching techniques to optimise the gain around the signal wavelength. A non-collinear geometry is often chosen because it makes it simpler to separate the signal from the idler, and usually allows a larger bandwidth. In particular, depending on the crystal and pump wavelength used, one can find a very broad and reasonably flat gain curve for a certain range of signal wavelengths. Out of that range it has already been shown that the bandwidth can be increased, but at the expense of a custom tailored, chirped pump beam [1].

The use of angular dispersion to bring each component of a broadband pulse closer to its phase-matching condition has already been shown for femtosecond laser Second Harmonic Generation [26], and more generally, for Sum Frequency Generation [79]. However, to our knowledge, the present paper is the first addressing the benefits of controlled angular dispersion applied to a Difference Frequency Generation process, such as Optical Parametric Chirped Pulse Amplification - OPCPA [10,11].

2. Model

Assuming a slowly varying signal envelope and flat top spatial and temporal pump profiles, with no depletion, the gain G and phase φ of the amplified signal in an OPA can be estimated using the analytical solution of the coupled wave equations defined in Reference [10]:

G=1+(γL)2(sinhBB)2
φ=tan1BsinAcoshBAcosAsinhBBcosAcoshBAsinAsinhB

where:

A=ΔkL2
B=[(γL)2(ΔkL2)2]12
γ=4πdeff(Ip2ε0npnsnicλsλi)12
ΔkL=(kpkski)L

with the restriction ωp=ωs+ωi. Here ε0 and c are the permittivity and speed of light in vacuum, γ is the gain coefficient, ΔkL the phase mismatch, L the amplifier length and d eff the effective nonlinear coefficient. I, n, ω, λ and k stand for the intensity, refractive index, frequency, wavelength and wave vector respectively, with the appropriate subscripts for pump (p), signal (s) and idler (i).

For calculation of the refractive indexes and related quantities we adopt the same Sellmeier equations as in the SNLO software [12].

As mentioned before, non-collinear phase matching helps increase the bandwidth. But at a fixed phase-matching angle θ PM, as the signal wavelength is changed the non-collinear angle must be readjusted in order to optimise the gain. That is the role of angular dispersion, providing that adjustment to some extent. The simulations are performed with OPCPA in mind, where one can assume a highly monochromatic pump beam and a chirped signal that can be instantaneously approximated to a single frequency; however this conceptual simplification should not deprive other setups to benefit from this study.

Following the treatment for non-collinear geometry used in Ref. [11] one finds that the ideal angle between a signal component and the pump beam for perfect phase-matching is given by

cos(θNC,ideal)=kp2+ks2ki22kpks

Figure 1 shows the behaviour of this function for BBO. This suggests that a wavelength-dependent angle properly added to a base non-collinear angle θ NC can extend the phase matched range around stationary points with positive 2nd derivative. As for the others, adding dispersion can only shift the wavelength they occur at.

 

Fig. 1. Dependence of the ideal non-collinear angle with the signal wavelength for some phase-matching angles, using a 532nm pump for a BBO crystal. The dots mark stationary points.

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In Fig. 2 we present the setup considered. In addition to the plane defined by the optical axis (OA) and the pump beam wave vector k p, whose relative angle is θ PM, we define a second plane, orthogonal to the first, at an angle θ NC with k p. In this plane the signal is dispersed angularly, and k s0 is the component that lies along the intersection of both planes. λ 0λ s0, the wavelength corresponding to k s0, is chosen from the inflexion point coordinates. Do not confuse λ 0 with the amplified signal central wavelength which, in general, turns out to be longer. This geometry was also chosen for clarity: actually, as long as the signal is an ordinary ray, the relative orientation of θ NC and θ PM in the crystal holds no constraint.

 

Fig. 2. Definition of the angles used in the simulation and wave vector matching. The idler gets dispersed in both directions.

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For this setup, assuming small angles so that Pythagoras theorem can be used, we write:

θNC,aprox2(λs)=θNC2(λs0)+[θdisp(λsλs0)]2

where θ disp refers to the rate of dispersion applied and θ NC(λ s0)=θ NC,ideal (λ s0). Differentiating twice, or analysing the slope in the vicinity δλ s of λ s0, we find how much dispersion is needed to match θ NC,ideal and θ NC,aprox up to second order around λ s0:

θdisp={[θNC,ideal(λs0).θNC,ideal(λs0)]12ifθNC,ideal(λs0)>0θNC,ideal(λs0±δλs)ifθNC,ideal(λs0)=0

This amount of dispersion gives perfect phase matching for wavelengths near λ s0, but its possible to manually optimise these results by tweaking the amount of dispersion and the phase matching angle so that a broader wavelength range can be amplified.

3. Results

In order to assess the benefits of this setup, simulations were run for a BBO crystal, pumped by a 532 nm, 1 GW.cm-2 beam, typical figures of a commercial Nd:YAG laser. These simulations were targeted at a gain of 1000, which sets a BBO crystal length of 8.7 mm. We then optimise the bandwidth by introducing a small phase mismatch; however, this must not affect the gain curve by more than ±10% to minimise signal degradation. θ NC is never set below half a degree for easy signal-idler separation. The dispersion source is modelled as the first diffraction order output of a grating with a normally incident beam. The improvement of this technique over the dispersionless non-collinear setup is shown in Fig. 3.

 

Fig. 3. (a) Spectral gain profile vs. λ s0 for an 8.7 mm long BBO crystal without dispersion. Pump is λ p=532 nm, I p=1 GW.cm-2. Some pixelation results from the step used in the simulation. (b) Similar to (a), but with optimised signal angular dispersion. The dashed line marks a turn point for abscissa counting and also signals the configuration detailed in Fig. 4 (c) Full width half maximum bandwidth of (a) and (b) in dotted and solid curves, respectively. Abscissas for the dotted line are as usual in literature, but for the solid line they are the effective central wavelengths for the gain bandwidth, not λ s0, because of the greater gain asymmetry around it.

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Fig. 4. Calculated gain (solid) and phase (dotted) for an 8.73 mm long BBO crystal, pumped by 1 GW.cm-2 at 532 nm; λ 0=1060 nm. The FWHM bandwidth is 724 nm, centred at 1170 nm.

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The largest bandwidth improvement lies on the long wavelength side of the degeneracy point, increasing by over an order of magnitude.

Figure 4 details Fig. 3(b) along the dashed line, where λ 0=1060 nm. The optimisation allowed a bandwidth exceeding 700 nm while leaving gain and phase very smooth.

Figure 5 shows the optimised and un-optimised configurations using dispersion. Optimisation consists of changing the amount of dispersion recommended by Eq. (9) and the initial value of θ PM. Solid lines there and in Fig. 3 refer to the same setup, but notice that dotted lines do not.

 

Fig. 5. BBO wavelength-dependence of the phase-matching angle (left), non-collinear angle (centre), and dispersion (right). The configurations with the flattest gain and phase for wavelengths near λ 0 are shown dotted; dispersion given by Eq. (9) is applied to flatten these curves up to second order. Solid lines show the optimised parameters to allow extended bandwidth.

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Figures 6 and 7 respectively illustrate the gain obtainable with un-optimised and optimised (dotted and solid) configurations of Fig. 5.

 

Fig. 6. (a) BBO’s spectral gain distribution using the parameters of the dotted curves in Fig. 5. (b), (c), (d), (e) Gain changes induced by perturbation of the critical parameters. Blue and yellow graphs show the effects of negative and positive deltas, respectively. They are presented colour-wise summed: white means both of them overlap, indicating the configurations with better error tolerance.

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Fig. 7. (a) BBO’s spectral gain distribution using the optimised parameters of the solid curves in Fig. 5. (b), (c), (d), (e) Gain changes induced by perturbation of the critical parameters. Colours are as in Fig. 6.

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From the previous graphics, it is evident the major bandwidth increase when angular dispersion is properly used. It is necessary, however, to control two more quantities: λ 0 and θ disp. Figures 6 and 7 assess their sensitivity, which are well within common alignment accuracy. One can also see that bandwidth un-optimised configurations, as they are working further from the limits of phase-matching, will drop relatively less bandwidth when slightly misaligned.

To illustrate the potential of this scheme we simulated an amplifier stage with a BBO crystal set for the maximum bandwidth configuration, like in Fig. 4. Inputting transform-limited pulses centred at 1053 nm, a common wavelength in high power lasers, these will experience minimal broadening even when phase terms down to the third order are allowed in the output:

- a 7 fs (FWHM) Gaussian pulse enlarges to <10 fs with contrast ratio >106;

- a 5 fs Gaussian pulse enlarges to <6.5 fs with contrast ratio >104.

The main application of this setup is therefore the amplification of ultra-broadband pulses at user selected wavelengths. Another possible use is as a tuneless amplifier for tunable laser sources with longer pulse durations.

4. Conclusion

We have demonstrated a new scheme that allows the use of a BBO crystal as an ultra broadband amplifier. Bandwidths from 200 nm to more than 700 nm are continuously available for central wavelengths ranging 850–1250 nm. Simulations presented in this work use the 2nd harmonic of a Nd:YAG laser as the pump source, but good results are also obtained using its 3rd harmonic and are expected for other pump wavelengths too. Other negative crystal types can also benefit from this scheme.

Users interested to further optimise around a specific wavelength could try a less linear dispersive element than a grating, which may grant them better non-collinearity curve matching.

References and links

1. K. Osvay and I. N. Ross, “Broadband sum-frequency generation by chirp-assisted group-velocity matching,” J. Opt. Soc. Am. B 13, 1431–1438 (1996). [CrossRef]  

2. S. Saikan, “Automatically tunable second harmonic generation of dye lasers,” Opt. Commun. 18, 439–443 (1976). [CrossRef]  

3. S. Saikan, D. Ouw, and F. P. Schafer, “Automatic phase-matched frequency-doubling system for the 240–350-nm region,” Appl. Opt. 18, 193–196 (1979). [CrossRef]   [PubMed]  

4. T. R. Zhang, H. R. Choo, and M. C. Downer, “Phase and group velocity matching for second harmonic generation of femtosecond pulses,” Appl. Opt. 29, 3927–3933 (1990). [CrossRef]   [PubMed]  

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

6. J. A. Fülöp, A. P. Kovács, and Z. Bor, “Broadband dispersion-compensated two-pass second harmonic generation of femtosecond pulses,” Opt. Commun. 188, 365–370 (2001). [CrossRef]  

7. G. Szabó and Zs. Bor, “Frequency conversion of ultrashort pulses,” Appl. Phys. B 58, 237–241 (1994). [CrossRef]  

8. M. Hacker, T. Feurer, R. Sauerbrey, T. Lucza, and G. Szabó, “Programmable femtosecond laser pulses in the ultraviolet,” J. Opt. Soc. Am. B 18, 866–871 (2001). [CrossRef]  

9. Y. Nabekawa and K. Midorikawa, “Broadband sum frequency mixing using noncollinear angularly dispersed geometry for indirect phase control of sub-20-femtosecond UV pulses,” Opt. Express 11, 324–338 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-324. [CrossRef]   [PubMed]  

10. I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125–133 (1997). [CrossRef]  

11. I. N. Ross, P. Matousek, G. New, and K. Osvay, “Analysis and optimization of optical parametric chirped pulse amplification,” J. Opt. Soc. Am. B 19, 2945–2956 (2002). [CrossRef]  

12. Sandia National Laboratories, http://www.sandia.gov/imrl/X1118/xxtal.htm

13. G. Cerullo and S. De Silvestri, “Review article - Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003). [CrossRef]  

14. I. Jovanovic, B. J. Comaskey, and D. M. Pennington, “Angular effects and beam quality in optical parametric amplification,” J. Appl. Phys. 90, 4328–4337 (2001) [CrossRef]  

References

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  1. K. Osvay and I. N. Ross, “Broadband sum-frequency generation by chirp-assisted group-velocity matching,” J. Opt. Soc. Am. B 13, 1431–1438 (1996).
    [Crossref]
  2. S. Saikan, “Automatically tunable second harmonic generation of dye lasers,” Opt. Commun. 18, 439–443 (1976).
    [Crossref]
  3. S. Saikan, D. Ouw, and F. P. Schafer, “Automatic phase-matched frequency-doubling system for the 240–350-nm region,” Appl. Opt. 18, 193–196 (1979).
    [Crossref] [PubMed]
  4. T. R. Zhang, H. R. Choo, and M. C. Downer, “Phase and group velocity matching for second harmonic generation of femtosecond pulses,” Appl. Opt. 29, 3927–3933 (1990).
    [Crossref] [PubMed]
  5. O. E. Martínez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989).
    [Crossref]
  6. J. A. Fülöp, A. P. Kovács, and Z. Bor, “Broadband dispersion-compensated two-pass second harmonic generation of femtosecond pulses,” Opt. Commun. 188, 365–370 (2001).
    [Crossref]
  7. G. Szabó and Zs. Bor, “Frequency conversion of ultrashort pulses,” Appl. Phys. B 58, 237–241 (1994).
    [Crossref]
  8. M. Hacker, T. Feurer, R. Sauerbrey, T. Lucza, and G. Szabó, “Programmable femtosecond laser pulses in the ultraviolet,” J. Opt. Soc. Am. B 18, 866–871 (2001).
    [Crossref]
  9. Y. Nabekawa and K. Midorikawa, “Broadband sum frequency mixing using noncollinear angularly dispersed geometry for indirect phase control of sub-20-femtosecond UV pulses,” Opt. Express 11, 324–338 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-324.
    [Crossref] [PubMed]
  10. I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125–133 (1997).
    [Crossref]
  11. I. N. Ross, P. Matousek, G. New, and K. Osvay, “Analysis and optimization of optical parametric chirped pulse amplification,” J. Opt. Soc. Am. B 19, 2945–2956 (2002).
    [Crossref]
  12. Sandia National Laboratories, http://www.sandia.gov/imrl/X1118/xxtal.htm
  13. G. Cerullo and S. De Silvestri, “Review article - Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
    [Crossref]
  14. I. Jovanovic, B. J. Comaskey, and D. M. Pennington, “Angular effects and beam quality in optical parametric amplification,” J. Appl. Phys. 90, 4328–4337 (2001)
    [Crossref]

2003 (2)

2002 (1)

2001 (3)

I. Jovanovic, B. J. Comaskey, and D. M. Pennington, “Angular effects and beam quality in optical parametric amplification,” J. Appl. Phys. 90, 4328–4337 (2001)
[Crossref]

J. A. Fülöp, A. P. Kovács, and Z. Bor, “Broadband dispersion-compensated two-pass second harmonic generation of femtosecond pulses,” Opt. Commun. 188, 365–370 (2001).
[Crossref]

M. Hacker, T. Feurer, R. Sauerbrey, T. Lucza, and G. Szabó, “Programmable femtosecond laser pulses in the ultraviolet,” J. Opt. Soc. Am. B 18, 866–871 (2001).
[Crossref]

1997 (1)

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125–133 (1997).
[Crossref]

1996 (1)

1994 (1)

G. Szabó and Zs. Bor, “Frequency conversion of ultrashort pulses,” Appl. Phys. B 58, 237–241 (1994).
[Crossref]

1990 (1)

1989 (1)

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

1979 (1)

1976 (1)

S. Saikan, “Automatically tunable second harmonic generation of dye lasers,” Opt. Commun. 18, 439–443 (1976).
[Crossref]

Bor, Z.

J. A. Fülöp, A. P. Kovács, and Z. Bor, “Broadband dispersion-compensated two-pass second harmonic generation of femtosecond pulses,” Opt. Commun. 188, 365–370 (2001).
[Crossref]

Bor, Zs.

G. Szabó and Zs. Bor, “Frequency conversion of ultrashort pulses,” Appl. Phys. B 58, 237–241 (1994).
[Crossref]

Cerullo, G.

G. Cerullo and S. De Silvestri, “Review article - Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[Crossref]

Choo, H. R.

Collier, J. L.

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125–133 (1997).
[Crossref]

Comaskey, B. J.

I. Jovanovic, B. J. Comaskey, and D. M. Pennington, “Angular effects and beam quality in optical parametric amplification,” J. Appl. Phys. 90, 4328–4337 (2001)
[Crossref]

De Silvestri, S.

G. Cerullo and S. De Silvestri, “Review article - Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[Crossref]

Downer, M. C.

Feurer, T.

Fülöp, J. A.

J. A. Fülöp, A. P. Kovács, and Z. Bor, “Broadband dispersion-compensated two-pass second harmonic generation of femtosecond pulses,” Opt. Commun. 188, 365–370 (2001).
[Crossref]

Hacker, M.

Jovanovic, I.

I. Jovanovic, B. J. Comaskey, and D. M. Pennington, “Angular effects and beam quality in optical parametric amplification,” J. Appl. Phys. 90, 4328–4337 (2001)
[Crossref]

Kovács, A. P.

J. A. Fülöp, A. P. Kovács, and Z. Bor, “Broadband dispersion-compensated two-pass second harmonic generation of femtosecond pulses,” Opt. Commun. 188, 365–370 (2001).
[Crossref]

Langley, A. J.

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125–133 (1997).
[Crossref]

Lucza, T.

Martínez, O. E.

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

Matousek, P.

I. N. Ross, P. Matousek, G. New, and K. Osvay, “Analysis and optimization of optical parametric chirped pulse amplification,” J. Opt. Soc. Am. B 19, 2945–2956 (2002).
[Crossref]

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125–133 (1997).
[Crossref]

Midorikawa, K.

Nabekawa, Y.

New, G.

Osvay, K.

Ouw, D.

Pennington, D. M.

I. Jovanovic, B. J. Comaskey, and D. M. Pennington, “Angular effects and beam quality in optical parametric amplification,” J. Appl. Phys. 90, 4328–4337 (2001)
[Crossref]

Ross, I. N.

Saikan, S.

Sauerbrey, R.

Schafer, F. P.

Szabó, G.

Towrie, M.

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125–133 (1997).
[Crossref]

Zhang, T. R.

Appl. Opt. (2)

Appl. Phys. B (1)

G. Szabó and Zs. Bor, “Frequency conversion of ultrashort pulses,” Appl. Phys. B 58, 237–241 (1994).
[Crossref]

IEEE J. Quantum Electron. (1)

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

J. Appl. Phys. (1)

I. Jovanovic, B. J. Comaskey, and D. M. Pennington, “Angular effects and beam quality in optical parametric amplification,” J. Appl. Phys. 90, 4328–4337 (2001)
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Commun. (3)

S. Saikan, “Automatically tunable second harmonic generation of dye lasers,” Opt. Commun. 18, 439–443 (1976).
[Crossref]

J. A. Fülöp, A. P. Kovács, and Z. Bor, “Broadband dispersion-compensated two-pass second harmonic generation of femtosecond pulses,” Opt. Commun. 188, 365–370 (2001).
[Crossref]

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125–133 (1997).
[Crossref]

Opt. Express (1)

Rev. Sci. Instrum. (1)

G. Cerullo and S. De Silvestri, “Review article - Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[Crossref]

Other (1)

Sandia National Laboratories, http://www.sandia.gov/imrl/X1118/xxtal.htm

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Figures (7)

Fig. 1.
Fig. 1. Dependence of the ideal non-collinear angle with the signal wavelength for some phase-matching angles, using a 532nm pump for a BBO crystal. The dots mark stationary points.
Fig. 2.
Fig. 2. Definition of the angles used in the simulation and wave vector matching. The idler gets dispersed in both directions.
Fig. 3.
Fig. 3. (a) Spectral gain profile vs. λ s0 for an 8.7 mm long BBO crystal without dispersion. Pump is λ p=532 nm, I p=1 GW.cm-2. Some pixelation results from the step used in the simulation. (b) Similar to (a), but with optimised signal angular dispersion. The dashed line marks a turn point for abscissa counting and also signals the configuration detailed in Fig. 4 (c) Full width half maximum bandwidth of (a) and (b) in dotted and solid curves, respectively. Abscissas for the dotted line are as usual in literature, but for the solid line they are the effective central wavelengths for the gain bandwidth, not λ s0, because of the greater gain asymmetry around it.
Fig. 4.
Fig. 4. Calculated gain (solid) and phase (dotted) for an 8.73 mm long BBO crystal, pumped by 1 GW.cm-2 at 532 nm; λ 0=1060 nm. The FWHM bandwidth is 724 nm, centred at 1170 nm.
Fig. 5.
Fig. 5. BBO wavelength-dependence of the phase-matching angle (left), non-collinear angle (centre), and dispersion (right). The configurations with the flattest gain and phase for wavelengths near λ 0 are shown dotted; dispersion given by Eq. (9) is applied to flatten these curves up to second order. Solid lines show the optimised parameters to allow extended bandwidth.
Fig. 6.
Fig. 6. (a) BBO’s spectral gain distribution using the parameters of the dotted curves in Fig. 5. (b), (c), (d), (e) Gain changes induced by perturbation of the critical parameters. Blue and yellow graphs show the effects of negative and positive deltas, respectively. They are presented colour-wise summed: white means both of them overlap, indicating the configurations with better error tolerance.
Fig. 7.
Fig. 7. (a) BBO’s spectral gain distribution using the optimised parameters of the solid curves in Fig. 5. (b), (c), (d), (e) Gain changes induced by perturbation of the critical parameters. Colours are as in Fig. 6.

Equations (9)

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G = 1 + ( γ L ) 2 ( sinh B B ) 2
φ = tan 1 B sin A cosh B A cos A sinh B B cos A cosh B A sin A sinh B
A = Δ kL 2
B = [ ( γ L ) 2 ( Δ kL 2 ) 2 ] 1 2
γ = 4 π d eff ( I p 2 ε 0 n p n s n i c λ s λ i ) 1 2
Δ kL = ( k p k s k i ) L
cos ( θ NC , ideal ) = k p 2 + k s 2 k i 2 2 k p k s
θ NC , aprox 2 ( λ s ) = θ NC 2 ( λ s 0 ) + [ θ disp ( λ s λ s 0 ) ] 2
θ disp = { [ θ NC , ideal ( λ s 0 ) . θ NC , ideal ( λ s 0 ) ] 1 2 if θ NC , ideal ( λ s 0 ) > 0 θ NC , ideal ( λ s 0 ± δ λ s ) if θ NC , ideal ( λ s 0 ) = 0

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