Abstract

Two typos are corrected, and the linear refractive index $n$ is removed from the expressions of the phase shift in Opt. Lett. 46, 560 (2021) [CrossRef]  . The removal of $n$ reduces the gate efficiency, but it does not affect the general findings. Here, we present the corrected equations and the corresponding new numerical results, showing that increasing the pulse energy from 1.8 nJ to 4 nJ leads to nearly the same results of Opt. Lett. 46, 560 (2021) [CrossRef]  .

© 2021 Optica Publishing Group

There was a typo in the expression of ${I_0}$, which should read ${I_0} = 2\sqrt {\ln 2} E/({\pi ^{3/2}}{\tau _0}w_0^2)$. There was a typo in the expression of the gate pulse temporal profile (Eq. (10) of [1]), which should read

$${I_{\rm{g}}}(t) = {I_0}\exp \left[{- 4\ln 2\frac{{{{(t - {t_0})}^2}}}{{\tau _0^2}}} \right].$$

The two typos refer to the text and do not affect the results in [1].

The phase shift induced by the optical Kerr effect should not contain the linear refractive index $n$ [2]. Hence, the phase shift reads

$${\rm{d}}\varphi = (2\pi {n_2}/\lambda){I_{\rm{g}}}(z){\rm{d}}z$$
in Eq. (3) of [1] and
$$\Delta \varphi (0,L) = \frac{{4\pi {n_2}{z_{{\rm{R}},0}}{I_0}}}{\lambda}\arctan \left({\frac{L}{{2{z_{{\rm{R}},0}}}}} \right)$$
in Eq. (7) of [1]. This correction affects the numerical results of [1], but it does not compromise the general findings, as shown in the following.

We recalculated Figs. 2, 3 and 4 of [1], which are plotted here as Figs. 1, 2 and 3, respectively. To obtain a large efficiency, we changed the pulse energy from 1.8 nJ to 4 nJ, which corresponds to a power increase from 1.8 W to 4 W, at a repetition rate of 1 GHz. The results for 1.8 W are also recalculated, and they are summarized in Table 1 for comparison with Table 1 of [1].

 figure: Fig. 1.

Fig. 1. The effect of dispersion on the phase shift $\Delta \varphi (0,L)$ is more apparent for thicker Kerr media or shorter pulses (not shown). Parameters: $\lambda = 800\;{\rm{nm}}$, ${w_0} = 10\;\unicode{x00B5}{\rm m}$, $E = 4\;{\rm{nJ}}$, ${\tau _0} = 80\;{\rm{fs}}$, $n = 2.45$, ${n_2} = 1.6 \times {10^{- 18}}\;{{\rm{m}}^2}/{\rm{W}}$, $\phi = 1057.19\;{\rm{f}}{{\rm{s}}^2}/{\rm{mm}}$.

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 figure: Fig. 2.

Fig. 2. Transmittance (gate efficiency) as a function of the pump and probe beam diameter for different lengths of the nonlinear crystal. (a) $L = {{1}}\;{\rm{mm}}$, (b) $L = {{2}}\;{\rm{mm}}$, (c) $L = {{3}}\;{\rm{mm}}$, (d) $L = {{5}}\;{\rm{mm}}$. Parameters: $\lambda = 800\;{\rm{nm}}$, $E = 4\;{\rm{nJ}}$, ${\tau _0} = 80\;{\rm{fs}}$, $n = 2.45$, ${n_2} = 1.6 \times {10^{- 18}}\;{{\rm{m}}^2}/{\rm{W}}$, $\phi = 1057.19\;{\rm{f}}{{\rm{s}}^2}/{\rm{mm}}$.

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 figure: Fig. 3.

Fig. 3. Transmittance as a function of time for different thicknesses of the nonlinear crystal. Parameters: $\lambda = 800\;{\rm{nm}}$, $E = 4\;{\rm{nJ}}$, ${\tau _0} = 80\;{\rm{fs}}$, $n = 2.45$, ${n_2} = 1.6 \times {10^{- 18}}\;{{\rm{m}}^2}/{\rm{W}}$, $\phi = 1057.19\;{\rm{f}}{{\rm{s}}^2}/{\rm{mm}}$.

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Tables Icon

Table 1. Efficiency and Time Resolution (FWHM) of the OKS for Different Kerr Media, Such as Bismuth Borosilicate (${\rm{B}}{{\rm{i}}_2}{{\rm{O}}_3}\! -\! {{\rm{B}}_2}{{\rm{O}}_3}\! -\! {\rm{Si}}{{\rm{O}}_2}$, BBS) [3,4] and Fused Silica (${\rm{Si}}{{\rm{O}}_2}$) [5] and Various Experimental Parameters: $\lambda = 800\;{\rm{nm}}$, Average Pump Power 1.8 W, and the Other Ones as Listed in the Tablea

Figure 1 shows the phase shift as a function of the crystal thickness with and without dispersion.

Figure 2 plots the transmittance as a function of different beam diameters and for different crystal thicknesses.

Figure 3 depicts the time resolution of the gate for different thicknesses of the nonlinear crystal.

Table 1 summarizes the gate efficiency of Kerr media under different experimental conditions for the optimized beam diameter, which has been calculated using our semi-analytical model.

We remark that the optimal diameters shown in Table 1 and in Fig. 2 do not change compared to Ref. [1], because the correction does not affect the diffraction problem, which depends on the beam waist ${w_0}$ and on the Rayleigh range ${z_{{\rm{R}},0}} = \pi nw_0^2/\lambda$, but it only removes a multiplication factor, the linear refractive index $n$, in the expression for the phase shift. Hence, a high efficiency can still be obtained by a moderate increase of the pulse energy (a factor of $n$ would give exactly the results shown in [1]). We also point out that the slightly better time resolution in Table 1 and Fig. 3, compared to Ref. [1], is due to the lower efficiency of the gate.

REFERENCES

1. A.-H. Fattah, A. M. Flatae, A. Farrag, and M. Agio, Opt. Lett. 46, 560 (2021). [CrossRef]  

2. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

3. W. Tan, H. Liu, J. Si, and X. Hou, Appl. Phys. Lett. 93, 051109 (2008). [CrossRef]  

4. “Optics toolbox - interactive calculators for scientists and engineers,” 2020, http://toolbox.lightcon.com/tools/dispersionparameters/.

5. C. Karras, D. Litzkendorf, S. Grimm, K. Schuster, W. Paa, and H. Stafast, Opt. Mater. Express 4, 2066 (2014). [CrossRef]  

References

  • View by:

  1. A.-H. Fattah, A. M. Flatae, A. Farrag, and M. Agio, Opt. Lett. 46, 560 (2021).
    [Crossref]
  2. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
  3. W. Tan, H. Liu, J. Si, and X. Hou, Appl. Phys. Lett. 93, 051109 (2008).
    [Crossref]
  4. “Optics toolbox - interactive calculators for scientists and engineers,” 2020, http://toolbox.lightcon.com/tools/dispersionparameters/ .
  5. C. Karras, D. Litzkendorf, S. Grimm, K. Schuster, W. Paa, and H. Stafast, Opt. Mater. Express 4, 2066 (2014).
    [Crossref]

2021 (1)

2014 (1)

2008 (1)

W. Tan, H. Liu, J. Si, and X. Hou, Appl. Phys. Lett. 93, 051109 (2008).
[Crossref]

Agio, M.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Farrag, A.

Fattah, A.-H.

Flatae, A. M.

Grimm, S.

Hou, X.

W. Tan, H. Liu, J. Si, and X. Hou, Appl. Phys. Lett. 93, 051109 (2008).
[Crossref]

Karras, C.

Litzkendorf, D.

Liu, H.

W. Tan, H. Liu, J. Si, and X. Hou, Appl. Phys. Lett. 93, 051109 (2008).
[Crossref]

Paa, W.

Schuster, K.

Si, J.

W. Tan, H. Liu, J. Si, and X. Hou, Appl. Phys. Lett. 93, 051109 (2008).
[Crossref]

Stafast, H.

Tan, W.

W. Tan, H. Liu, J. Si, and X. Hou, Appl. Phys. Lett. 93, 051109 (2008).
[Crossref]

Appl. Phys. Lett. (1)

W. Tan, H. Liu, J. Si, and X. Hou, Appl. Phys. Lett. 93, 051109 (2008).
[Crossref]

Opt. Lett. (1)

Opt. Mater. Express (1)

Other (2)

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

“Optics toolbox - interactive calculators for scientists and engineers,” 2020, http://toolbox.lightcon.com/tools/dispersionparameters/ .

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Figures (3)

Fig. 1.
Fig. 1. The effect of dispersion on the phase shift $\Delta \varphi (0,L)$ is more apparent for thicker Kerr media or shorter pulses (not shown). Parameters: $\lambda = 800\;{\rm{nm}}$, ${w_0} = 10\;\unicode{x00B5}{\rm m}$, $E = 4\;{\rm{nJ}}$, ${\tau _0} = 80\;{\rm{fs}}$, $n = 2.45$, ${n_2} = 1.6 \times {10^{- 18}}\;{{\rm{m}}^2}/{\rm{W}}$, $\phi = 1057.19\;{\rm{f}}{{\rm{s}}^2}/{\rm{mm}}$.
Fig. 2.
Fig. 2. Transmittance (gate efficiency) as a function of the pump and probe beam diameter for different lengths of the nonlinear crystal. (a) $L = {{1}}\;{\rm{mm}}$, (b) $L = {{2}}\;{\rm{mm}}$, (c) $L = {{3}}\;{\rm{mm}}$, (d) $L = {{5}}\;{\rm{mm}}$. Parameters: $\lambda = 800\;{\rm{nm}}$, $E = 4\;{\rm{nJ}}$, ${\tau _0} = 80\;{\rm{fs}}$, $n = 2.45$, ${n_2} = 1.6 \times {10^{- 18}}\;{{\rm{m}}^2}/{\rm{W}}$, $\phi = 1057.19\;{\rm{f}}{{\rm{s}}^2}/{\rm{mm}}$.
Fig. 3.
Fig. 3. Transmittance as a function of time for different thicknesses of the nonlinear crystal. Parameters: $\lambda = 800\;{\rm{nm}}$, $E = 4\;{\rm{nJ}}$, ${\tau _0} = 80\;{\rm{fs}}$, $n = 2.45$, ${n_2} = 1.6 \times {10^{- 18}}\;{{\rm{m}}^2}/{\rm{W}}$, $\phi = 1057.19\;{\rm{f}}{{\rm{s}}^2}/{\rm{mm}}$.

Tables (1)

Tables Icon

Table 1. Efficiency and Time Resolution (FWHM) of the OKS for Different Kerr Media, Such as Bismuth Borosilicate ( B i 2 O 3 B 2 O 3 S i O 2 , BBS) [3,4] and Fused Silica ( S i O 2 ) [5] and Various Experimental Parameters: λ = 800 n m , Average Pump Power 1.8 W, and the Other Ones as Listed in the Tablea

Equations (3)

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

I g ( t ) = I 0 exp [ 4 ln 2 ( t t 0 ) 2 τ 0 2 ] .
d φ = ( 2 π n 2 / λ ) I g ( z ) d z
Δ φ ( 0 , L ) = 4 π n 2 z R , 0 I 0 λ arctan ( L 2 z R , 0 )