Sum-frequency nonlinear Cherenkov radiation generated on the boundary of bulk medium crystal

Xiaojing Wang; Jianjun Cao; Xiaohui Zhao; Yuanlin Zheng; Huaijin Ren; Xuewei Deng; Xianfeng Chen

doi:10.1364/OE.23.031838

1. Introduction

As is known, Cherenkov radiation (CR) [1] is the coherent light in a forward-pointing conical wave-front form emitted by a charged particle which is moving faster than the phase velocity of light in that medium. Similarly, nonlinear Cherenkov radiation (NCR) is the coherent harmonic wave which occurs as the phase velocity of the nonlinear polarization wave surpasses that of the harmonic wave [2]. Due to the tolerant requirement for auto-phase-matching, NCR has attracted lots of attention in the past few years. Many different type of NCR phenomenon have been observed, for example, Cherenkov type sum-frequency generation [3, 4 ], third and higher order harmonic generation [5, 6 ], difference frequency generation [7], and so on. Since NCR provides a tolerant phase-matching approach for frequency conversion, it has a great application prospect for nondestructive diagnostics, microscopy [8] and ultra-short pulse characterization [3].

Researchers have found that the intensity of NCR can be greatly enhanced by the domain wall of nonlinear photonic structures like periodically poled lithium niobate (PPLN) [2, 9, 10 ]. From another perspective, nonlinear polarization wave is confined along the domain walls, so NCR can be realized by modulating the phase velocity of the polarization in the anomalous dispersion medium which used to be the forbidden region of NCR [11]. The mechanism of this enhancement effect can be attributed to two main reasons. One is that the symmetry breaking in domain walls changes domain walls’ local physics properties, the other is that the sharp −1 to 1 second order susceptibility modulation at the inverted domains boundaries provides broad spectra of transverse reciprocal vectors [12]. Except for domain wall as a nonlinear interface with discontinuous nonlinear coefficient $χ^{(2)}$ changing from 1 to −1, there are other nonlinear interfaces where the nonlinear coefficients are also discontinuous. For example, the crystal boundaries is such discontinuity interface from 1 to 0. In our previous research on the boundary of nonlinear crystal we can observe the enhanced NCR which is generated by the sum-frequency polarization wave stimulated by the incident light and the reflected light [13, 14 ]. Furthermore, Roppo et al. have also studied on the origin of this effect [15]. In previous studies, sum-frequency polarization wave is generally stimulated by two different waves at domain walls [4, 16 ], or is stimulated on the boundary of crystal by the incident and reflected light with the same wavelength and along the direction of crystal surface naturally [13]. So it is worth discussing the SF NCR generated by pump with different wavelengths on crystal surface to find more evidence that the polarization wave is always confined to the boundary.

In this work, we study the surface sum-frequency NCR process, which is still a vacancy in previous studies. Theoretical analysis are made to describe the universal conditions of surface sum-frequency NCR, which is demonstrated experimentally. Our studies demonstrated that sum-frequency polarization wave is confined in the surface of the nonlinear crystal, which suggests potential applications in surface detection and other surface physics studies. In addition, we discussed critical conditions of surface sum-frequency NCR using any two laser wavelengths under normal and anomalous dispersion condition.

2. Experiment setup and results

We use two collinear laser beams with different frequencies to illuminate the inner surface of the crystal. The experimental layout is shown in Fig. 1(a) . The light source is an optical parametric amplifier (OPA, TOPAS, Coherent Inc.) pumped by a Ti:Sapphire femtosecond regenerative amplifier system (50fs duration, 1KHz repetition rate). We used the pump centered at 800nm and the signal centered at 1300nm (both extraordinarily polarized) to illuminate the crystal as the fundamental wave. The two laser beams are synchronized and loosely focused into the sample at the angle of $θ$ with respect to the X-axis. The crystal is a z-cut 5%/mol MgO:LiNbO₃ of the size $3 \times 10 \times 2 m m^{3} (x \times y \times z)$ , and all of the six surfaces are polished. When the two synchronized laser beams oblique illuminate the boundary of the crystal, we can observe a series of colorful laser beams as shown in Figs. 1(b) and 1(c).

Fig. 1 (a) Schematic of the experimental setup. (b)-(c) Photographs of NCR with different incident angles.

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Spot marked with 0 is second harmonics (SH) propagating along the crystal surface, which is generated by the incidence and its reflection, as mentioned in [17]. In fact, this SH spot contains two frequency component coming from 1300nm and 800nm nearly overlapped in space. The spot on the right side of spot 0 on the screen in Fig. 1(a) is the fundamental wave reflected by the opposite boundary, which is also recorded to calculate the incident angle of the fundamental wave. Spot 1 represents two phase-mismatched SHG beams which are collinear with two fundamental beams. They are generated by the two fundamental incident waves respectively. Spots 2 and 5 are the NCR with the wavelength of 650nm and 400nm (e-polarized). The phase-matching geometries of these two NCR are shown in Figs. 2(b) and 2(c) , respectively. They are generated by nonlinear polarization waves which are stimulated by the incident wave and the reflected wave of the two fundamental wave respectively. The phase-matching is ee-e type, which satisfies normal dispersion condition. Polarization waves which generate NCR spots 2 and 5 can be expressed as

{\vec{P_{1}}}^{N L} = 4 ε_{0} d_{e f f} {\vec{E}}_{1}^{2} e^{i [({\vec{k}}_{1} + {\vec{k}}_{1}^{'}) \cdot \vec{r} - 2 ω_{1} t]}

{\vec{P}}_{2}^{N L} = 4 ε_{0} d_{e f f} {\vec{E}}_{2}^{2} e^{i [({\vec{k}}_{2} + {\vec{k}}_{2}^{'}) \cdot \vec{r} - 2 ω_{2} t]}

where

{\vec{k}}_{1}

,

{\vec{k}}_{1}^{'}

,

{\vec{k}}_{2}

and

{\vec{k}}_{2}^{'}

are wave vectors of the incident wave and reflected wave, subscript 1, 2 mark the fundamental wave of 1300nm and 800nm, respectively. The wave vectors of sum-frequency polarization wave of spots 2 and 5 are

{\vec{k}}_{p 1} = {\vec{k}}_{1} + {\vec{k}}_{1}^{'}

,

{\vec{k}}_{p 2} = {\vec{k}}_{2} + {\vec{k}}_{2}^{'}

, respectively. The two polarization waves propagate along x-axis of the nonlinear crystal with value of

k_{p 1} = 2 k_{1} \cos θ

and

k_{p 2} = 2 k_{2} \cos θ

, where

θ

is the angle between incident wave and x-axis. The NCR spots 2 and 5 appear due to the stimulation of these polarization waves. Under the normal dispersion environment, it is very easy to realize NCR even with a very small incident angle.

Fig. 2 (a) Through internal reflection on the crystal boundary, NCR can be generated by sum-frequency polarization waves. (b) Phase-matching geometries of generation of NCR 650nm with the fundamental wave 1300nm. (c) Phase-matching geometries of NCR 400nm generated by the fundamental wave 800nm. (d) Phase-matching geometries of NCR 495nm generated by the fundamental wave 1300nm and 800nm.

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The situation of spot 3 has significant differences. By measuring spectrum we know that the central wavelength of spot 3 is 495nm, which is the sum-frequency of 1300nm and 800nm. As is shown in Fig. 2(d), the direction of ${\vec{k}}_{1} + \vec{k} {}_{2}^{'}$ is not along the crystal surface, but has an angle with $γ$ respect to the boundary. Will the polarization wave still propagate along the direction of ${\vec{k}}_{1} + \vec{k} {}_{2}^{'}$ as in the case of bulk medium, or along the crystal surface as in the case of domain wall [11, 16 ]? The sum-frequency polarization wave can be expressed as

{\vec{P}}_{3}^{N L} = 4 ε_{0} d_{e f f} {\vec{E}}_{1} {\vec{E}}_{2} e^{i [({\vec{k}}_{1} + {\vec{k}}_{2}^{'}) \cdot \vec{r} - (ω_{1} + ω_{2}) t]}

If the nonlinear polarization wave is confined to the boundary as domain wall, the Eq. (3) will turn into other form:

{\vec{P}}_{3}^{N L} = 4 ε_{0} d_{e f f} {\vec{E}}_{1} {\vec{E}}_{2} e^{i [k_{p 3} r \cos γ - (ω_{1} + ω_{2}) t]}

From the experimental results in later part, we will see that this assumption is correct, i.e., polarization wave propagates along the interface.

Due to the Cherenkov phase matching condition, the bandwidth and the intensities of the harmonics completely depend on the spectrum and power of the pumps. In this experiment, the bandwidths of the two pumps of 1300nm and 800nm are 80nm and 30nm respectively, the bandwidths of the second harmonics are measured to be about 40nm and 15nm, and the sum-frequency is about 23nm. In agreement with previous studies, the spectra of the Cherenkov harmonics exactly reflect that of the pump waves [10], and the conversion efficiency does not change with the fundamental waves [17]. In Figs. 1(b) and 1(c), we can see another spot which is marked with 4, the position of which changes quickly with the angle of the incident wave. Its wavelength is measured to be 578nm coming from a four-wave mixing process, but we will not discuss its generation mechanism in detail here.

According to Figs. 2(b)-2(d), the emergence angle of NCR and the incident angle of fundamental waves must satisfy

n_{1300 e} (π / 2 - θ) \cos θ = n_{650 e} (π / 2 - α_{1}) \cos α_{1}

n_{800 e} (π / 2 - θ) \cos θ = n_{400 e} (π / 2 - α_{2}) \cos α_{2}

ω_{1} n_{1300 e} (π / 2 - θ) \cos θ + ω_{2} n_{800 e} (π / 2 - θ) = (ω_{1} + ω_{2}) n_{495 e} (π / 2 - α_{3}) \cos α_{3}

in order to satisfy ee-e type phase-matching relationship. The equation also can be further written as

\frac{n_{650 o} \cdot n_{650 e}}{\sqrt{n_{650 o}^{2} + n_{650 e}^{2} \tan^{2} α_{1}}} = \frac{n_{1300 o} \cdot n_{1300 e}}{\sqrt{n_{1300 o}^{2} + n_{1300 e}^{2} \tan^{2} θ}}

\frac{n_{400 o} \cdot n_{400 e}}{\sqrt{n_{400 o}^{2} + n_{400 e}^{2} \tan^{2} α_{2}}} = \frac{n_{800 o} \cdot n_{800 e}}{\sqrt{n_{800 o}^{2} + n_{800 e}^{2} \tan^{2} θ}}

(ω_{1} + ω_{2}) \cdot \frac{n_{495 o} \cdot n_{495 e}}{\sqrt{n_{495 o}^{2} + n_{495 e}^{2} \tan^{2} α_{3}}} = ω_{1} \cdot \frac{n_{1300 o} \cdot n_{1300 e}}{\sqrt{n_{1300 o}^{2} + n_{1300 e}^{2} \tan^{2} θ}} + ω_{2} \cdot \frac{n_{800 o} \cdot n_{800 e}}{\sqrt{n_{800 o}^{2} + n_{800 e}^{2} \tan^{2} θ}}

Figure 3(a) displays the experimental data and theoretical calculation of external emergence angle of NCR, and the experimental data fit well with the theoretical curves. With the increase of the incident angle of the fundamental wave, the NCR emergence angle become larger as well. From the calculation results, we can see that there is no limitation for the minimum incident angle. But the maximum incident angles for the fundamental waves exist, which are 30.5° for 400nm, 61.5° for 650nm, and 50° for 495nm. After exceeding this angles, NCR will be totally reflected on end face of the crystal. The crystal boundary plays a very important role on the enhancement of the sum-frequency NCR. We find that sum-frequency NCR only occurs when the fundamental wave illuminate the crystal boundary, although it is theoretically possible to be obtained even when the incident angle is zero in the normal dispersion region. So the enhancement is due to the boundary, which provides a novel mechanism to generate the sum-frequency waves.

Fig. 3 (a) Theoretical calculation curve and experimental measured data of three emergence angles of NCR with the wavelength of 400nm, 495nm and 650nm. (b) The photographs of the collinear phase-mismatched SHG and sum-frequency wave with the o-polarized fundamental wave incident at four different angles (7.5°, 10°, 12.5°, 15°).

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We also conducted a comparative experiment by exploiting the ordinary-polarized fundamental wave of 1300nm and 800nm to study the oo-e type nonlinear process on the crystal boundary. We observed the collinear phase-mismatched SHG of the reflected wave 1300nm and 800nm and their phase-mismatched sum-frequency wave stimulated along the crystal surface. The position of these spots demonstrate position-dependence of the angle between the incident wave and the boundary of the crystal. The experimental photographs of the process are schematically shown in Fig. 3(b).

Next we will discuss limitations for the wavelengths and the critical incident angles of two pumps to realize sum-frequency NCR. As is well known, the basic requirement for nonlinear Cherenkov radiation is that the phase velocity of the nonlinear polarization must exceed that of the Cherenkov harmonics, i.e., $v_{n p} > v_{N C R}$ . It can be derived that the surface sum-frequency NCR should meet the requirements of $\cos θ \cdot (k_{1} + k_{2}) < k_{3}$ , or in other words, $\cos θ \cdot (ω_{1} n_{1} + ω_{2} n_{2}) < (ω_{1} + ω_{2}) n_{3}$ . Since we can change the incident angle $θ$ , no matter how large $k_{1} + k_{2}$ is, we can find the proper $θ$ satisfying the Cherenkov condition.

But without angle modulation, there is limitations for the wavelengths, which is more complicated with two pumps at different frequencies. The criterion for SF NCR becomes:

ee - e ： ω_{1} n_{1 e} + ω_{2} n_{2 e} < (ω_{1} + ω_{2}) n_{3 e}

oo - e ： ω_{1} n_{1 o} + ω_{2} n_{2 o} < (ω_{1} + ω_{2}) n_{3 e}

Under the condition of normal dispersion, we have

n_{3 e} > n_{1 e}

,

n_{3 e} > n_{2 e}

,so the inequality (11) always stands up and there is no cut-off angle of sum-frequency NCR for ee-e type phase-matching. However, oo-e type is more complicated, because either

n_{3 e} < n_{1 o}

or

n_{3 e} < n_{2 o}

can make inequality (12) false. We calculate inequality (12) with the wavelength of pump1 and pump 2 from 600nm to 1800nm. From the calculation results, we can see that when the two fundamental waves are located on the yellow area in Fig. 4 , there is a cut-off angle, that means, sum-frequency NCR only appears as

θ > arc \cos [(ω_{1} + ω_{2}) n_{3 e} / (ω_{1} n_{1 o} + ω_{2} n_{2 o})]

. But if , i.e., the wavelength is located in the green part of Fig. 4, there is no critical angle, that means NCR can be generated while the incident angle of fundamental waves is arbitrary.

Fig. 4 Theoretical calculation of criterion for cut-off angle with different fundamental waves under the oo-e type phase-matching condition.

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3. Conclusion

In summary, we experimentally demonstrated that sum-frequency polarization wave generated by two synchronized laser beams on the nonlinear crystal boundary can realize enhanced sum-frequency NCR. This is a method to generate sum-frequency NCR on the interface with 0-1 abrupt change of nonlinear second coefficient, in addition to the waveguide and the domain wall structure. Without angle modulation, there is limitations for the pump wavelengths to generate sum-frequency NCR exploiting o-polarized pumps, but no limitations for e-polarized pumps.

Acknowledgments

This work was supported in part by the National Basic Research Program 973 of China under Grant No.2011CB808101, the National Natural Science Foundation of China under Grant Nos. 61125503, 61235009, 61505189 and 61205110, the Foundation for Development of Science and Technology of Shanghai under Grant No. 13JC1408300, Presidential Foundation of the China Academy of Engineering Physics (Grant No.201501023), and in part by the Innovative Foundation of Laser Fusion Research Center.

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Sum-frequency nonlinear Cherenkov radiation generated on the boundary of bulk medium crystal

Abstract

1. Introduction

2. Experiment setup and results

3. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (12)

Optics Express