Birefringence and Anisotropy of the Losses Due to Two-Photon Absorption of Femtosecond Pulses in Crystals

: It is shown that the two-photon absorption (TPA)-induced losses in a non-centrosymmetric crystal reduce considerably when the angle θ between the polarization vector of incident radiation and the optical c -axis is 45 ◦ . In such a geometry of interaction in a Ca 3 (VO 4 ) 2 crystal, the effective TPA coefficient for 300 fs laser pulses at θ = 45 ◦ is 3.5 ± 0.5 times lower than its maximum at θ = 0 ◦ , which is more than two times higher than reported earlier in the literature.


Introduction
Nonlinear absorption, and two-photon absorption (TPA) in particular, plays an important role in high-power laser and nonlinear optical technologies because it limits the transparency of involved optical materials at high intensities of radiation.It may be used, for example, as a negative-feedback technology for preventing the laser-induced damage of optical elements, for controlling generated pulse intensities and durations when placed into laser cavities, etc. [1].On the other hand, it may be detrimental, limiting the output power and efficiency of corresponding devices and systems [2][3][4].For these reasons, the development of methods and techniques for a controlled variation in TPA-induced losses is a relevant and important task.
The TPA-induced losses in a medium length L with low linear losses can be conventionally described using the well-known relation [5] where T and T 0 are the sample's nonlinear and linear transmittance, β ef is the effective TPA coefficient of a medium and I p0 is the incident radiation intensity.So, T is variable through variation in incident radiation intensity and sample length.Another option for variation in T exists in crystals, where the TPA coefficient β depends on angle θ between a polarization vector of incident radiation and crystal axes [6][7][8][9].The common features of β(θ) in centrosymmetric crystals and non-birefringent directions of propagation inside of non-centrosymmetric crystals are the following: (i) from ~1.1 to ~1.5 times difference in their ratio for radiation polarization along (θ = 0 • ) and across (θ = 90 • ) a crystal axis, and (ii) its smooth variation between 0 • and 90 • with either the maximum or minimum at ~45 • with an amplitude of variation β(β max /β min ) up to ~1.6 [7][8][9].Studies in [3,6,10] revealed that the difference in β(θ) at θ = 0 • and 90 • is much higher (up to ~2.6 times) in uniaxial crystals when radiation propagates perpendicular to the c-axis and is polarized along and across the optical c-axis.The behavior of β(θ) at θ between 0 • and 90 • was not studied there.In this work, we experimentally studied and theoretically substantiated the observed dependence of T(θ) and β(θ) in a promising nonlinear optical crystal Ca 3 (VO 4 ) 2 [11].

Materials and Methods
A femtosecond laser-Satsuma (Amplitude Systems, Pessac, France)-was used as the source of the following, with a repetition rate of a 10 kHz train of t p ∼ = 300 fs laser pulses at a wavelength λ = 515 nm.A sub-picosecond duration of pulses ensured the absence of the accumulation of photo-generated carriers [9].A linearly polarized TEM 00 radiation beam was focused into the bulk of the 13 mm long Ca 3 (VO 4 ) 2 crystal (E g ∼ = 3.4 eV [10]) sample using a 50 mm focal length plano-convex lens.The resulting 1/e 2 level spot radius was ρ 0 ∼ = 13 µm at the focal plane-giving an effective TPA interaction length inside the sample, i.e., for a Gaussian beam focused into a medium with the refractive index n, of L ef = 2πρ 0 2 n/λ ∼ = 4 mm.The intensity of radiation in each pulse, I = 2E/πρ 0 2 t p , was calculated from either the incident or transmitted pulse energy, E, as measured using an Ophir PD10-C photodiode detector.Because the output of the laser was not stabilized, the magnitudes of E were obtained with an uncertainty of 0.5% through averaging the detector response using a computer program.Consequently, the uncertainty of the sample transmittance was ~1% of its absolute value, as indicated by the error bars in the graphs below.The relative uncertainty of intensity I was evaluated as ~5%.The intensity of the incident (pump) pulses, I p = 2E p /πρ 0 2 t p , varied from ~2 GW/cm 2 to ~50 GW/cm 2 using a polarizing attenuator made of a half-wave plate and a polarization-selective prism.The polished flat entrance and exit surfaces of the Ca 3 (VO 4 ) 2 crystal sample were oriented for the c-axis laying in the plane of these surfaces.The polarization and propagation direction of the pump beam were in plane and perpendicular to these surfaces respectively.The angle θ between the radiation polarization and c-axis of the sample was varied by rotating it around the axis, along which the pump beam was propagated.

Materials and Methods
A femtosecond laser-Satsuma (Amplitude Systems, Pessac, France)-was used as the source of the following, with a repetition rate of a 10 kHz train of tp ≅ 300 fs laser pulses at a wavelength λ = 515 nm.A sub-picosecond duration of pulses ensured the absence of the accumulation of photo-generated carriers [9].A linearly polarized TEM00 radiation beam was focused into the bulk of the 13 mm long Ca3(VO4)2 crystal (Eg ≅ 3.4 eV [10]) sample using a 50 mm focal length plano-convex lens.The resulting 1/e 2 level spot radius was ρ0 ≅ 13 µm at the focal plane-giving an effective TPA interaction length inside the sample, i.e., for a Gaussian beam focused into a medium with the refractive index n, of Lef = 2πρ0 2 n/λ ≅ 4 mm.The intensity of radiation in each pulse, I = 2E/πρ0 2 tp, was calculated from either the incident or transmitted pulse energy, E, as measured using an Ophir PD10-C photodiode detector.Because the output of the laser was not stabilized, the magnitudes of E were obtained with an uncertainty of 0.5% through averaging the detector response using a computer program.Consequently, the uncertainty of the sample transmittance was ~1% of its absolute value, as indicated by the error bars in the graphs below.The relative uncertainty of intensity I was evaluated as ~5%.The intensity of the incident (pump) pulses, Ip = 2Ep/πρ0 2 tp, varied from ~2 GW/cm 2 to ~50 GW/cm 2 using a polarizing attenuator made of a half-wave plate and a polarization-selective prism.The polished flat entrance and exit surfaces of the Ca3(VO4)2 crystal sample were oriented for the c-axis laying in the plane of these surfaces.The polarization and propagation direction of the pump beam were in plane and perpendicular to these surfaces respectively.The angle θ between the radiation polarization and c-axis of the sample was varied by rotating it around the axis, along which the pump beam was propagated.
The absolute values of T in Figure 2 were determined as the ratio of the measured transmitted energy E tr to the product of a corresponding E i and linear transmittance of the sample at a particular θ (Figure 1).Then, using Equation (1), we obtained the effective TPA coefficient β ef (θ), which is related to the absolute value of β for this crystal, obtained in [9], as The absolute values of T in Figure 2 were determined as the ratio of the measured transmitted energy Etr to the product of a corresponding Ei and linear transmittance of the sample at a particular θ (Figure 1).Then, using Equation (1), we obtained the effective TPA coefficient βef(θ), which is related to the absolute value of β for this crystal, obtained in [9], as βef ≅ β/4.
The presented in Figure 3 dependence βef(θ) is the inverted replica of T(θ) (Figure 2).It follows from Figure 3 that the ratio of values of βef at θ ≅ 0° and 90°, which is known as the dichroism of β, is 2±0.4.This result is consistent with the ratio (2.6 ± 1) of the absolute magnitudes of β⊥ and β‖ obtained for this crystal in [10] and for several other crystals in the same geometry of interaction [3,6].These values are 1.5 times higher than the dichroism observed in [7][8][9] in other geometries of interaction.The ratio of the highest βef(0°) and of the lower βef(90°) to the lowest βef(θ) at θ = 45°, which is 3.5 ± 0.5 and 1.8 ± 0.5 (see Figure 3), is the record high for crystals too.These magnitudes are two times higher than ones observed in [7][8][9], and three times higher than that predicted in [13] for the linear-circular dichroism in semiconductors.
In an attempt to account for the observed features of T(θ) and βef(θ), we looked into the effect of sample birefringence.It is known that Ca3(VO4)2 is the uniaxial negative crystal (no > ne).Considering that β is directly linked to the complex third-order nonlinear susceptibility χ (3) of a material [14], and taking into account that the higher n in a material the higher χ (3) is in it [15], it is logical that βef(0°) > βef(90°) in Ca3(VO4)2.The θ ≠ 0° and 90° of the electric field p E  of a linearly polarized incident radiation splits into one compo- The presented in Figure 3 dependence β ef (θ) is the inverted replica of T(θ) (Figure 2).It follows from Figure 3 that the ratio of values of β ef at θ ∼ = 0 • and 90 • , which is known as the dichroism of β, is 2 ± 0.4.This result is consistent with the ratio (2.6 ± 1) of the absolute magnitudes of β ⊥ and β ∥ obtained for this crystal in [10] and for several other crystals in the same geometry of interaction [3,6].These values are 1.5 times higher than the dichroism observed in [7][8][9] in other geometries of interaction.The absolute values of T in Figure 2 were determined as the ratio of the measured transmitted energy Etr to the product of a corresponding Ei and linear transmittance of the sample at a particular θ (Figure 1).Then, using Equation (1), we obtained the effective TPA coefficient βef(θ), which is related to the absolute value of β for this crystal, obtained in [9], as βef ≅ β/4.
The presented in Figure 3 dependence βef(θ) is the inverted replica of T(θ) (Figure 2).It follows from Figure 3 that the ratio of values of βef at θ ≅ 0° and 90°, which is known as the dichroism of β, is 2±0.4.This result is consistent with the ratio (2.6 ± 1) of the absolute magnitudes of β⊥ and β‖ obtained for this crystal in [10] and for several other crystals in the same geometry of interaction [3,6].These values are 1.5 times higher than the dichroism observed in [7][8][9] in other geometries of interaction.The ratio of the highest βef(0°) and of the lower βef(90°) to the lowest βef(θ) at θ = 45°, which is 3.5 ± 0.5 and 1.8 ± 0.5 (see Figure 3), is the record high for crystals too.These magnitudes are two times higher than ones observed in [7][8][9], and three times higher than that predicted in [13] for the linear-circular dichroism in semiconductors.
In an attempt to account for the observed features of T(θ) and βef(θ), we looked into the effect of sample birefringence.It is known that Ca3(VO4)2 is the uniaxial negative crystal (no > ne).Considering that β is directly linked to the complex third-order nonlinear susceptibility χ (3) of a material [14], and taking into account that the higher n in a material the higher χ (3) is in it [15], it is logical that βef(0°) > βef(90°) in Ca3(VO4)2.The θ ≠ 0° and 90° of the electric field p E  of a linearly polarized incident radiation splits into one compo- The ratio of the highest β ef (0 • ) and of the lower β ef (90 • ) to the lowest β ef (θ) at θ = 45 • , which is 3.5 ± 0.5 and 1.8 ± 0.5 (see Figure 3), is the record high for crystals too.These magnitudes are two times higher than ones observed in [7][8][9], and three times higher than that predicted in [13] for the linear-circular dichroism in semiconductors.
In an attempt to account for the observed features of T(θ) and β ef (θ), we looked into the effect of sample birefringence.It is known that Ca 3 (VO 4 ) 2 is the uniaxial negative crystal (n o > n e ).Considering that β is directly linked to the complex third-order nonlinear susceptibility χ (3) of a material [14], and taking into account that the higher n in a material the higher χ (3) is in it [15], it is logical that β ef (0 • ) > β ef (90 • ) in Ca 3 (VO 4 ) 2 .The θ ̸ = 0 • and 90 • of the electric field → E p of a linearly polarized incident radiation splits into one component directed along the c-axis component with the amplitude E o = E p cosθ, which propagates and interacts with the medium as an ordinary wave, and into one perpendicular to the c-axis component with the amplitude E e = E p sinθ, which propagates and interacts with the medium as an extraordinary wave.Because these two components are mutually perpendicular, they cannot interfere.Consequently, the TPA of these two components may be considered independently, giving the transmitted intensity of each component as where i = o or e, I (i) = cn i (E i ) 2 /8π is the intensity of corresponding component, and n i and β (i) are the refractive index and TPA coefficient for the corresponding component.The resulting transmitted intensity would then be I(θ) = I o (θ) + I e (θ) and the total transmittance The dependence of T(θ) for the two different I p s is presented in Figure 2 by solid lines.A good agreement between the experimental data and Equation ( 3) is clear.This in particular means that the detrimental effect of TPA on stimulated Raman scattering (SRS) in a Ca 3 (VO 4 ) 2 crystal [4] may be substantially reduced at θ = 45 • .Taking into account that the magnitude of the TPA coefficient is uniquely linked to the third-order nonlinear susceptibility in a material [8,14], the observed features of β ef (θ) open possibilities for controlling self-focusing, self-and cross-phase modulation, and other nonlinear effects associated with third-order nonlinear susceptibility.Since numerous other SRS-active materials, [16,17], are non-centrosymmetric crystals, some, if not all, may also exhibit similar features.
In conclusion, it is shown that using a birefringent direction of incident radiation propagation in non-centrosymmetric crystals allows us to achieve a significantly higher, than reported earlier, amplitude of smooth control of the TPA-induced losses, and of the effective TPA coefficient respectively, through variation in the angle θ between the radiation polarization vector and optical axes.This is particularly relevant in a Ca 3 (VO 4 ) 2 crystal, where in such a geometry of interaction a 3.5 ± 0.5 times reduction in the effective TPA coefficient was spotted for 300 fs laser pulses at θ = 45 • , compared to its maximum at θ = 0 • , which is more than two times higher than reported earlier in the literature for other crystals.

Figure 1 .
Figure 1.Linear transmittance of the sample vs. angle θ (dots); the dashed line is the sinθ dependence.

Figure 1 .
Figure 1.Linear transmittance of the sample vs. angle θ (dots); the dashed line is the sinθ dependence.

Figure 3 .
Figure 3.The effective TPA coefficient vs. angle θ.The dashed line is the fitting of the experimental data.

Figure 2 .
Figure 2. Transmittance of the sample vs. angle θ at I p = 22 GW/cm 2 (dots) and 50 GW/cm 2 (triangles).Solid lines are the result of calculations using Equation (3) for correspondent I p .

Figure 3 .
Figure 3.The effective TPA coefficient vs. angle θ.The dashed line is the fitting of the experimental data.

Figure 3 .
Figure 3.The effective TPA coefficient vs. angle θ.The dashed line is the fitting of the experimental data.