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Numerical simulation of laser focusing properties inside birefringent crystal LEIMIN DENG, PENG LIU, JUN DUAN,* XIAOYAN ZENG, BAOYE WU,
AND
XIZHAO WANG
Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China *Corresponding author: [email protected] Received 2 November 2015; revised 30 November 2015; accepted 23 December 2015; posted 23 December 2015 (Doc. ID 253010); published 1 February 2016
The transmission properties of a focused laser inside anisotropic material are complex due to the birefringent effect, which has remarkable influence on the light distribution and frequency multiplication efficiency of crystals. Meanwhile, it will also affect the laser micromachining precision of birefringent materials with random polarization. In this study, ray tracing and diffraction integral methods were proposed to develop the mathematical model of a laser focused through an isotropic medium into a KDP crystal. Using these models, the focusing properties and 3D light intensity distribution of a focused laser inside a KDP crystal at different orientations were investigated. The research shows that the size and shape of the E-ray focus will distort, and its peak power density decreases rapidly with the decrease of the angle between the optical axis and the crystal surface. Meanwhile, the focal position of the E-ray will also move with the change of optical axis orientation. Based on the simulated results, an approximate 3D light intensity equation of a laser focused into birefringent material was also proposed, which is in good agreement with the theoretical analysis. The related simulated results have an important engineering value for nonlinear optics and laser processing of birefringent materials. © 2016 Optical Society of America OCIS codes: (010.3310) Laser beam transmission; (260.1180) Crystal optics; (260.1440) Birefringence; (000.4430) Numerical approximation and analysis. http://dx.doi.org/10.1364/AO.55.000853
1. INTRODUCTION Optical crystals play an important role in laser systems [1], especially in frequency multiplication and optical switching devices. In order to achieve high-frequency multiplication efficiency, lasers should be focused into optical crystals. Due to the birefringent effect, the transmission properties of a focused laser inside birefringent crystal are complex and the light distribution of an E-ray has a significant difference compared with an O-ray, which has remarkable influence on II-type frequency multiplication efficiency [2]. Therefore, studying the 3D light distribution of a focused laser inside crystal at different optical axis orientations has an important engineering value for improving the performance of crystals. Meanwhile, laser processing microstructures inside crystals has aroused much attention [3–5] such as in fabricating optical waveguides, optical lenses, photonic crystals, microfluidic channels, and micro-explosion voids. The birefringent effect will cause a change in the size, shape, and position of the E-ray focus, which will deteriorate the laser micromachining accuracy of anisotropic materials such as CDs, semiconductor materials, and optical crystal devices. Even though shielding the E-ray of the laser will solve these problems, it is a very costly and 1559-128X/16/040853-08$15/0$15.00 © 2016 Optical Society of America
complicated method for some industry equipment such as rotary processing. Zhou and Gu [6] researched the laser-driven micro-explosion technology of LiNbO3 crystal and found that the fabrication parameters varied dramatically with the change of crystal direction due to the anisotropy. Deng [7] et al. researched the laser cutting of anisotropic materials and also found that the complex light distribution of the E-ray caused by the birefringent effect had a remarkably bad influence on the cutting quality. Many similar experiments have been carried out, but there is almost no specific theoretical study of this issue. Stamnes and Sithambaranathan [8] have studied electronic waves focused into uniaxial and biaxial crystals in a series of papers based on the exact solution of the reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating two mediums. They presented consistent numerical and experimental results for the focusing of two/threedimensional electromagnetic waves into uniaxial crystal [9] and made comparisons between numerical, asymptotic, and experimental results for the two-dimensional situation. The approach was also used to study the transmission of Gaussian beams [10]. Jain et al. studied the effects of aperture size on
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the focusing of electromagnetic waves into a biaxial crystal [11,12], showing that two foci appeared when the aperture is large enough. Li et al. [13], Stallinga [14], Yonezawa et al. [15], and Gajdátsy and Erdélyi [16] have also researched the focusing properties of lasers in birefringent materials. However, their works focused on revealing the mechanism rather than engineering applications. Meanwhile, the results were either achieved for some specific angles or solved under 2D conditions and did not provide a universal 3D light intensity distribution. In this study, both the ray tracing and diffraction integral methods were adopted to develop the mathematical model of a laser focused through an isotropic medium into a KDP crystal. The ray tracing method can reveal clearly the formative mechanism of light distribution, but its simulated results near the focus are imprecise because the interactions between the rays are ignored. On the contrary, the diffraction integral method can obtain an accurate 3D light distribution, but the formative mechanism is blurred. Therefore, in this paper, the focusing properties and 3D light distribution of a focused laser inside a KDP crystal at different optical axis orientations were investigated by using a combination of these two methods. The distorted rules and spot morphologies of an E-ray focus were also analyzed, and the changing rules of focal intensity were also investigated. Based on the simulated results, an approximate 3D light distribution equation of a laser focused into birefringent material was also proposed, which is in good agreement with the theoretical analysis. The related simulated results have an important engineering value for nonlinear optics and the laser processing of birefringent materials. 2. NUMERICAL SIMULATION METHOD ON LASER TRANSMISSION CHARACTERISTICS A. Model Building and Computing Method
The schematic diagram of a laser focused into a KDP crystal is shown in Fig. 1. A randomly polarized laser will be separated into an O-ray and an E-ray after transmission into the KDP crystal from air. When the laser is not focused and at normal incidence, the angle between two rays (η0 ) can be expressed as 2 1 1 1 sin β cos2 β −1 tan η0 sin 2β 2 − 2 2 ; (1) 2 no ne n2o ne
and the crystal surface. However, when the incident laser is focused, the rays are no longer at normal incidence, and the angles between the rays and the optical axis are also unfixed, which will lead to complex laser beam transmission characteristics. The optical axis has more influence on the meridian plane rays than the sagittal plane rays, which will make the E-ray focus split, as shown in Fig. 1. 1. Ray Tracing Model
When a fundamental-mode laser is focused into an isotropic medium, the equation of light intensity distribution in air can be expressed as I x; y; z Meanwhile,
2P ω−2r2 z2 e : πω2 z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi λz − F H 2 1 ωz ω0 ; πω20
(3)
where Ix; y; z is the light intensity distribution, P is the output power of the laser, λ is the laser wavelength, H is the distance between the lens and the crystal, F is the focal length, and ω0 is the spot radius of the laser at the focal position. In this model, the laser beam is divided into N parts along its radius, as shown in Fig. 2. For the divided nth circular ring r n , it can be also separated into M n 6 1 elements. Hence, the whole laser spot is split into many small elements. The power P nm that each element Q nm carries can be expressed as Z 2πm1 Z 2n1ω0 2P 1 ω−2r2 02 6n1 N e rdr: dθ (4) P nm 2nω0 2πm π 6n1 ω2 0 N When an unpolarized laser ray (element Q nm ) transmits into a KDP crystal, the wavefront of its O-ray part inside the crystal is spherical (Ω1 ), as shown in Fig. 3, and can be expressed as F o x 2 y2 z 2 −
1 0: n2o
(5)
The wavefront of its E-ray part is ellipsoidal (Ω2 ) and can be expressed as
where no and ne are the refractive indices of the O-ray and E-ray, respectively, and β is the angle between the optical axis
Fig. 1. Schematic diagram of a laser focused into a KDP crystal.
(2)
Fig. 2. Schematic diagram of a spot segmentation.
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element. The light plural amplitude E˜ of element Q nm before transmission can be approximated as a plane wave, and the equation is i nh o qffiffiffiffi 2 2 E˜
2P i k H −F 2n ω 0 2 −arctan f π −4n2 N 2 H −F 2 H −F
ω0
e
N
H −F f
e
;
(10)
in which 2π πω2 (11) ; f 0: λ λ According to the Helmholtz–Kirchhoff diffraction integra˜ tion principle, the light field intensity EG of any point (G) in a crystal generated by element Q nm can be deduced as Z Z ˜ ikni ⃗s 1 ∂E e ∂ e ikni ⃗s ˜ − E˜ dσ; (12) EG ⃗s ⃗s 4π Σ ∂d⃗ ∂d⃗ where Σ is the surface element of Q nm , d⃗ is the normal vector of the surface element, ⃗s is the vector from the integral point to the G point, and ni is the refractive index of the crystal. Substituting Eq. (10) into Eq. (12), the equation can be expressed as ikn s ZZ i e i cosd⃗ ; ⃗s − cosd⃗ ; v⃗ ˜ ˜ dσ; EG − E· λ Σ 2 s (13) k
Fig. 3. Schematic diagram of Huygens’ principle [17].
y2 x sin β z cos β2 x cos β − z sin β2 2 2 F e 2 −1 1 ne
1 ne
0:
1 no
(6)
Assuming the optical path difference of BC in Fig. 3 is 1, the equations of the tangent plane passing through point B and point x; y; z on the wavefront surface can be expressed as ( ∂F cosθx ∂F sinθx ∂F ∂x sinθz − x ∂y sinθz − y ∂z −z 0 ; (7) ∂F ∂F ∂x sinθ − ∂y cosθx 0 where θx is the angle between the incident ray Q nm and the X axis, and θz is the angle between the incident ray Q nm and the Z axis. For the O-ray, we have ∂F ∂F o 2x; ∂x ∂x
∂F ∂F o 2y; ∂y ∂y
∂F ∂F o 2z; ∂z ∂z (8)
where v⃗ is the transmission vector of ray Q nm, and z cosd⃗ ; ⃗s ≈ − ; s s
(14)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos θ − ρ cos φ2 r sin θ − ρ sin φ2 z 2 ; (15)
where r, θ and ρ, φ are the cylindrical coordinate values of the integral point and the G point, respectively. For the O-ray, we have cosd⃗ ; v⃗ −z o :
(16)
while for the E-ray, we have 8 ∂F 2 2 e > ∂F > ∂x 2x sin β z cos β · ne sin β 2x cos β − z sin β · no cos β < ∂x ∂F o ∂F 2 : ∂y ∂y 2yne > > : ∂F ∂F o 2 2 ∂z ∂z −2x sin β z cos β · ne cos β 2x cos β − z sin β · no sin β
Combined with Eqs. (2)–(9), the O-ray exit direction vector v⃗ o x o ; y o ; z o and the E-ray exit direction vector v⃗ e x e ; y e ; z e of ray Q nm can be worked out. Similarly, the transmission characteristics of the whole laser beam can be simulated by using MATLAB software. 2. Diffraction Integral Model
According to Babinet’s principle, the diffraction pattern of a laser beam is identical to the superposition of all separated elements. Therefore, the light field intensity of any point can be simulated by calculating the diffraction transmission of each
(9)
For the E-ray, we have cosd⃗ ; v⃗ −z e :
(17)
Combined with Eqs. (10)–(17), the light field intensity of the G point generated by element Q nm can be expressed as Z 2πm1 Z 2n1ω0 i E˜ z 6n1 N ˜ q− e ikni s · ρdφdρ; EG − 2nω0 2mπ 2λs0 s N 6n1 (18)
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where q z o for the O-ray, and q z e for the E-ray. Meanwhile, the transmission characteristic of the O-ray inside the crystal is isotropous, so ni no . For the E-ray, ni ne0 and we have 1 cos2 θv sin2 θv ; 0 2 n2o n2e ne
(19)
in which θv is the angle between the ray Q nm and the transmission vector v⃗ , and cosβ · r cos θ − ρ cos φ z sin β : (20) s Combined with Eqs. (18)–(20), the light field intensity of the G point generated by the element Q nm can be calculated. Then by calculating all the elements of the laser beam by computer circulation, the diffraction integral transmission model of the laser inside the KDP crystal can be worked out. cos θv
B. Accuracy Verification of the Models
The above models are adopted to simulate the transmission characteristics of a focused laser inside a KDP crystal. The thickness of the crystal is 12.2 mm, and the angle between the optical axis and the crystal surface is β 30.9°. Meanwhile, the wavelength of the laser is λ 355 nm, and the refractive indices of the O-ray and E-ray inside the KDP crystal are 1.5311 and 1.4858, respectively. The incident laser is a fundamental-mode Gaussian beam with a polarization ratio of O∶E 7∶3 and a total output power of 4 W. The beam
Fig. 4. Contrast of the experimental result and the simulated result. (a),(b) Diffraction integral results; (c),(d) ray tracing results; and (e) experimental results.
diameter is D 10 mm, and the focal length of the lens is F 73 mm. The distance between the lens and the crystal is H 64.3 mm, so the laser focus is near the back of the KDP crystal. Photographic paper was placed on the back of the crystal to record the spot pattern, and the experimental and simulated results are shown in Fig. 4. Because the laser spot is far above the diffraction limit, the simulated results of the ray tracing and diffraction integral methods are identical. The center distance of the two spots is 308.2 μm by the diffraction integral simulated result and 308.7 μm by the ray tracing simulated result, where both agree well with the experimental result of 308 μm. The 0.5 μm deviation between the two simulated results was generated because the initial phase was neglected in the ray tracing method. The morphology of the simulated spots also agreed well with the experimental results. Therefore, the accuracy of the two models are both fine. 3. RESULTS AND ANALYSES OF NUMERICAL SIMULATIONS Using these models, the laser focusing properties in the KDP crystal were simulated. The laser is a 1064 nm randomly polarized fiber laser with 20 W output power, and its beam diameter is D 15 mm. A. Effect of Crystal Orientation β
Figure 5 shows the relationship between the focus position and the crystal orientation when the focal length of the optical lens is F 75 mm, and the lens surface is H 60 mm above the crystal surface. For the O-ray, its focal position along the Z axis is always 22.416 mm under the crystal surface, because its transmission is isotropic, as shown in Fig. 5 (red line). For E-ray, with the decrease of β, the impact of the crystal orientation on the focus properties becomes more apparent. The crystal orientation has the same influence on the focus of the meridian plane and the sagittal plane when β 90°. Therefore, only one focus is formed at the 21.409 mm position. When β ≠ 90°, the crystal orientation has a greater impact on the meridian plane ray, while it has a relatively small impact on the sagittal plane ray. As a result, the meridian plane beam and the sagittal plane beam each produce their own focus. The distance of these two foci increases with the decrease of the angle β, which will gradually lead the shape of the E-ray focus to be an ellipse and will
Fig. 5. Relationship between the focus position and the crystal orientation.
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B. Effect of Focal Length F
Fig. 6. Morphology of laser foci when β 60° (simulated results). (a) Sagittal plane focus morphology of the E-ray, (b) meridian plane focus morphology of the E-ray, and (c) focus morphology of the O-ray.
The focus positions of the E-ray at different focal lengths (F 70–85 mm) were also simulated with the conditions H 61.3 mm and D 15 mm. The results indicate that with an increase in focal length, the focus position of the Eray will also increase, as shown in Figs. 8(a) and 8(c). All the curves coincide, as shown in Figs. 8(b) and 8(d). Therefore, the focus positions of the E-ray are proportional to F -H . As a result, the Z axis position of the sagittal focus (Z es ) can be revised as Z es
evolve into two vertical focus lines when β is large enough, as shown in Fig. 6. MATLAB software is used to analyze the relationship between the E-ray focus position and β with the polynomial fitting method, and the results indicate that coefficients greater than 4th degree tend to 0. Therefore, the relationship between the Z axis position (Z es ) of the sagittal focus and the crystal orientation β can be expressed as
F −H p1 · β3 p2 · β2 p3 · β p4: 15
(24)
In the same way, the Z axis position (Z em ) of the meridian focus can be revised as Z em
F −H p5 · β3 p6 · β2 p7 · β p8: 15
(25)
Meanwhile, the simulated results also indicate that the relative deviation (ε) is irrelevant to the focal length F , so Eq. (23) is also effective while the focal length is changed.
Z es p1 · β3 p2 · β2 p3 · β p4 : p1 1.4767 × 10−6 ; p2 −2.0087 × 10−4 ; p3 6.1914 × 10−4 ; p4 21.903
In the same way, the Z axis position (Z em ) of the meridian focus can be expressed as Z es p5 · β3 p6 · β2 p7 · β p8 : p5 4.6496 × 10−6 ; p6 −6.252 × 10−4 ; p7 1.6801 × 10−3 ; p8 22.932
(21)
(22)
Meanwhile, the simulated results also indicate that the anisotropy of the crystal will change the angle between the E-ray and the O-ray, and the relative deviation (ε) compared with Eq. (1) is shown in Fig. 7. The relative deviation is a sine function of 2β. Therefore, the angle between the E-ray and the O-ray can be expressed as η η0 ε η0 − 5.5 × 10−4 sin2β:
(23)
Fig. 7. Relationship between the relative angle deviation ε and the crystal orientation.
Fig. 8. Relationship between the focus position and the optical axis orientation of a crystal with different focal lengths. (a),(b) Meridian plane focus. (c), (d) Sagittal plane focus.
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C. Approximate 3D Light Intensity Equation of a Laser Focused into a KDP Crystal
Based on the simulated results, the 3D light intensity of the O-ray is the same as isotropic materials and can be expressed as 2
I o χ1
2
2P −2xω2y n z ; e πω2n z
(26)
in which χ 1 is the ratio of the O-ray in the laser, and ωn z is the beam diameter of the O-ray at the z position, expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi < ω z ω 1 z−no F no H 2 n 0 f : (27) : no πω20 f λ Although the transmission properties of the E-ray present as a deflection of the laser beam and a split of the laser focus, the 3D light intensity of the E-ray can be approximated still as an elliptical Gaussian function. According to the geometry theory, the coordinate axis after revolving an angle η along the Z → X direction can be expressed as x p x cos η z sin η; y p y : (28) z p z cos η − x sin η Thus, the 3D light intensity of the E-ray can be expressed as 2x 2
2y 2
p p − 2 − 2 2P I e χ2 e ω1 zp ω2 z p ; πω1 z p ω2 z p
(29)
in which χ 2 is the ratio of the E-ray in the laser beam. Meanwhile, ω1 z p and ω2 z p are the beam diameters of the E-ray in the meridian and sagittal planes at the z p position, respectively. There are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u Z em 2 u z − p cos η ω z ω t1 ; 1
p
m
ne0 πω2m λ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u Z es u z − p cos η ; ω2 z p ωs t1 n 0 πω2 e
λ
(30)
4. DISCUSSION
As a result, the 3D light intensity of the E-ray can be finally expressed as 2
Meanwhile, according to Eq. (32), Z em 13.074 mm, Z es 12.63 mm, ne0 1.467611, ω0 3 μm, η 0.0191953 rad, ωm 3.065 μm, and ωs 2.961 μm, and the light intensity that Eq. (32) describes is shown in Fig. 9(b). The comparison of these two results indicate that there is only some negligible part different and that the veracity of Eq. (32) is fine. Even though the diffraction integral model can achieve an accurate light intensity distribution of the focused laser into the birefringent crystal, it needs a very long time for simulation. Meanwhile, it is too complicated for low-precision engineering applications. Therefore, the diffraction integral model is used for some high-precision applications, for example, in analyzing the light intensity distribution inside frequency-switching crystals. Meanwhile, the approximate 3D light intensity equation in Eq. (32) is simple and rapid in some low-precision applications, such as in laser cutting, and modifying and micro-explosion processing of birefringent crystals. Therefore, this equation can be adopted in optical system design software directly to estimate the 3D light intensity and processing results of a focused laser inside crystal.
s
In which ωm and ωs are the equivalent focus radii of the meridian and sagittal planes, respectively. According to the optical path reversibility principle, ω1 0 and ω2 0 should equal to the input laser beam radius. Therefore, ω0 , ωm , and ωs can be solved by the function sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 λF − λH 2 D ω0 1 ω1 0 ω2 0: (31) 2 πω20
Ix; y; z χ 1
Fig. 9. Verification of the approximate 3D light intensity equation. (a) Simulation results of the diffraction integral model. (b) Simulation results of the approximate 3D light intensity equation.
2
2P −2xω2y n z e πω2n z 2
sin η −2x cos ηz 2P ω2 z p 1 e χ2 πω1 z p · ω2 z p
2y 2 ω2 z p 2
−
: (32)
The diffraction integral model was used to test the veracity of this 3D light intensity equation with the parameters F 15 mm, D 3.4 mm, H 6.3 mm, λ 1064 nm, and P 20 W, and the simulation results are shown in Fig. 9(a).
Assuming the input laser power was 20 W, the light intensity of the focus at different crystal orientations was simulated, as shown in Fig. 10. The light intensity of the O-ray focus is constant regardless of the crystal orientation variation because the birefringent effect is noneffective for the O-ray, while the E-ray focus is changed substantially with it. The results indicate that when β 90°, the E-ray focus is not split and has the highest light intensity which even exceeds the value of the O-ray due to the focusing enhancement effect of birefringence. With the decrease of β, the energy dispersion caused by the focus split will weaken the light intensity of the E-ray focus rapidly, and the light intensity of the E-ray focus is only 1/3 of the O-ray focus when β 75°. With a further decrease in β, the downtrend of the intensity will become gentle and almost does not change when β < 50°. At this time, the intensity of the E-ray focus is only 2%–5% of the O-ray. In order to reach a highly controllable light intensity distribution, the laser is always focused during the laser-induced thermal stress cutting process of the transparent birefringent materials such as KDP, quartz, and sapphire crystals. However, the
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Fig. 10. Relationship between the focus power density and the crystal orientation.
Fig. 11. Relationship between the focus power density and the focal length.
high power density at the focus position could easily arouse cracks or fragmentation, especially for KDP crystals [1,7]. Since the commonly used orientations of KDP crystals are β 30.9° (II type), 42.3° (I type), and 48.5° (I type), according to the simulation results of Eq. (32) and Fig. 10, the focused E-ray beam can not only achieve a light distribution similar to the O-ray beam to produce a uniform cutting stress, but also has a weaker power density at the focus position which is beneficial in reducing stress cracking. Therefore, the focused E-ray is an ideal beam for the thermal stress cutting of a KDP crystal. In the laser damage processing of birefringent material such as producing micro-explosion voids, waveguides, or 3D property modifications, the E-ray focus could produce some additional damage which deteriorates the processing precision. Taking the KDP crystal (β 30.9°, II type) as an example, since its damage threshold is about 2.3 × 1016 W · m−2 [18], according to the simulation results, the E-ray focus could cause damage effects when the peak power of the laser is over 5.9 × 107 W, while the O-ray focus only needs 6.5 × 104 W. Therefore, in the damage processing of the birefringent material, the effect of the E-ray can be ignored when the laser peak power density is low. For conditions that need laser power densities over 5.9 × 107 W, the effect of the E-ray should not be neglected as it will produce two vertical damaged lines inside the crystal (Fig. 6) and lead to crystal crack easily. At this moment, the E-ray has to be shielded in order to achieve high processing quality and safety. The power densities of the laser focus at different focal lengths also have been simulated, as shown in Fig. 11. There are two mechanisms dominating the power density of the E-ray focus: the diffraction effect and spot distortion caused by birefringence. When β 90°, the power density variation rule of the E-ray focus is similar to the O-ray, which indicates that the diffraction effect is the main mechanism leading to power density decreasing. With the reduction of β, the influence of the birefringent effect becomes stronger and makes the light intensity of the E-ray focus drop rapidly. When β < 50°, the birefringent effect is the major factor that influences the
E-ray focusing properties, and the light intensity of the E-ray focus at different focal lengths is almost unchanged. In the laser-induced thermal stress cutting of the birefringent material with small β, the processing safety of the Eray beam is still high no matter how the focal length changes. Therefore, the E-ray beam is beneficial for laser-induced thermal stress cutting under the conditions of β < 80°. In the laser damage process of the birefringent material, when β is small, using a shorter focus length lens can achieve a high contrast of power density between the O-ray and E-ray focus, which is beneficial for laser damaging processes. However, when β is large, the E-ray focus has a higher power density which can be used to produce an additional explosion point to improve processing efficiency. 5. CONCLUSION In this work, a ray tracing and diffraction integral method were proposed to develop the mathematical model of a laser focused through an isotropic medium into a KDP crystal. The laser focusing properties inside the KDP crystal at different orientations were investigated. The simulated results indicate that the birefringent has a great impact on the E-ray focus position, peak power density, and shape. Based on the simulated results, an approximate 3D light intensity equation of the laser focused into the birefringent material was also proposed, which is in good agreement with the theoretical analysis and has an important engineering value for nonlinear optics and the laser processing of birefringent materials. The analyzed results of the birefringence effects on laser processing indicate that the E-ray beam is beneficial for laser-induced thermal stress cutting of birefringent materials under the conditions of β < 80°. The E-ray beam can be ignored during low-power damage processing when β and the focal length are small, while it can be used to improve the processing efficiency when β is large. Funding. National Natural Science Foundation of China (NSFC) (51135005, 51175205); Specialized Research Fund
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for the Doctoral (20130142110059).
Program
of
Higher
Education
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Research Article 9. D. Jiang and J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 174, 321–334 (2000). 10. G. S. Sithambaranathan and J. J. Stamnes, “Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,” Opt. Commun. 209, 55–67 (2002). 11. M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Effects of aperture size on focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 266, 438–447 (2006). 12. M. Jain, J. K. Lotsberg, and J. J. Stamnes, “Comparisons of exact and paraxial intensities of electromagnetic waves focused into uniaxial crystals,” J. Opt. A 8, 709–719 (2006). 13. J. Li, H. Jiang, J. Xiao, and Q. Gong, “The mechanism of multifocusing of lasers into uniaxial crystals,” J. Opt. A 9, 664–672 (2007). 14. S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001). 15. K. Yonezawa, Y. Kozawa, and S. Sato, “Focusing of radially and azimuthally polarized beams through a uniaxial crystal,” J. Opt. Soc. Am. A 25, 469–472 (2008). 16. G. Gajdátsy and M. Erdélyi, “Analysis of focus distortion based on birefringence,” J. Opt. A 9, 982–987 (2007). 17. L. Bergstein and T. Zachos, “A Huygens’ principle for uniaxially anisotropic media,” J. Opt. Soc. Am. A 56, 931–934 (1966). 18. J. A. Menapace, “Developing magnetorheological finishing (MRF) technology for the manufacture of large-aperture optics in megajoule class laser systems,” Proc. SPIE 7842, 78421W (2010).
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Numerical simulation of laser focusing properties inside birefringent crystal LEIMIN DENG, PENG LIU, JUN DUAN,* XIAOYAN ZENG, BAOYE WU,
AND
XIZHAO WANG
Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China *Corresponding author: [email protected] Received 2 November 2015; revised 30 November 2015; accepted 23 December 2015; posted 23 December 2015 (Doc. ID 253010); published 1 February 2016
The transmission properties of a focused laser inside anisotropic material are complex due to the birefringent effect, which has remarkable influence on the light distribution and frequency multiplication efficiency of crystals. Meanwhile, it will also affect the laser micromachining precision of birefringent materials with random polarization. In this study, ray tracing and diffraction integral methods were proposed to develop the mathematical model of a laser focused through an isotropic medium into a KDP crystal. Using these models, the focusing properties and 3D light intensity distribution of a focused laser inside a KDP crystal at different orientations were investigated. The research shows that the size and shape of the E-ray focus will distort, and its peak power density decreases rapidly with the decrease of the angle between the optical axis and the crystal surface. Meanwhile, the focal position of the E-ray will also move with the change of optical axis orientation. Based on the simulated results, an approximate 3D light intensity equation of a laser focused into birefringent material was also proposed, which is in good agreement with the theoretical analysis. The related simulated results have an important engineering value for nonlinear optics and laser processing of birefringent materials. © 2016 Optical Society of America OCIS codes: (010.3310) Laser beam transmission; (260.1180) Crystal optics; (260.1440) Birefringence; (000.4430) Numerical approximation and analysis. http://dx.doi.org/10.1364/AO.55.000853
1. INTRODUCTION Optical crystals play an important role in laser systems [1], especially in frequency multiplication and optical switching devices. In order to achieve high-frequency multiplication efficiency, lasers should be focused into optical crystals. Due to the birefringent effect, the transmission properties of a focused laser inside birefringent crystal are complex and the light distribution of an E-ray has a significant difference compared with an O-ray, which has remarkable influence on II-type frequency multiplication efficiency [2]. Therefore, studying the 3D light distribution of a focused laser inside crystal at different optical axis orientations has an important engineering value for improving the performance of crystals. Meanwhile, laser processing microstructures inside crystals has aroused much attention [3–5] such as in fabricating optical waveguides, optical lenses, photonic crystals, microfluidic channels, and micro-explosion voids. The birefringent effect will cause a change in the size, shape, and position of the E-ray focus, which will deteriorate the laser micromachining accuracy of anisotropic materials such as CDs, semiconductor materials, and optical crystal devices. Even though shielding the E-ray of the laser will solve these problems, it is a very costly and 1559-128X/16/040853-08$15/0$15.00 © 2016 Optical Society of America
complicated method for some industry equipment such as rotary processing. Zhou and Gu [6] researched the laser-driven micro-explosion technology of LiNbO3 crystal and found that the fabrication parameters varied dramatically with the change of crystal direction due to the anisotropy. Deng [7] et al. researched the laser cutting of anisotropic materials and also found that the complex light distribution of the E-ray caused by the birefringent effect had a remarkably bad influence on the cutting quality. Many similar experiments have been carried out, but there is almost no specific theoretical study of this issue. Stamnes and Sithambaranathan [8] have studied electronic waves focused into uniaxial and biaxial crystals in a series of papers based on the exact solution of the reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating two mediums. They presented consistent numerical and experimental results for the focusing of two/threedimensional electromagnetic waves into uniaxial crystal [9] and made comparisons between numerical, asymptotic, and experimental results for the two-dimensional situation. The approach was also used to study the transmission of Gaussian beams [10]. Jain et al. studied the effects of aperture size on
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the focusing of electromagnetic waves into a biaxial crystal [11,12], showing that two foci appeared when the aperture is large enough. Li et al. [13], Stallinga [14], Yonezawa et al. [15], and Gajdátsy and Erdélyi [16] have also researched the focusing properties of lasers in birefringent materials. However, their works focused on revealing the mechanism rather than engineering applications. Meanwhile, the results were either achieved for some specific angles or solved under 2D conditions and did not provide a universal 3D light intensity distribution. In this study, both the ray tracing and diffraction integral methods were adopted to develop the mathematical model of a laser focused through an isotropic medium into a KDP crystal. The ray tracing method can reveal clearly the formative mechanism of light distribution, but its simulated results near the focus are imprecise because the interactions between the rays are ignored. On the contrary, the diffraction integral method can obtain an accurate 3D light distribution, but the formative mechanism is blurred. Therefore, in this paper, the focusing properties and 3D light distribution of a focused laser inside a KDP crystal at different optical axis orientations were investigated by using a combination of these two methods. The distorted rules and spot morphologies of an E-ray focus were also analyzed, and the changing rules of focal intensity were also investigated. Based on the simulated results, an approximate 3D light distribution equation of a laser focused into birefringent material was also proposed, which is in good agreement with the theoretical analysis. The related simulated results have an important engineering value for nonlinear optics and the laser processing of birefringent materials. 2. NUMERICAL SIMULATION METHOD ON LASER TRANSMISSION CHARACTERISTICS A. Model Building and Computing Method
The schematic diagram of a laser focused into a KDP crystal is shown in Fig. 1. A randomly polarized laser will be separated into an O-ray and an E-ray after transmission into the KDP crystal from air. When the laser is not focused and at normal incidence, the angle between two rays (η0 ) can be expressed as 2 1 1 1 sin β cos2 β −1 tan η0 sin 2β 2 − 2 2 ; (1) 2 no ne n2o ne
and the crystal surface. However, when the incident laser is focused, the rays are no longer at normal incidence, and the angles between the rays and the optical axis are also unfixed, which will lead to complex laser beam transmission characteristics. The optical axis has more influence on the meridian plane rays than the sagittal plane rays, which will make the E-ray focus split, as shown in Fig. 1. 1. Ray Tracing Model
When a fundamental-mode laser is focused into an isotropic medium, the equation of light intensity distribution in air can be expressed as I x; y; z Meanwhile,
2P ω−2r2 z2 e : πω2 z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi λz − F H 2 1 ωz ω0 ; πω20
(3)
where Ix; y; z is the light intensity distribution, P is the output power of the laser, λ is the laser wavelength, H is the distance between the lens and the crystal, F is the focal length, and ω0 is the spot radius of the laser at the focal position. In this model, the laser beam is divided into N parts along its radius, as shown in Fig. 2. For the divided nth circular ring r n , it can be also separated into M n 6 1 elements. Hence, the whole laser spot is split into many small elements. The power P nm that each element Q nm carries can be expressed as Z 2πm1 Z 2n1ω0 2P 1 ω−2r2 02 6n1 N e rdr: dθ (4) P nm 2nω0 2πm π 6n1 ω2 0 N When an unpolarized laser ray (element Q nm ) transmits into a KDP crystal, the wavefront of its O-ray part inside the crystal is spherical (Ω1 ), as shown in Fig. 3, and can be expressed as F o x 2 y2 z 2 −
1 0: n2o
(5)
The wavefront of its E-ray part is ellipsoidal (Ω2 ) and can be expressed as
where no and ne are the refractive indices of the O-ray and E-ray, respectively, and β is the angle between the optical axis
Fig. 1. Schematic diagram of a laser focused into a KDP crystal.
(2)
Fig. 2. Schematic diagram of a spot segmentation.
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element. The light plural amplitude E˜ of element Q nm before transmission can be approximated as a plane wave, and the equation is i nh o qffiffiffiffi 2 2 E˜
2P i k H −F 2n ω 0 2 −arctan f π −4n2 N 2 H −F 2 H −F
ω0
e
N
H −F f
e
;
(10)
in which 2π πω2 (11) ; f 0: λ λ According to the Helmholtz–Kirchhoff diffraction integra˜ tion principle, the light field intensity EG of any point (G) in a crystal generated by element Q nm can be deduced as Z Z ˜ ikni ⃗s 1 ∂E e ∂ e ikni ⃗s ˜ − E˜ dσ; (12) EG ⃗s ⃗s 4π Σ ∂d⃗ ∂d⃗ where Σ is the surface element of Q nm , d⃗ is the normal vector of the surface element, ⃗s is the vector from the integral point to the G point, and ni is the refractive index of the crystal. Substituting Eq. (10) into Eq. (12), the equation can be expressed as ikn s ZZ i e i cosd⃗ ; ⃗s − cosd⃗ ; v⃗ ˜ ˜ dσ; EG − E· λ Σ 2 s (13) k
Fig. 3. Schematic diagram of Huygens’ principle [17].
y2 x sin β z cos β2 x cos β − z sin β2 2 2 F e 2 −1 1 ne
1 ne
0:
1 no
(6)
Assuming the optical path difference of BC in Fig. 3 is 1, the equations of the tangent plane passing through point B and point x; y; z on the wavefront surface can be expressed as ( ∂F cosθx ∂F sinθx ∂F ∂x sinθz − x ∂y sinθz − y ∂z −z 0 ; (7) ∂F ∂F ∂x sinθ − ∂y cosθx 0 where θx is the angle between the incident ray Q nm and the X axis, and θz is the angle between the incident ray Q nm and the Z axis. For the O-ray, we have ∂F ∂F o 2x; ∂x ∂x
∂F ∂F o 2y; ∂y ∂y
∂F ∂F o 2z; ∂z ∂z (8)
where v⃗ is the transmission vector of ray Q nm, and z cosd⃗ ; ⃗s ≈ − ; s s
(14)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos θ − ρ cos φ2 r sin θ − ρ sin φ2 z 2 ; (15)
where r, θ and ρ, φ are the cylindrical coordinate values of the integral point and the G point, respectively. For the O-ray, we have cosd⃗ ; v⃗ −z o :
(16)
while for the E-ray, we have 8 ∂F 2 2 e > ∂F > ∂x 2x sin β z cos β · ne sin β 2x cos β − z sin β · no cos β < ∂x ∂F o ∂F 2 : ∂y ∂y 2yne > > : ∂F ∂F o 2 2 ∂z ∂z −2x sin β z cos β · ne cos β 2x cos β − z sin β · no sin β
Combined with Eqs. (2)–(9), the O-ray exit direction vector v⃗ o x o ; y o ; z o and the E-ray exit direction vector v⃗ e x e ; y e ; z e of ray Q nm can be worked out. Similarly, the transmission characteristics of the whole laser beam can be simulated by using MATLAB software. 2. Diffraction Integral Model
According to Babinet’s principle, the diffraction pattern of a laser beam is identical to the superposition of all separated elements. Therefore, the light field intensity of any point can be simulated by calculating the diffraction transmission of each
(9)
For the E-ray, we have cosd⃗ ; v⃗ −z e :
(17)
Combined with Eqs. (10)–(17), the light field intensity of the G point generated by element Q nm can be expressed as Z 2πm1 Z 2n1ω0 i E˜ z 6n1 N ˜ q− e ikni s · ρdφdρ; EG − 2nω0 2mπ 2λs0 s N 6n1 (18)
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where q z o for the O-ray, and q z e for the E-ray. Meanwhile, the transmission characteristic of the O-ray inside the crystal is isotropous, so ni no . For the E-ray, ni ne0 and we have 1 cos2 θv sin2 θv ; 0 2 n2o n2e ne
(19)
in which θv is the angle between the ray Q nm and the transmission vector v⃗ , and cosβ · r cos θ − ρ cos φ z sin β : (20) s Combined with Eqs. (18)–(20), the light field intensity of the G point generated by the element Q nm can be calculated. Then by calculating all the elements of the laser beam by computer circulation, the diffraction integral transmission model of the laser inside the KDP crystal can be worked out. cos θv
B. Accuracy Verification of the Models
The above models are adopted to simulate the transmission characteristics of a focused laser inside a KDP crystal. The thickness of the crystal is 12.2 mm, and the angle between the optical axis and the crystal surface is β 30.9°. Meanwhile, the wavelength of the laser is λ 355 nm, and the refractive indices of the O-ray and E-ray inside the KDP crystal are 1.5311 and 1.4858, respectively. The incident laser is a fundamental-mode Gaussian beam with a polarization ratio of O∶E 7∶3 and a total output power of 4 W. The beam
Fig. 4. Contrast of the experimental result and the simulated result. (a),(b) Diffraction integral results; (c),(d) ray tracing results; and (e) experimental results.
diameter is D 10 mm, and the focal length of the lens is F 73 mm. The distance between the lens and the crystal is H 64.3 mm, so the laser focus is near the back of the KDP crystal. Photographic paper was placed on the back of the crystal to record the spot pattern, and the experimental and simulated results are shown in Fig. 4. Because the laser spot is far above the diffraction limit, the simulated results of the ray tracing and diffraction integral methods are identical. The center distance of the two spots is 308.2 μm by the diffraction integral simulated result and 308.7 μm by the ray tracing simulated result, where both agree well with the experimental result of 308 μm. The 0.5 μm deviation between the two simulated results was generated because the initial phase was neglected in the ray tracing method. The morphology of the simulated spots also agreed well with the experimental results. Therefore, the accuracy of the two models are both fine. 3. RESULTS AND ANALYSES OF NUMERICAL SIMULATIONS Using these models, the laser focusing properties in the KDP crystal were simulated. The laser is a 1064 nm randomly polarized fiber laser with 20 W output power, and its beam diameter is D 15 mm. A. Effect of Crystal Orientation β
Figure 5 shows the relationship between the focus position and the crystal orientation when the focal length of the optical lens is F 75 mm, and the lens surface is H 60 mm above the crystal surface. For the O-ray, its focal position along the Z axis is always 22.416 mm under the crystal surface, because its transmission is isotropic, as shown in Fig. 5 (red line). For E-ray, with the decrease of β, the impact of the crystal orientation on the focus properties becomes more apparent. The crystal orientation has the same influence on the focus of the meridian plane and the sagittal plane when β 90°. Therefore, only one focus is formed at the 21.409 mm position. When β ≠ 90°, the crystal orientation has a greater impact on the meridian plane ray, while it has a relatively small impact on the sagittal plane ray. As a result, the meridian plane beam and the sagittal plane beam each produce their own focus. The distance of these two foci increases with the decrease of the angle β, which will gradually lead the shape of the E-ray focus to be an ellipse and will
Fig. 5. Relationship between the focus position and the crystal orientation.
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B. Effect of Focal Length F
Fig. 6. Morphology of laser foci when β 60° (simulated results). (a) Sagittal plane focus morphology of the E-ray, (b) meridian plane focus morphology of the E-ray, and (c) focus morphology of the O-ray.
The focus positions of the E-ray at different focal lengths (F 70–85 mm) were also simulated with the conditions H 61.3 mm and D 15 mm. The results indicate that with an increase in focal length, the focus position of the Eray will also increase, as shown in Figs. 8(a) and 8(c). All the curves coincide, as shown in Figs. 8(b) and 8(d). Therefore, the focus positions of the E-ray are proportional to F -H . As a result, the Z axis position of the sagittal focus (Z es ) can be revised as Z es
evolve into two vertical focus lines when β is large enough, as shown in Fig. 6. MATLAB software is used to analyze the relationship between the E-ray focus position and β with the polynomial fitting method, and the results indicate that coefficients greater than 4th degree tend to 0. Therefore, the relationship between the Z axis position (Z es ) of the sagittal focus and the crystal orientation β can be expressed as
F −H p1 · β3 p2 · β2 p3 · β p4: 15
(24)
In the same way, the Z axis position (Z em ) of the meridian focus can be revised as Z em
F −H p5 · β3 p6 · β2 p7 · β p8: 15
(25)
Meanwhile, the simulated results also indicate that the relative deviation (ε) is irrelevant to the focal length F , so Eq. (23) is also effective while the focal length is changed.
Z es p1 · β3 p2 · β2 p3 · β p4 : p1 1.4767 × 10−6 ; p2 −2.0087 × 10−4 ; p3 6.1914 × 10−4 ; p4 21.903
In the same way, the Z axis position (Z em ) of the meridian focus can be expressed as Z es p5 · β3 p6 · β2 p7 · β p8 : p5 4.6496 × 10−6 ; p6 −6.252 × 10−4 ; p7 1.6801 × 10−3 ; p8 22.932
(21)
(22)
Meanwhile, the simulated results also indicate that the anisotropy of the crystal will change the angle between the E-ray and the O-ray, and the relative deviation (ε) compared with Eq. (1) is shown in Fig. 7. The relative deviation is a sine function of 2β. Therefore, the angle between the E-ray and the O-ray can be expressed as η η0 ε η0 − 5.5 × 10−4 sin2β:
(23)
Fig. 7. Relationship between the relative angle deviation ε and the crystal orientation.
Fig. 8. Relationship between the focus position and the optical axis orientation of a crystal with different focal lengths. (a),(b) Meridian plane focus. (c), (d) Sagittal plane focus.
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C. Approximate 3D Light Intensity Equation of a Laser Focused into a KDP Crystal
Based on the simulated results, the 3D light intensity of the O-ray is the same as isotropic materials and can be expressed as 2
I o χ1
2
2P −2xω2y n z ; e πω2n z
(26)
in which χ 1 is the ratio of the O-ray in the laser, and ωn z is the beam diameter of the O-ray at the z position, expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi < ω z ω 1 z−no F no H 2 n 0 f : (27) : no πω20 f λ Although the transmission properties of the E-ray present as a deflection of the laser beam and a split of the laser focus, the 3D light intensity of the E-ray can be approximated still as an elliptical Gaussian function. According to the geometry theory, the coordinate axis after revolving an angle η along the Z → X direction can be expressed as x p x cos η z sin η; y p y : (28) z p z cos η − x sin η Thus, the 3D light intensity of the E-ray can be expressed as 2x 2
2y 2
p p − 2 − 2 2P I e χ2 e ω1 zp ω2 z p ; πω1 z p ω2 z p
(29)
in which χ 2 is the ratio of the E-ray in the laser beam. Meanwhile, ω1 z p and ω2 z p are the beam diameters of the E-ray in the meridian and sagittal planes at the z p position, respectively. There are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u Z em 2 u z − p cos η ω z ω t1 ; 1
p
m
ne0 πω2m λ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u Z es u z − p cos η ; ω2 z p ωs t1 n 0 πω2 e
λ
(30)
4. DISCUSSION
As a result, the 3D light intensity of the E-ray can be finally expressed as 2
Meanwhile, according to Eq. (32), Z em 13.074 mm, Z es 12.63 mm, ne0 1.467611, ω0 3 μm, η 0.0191953 rad, ωm 3.065 μm, and ωs 2.961 μm, and the light intensity that Eq. (32) describes is shown in Fig. 9(b). The comparison of these two results indicate that there is only some negligible part different and that the veracity of Eq. (32) is fine. Even though the diffraction integral model can achieve an accurate light intensity distribution of the focused laser into the birefringent crystal, it needs a very long time for simulation. Meanwhile, it is too complicated for low-precision engineering applications. Therefore, the diffraction integral model is used for some high-precision applications, for example, in analyzing the light intensity distribution inside frequency-switching crystals. Meanwhile, the approximate 3D light intensity equation in Eq. (32) is simple and rapid in some low-precision applications, such as in laser cutting, and modifying and micro-explosion processing of birefringent crystals. Therefore, this equation can be adopted in optical system design software directly to estimate the 3D light intensity and processing results of a focused laser inside crystal.
s
In which ωm and ωs are the equivalent focus radii of the meridian and sagittal planes, respectively. According to the optical path reversibility principle, ω1 0 and ω2 0 should equal to the input laser beam radius. Therefore, ω0 , ωm , and ωs can be solved by the function sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 λF − λH 2 D ω0 1 ω1 0 ω2 0: (31) 2 πω20
Ix; y; z χ 1
Fig. 9. Verification of the approximate 3D light intensity equation. (a) Simulation results of the diffraction integral model. (b) Simulation results of the approximate 3D light intensity equation.
2
2P −2xω2y n z e πω2n z 2
sin η −2x cos ηz 2P ω2 z p 1 e χ2 πω1 z p · ω2 z p
2y 2 ω2 z p 2
−
: (32)
The diffraction integral model was used to test the veracity of this 3D light intensity equation with the parameters F 15 mm, D 3.4 mm, H 6.3 mm, λ 1064 nm, and P 20 W, and the simulation results are shown in Fig. 9(a).
Assuming the input laser power was 20 W, the light intensity of the focus at different crystal orientations was simulated, as shown in Fig. 10. The light intensity of the O-ray focus is constant regardless of the crystal orientation variation because the birefringent effect is noneffective for the O-ray, while the E-ray focus is changed substantially with it. The results indicate that when β 90°, the E-ray focus is not split and has the highest light intensity which even exceeds the value of the O-ray due to the focusing enhancement effect of birefringence. With the decrease of β, the energy dispersion caused by the focus split will weaken the light intensity of the E-ray focus rapidly, and the light intensity of the E-ray focus is only 1/3 of the O-ray focus when β 75°. With a further decrease in β, the downtrend of the intensity will become gentle and almost does not change when β < 50°. At this time, the intensity of the E-ray focus is only 2%–5% of the O-ray. In order to reach a highly controllable light intensity distribution, the laser is always focused during the laser-induced thermal stress cutting process of the transparent birefringent materials such as KDP, quartz, and sapphire crystals. However, the
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Fig. 10. Relationship between the focus power density and the crystal orientation.
Fig. 11. Relationship between the focus power density and the focal length.
high power density at the focus position could easily arouse cracks or fragmentation, especially for KDP crystals [1,7]. Since the commonly used orientations of KDP crystals are β 30.9° (II type), 42.3° (I type), and 48.5° (I type), according to the simulation results of Eq. (32) and Fig. 10, the focused E-ray beam can not only achieve a light distribution similar to the O-ray beam to produce a uniform cutting stress, but also has a weaker power density at the focus position which is beneficial in reducing stress cracking. Therefore, the focused E-ray is an ideal beam for the thermal stress cutting of a KDP crystal. In the laser damage processing of birefringent material such as producing micro-explosion voids, waveguides, or 3D property modifications, the E-ray focus could produce some additional damage which deteriorates the processing precision. Taking the KDP crystal (β 30.9°, II type) as an example, since its damage threshold is about 2.3 × 1016 W · m−2 [18], according to the simulation results, the E-ray focus could cause damage effects when the peak power of the laser is over 5.9 × 107 W, while the O-ray focus only needs 6.5 × 104 W. Therefore, in the damage processing of the birefringent material, the effect of the E-ray can be ignored when the laser peak power density is low. For conditions that need laser power densities over 5.9 × 107 W, the effect of the E-ray should not be neglected as it will produce two vertical damaged lines inside the crystal (Fig. 6) and lead to crystal crack easily. At this moment, the E-ray has to be shielded in order to achieve high processing quality and safety. The power densities of the laser focus at different focal lengths also have been simulated, as shown in Fig. 11. There are two mechanisms dominating the power density of the E-ray focus: the diffraction effect and spot distortion caused by birefringence. When β 90°, the power density variation rule of the E-ray focus is similar to the O-ray, which indicates that the diffraction effect is the main mechanism leading to power density decreasing. With the reduction of β, the influence of the birefringent effect becomes stronger and makes the light intensity of the E-ray focus drop rapidly. When β < 50°, the birefringent effect is the major factor that influences the
E-ray focusing properties, and the light intensity of the E-ray focus at different focal lengths is almost unchanged. In the laser-induced thermal stress cutting of the birefringent material with small β, the processing safety of the Eray beam is still high no matter how the focal length changes. Therefore, the E-ray beam is beneficial for laser-induced thermal stress cutting under the conditions of β < 80°. In the laser damage process of the birefringent material, when β is small, using a shorter focus length lens can achieve a high contrast of power density between the O-ray and E-ray focus, which is beneficial for laser damaging processes. However, when β is large, the E-ray focus has a higher power density which can be used to produce an additional explosion point to improve processing efficiency. 5. CONCLUSION In this work, a ray tracing and diffraction integral method were proposed to develop the mathematical model of a laser focused through an isotropic medium into a KDP crystal. The laser focusing properties inside the KDP crystal at different orientations were investigated. The simulated results indicate that the birefringent has a great impact on the E-ray focus position, peak power density, and shape. Based on the simulated results, an approximate 3D light intensity equation of the laser focused into the birefringent material was also proposed, which is in good agreement with the theoretical analysis and has an important engineering value for nonlinear optics and the laser processing of birefringent materials. The analyzed results of the birefringence effects on laser processing indicate that the E-ray beam is beneficial for laser-induced thermal stress cutting of birefringent materials under the conditions of β < 80°. The E-ray beam can be ignored during low-power damage processing when β and the focal length are small, while it can be used to improve the processing efficiency when β is large. Funding. National Natural Science Foundation of China (NSFC) (51135005, 51175205); Specialized Research Fund
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Vol. 55, No. 4 / February 1 2016 / Applied Optics
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