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Nonparaxial propagation of Lorentz–Gauss beams in uniaxial crystal orthogonal to the optical axis Xun Wang,1 Zhirong Liu,1,* and Daomu Zhao2 1
Department of Applied Physics, East China Jiaotong University, Nanchang 330013, China 2 Department of Physics, Zhejiang University, Hangzhou 310027, China *Corresponding author: [email protected] Received January 6, 2014; revised February 25, 2014; accepted February 25, 2014; posted February 26, 2014 (Doc. ID 204184); published March 31, 2014
Analytical expressions for the three components of nonparaxial propagation of a polarized Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis are derived and used to investigate its propagation properties in uniaxial crystal. The influences of the initial beam parameters and the parameters of the uniaxial crystal on the evolution of the beam-intensity distribution in the uniaxial crystal are examined in detail. Results show that the statistical properties of a nonparaxial Lorentz–Gauss beam in a uniaxial crystal orthogonal to the optical axis are closely determined by the initial beam’s parameters and the parameters of the crystal: the beam waist sizes—w0 , w0x , and w0y —not only affect the size and shape of the beam profile in uniaxial crystal but also determine the nonparaxial effect of a Lorentz–Gauss beam; the beam profile of a Lorentz–Gauss beam in uniaxial crystal is elongated in the x or y direction, which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index; with increasing deviation of the ratio from unity, the extension of the beam profile augments. The results indicate that uniaxial crystal provides an effective and convenient method for modulating the Lorentz–Gauss beams. Our results may be valuable in some fields, such as optical trapping and nonlinear optics, where a light beam with a special profile and polarization is required. © 2014 Optical Society of America OCIS codes: (260.1180) Crystal optics; (260.1960) Diffraction theory; (350.5500) Propagation. http://dx.doi.org/10.1364/JOSAA.31.000872
1. INTRODUCTION Due to high angular spreading, appropriate models named Lorentz–Gauss beams were introduced to describe the radiation emitted by a single-model diode laser [1–3]. In recent years, the properties and applications of Lorentz–Gauss beams were extensively investigated [4–7]. Propagation of coherent and partially coherent Lorentz Gauss beams was considered in paraxial and nonparaxial cases, respectively [8,9]. Propagation of Lorentz–Gauss beams in free space, in turbulent atmosphere, and through an optical system [10–12] was, respectively, examined. Beam propagation factors, vectorial structure, focal shift, fractional Fourier transform, and Wigner distribution function of Lorentz–Gauss beams were also studied [13–17]. On the other hand, much attention was given to the research on the propagation of laser beams in uniaxial crystal since the construction of the vectorial theory of propagation in uniaxially anisotropic media [18]. The basic idea of this method is that the optical field inside the anisotropic medium can be evaluated as the superposition of ordinary and extraordinary beams. Thus, through solving the boundary value problems of Maxwell’s equations, the beam propagation equations in uniaxial crystals can be derived. The method has been applied to investigate the propagation of various beams in uniaxial crystals [19–26]. To the best of our knowledge, the nonparaxial propagation of Lorentz–Gauss beams in uniaxial crystal orthogonal to the optical axis has not yet been reported. In this work, based on the nonparaxial propagation theory in uniaxial crystals [18], we investigate the propagation of polarized Lorentz–Gauss beams in uniaxial crystal 1084-7529/14/040872-07$15.00/0
orthogonal to the optical axis in the nonparaxial case. Analytical expressions for the three components of the nonparaxial propagation of a polarized Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis are derived and used to investigate its propagation properties in uniaxial crystal. The influences of both the initial beam parameters and the parameters of the uniaxial crystal on the evolution of the beam intensity distribution in the uniaxial crystal are examined in detail. Results show that the statistical properties of a nonparaxial Lorentz–Gauss beam are closely determined by the initial beam parameters and the parameters of the crystal.
2. NONPARAXIAL PROPAGATION OF A LORENTZ–GAUSS BEAM IN UNIAXIAL CRYSTAL In the Cartesian coordinate system, the light beam considered here travels along the z axis, the geometry of the propagation of a beam in a uniaxial crystal orthogonal to the optical axis is shown in Fig. 1, and the relative dielectric tensor of the crystal is described by 0
n2e
0
B εB @ 0
n2o
0
0
0
1
C 0 C A; 2 no
(1)
where no and ne are the ordinary and extraordinary refractive index, respectively. © 2014 Optical Society of America
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expressed as the sum of the paraxial field and a nonparaxial correction term to describe the nonparaxial propagation of a Lorentz–Gauss beam in uniaxial crystal [18]: ZZ Ex; y; z expik0 ne z Fig. 1. Geometry of the propagation of a laser beam in a uniaxial crystal orthogonal to the optical axis.
E x r 0 ; 0
Ex x0 ; y0 ; 0
E y r 0 ; 0
0 @
x2 y2 ; E0 exp − 0 2 0 w 0 w0x w0y 1x0 ∕w0x 2 1y0 ∕w0y 2
−∞
1 0 0 iC hk k k2x k2y B ~ C B x y~ × exp −i z B k20 n2o E x k Ey k C: A @ 2k0 no k − ko ny o E~ y k
1 A;
After some operation, Eq. (6) can be rewritten as ZZ
where E 0 is a constant, w0 is the waist wide of the Gauss part, and w0x and w0y are the parameters related to the beam waist widths of the Lorentz part in x and y directions, respectively. According to the theory of optical nonparaxial propagation in uniaxial crystals orthogonal to the optical axis [18], the electric field can be conveniently interpreted as a linear superposition of plane waves. The propagation of the electric field of a beam in a uniaxial crystal orthogonal to the optical axis obeys the following equation [18]:
Ex x; y; z
k0 no 2πiz
E y x; y; z
ik0 no 2πz3
1 E~ x k C B kx ky ∞ ~ C B Ex; y; z d2 k expikr expikez zB − k20 n2o −k2x E x k C A @ −∞ − k2knez2k−kx 2 E~ x k o x 0 ZZ ∞ 2 d k expikr expikoz z
E z x; y; z
0
−∞
B B ×B @
h
kx ky k20 n2o −k2x
h
E~ x k E~ y k
k k ky − kozy k2 nx2 −k 2 x 0 o
i
E~ x k E~ y k
1 2π2
ZZ
∞
−∞
ZZ
(3)
2
d r 0 exp−ik · rE α r 0 ; 0 α x; y; (4)
koz
− k2x 1∕2 ;
kez
k20 n2o
− n2e ∕n2o k2x
− k2y 1∕2 :
∞ −∞
ik0 no 2πz2
ZZ
∞
−∞
(7)
x − x0 y − y0 E x x0 ; y0 ; 0
(8)
x − x0 Ex x0 ; y0 ; 0 × Πe r; r o
k Πe r;r 0 expik0 ne zexp − 0 n2o x − x0 2 n2e y − y0 2 ; 2izne (10) k n Πo r; r 0 expik0 no z exp − 0 o x − x0 2 y − y0 2 : 2iz (11) Substituting Eq. (2) into Eq. (7), we obtain E x x; y; z
k0 no E 0 expik0 ne zΓx x; zΓx y; z; 2πiz
(5)
When the beam waist radius of a Lorentz–Gauss beam is comparable with the wavelength, Eq. (3) can be approximately
(12)
where 2 x w0x exp − 02 2 x2 w w −∞ 0x 0 0 k0 n2o × exp − x − x0 2 dx0 ; 2izne
Z k20 n2o
E x x0 ; y0 ; 0Πe r; r 0 ; 0dx0 dy0 ;
with Πe r; r 0 and Πo r; r 0 being given by
where k⃗ kx e⃗ x ky e⃗ y , ⃗r x⃗ex y⃗ey , and k0 2π∕λ are the wavenumbers with λ being the wavelength of the incident beam in vacuum. d2 k⃗ dkx dky is the position vector in the spatial-frequency domain. E~ α k is the 2D Fourier transform of the transverse components of the electric field in the plane z 0 and is given by E~ α k
−∞
y − y0 E y x0 ; y0 ; 0 × Πo r; r 0 dx0 dy0 ; (9)
1 C C C; iA
∞
× Πe r; r o − Πo r; r o dx0 dy0 ZZ ∞ k n 0 o E x ; y ; 0Πo r; r 0 dx0 dy0 ; 2πiz −∞ y 0 0
ZZ
0
(6)
(2)
0
0
d2 k expik · r
1 0 E~ x k B n2e k2x n2o k2y B − kx ky E~ k C C × exp −i z B k20 n2o x C A @ 2k0 ne n2o − nk enkx2 E~ x k 0 o ZZ ∞ expik0 no z d2 k expik · r
!
Ey x0 ; y0 ; 0
−∞
Assume that the Lorentz–Gauss beam considered here is linearly polarized in the x direction and is incident on a uniaxial crystal in the plane z 0. The electric field of a Lorentz– Gauss beam in the input plane z 0 takes the form [1] !
∞
Γx x; z
∞
(13)
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2 w0y y exp − 02 2 2 w0 −∞ w0y y0 2 k n × exp − 0 o y − y0 2 dy0 : 2izne Z
Γx y; z
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∞
π 2 k 0 no E 0 k expik0 ne z exp − 0 n2o x2 n2e y2 4 2πiz 2izne 2 2 k no 2 ne 2 × exp − 0 x y 2izne A B
E x x; y; z (14)
− − × V x V x V y V y ;
Equations (13) and (14) can be rewritten as k0 n2o 2 k0 n2o x2 Γx x; z exp − x exp 2izne 2izne A Z∞ w0x k0 n2o x 2 dx0 ; × A x − exp 0 2 2 A 2izne −∞ w0x x0 (15)
(22)
with V x and V y given by
k0 n2o x 2 exp A w0x i A 2izne 2 2s 33 2A k n x o 0 55; w0x i × 41 − erf 4 2izne A
V x
k0 ne y 2 B w0y i B 2iz " "r ## k0 ne B y × 1 − erf w0y i ; 2iz B
(23a)
k n k n y2 Γx y; z exp − 0 e y2 exp 0 e 2iz 2iz B Z∞ w0y k0 ne y 2 dy0 ; (16) exp B y0 − × 2 2 B 2iz −∞ w0y y0
V y exp
(23b)
where
where 2izne ; A1 k0 n2o w20
2iz B1 : k0 ne w20
2 erfx p π
(17)
Z 0
x
exp−s2 ds:
(24)
Substituting Eq. (2) into Eqs. (8) and (9), we obtain By using the following convolution of functions Z f 1 τ f 2 τ
E y x; y; z ∞
−∞
f 1 ηf 2 τ − ηdη:
ZZ
∞
−∞
x − x0 y − y0 E x x0 ; y0 ; 0
× Πe r; r o − Πo r; r o dx0 dy0
(18)
ik0 no expik0 ne zx − x0 y − y0 Γx x; zΓx y; z 2πz3 ik n − 0 3o expik0 no zx − x0 y − y0 Γy x; zΓy y; z; 2πz (25)
Equations (15) and (16) can be expressed as follows: k n2 Γx x; z w0x exp − 0 o x2 2izne k0 n2o x2 x x f2 ; f1 × exp A A 2izne A
ik0 no 2πz3
(19)
k n k n y2 y y f 2 ; f1 Γx y;z w0y exp − 0 e y2 exp 0 e B B 2iz 2iz B (20)
ZZ ik0 no ∞ x − x0 E x x0 ; y0 ; 0 × Πe r; r o dx0 dy0 2πz2 −∞ ik n 0 2o expik0 ne zx − x0 Γx x; zΓx y; z. 2πz (26)
E z x; y; z
Similarly, 2 α0 w0α k0 no 2 exp − 2 × exp − α − α0 dα0 Γy α;z 2 2 2iz w0 −∞ w0α α0 k n k n α2 exp − 0 o α2 exp 0 o 2iz 2iz C Z∞ w0α k 0 no C α 2 α dα0 exp − − × 0 2 2 2iz C −∞ w0α α0 k n k n α2 α α f 2 ; w0α exp − 0 o α2 exp 0 o f1 C C 2iz 2iz C Z
where f 1 τ and f 2 τ are expressed as f 1 τ
1 ; w20x τ2
k n2 A f 2 τ exp − 0 o τ2 ; 2izne
(21a)
f 1 τ
1 ; w20y τ2
k nB f 2 τ exp − 0 e τ2 : 2iz
(21b)
∞
(27) By using the convolution theorem of the Fourier transform, Eq. (12) can be expressed as
where
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α x; y;
C 1
2iz : k0 no w20
(28)
By using the convolution theory of the Fourier transform, Eqs. (25) and (26) can be expressed as π 2 ik0 no k0 2 2 2 2 expik0 ne z exp − E y x; y; z n x ne y 4 2πz3 2izne o 2 k no 2 n2e 2 x − iw0x V x y × exp − 0 x 2izne A B − x iw0x V −x × y − iw0y V y y iw0y V y k n − expik0 no z exp − 0 o x2 y2 2iz k n × exp − 0 o x2 y2 × x − iw0x U x 2izC − x iw0x U −x y − iw0y U y y iw0y U y ;
(29) π 2 ik0 no E0 k0 2 x2 n2 y2 expik n z exp − n o e 0 e 4 2πz2 2izne 2 2 k0 no 2 ne 2 x − iw0x V x y × exp x 2izne A B
E z x; y; z
− x iw0x V −x × V y V y ;
(30)
where U x and U y are given by
k0 no x 2 C w0x i C 2iz " "r ## k0 no C x × 1 − erf w0x i ; 2iz C
U x exp
k0 no y 2 C w0y i C 2iz " "r ## k0 no C y × 1 − erf w0y i : 2iz C
875
distributions of nonparaxial Lorentz–Gauss beams orthogonal to the optical axis of a uniaxial crystal are depicted in Figs. 2–7. For the convenience of comparison, all the contour graphs in Figs. 2–7 have been normalized to the peak intensity value in the input plane z 0. All the following numerical calculations are carried out by using Eqs. (22), (29), and (30). Figure 2 shows a contour graph of the x component of a Lorentz–Gauss beam in several observation planes upon propagation orthogonal to the optical axis of the uniaxial crystal. The calculation parameters are chosen as: ordinary refractive index no 2.616, extraordinary refractive index ne is given by e ne ∕no , where e is set to be 1.5, and the waist widths are chosen as w0 1λ, w0x w0y 0.5λ. And several observation planes are chosen as z 0.1zr , z 0.5zr , z zr , z 3zr , respectively, where zr k0 w20 ∕2 is the Rayleigh length of the Gaussian part. As the w0x is equal to w0y , the beam profile is a symmetrical circular shape in the incident plane. But with the anisotropic effect of the crystal, for positive crystal with e > 1, the beam profile in the x direction spreads faster than in the y direction. Therefore, with the increase of propagation distance z, the size of the beam spot is enlarged observably in the x direction, and meanwhile the intensity decreased, see Figs. 2(a) to 2(d). Figures 3 and 4 depict contour graphs of the y component and the longitudinal component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal, respectively. The calculation parameters are the same as those in Fig. 2. Interestingly, we can notice that, although the y component of a Lorentz–Gauss beam is zero in the original input plane, after propagation in the uniaxial crystal, both the y and the longitudinal components emerge, and they are no longer equal to zero any more. This is due to a change of polarization state of the radiation that occurs during propagation in the uniaxial crystal [18]. From Figs. (3) and (4), we can see that
(31a)
U y exp
(31b)
Equations (22), (29), and (30) are the main analytical results for the nonparaxial propagation of a Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis. The y component of the nonparaxial propagation correction is of the second order in the paraxiality degree, while the z component is of the first order [18]. It shows that the z component in the slightly nonparaxial regime is stronger than the y component. In the paraxial case, the propagation of a Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis is just given by Eq. (22), and the other two components are equal to zero.
3. NUMERICAL SIMULATIONS AND ANALYSIS Based on the analytical expressions obtained in Section 2, in the following we numerically investigate the nonparaxial propagation properties of Lorentz–Gauss beams in uniaxial crystal orthogonal to the optical axis. The light intensity
Fig. 2. Contour graph of the intensity of the x component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e 1.5 (a) z 0.1zr , (b) z 0.5zr , (c) z zr , and (d) z 3zr .
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Fig. 3. Contour graph of the intensity of the y component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e 1.5 (a) z 0.1zr , (b) z 0.5zr , (c) z zr , and (d) z 3zr .
Fig. 4. Contour graph of the intensity of the longitudinal component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e 1.5 (a) z 0.1zr , (b) z 0.5zr , (c) z zr , and (d) z 3zr .
the magnitude of the intensity of the longitudinal component is smaller than that of the x component, and the magnitude of the intensity of the y component is far smaller than that of the x component, and when propagation distance z increases, both the intensity of the y and the longitudinal components decrease. Obviously, in the initial plane the beam profile of the y component is composed of four asymmetric lobes, while the beam profile of the longitudinal component is made up of
Wang et al.
Fig. 5. Contour graph of the intensity of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e 1.5 (a) z 0.1zr , (b) z 0.5zr , (c) z zr , and (d) z 3zr .
Fig. 6. Contour graph of the intensity of a Lorentz–Gauss beam in the observation plane z zr of different uniaxial crystal: (a) e 0.6, (b) e 0.8, (c) e 1, and (d) e 1.5.
two asymmetric lobes [see Figs. 3(a) and 4(a)]. When the observation plane ranges from z 0.1zr to the z 3zr , the sizes of spots of both the y and longitudinal components are enlarged with the elongation in the x direction. Meanwhile, the four lobes of the beam profiles of the y component finally split into eight ones, but the four lobes of beam profiles located in the central area are dominant [see Figs. 3(b) to 3(d) and Figs. 4(b) to 4(d)]. Figure 5 presents contour graphs of the intensity distributions of a Lorentz–Gauss beam in the uniaxial crystal in
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Fig. 7. Contour graph of the intensity of Lorentz–Gauss beams of different waist widths in the observation plane z zr , e 1.5: (a) w0 w0x w0y 0.5λ, (b) w0 1λ, w0x w0y 0.5λ, (c) w0 1λ, w0x 0.5λ, w0y 1.5λ, (d) w0 1λ, w0x 1.5λ, w0y 0.5λ, (e) w0 5λ, w0x 0.5λ, w0y 1.5λ, and (f) w0 5λ, w0x 1.5λ, w0y 0.5λ.
several observation planes. The calculation parameters are the same as those in Fig. 2. Because of jE x j2 > jEz j2 > jE y j2 , the contour graphs of the intensity distributions of a Lorentz–Gauss beam is close to the x component. Additionally, the positions of peak intensity value of the three components do not overlap in the coordinate system, so the peak intensity value of a Lorentz–Gauss beam is not the simple summation of the three components. Figure 6 shows contour graphs of the intensity of a Lorentz– Gauss beam in the observation plane z zr in different uniaxial crystal, i.e., (a) e 0.6, (b) e 0.8, (c) e 1, and (d) e 1.5. Other calculation parameters are the same as those in Fig. 2. It is obvious from Fig. (6) that for the case e < 1, i.e., negative crystal, the beam profile of a Lorentz– Gauss beam is elongated in the y direction, see Figs. 6(a) and 6(b); for the case e > 1, i.e., positive crystal, the beam profile of a Lorentz–Gauss beam is elongated in the x direction [see Fig. 6(d)]. Furthermore, with the increase of the deviation of e from unity, the extension of the beam spot augments. And for the case e 1, i.e., isotropic medium, the beam profile of a Lorentz–Gauss beam is symmetrical, see Fig. 6(c). Figure 7 presents contour graphs of the intensity of Lorentz–Gauss beams of different waist widths—w0 , w0x , and w0y —in the observation plane z zr , where e is set to be 1.5. From Figs. 7(a) and 7(b), we can find that with the increase of w0 , the sizes of the spots enlarged significantly. And due to the anisotropic influence of the positive crystal, the beam profiles extend in the x direction, even though w0x is equal to w0y . From Figs. 7(c), and 7(d), we can find that for little w0 , the contour graph of intensity of the Lorentz–Gauss
beams is insensitive to the ratio of w0x to w0y . However, for large w0 [see Figs. 7(e) and 7(f)], the ratio of w0x to w0y greatly affects the intensity distribution. Therefore the size of the spot increases with w0 , and the shape of the beam profile is related to w0x and w0y .
4. CONCLUSIONS In conclusion, analytical expressions for the three components of nonparaxial propagation of a polarized Lorentz– Gauss beam in uniaxial crystal orthogonal to the optical axis are derived, and the corresponding light intensity distributions have been illustrated with numerical examples. Results show that even though the y component of a Lorentz–Gauss beam in the input plane is zero, the y and longitudinal components arise upon propagation in uniaxial crystal, which cannot be ignored. And the beam waist sizes—w0 , w0x , and w0y —not only affect both the size and shape of the beam profile in uniaxial crystal but also determine the nonparaxial effect of a Lorentz–Gauss beam; even if only one of the three parameters is comparable with the light wavelength, the Lorentz–Gauss beam should be treated beyond the paraxial approximation. Furthermore, the beam profile of a Lorentz– Gauss beam in uniaxial crystal is elongated in the x or y direction, which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index; with increasing the deviation of the ratio from unity, the extension of the beam profile augments. Our results indicate that uniaxial crystal provides an effective and convenient method for modulating the Lorentz–Gauss beams where a light beam with special profile and polarization is required.
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ACKNOWLEDGMENTS This work is supported by the National Science Foundation of China (grant no. 11274273 and 11365009), the Jiangxi Provincial Natural Science Foundation of China (grant no. 20114BAB211014), and the East China Jiaotong University Startup Outlay for Doctor Scientific Research (grant no. 09132007). The author is indebted to the reviewers for their valuable comments.
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Nonparaxial propagation of Lorentz–Gauss beams in uniaxial crystal orthogonal to the optical axis Xun Wang,1 Zhirong Liu,1,* and Daomu Zhao2 1
Department of Applied Physics, East China Jiaotong University, Nanchang 330013, China 2 Department of Physics, Zhejiang University, Hangzhou 310027, China *Corresponding author: [email protected] Received January 6, 2014; revised February 25, 2014; accepted February 25, 2014; posted February 26, 2014 (Doc. ID 204184); published March 31, 2014
Analytical expressions for the three components of nonparaxial propagation of a polarized Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis are derived and used to investigate its propagation properties in uniaxial crystal. The influences of the initial beam parameters and the parameters of the uniaxial crystal on the evolution of the beam-intensity distribution in the uniaxial crystal are examined in detail. Results show that the statistical properties of a nonparaxial Lorentz–Gauss beam in a uniaxial crystal orthogonal to the optical axis are closely determined by the initial beam’s parameters and the parameters of the crystal: the beam waist sizes—w0 , w0x , and w0y —not only affect the size and shape of the beam profile in uniaxial crystal but also determine the nonparaxial effect of a Lorentz–Gauss beam; the beam profile of a Lorentz–Gauss beam in uniaxial crystal is elongated in the x or y direction, which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index; with increasing deviation of the ratio from unity, the extension of the beam profile augments. The results indicate that uniaxial crystal provides an effective and convenient method for modulating the Lorentz–Gauss beams. Our results may be valuable in some fields, such as optical trapping and nonlinear optics, where a light beam with a special profile and polarization is required. © 2014 Optical Society of America OCIS codes: (260.1180) Crystal optics; (260.1960) Diffraction theory; (350.5500) Propagation. http://dx.doi.org/10.1364/JOSAA.31.000872
1. INTRODUCTION Due to high angular spreading, appropriate models named Lorentz–Gauss beams were introduced to describe the radiation emitted by a single-model diode laser [1–3]. In recent years, the properties and applications of Lorentz–Gauss beams were extensively investigated [4–7]. Propagation of coherent and partially coherent Lorentz Gauss beams was considered in paraxial and nonparaxial cases, respectively [8,9]. Propagation of Lorentz–Gauss beams in free space, in turbulent atmosphere, and through an optical system [10–12] was, respectively, examined. Beam propagation factors, vectorial structure, focal shift, fractional Fourier transform, and Wigner distribution function of Lorentz–Gauss beams were also studied [13–17]. On the other hand, much attention was given to the research on the propagation of laser beams in uniaxial crystal since the construction of the vectorial theory of propagation in uniaxially anisotropic media [18]. The basic idea of this method is that the optical field inside the anisotropic medium can be evaluated as the superposition of ordinary and extraordinary beams. Thus, through solving the boundary value problems of Maxwell’s equations, the beam propagation equations in uniaxial crystals can be derived. The method has been applied to investigate the propagation of various beams in uniaxial crystals [19–26]. To the best of our knowledge, the nonparaxial propagation of Lorentz–Gauss beams in uniaxial crystal orthogonal to the optical axis has not yet been reported. In this work, based on the nonparaxial propagation theory in uniaxial crystals [18], we investigate the propagation of polarized Lorentz–Gauss beams in uniaxial crystal 1084-7529/14/040872-07$15.00/0
orthogonal to the optical axis in the nonparaxial case. Analytical expressions for the three components of the nonparaxial propagation of a polarized Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis are derived and used to investigate its propagation properties in uniaxial crystal. The influences of both the initial beam parameters and the parameters of the uniaxial crystal on the evolution of the beam intensity distribution in the uniaxial crystal are examined in detail. Results show that the statistical properties of a nonparaxial Lorentz–Gauss beam are closely determined by the initial beam parameters and the parameters of the crystal.
2. NONPARAXIAL PROPAGATION OF A LORENTZ–GAUSS BEAM IN UNIAXIAL CRYSTAL In the Cartesian coordinate system, the light beam considered here travels along the z axis, the geometry of the propagation of a beam in a uniaxial crystal orthogonal to the optical axis is shown in Fig. 1, and the relative dielectric tensor of the crystal is described by 0
n2e
0
B εB @ 0
n2o
0
0
0
1
C 0 C A; 2 no
(1)
where no and ne are the ordinary and extraordinary refractive index, respectively. © 2014 Optical Society of America
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expressed as the sum of the paraxial field and a nonparaxial correction term to describe the nonparaxial propagation of a Lorentz–Gauss beam in uniaxial crystal [18]: ZZ Ex; y; z expik0 ne z Fig. 1. Geometry of the propagation of a laser beam in a uniaxial crystal orthogonal to the optical axis.
E x r 0 ; 0
Ex x0 ; y0 ; 0
E y r 0 ; 0
0 @
x2 y2 ; E0 exp − 0 2 0 w 0 w0x w0y 1x0 ∕w0x 2 1y0 ∕w0y 2
−∞
1 0 0 iC hk k k2x k2y B ~ C B x y~ × exp −i z B k20 n2o E x k Ey k C: A @ 2k0 no k − ko ny o E~ y k
1 A;
After some operation, Eq. (6) can be rewritten as ZZ
where E 0 is a constant, w0 is the waist wide of the Gauss part, and w0x and w0y are the parameters related to the beam waist widths of the Lorentz part in x and y directions, respectively. According to the theory of optical nonparaxial propagation in uniaxial crystals orthogonal to the optical axis [18], the electric field can be conveniently interpreted as a linear superposition of plane waves. The propagation of the electric field of a beam in a uniaxial crystal orthogonal to the optical axis obeys the following equation [18]:
Ex x; y; z
k0 no 2πiz
E y x; y; z
ik0 no 2πz3
1 E~ x k C B kx ky ∞ ~ C B Ex; y; z d2 k expikr expikez zB − k20 n2o −k2x E x k C A @ −∞ − k2knez2k−kx 2 E~ x k o x 0 ZZ ∞ 2 d k expikr expikoz z
E z x; y; z
0
−∞
B B ×B @
h
kx ky k20 n2o −k2x
h
E~ x k E~ y k
k k ky − kozy k2 nx2 −k 2 x 0 o
i
E~ x k E~ y k
1 2π2
ZZ
∞
−∞
ZZ
(3)
2
d r 0 exp−ik · rE α r 0 ; 0 α x; y; (4)
koz
− k2x 1∕2 ;
kez
k20 n2o
− n2e ∕n2o k2x
− k2y 1∕2 :
∞ −∞
ik0 no 2πz2
ZZ
∞
−∞
(7)
x − x0 y − y0 E x x0 ; y0 ; 0
(8)
x − x0 Ex x0 ; y0 ; 0 × Πe r; r o
k Πe r;r 0 expik0 ne zexp − 0 n2o x − x0 2 n2e y − y0 2 ; 2izne (10) k n Πo r; r 0 expik0 no z exp − 0 o x − x0 2 y − y0 2 : 2iz (11) Substituting Eq. (2) into Eq. (7), we obtain E x x; y; z
k0 no E 0 expik0 ne zΓx x; zΓx y; z; 2πiz
(5)
When the beam waist radius of a Lorentz–Gauss beam is comparable with the wavelength, Eq. (3) can be approximately
(12)
where 2 x w0x exp − 02 2 x2 w w −∞ 0x 0 0 k0 n2o × exp − x − x0 2 dx0 ; 2izne
Z k20 n2o
E x x0 ; y0 ; 0Πe r; r 0 ; 0dx0 dy0 ;
with Πe r; r 0 and Πo r; r 0 being given by
where k⃗ kx e⃗ x ky e⃗ y , ⃗r x⃗ex y⃗ey , and k0 2π∕λ are the wavenumbers with λ being the wavelength of the incident beam in vacuum. d2 k⃗ dkx dky is the position vector in the spatial-frequency domain. E~ α k is the 2D Fourier transform of the transverse components of the electric field in the plane z 0 and is given by E~ α k
−∞
y − y0 E y x0 ; y0 ; 0 × Πo r; r 0 dx0 dy0 ; (9)
1 C C C; iA
∞
× Πe r; r o − Πo r; r o dx0 dy0 ZZ ∞ k n 0 o E x ; y ; 0Πo r; r 0 dx0 dy0 ; 2πiz −∞ y 0 0
ZZ
0
(6)
(2)
0
0
d2 k expik · r
1 0 E~ x k B n2e k2x n2o k2y B − kx ky E~ k C C × exp −i z B k20 n2o x C A @ 2k0 ne n2o − nk enkx2 E~ x k 0 o ZZ ∞ expik0 no z d2 k expik · r
!
Ey x0 ; y0 ; 0
−∞
Assume that the Lorentz–Gauss beam considered here is linearly polarized in the x direction and is incident on a uniaxial crystal in the plane z 0. The electric field of a Lorentz– Gauss beam in the input plane z 0 takes the form [1] !
∞
Γx x; z
∞
(13)
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2 w0y y exp − 02 2 2 w0 −∞ w0y y0 2 k n × exp − 0 o y − y0 2 dy0 : 2izne Z
Γx y; z
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∞
π 2 k 0 no E 0 k expik0 ne z exp − 0 n2o x2 n2e y2 4 2πiz 2izne 2 2 k no 2 ne 2 × exp − 0 x y 2izne A B
E x x; y; z (14)
− − × V x V x V y V y ;
Equations (13) and (14) can be rewritten as k0 n2o 2 k0 n2o x2 Γx x; z exp − x exp 2izne 2izne A Z∞ w0x k0 n2o x 2 dx0 ; × A x − exp 0 2 2 A 2izne −∞ w0x x0 (15)
(22)
with V x and V y given by
k0 n2o x 2 exp A w0x i A 2izne 2 2s 33 2A k n x o 0 55; w0x i × 41 − erf 4 2izne A
V x
k0 ne y 2 B w0y i B 2iz " "r ## k0 ne B y × 1 − erf w0y i ; 2iz B
(23a)
k n k n y2 Γx y; z exp − 0 e y2 exp 0 e 2iz 2iz B Z∞ w0y k0 ne y 2 dy0 ; (16) exp B y0 − × 2 2 B 2iz −∞ w0y y0
V y exp
(23b)
where
where 2izne ; A1 k0 n2o w20
2iz B1 : k0 ne w20
2 erfx p π
(17)
Z 0
x
exp−s2 ds:
(24)
Substituting Eq. (2) into Eqs. (8) and (9), we obtain By using the following convolution of functions Z f 1 τ f 2 τ
E y x; y; z ∞
−∞
f 1 ηf 2 τ − ηdη:
ZZ
∞
−∞
x − x0 y − y0 E x x0 ; y0 ; 0
× Πe r; r o − Πo r; r o dx0 dy0
(18)
ik0 no expik0 ne zx − x0 y − y0 Γx x; zΓx y; z 2πz3 ik n − 0 3o expik0 no zx − x0 y − y0 Γy x; zΓy y; z; 2πz (25)
Equations (15) and (16) can be expressed as follows: k n2 Γx x; z w0x exp − 0 o x2 2izne k0 n2o x2 x x f2 ; f1 × exp A A 2izne A
ik0 no 2πz3
(19)
k n k n y2 y y f 2 ; f1 Γx y;z w0y exp − 0 e y2 exp 0 e B B 2iz 2iz B (20)
ZZ ik0 no ∞ x − x0 E x x0 ; y0 ; 0 × Πe r; r o dx0 dy0 2πz2 −∞ ik n 0 2o expik0 ne zx − x0 Γx x; zΓx y; z. 2πz (26)
E z x; y; z
Similarly, 2 α0 w0α k0 no 2 exp − 2 × exp − α − α0 dα0 Γy α;z 2 2 2iz w0 −∞ w0α α0 k n k n α2 exp − 0 o α2 exp 0 o 2iz 2iz C Z∞ w0α k 0 no C α 2 α dα0 exp − − × 0 2 2 2iz C −∞ w0α α0 k n k n α2 α α f 2 ; w0α exp − 0 o α2 exp 0 o f1 C C 2iz 2iz C Z
where f 1 τ and f 2 τ are expressed as f 1 τ
1 ; w20x τ2
k n2 A f 2 τ exp − 0 o τ2 ; 2izne
(21a)
f 1 τ
1 ; w20y τ2
k nB f 2 τ exp − 0 e τ2 : 2iz
(21b)
∞
(27) By using the convolution theorem of the Fourier transform, Eq. (12) can be expressed as
where
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α x; y;
C 1
2iz : k0 no w20
(28)
By using the convolution theory of the Fourier transform, Eqs. (25) and (26) can be expressed as π 2 ik0 no k0 2 2 2 2 expik0 ne z exp − E y x; y; z n x ne y 4 2πz3 2izne o 2 k no 2 n2e 2 x − iw0x V x y × exp − 0 x 2izne A B − x iw0x V −x × y − iw0y V y y iw0y V y k n − expik0 no z exp − 0 o x2 y2 2iz k n × exp − 0 o x2 y2 × x − iw0x U x 2izC − x iw0x U −x y − iw0y U y y iw0y U y ;
(29) π 2 ik0 no E0 k0 2 x2 n2 y2 expik n z exp − n o e 0 e 4 2πz2 2izne 2 2 k0 no 2 ne 2 x − iw0x V x y × exp x 2izne A B
E z x; y; z
− x iw0x V −x × V y V y ;
(30)
where U x and U y are given by
k0 no x 2 C w0x i C 2iz " "r ## k0 no C x × 1 − erf w0x i ; 2iz C
U x exp
k0 no y 2 C w0y i C 2iz " "r ## k0 no C y × 1 − erf w0y i : 2iz C
875
distributions of nonparaxial Lorentz–Gauss beams orthogonal to the optical axis of a uniaxial crystal are depicted in Figs. 2–7. For the convenience of comparison, all the contour graphs in Figs. 2–7 have been normalized to the peak intensity value in the input plane z 0. All the following numerical calculations are carried out by using Eqs. (22), (29), and (30). Figure 2 shows a contour graph of the x component of a Lorentz–Gauss beam in several observation planes upon propagation orthogonal to the optical axis of the uniaxial crystal. The calculation parameters are chosen as: ordinary refractive index no 2.616, extraordinary refractive index ne is given by e ne ∕no , where e is set to be 1.5, and the waist widths are chosen as w0 1λ, w0x w0y 0.5λ. And several observation planes are chosen as z 0.1zr , z 0.5zr , z zr , z 3zr , respectively, where zr k0 w20 ∕2 is the Rayleigh length of the Gaussian part. As the w0x is equal to w0y , the beam profile is a symmetrical circular shape in the incident plane. But with the anisotropic effect of the crystal, for positive crystal with e > 1, the beam profile in the x direction spreads faster than in the y direction. Therefore, with the increase of propagation distance z, the size of the beam spot is enlarged observably in the x direction, and meanwhile the intensity decreased, see Figs. 2(a) to 2(d). Figures 3 and 4 depict contour graphs of the y component and the longitudinal component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal, respectively. The calculation parameters are the same as those in Fig. 2. Interestingly, we can notice that, although the y component of a Lorentz–Gauss beam is zero in the original input plane, after propagation in the uniaxial crystal, both the y and the longitudinal components emerge, and they are no longer equal to zero any more. This is due to a change of polarization state of the radiation that occurs during propagation in the uniaxial crystal [18]. From Figs. (3) and (4), we can see that
(31a)
U y exp
(31b)
Equations (22), (29), and (30) are the main analytical results for the nonparaxial propagation of a Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis. The y component of the nonparaxial propagation correction is of the second order in the paraxiality degree, while the z component is of the first order [18]. It shows that the z component in the slightly nonparaxial regime is stronger than the y component. In the paraxial case, the propagation of a Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis is just given by Eq. (22), and the other two components are equal to zero.
3. NUMERICAL SIMULATIONS AND ANALYSIS Based on the analytical expressions obtained in Section 2, in the following we numerically investigate the nonparaxial propagation properties of Lorentz–Gauss beams in uniaxial crystal orthogonal to the optical axis. The light intensity
Fig. 2. Contour graph of the intensity of the x component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e 1.5 (a) z 0.1zr , (b) z 0.5zr , (c) z zr , and (d) z 3zr .
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Fig. 3. Contour graph of the intensity of the y component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e 1.5 (a) z 0.1zr , (b) z 0.5zr , (c) z zr , and (d) z 3zr .
Fig. 4. Contour graph of the intensity of the longitudinal component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e 1.5 (a) z 0.1zr , (b) z 0.5zr , (c) z zr , and (d) z 3zr .
the magnitude of the intensity of the longitudinal component is smaller than that of the x component, and the magnitude of the intensity of the y component is far smaller than that of the x component, and when propagation distance z increases, both the intensity of the y and the longitudinal components decrease. Obviously, in the initial plane the beam profile of the y component is composed of four asymmetric lobes, while the beam profile of the longitudinal component is made up of
Wang et al.
Fig. 5. Contour graph of the intensity of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e 1.5 (a) z 0.1zr , (b) z 0.5zr , (c) z zr , and (d) z 3zr .
Fig. 6. Contour graph of the intensity of a Lorentz–Gauss beam in the observation plane z zr of different uniaxial crystal: (a) e 0.6, (b) e 0.8, (c) e 1, and (d) e 1.5.
two asymmetric lobes [see Figs. 3(a) and 4(a)]. When the observation plane ranges from z 0.1zr to the z 3zr , the sizes of spots of both the y and longitudinal components are enlarged with the elongation in the x direction. Meanwhile, the four lobes of the beam profiles of the y component finally split into eight ones, but the four lobes of beam profiles located in the central area are dominant [see Figs. 3(b) to 3(d) and Figs. 4(b) to 4(d)]. Figure 5 presents contour graphs of the intensity distributions of a Lorentz–Gauss beam in the uniaxial crystal in
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Fig. 7. Contour graph of the intensity of Lorentz–Gauss beams of different waist widths in the observation plane z zr , e 1.5: (a) w0 w0x w0y 0.5λ, (b) w0 1λ, w0x w0y 0.5λ, (c) w0 1λ, w0x 0.5λ, w0y 1.5λ, (d) w0 1λ, w0x 1.5λ, w0y 0.5λ, (e) w0 5λ, w0x 0.5λ, w0y 1.5λ, and (f) w0 5λ, w0x 1.5λ, w0y 0.5λ.
several observation planes. The calculation parameters are the same as those in Fig. 2. Because of jE x j2 > jEz j2 > jE y j2 , the contour graphs of the intensity distributions of a Lorentz–Gauss beam is close to the x component. Additionally, the positions of peak intensity value of the three components do not overlap in the coordinate system, so the peak intensity value of a Lorentz–Gauss beam is not the simple summation of the three components. Figure 6 shows contour graphs of the intensity of a Lorentz– Gauss beam in the observation plane z zr in different uniaxial crystal, i.e., (a) e 0.6, (b) e 0.8, (c) e 1, and (d) e 1.5. Other calculation parameters are the same as those in Fig. 2. It is obvious from Fig. (6) that for the case e < 1, i.e., negative crystal, the beam profile of a Lorentz– Gauss beam is elongated in the y direction, see Figs. 6(a) and 6(b); for the case e > 1, i.e., positive crystal, the beam profile of a Lorentz–Gauss beam is elongated in the x direction [see Fig. 6(d)]. Furthermore, with the increase of the deviation of e from unity, the extension of the beam spot augments. And for the case e 1, i.e., isotropic medium, the beam profile of a Lorentz–Gauss beam is symmetrical, see Fig. 6(c). Figure 7 presents contour graphs of the intensity of Lorentz–Gauss beams of different waist widths—w0 , w0x , and w0y —in the observation plane z zr , where e is set to be 1.5. From Figs. 7(a) and 7(b), we can find that with the increase of w0 , the sizes of the spots enlarged significantly. And due to the anisotropic influence of the positive crystal, the beam profiles extend in the x direction, even though w0x is equal to w0y . From Figs. 7(c), and 7(d), we can find that for little w0 , the contour graph of intensity of the Lorentz–Gauss
beams is insensitive to the ratio of w0x to w0y . However, for large w0 [see Figs. 7(e) and 7(f)], the ratio of w0x to w0y greatly affects the intensity distribution. Therefore the size of the spot increases with w0 , and the shape of the beam profile is related to w0x and w0y .
4. CONCLUSIONS In conclusion, analytical expressions for the three components of nonparaxial propagation of a polarized Lorentz– Gauss beam in uniaxial crystal orthogonal to the optical axis are derived, and the corresponding light intensity distributions have been illustrated with numerical examples. Results show that even though the y component of a Lorentz–Gauss beam in the input plane is zero, the y and longitudinal components arise upon propagation in uniaxial crystal, which cannot be ignored. And the beam waist sizes—w0 , w0x , and w0y —not only affect both the size and shape of the beam profile in uniaxial crystal but also determine the nonparaxial effect of a Lorentz–Gauss beam; even if only one of the three parameters is comparable with the light wavelength, the Lorentz–Gauss beam should be treated beyond the paraxial approximation. Furthermore, the beam profile of a Lorentz– Gauss beam in uniaxial crystal is elongated in the x or y direction, which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index; with increasing the deviation of the ratio from unity, the extension of the beam profile augments. Our results indicate that uniaxial crystal provides an effective and convenient method for modulating the Lorentz–Gauss beams where a light beam with special profile and polarization is required.
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ACKNOWLEDGMENTS This work is supported by the National Science Foundation of China (grant no. 11274273 and 11365009), the Jiangxi Provincial Natural Science Foundation of China (grant no. 20114BAB211014), and the East China Jiaotong University Startup Outlay for Doctor Scientific Research (grant no. 09132007). The author is indebted to the reviewers for their valuable comments.
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