- Email: [email protected]
Focusing of linearly polarized Lorentz–Gauss beam with one optical vortex
Optik 124 (2013) 2969–2973
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Focusing of linearly polarized Lorentz–Gauss beam with one optical vortex Fu Rui a , Dawei Zhang b,c,∗ , Mei Ting d , Xiumin Gao a,b , Songlin Zhuang b a
Electronics and Information College, Hangzhou Dianzi University, Hangzhou 310018, China Engineering Research Center of Optical Instrument and System, Ministry of Education and Shanghai Key Lab of Modern Optical System, University of Shanghai for Science and Technology, No. 516 JunGong Road, Shanghai 200093, China c School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore d Laboratory of Nanophotonic Functional Materials and Devices Institute of Optoelectronic Materials and Technology, South China Normal University, Guangzhou 510631, China b
a r t i c l e
i n f o
Article history: Received 27 April 2012 Accepted 4 September 2012
Keywords: Lorentz–Gaussian beam Focusing properties Optical vortex
a b s t r a c t Focusing properties of linearly polarized Lorentz–Gauss beam with one on-axis optical vortex was investigated by vector diffraction theory. Results show that the focal pattern can be altered considerably by charge number of the optical vortex and the beam parameters. Many novel focal patterns may occur, such as dark hollow focus, multiple-peak focal pattern, and dark hollow focus chain. For certain beam parameter, the focal pattern changes more remarkably in the direction perpendicular to the incident polarized direction than that in polarized direction. Especially, there is one clear relation between the intensity distributions along the direction perpendicular to the polarized direction and the charge number of the optical vortex, which may be used as one measurement method of charge number. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Diode lasers are rapidly advancing recently, and the commercial and industrial use of laser diodes has dramatically increased, such as in fiber communication, optical sensors, and optical measuring instruments [1–8]. The beam shape is very important characteristics in almost all these applications. Hence the availability of diode lasers is determined not only by its maximum output but also by its beam shape [6]. Analytical descriptions for the far field of the emitted light beam of a diode laser, based on Maxwell’s equations, are available in literature. Casey and Pannish have given such expressions that agree well with the measured intensity patterns [2,3]. However, owing to mathematical complexity, these representations are seldom used for actual design calculations. Instead, a simple Gaussian model of the diode laser beam has been used extensively for certain time [4–6]. However, Gaussian approximation does not represent the field distribution perpendicular to the junction of a monomode diode laser. The Lorentzian approximation is valid for a variety of commercially available double heterojunction (DH) Ga1−x Alx As lasers, whose active regions are as narrow as 0.1 m for a typical emission wavelength of ∼0.8 m, furthermore, the divergence of the field normal to the junction is generally so large that the beam is truncated [1]. And a simple mathematical model for the far field of a monodiode laser is employed for easy but fairly accurate computations of the optical field in the focal
∗ Corresponding author. Tel.: +86 21 55272096. E-mail address: [email protected] (D. Zhang). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.09.011
region, in which the amplitude distributions of the fields in the directions parallel and normal to the junction are the Lorentzian and Gaussian functions, respectively [1,9]. Recently, the experimental data from a double heterostructure laser were used to demonstrate that the far field distribution in the direction normal to the junction plane approaches a Lorentzian function, and in the direction parallel to the junction it may be approximated by a Gaussian function [1–3]. Experimental measurements of far field of diode lasers show that the field distribution in the directions normal and parallel to the junction plane agree well with Lorentzian and Gaussian function, respectively [10]. Since then, the Lorentz–Gauss beams have been studied extensively. The vectorial structures of Lorentz–Gauss beams have been examined in the far field [11]. The analytical propagation expressions of Lorentz–Gauss beams beyond the paraxial approximation have been derived [12,13]. The fractional Fourier transform has been applied to treat the propagation of the Lorentz–Gauss beam [14]. The focal shift of a Lorentz–Gauss beam focused by an aperture-lens system has been numerically investigated [15]. Based on the second-order moments, the beam propagation factors of Lorentz–Gauss beams have been investigated [16]. The average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere and propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system were also investigated [17,18]. The propagation of Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis and through an apertured fractional Fourier transform optical system were also studied [19,20]. In addition, Radiation force of highly focused Lorentz–Gauss beams on a Rayleigh particle was also investigated theoretically [21].
2970
F. Rui et al. / Optik 124 (2013) 2969–2973
Fig. 1. Schematic of laser chip and rectangular coordinate system for laser diode.
More attention has been paid to the propagating properties of the Lorentz–Gauss beams. In fact, their focusing characteristics should be noticed and play an important role in many focusing optical systems, especially when diode lasers are used in these systems. In addition, researches on optical vortices have grown rapidly recently because optical vortices have some interesting properties and promising applications [21–26]. For instance, optical vortices can be used to construct highly versatile optical tweezers and arrays vortices can also assemble micro-particles into dynamically optical pumps [27]. However, to the best of our knowledge, the focusing of Lorentz–Gauss beams containing optical vortex has not been studied so far. In order to get deep insight into the properties of Lorentz–Gauss beams and extend their applications, focusing properties of linearly polarized Lorentz–Gaussian beam with one on-axis optical vortex was investigated by vector diffraction theory. The focusing principle is given in Section 2. And Section 3 shows the results and discussions. The conclusions are summarized in Section 4.
radius of optical aperture in focusing system, f is focal length of the focusing system. NA is the numerical aperture of the focusing system, defined as the multiple of refractive index of surrounding medium and sine value of aperture angle, NA = n sin(˛) =
rp f
In the focusing systems, the incident beam is Lorentz–Gauss beam, whose geometric parameters and coordinate system are shown in Fig. 1. The amplitude distributions of the electric field in the directions parallel and normal to the junction are the Lorentzian and Gaussian functions, respectively [1,9]
−
x02
ω02
02
(1)
02 + y02
where parameters ω0 and 0 are the 1/e-width of the Gaussian distribution and the half width of the Lorentzian distribution, respectively [1,9]. In order to calculate the focusing properties easily and clearly, the Eq. (1) can be rewritten as,
2 x0
E0 (x0 , y0 ) = E0 (x0 )E0 (y0 ) = exp −
ω0
1 1 + (y0 /0 )
2
(2)
(6)
n is refractive index of surrounding medium, the system we investigated is in air, so n ≈ 1, ˛ is aperture angle. By similar deviation r0 / 0 can be written as r0 r0 f rp sin() = = 0 NAy f rp 0
(7)
where y = 0 /rp is called relative beam waist in x coordinate direction, and can be called relative Lorentz parameter. Therefore, the electric field distribution can be rewritten as,
2
E0 (, ϕ) = exp −
2. Principle of focusing system
E0 (x0 , y0 ) = E0 (x0 )E0 (y0 ) = exp
Fig. 2. Intensity distributions of Lorentz–Gauss beam under condition of 0 = 0.6, and (a) ω0 = 0.6, (b) ω0 = 0.3, respectively.
cos2 (ϕ)sin ()
NA2 wx2
1 1+
sin2 (ϕ) sin2 () NA2 y2
exp(imϕ)
(8)
It is assumed that the incident Lorentz–Gauss beam is linearly polarized along the x axis. According to vector diffraction theory, the electric field in focal region can be written in the follow form [28,29] ៝ E(,
, z) =
1
2
cos + sin ϕ 1 − cos
x
˝
+ cos ϕ sin ϕ(cos − 1)y + cos ϕ sin z}
× exp −
2
cos2 (ϕ)sin ()
NA2 wx2
× exp −ik sin cos(ϕ −
1 2
2
1 + sin (ϕ) · sin ()⁄NA2 · y2
exp(imϕ)
) exp(−ikz cos ) sin ddϕ
(9)
Now the radial coordinate in z = 0 plane is introduced, therefore, x0 = r0 cos(ϕ),
y0 = r0 sin(ϕ)
(3)
where r0 is the radial coordinate, and ϕ is the azimuthal angle. When the incident Lorentz–Gauss beam contains one on-axis optical vortex, Eq. (1) can be rewritten as
r 2
E0 (r0 , ϕ) = exp − cos2 (ϕ)
1
0
ω0
2
r0 2 exp(imϕ)
1 + sin (ϕ)
0
(4)
where, m is the charge number of the optical vortex, and the r0 /ω0 can be written in the form, r0 r0 f rp sin() = = ω0 NAwx f rp ω0
(5)
where wx = ω0 /rp is called relative beam waist in y coordinate direction and also called as relative Gauss parameter. rp is the outer
where ϕ ∈ [0, 2) , ∈ [0, arcsin(NA)]. Vectors x, y, and z are the unit vectors in the x, y, and z directions, respectively. It is clear that the incident beams is depolarized and has three components (Ei , Ej and Ek ) in x, y, and z directions, respectively. The variables , , and z are the cylindrical coordinates of an observation point in focal region. The optical intensity in focal region is proportional to the modulus square of Eq. (9). 3. Results and discussions It is usually deemed that the Lorentz parameter 0 should be bigger that the Gauss parameter ω0 , which is corresponding to the condition of the laser diode. Fig. 2 illustrates the spot patterns for 0 = 0.6 and two cases of ω0 . It can be seen that the outer contour curves approaches to diamond shape. Now the focusing properties of Lorentz–Gauss beams are investigated. Without loss of validity and generality, it should be noted that the distance unit in all figures
F. Rui et al. / Optik 124 (2013) 2969–2973
2971
Fig. 3. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, and y = 1.2.
is wavelength of incident beam in vacuum. Fig. 3 gives the one focal patterns without any optical vortex under condition of NA = 0.95, ωx = 0.3, and y = 1.2, from which we can see that there is only one focal spot with shorter transverse size along y coordinate direction than that along x coordinate direction. the Now focusing patterns of linearly polarized Lorentz–Gaussian beam with one on-axis optical vortex were calculated by vector diffraction. Figs. 4 and 5 illustrate the focal patterns under condition of NA = 0.95, ωx = 0.3, y = 0.3, and different charge number m. There is one annular focal pattern with one center dark hollow focus for case of m = 1, as shown in Fig. 4(a). Optical tweezers technique has become valuable tools and accelerated many major advances in numerous areas of science since Ashkin and co-workers accomplished optical tweezers experimentally [30–35], and it is usually deemed that the forces exerted on the particle in light field consist of two kinds of forces, one is the optical gradient force, which plays a crucial role in constructing optical trap and its intensity is proportional to the optical intensity gradient; the other kind of force is scattering force, which usually has complex forms because this kind of force is related to the properties of the trapped particles, and whose intensity is proportional to the optical intensity [36]. Dark focal spot refers to those
Fig. 4. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, y = 0.3, and (a) m = 1, (b) m = 1.5, (c) m = 2, and (d) m = 2.5, respectively.
Fig. 5. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, y = 0.3, and (a) m = 3, (b) m = 3.5, (c) m = 4, and (d) m = 4.5, respectively.
focuses whose optical intensity is weaker than that around it and is stable optical trap for those particles whose refractive index is smaller than that of surrounding media, and this condition is very common, especially in life science. Therefore, the dark focus of the Lorentz–Gauss beams with one on-axis optical vortex may be used in optical tweezers technology. On increasing change number, the whole focal pattern changes very considerably. There occur two dark hollow foci along y coordinate under condition of m = 2. And when charge number changes between 1 and 2, the optical energy shifts along y coordinate and the whole focal pattern is asymmetric, as shown in Fig. 4(b). When charge number increases continuously, the focal pattern changes very regularly, the number of dark hollow foci equals to the change number of optical vortex when the charge number is integral number. There occur two big intensity focal spots along x coordinate. In addition, when charge number of optical vortex changes between the contiguous two
Fig. 6. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, y = 1.2, and (a) m = 1, (b) m = 1.5, (c) m = 2, and (d) m = 2.5, respectively.
2972
F. Rui et al. / Optik 124 (2013) 2969–2973
Fig. 7. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, y = 1.2, and (a) m = 3, (b) m = 3.5, (c) m = 4, and (d) m = 4.5, respectively.
Fig. 9. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.7, ωx = 0.3, y = 0.3, and (a) m = 3, (b) m = 3.5, (c) m = 4, and (d) m = 4.5, respectively.
integral numbers, the focal pattern also changes very remarkably. So the focusing of Lorentz–Gauss beams not only can be used to construct tunable optical traps, but also may be used as one method to measure charge number of optical vortex. Figs. 6 and 7 illustrate the focal patterns under condition of NA = 0.95, ωx = 0.3, y = 1.2, and different charge number m, which is more general for commercial laser diode beams, whose output beam is usually wider in the direction perpendicular its junction. Under condition of m = 1, there are two overlapping intensity peaks along y coordinate, as shown in Fig. 6(a). On increasing m, focal pattern changes considerably. There are one center max intensity peak with two small peaks outside along y coordinate direction for case of m = 2, as shown in Fig. 6(c). On increasing charge number, the focal pattern goes on changing, as shown in Fig. 7. When charge number is 3, there is one annular focal pattern, and there are also
three hallow dark hollow foci. And for case of m = 4, one main pattern turns on one discal focal spot containing two dark hollow foci, as shown in Fig. 7(c), in addition, there are also two hallow dark hollow foci, so there are total four dark hollow foci that equals to the charge number. And under condition of non-integral charge number, the whole focal pattern evolves asymmetrically. By comparing focal patterns under condition of different relative Lorentz parameter, we can see that the effect of this parameter on the whole focal pattern is very considerably. And under all these two condition of Lorentz parameters, the focal pattern is related to the charge number, which may be used as a method to measuring the charge number of optical vortex. Now we investigate the effect of numerical aperture on focal pattern. Figs. 8 and 9 illustrate the focal patterns under condition of ωx = 0.3, y = 0.3, NA = 0.7, and different charge number of the
Fig. 8. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.7, ωx = 0.3, y = 0.3, and (a) m = 1, (b) m = 1.5, (c) m = 2, and (d) m = 2.5, respectively.
Fig. 10. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.7, ωx = 0.3, y = 1.2, and (a) m = 1, (b) m = 1.5, (c) m = 2, and (d) m = 2.5, respectively.
F. Rui et al. / Optik 124 (2013) 2969–2973
2973
References
Fig. 11. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.7, ωx = 0.3, y = 1.2, and (a) m = 3, (b) m = 3.5, (c) m = 4, and (d) m = 4.5, respectively.
optical vortex. By comparing the focal patterns under NA = 0.7 with those under NA = 0.95, we can see that focal pattern evolution on increasing charge number is very similar, the whole focal pattern under the same charge are also very similar, the considerable difference is the size of the focal pattern. Figs. 10 and 11 give the focal patterns for case of NA = 0.7, ωx = 0.3, y = 1.2, and different charge number, it was also be seen that the effect of numerical aperture on focal pattern evolution is not remarkable. 4. Conclusions Focusing of linearly polarized Lorentz–Gauss beam containing one optical vortex was investigated numerically by the vector diffraction theory. Results show that the focal pattern can be altered considerably by charge number of the optical vortex and the beam parameters. And many novel focal patterns may occur, such as dark hollow focus, multiple-peak focal pattern, and dark hollow focus chain. More interesting, under certain beam parameters, the focal pattern changes more remarkably in the direction perpendicular to the incident polarized direction than that in polarized direction, and there is one clear relation between the intensity distributions along the direction perpendicular to the polarized direction and the charge number of the optical vortex, namely, the local intensity minimum number equals charge number, which may be developed as one measurement method of charge number. Acknowledgement The authors gratefully acknowledge the financial support from the Singapore National Research Foundation (CRP Award No. NRF-G-CRP 2007-01). And this work was partly supported by the National Natural Science Foundation of China (grant no. 61176085, 60908021), the Department of Education of Guangdong Province, China (grant no. gjhz1103), the Bureau of Science and Information Technology of Guangzhou Municipality (grant no. 2010U1-D00131), and Education Commission of Zhejiang Province of China (Y201120426).
[1] A. Naqwi, F. Durst, Focusing of diode laser beams: a simple mathematical model, Appl. Opt. 29 (1990) 1780–1785. [2] W.P. Dumke, The angular beam divergence in double-heterojunction lasers with very thin active regions, IEEE J. Quantum Electron. 7 (1975) 400–402. [3] H.C. Casey Jr., M.B. Pannish, Heterostructure Lasers, Academic Press, New York, 1978. [4] W.B. Joyce, B.C. Deloach, Alignment of Gaussian Beams, Appl. Opt. 23 (1984) 4187–4196. [5] M. Saruwatari, K. Nawata, Semiconductor laser to single-mode fiber coupler, Appl. Opt. 18 (1979) 1847–1856. [6] D.D. Cook, F.R. Nash, Gain-induced guiding and astigmatic output beam of GaAs laser, J. Appl. Phys. 46 (1975) 1660–1672. [7] T. Ohtomo, S.C. Chu, K. Otsuka, Generation of vortex beams from lasers with controlled Hermite- and Ince-Gaussian modes, Opt. Express 16 (2008) 5082–5094. [8] A. Agnesi, P. Uggetti, Mesurement of thermal diffractive losses in end-pumped solid-state lasers, Opt. Commun. 212 (2002) 371–376. [9] X. Dong, A. Naqwi, Far-field distribution of double-heterostruture diode laser beams, Appl. Opt. 32 (1993) 4491–4494. [10] S. Nemoto, Experimental evaluation of a new expression for the far field of a diode laser beam, Appl. Opt. 33 (1994) 6387–6392. [11] G. Zhou, Analytical vectorial structure of a Lorentz–Gauss beam in the far field, Appl. Phys. B 93 (2008) 891–899. [12] G. Zhou, Nonparaxial propagation of a Lorentz–Gauss beam, J. Opt. Soc. Am. B 26 (2009) 141–147. [13] H. Yu, L. Xiong, B. Lü, Nonparaxial Lorentz and Lorentz–Gauss beams, Optik 121 (2010) 1455–1461. [14] G. Zhou, Fractional Fourier transform of Lorentz–Gauss beams, J. Opt. Soc. Am. A 26 (2009) 350–355. [15] G. Zhou, Focal shift of focused truncated Lorentz–Gauss beam, J. Opt. Soc. Am. A 25 (2008) 2594–2599. [16] G. Zhou, Beam propagation factors of a Lorentz–Gauss beam, Appl. Phys. B 96 (2009) 149–153. [17] G. Zhou, X. Chu, Average intensity and spreading of a Lorentz–Gauss Beam in turbulent atmosphere, Opt. Express 18 (2010) 726–731. [18] G. Zhou, Propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system, Opt. Express 18 (2010) 4637–4643. [19] C. Zhao, Y. Cai, Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis, J. Mod. Opt. 57 (2010) 375–384. [20] W. Du, C. Zhao, Y. Cai, Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system, Opt. Lasers Eng. 49 (2011) 25–31. [21] Y. Jiang, K. Huang, X. Lu, Radiation force of highly focused Lorentz–Gauss beams on a Rayleigh particle, Opt. Express 19 (2011) 9708–9713. [22] D. Rozas, Z.S. Sacks, G.A. Swartzlander Jr., Experimental observation of fluidlike motion of optical vortices, Phys. Rev. Lett. 79 (1997) 3399–3402. [23] W.M. Lee, X.C. Yuan, W.C. Cheong, Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation, Opt. Lett. 29 (2004) 1796–1798. [24] Kosta Ladavac, G. Grier David, Microoptomechanical pumps assembled and driven by holographic optical vortex arrays, Opt. Express 12 (2004) 1144–1149. [25] P.F. Chimento, G.W. Hooft, E.R. Eliel, Plasmonic tomography of optical vortices, Opt. Lett. 35 (2010) 3775–3777. [26] N. Hermosa, A. Aiello, J.P. Woerdman, Quadrant detector calculation for vortex beams, Opt. Lett. 36 (2011) 409–411. [27] D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, E. Di Fabrizio, Laser trapping and micro-manipulation using optical vortices, Microelectron. Eng. 77 (2005) 125–131. [28] M. Gu, Advanced Optical Imaging Theory, Springer, Heidelberg, 2000. [29] D. Ganic, X. Gan, M. Gu, Focusing of doughnut laser beams by a high numericalaperture objective in free space, Opt. Express 11 (2003) 2747–2752. [30] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Observation of a singlebeam gradient force optical trap for dielectric particles, Opt. Lett. 11 (1986) 288–290. [31] A. Ashkin, J.M. Dziedzic, T. Yamane, Optical trapping and manipulation of single cells using infrared laser beams, Nature 330 (1989) 769–771. [32] D.G. Grier, A revolution in optical manipulation, Nature 424 (2003) 810–816. [33] M.P. MacDonald, G.C. Spalding, K. Dholakia, Microfluidic sorting in an optical lattice, Nature 426 (2003) 421–424. [34] V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, Simultaneous micromanipulation in multiple planes using a self reconstructing light beam, Nature 419 (2002) 145–147. [35] L. Paterson, M.P. MacDonald, J. Arlt, W. Sibbett, P.E. Bryant, K. Dholakia, Controlled rotation of optical trapped microscopic particles, Science 292 (2001) 912–914. [36] K. Visscher, G.J. Brakenhoff, Theoretical study of optically induced forces on spherical particles in a single beam trap. I. Rayleigh scatterers, Optik 89 (1992) 174–180.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Focusing of linearly polarized Lorentz–Gauss beam with one optical vortex Fu Rui a , Dawei Zhang b,c,∗ , Mei Ting d , Xiumin Gao a,b , Songlin Zhuang b a
Electronics and Information College, Hangzhou Dianzi University, Hangzhou 310018, China Engineering Research Center of Optical Instrument and System, Ministry of Education and Shanghai Key Lab of Modern Optical System, University of Shanghai for Science and Technology, No. 516 JunGong Road, Shanghai 200093, China c School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore d Laboratory of Nanophotonic Functional Materials and Devices Institute of Optoelectronic Materials and Technology, South China Normal University, Guangzhou 510631, China b
a r t i c l e
i n f o
Article history: Received 27 April 2012 Accepted 4 September 2012
Keywords: Lorentz–Gaussian beam Focusing properties Optical vortex
a b s t r a c t Focusing properties of linearly polarized Lorentz–Gauss beam with one on-axis optical vortex was investigated by vector diffraction theory. Results show that the focal pattern can be altered considerably by charge number of the optical vortex and the beam parameters. Many novel focal patterns may occur, such as dark hollow focus, multiple-peak focal pattern, and dark hollow focus chain. For certain beam parameter, the focal pattern changes more remarkably in the direction perpendicular to the incident polarized direction than that in polarized direction. Especially, there is one clear relation between the intensity distributions along the direction perpendicular to the polarized direction and the charge number of the optical vortex, which may be used as one measurement method of charge number. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Diode lasers are rapidly advancing recently, and the commercial and industrial use of laser diodes has dramatically increased, such as in fiber communication, optical sensors, and optical measuring instruments [1–8]. The beam shape is very important characteristics in almost all these applications. Hence the availability of diode lasers is determined not only by its maximum output but also by its beam shape [6]. Analytical descriptions for the far field of the emitted light beam of a diode laser, based on Maxwell’s equations, are available in literature. Casey and Pannish have given such expressions that agree well with the measured intensity patterns [2,3]. However, owing to mathematical complexity, these representations are seldom used for actual design calculations. Instead, a simple Gaussian model of the diode laser beam has been used extensively for certain time [4–6]. However, Gaussian approximation does not represent the field distribution perpendicular to the junction of a monomode diode laser. The Lorentzian approximation is valid for a variety of commercially available double heterojunction (DH) Ga1−x Alx As lasers, whose active regions are as narrow as 0.1 m for a typical emission wavelength of ∼0.8 m, furthermore, the divergence of the field normal to the junction is generally so large that the beam is truncated [1]. And a simple mathematical model for the far field of a monodiode laser is employed for easy but fairly accurate computations of the optical field in the focal
∗ Corresponding author. Tel.: +86 21 55272096. E-mail address: [email protected] (D. Zhang). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.09.011
region, in which the amplitude distributions of the fields in the directions parallel and normal to the junction are the Lorentzian and Gaussian functions, respectively [1,9]. Recently, the experimental data from a double heterostructure laser were used to demonstrate that the far field distribution in the direction normal to the junction plane approaches a Lorentzian function, and in the direction parallel to the junction it may be approximated by a Gaussian function [1–3]. Experimental measurements of far field of diode lasers show that the field distribution in the directions normal and parallel to the junction plane agree well with Lorentzian and Gaussian function, respectively [10]. Since then, the Lorentz–Gauss beams have been studied extensively. The vectorial structures of Lorentz–Gauss beams have been examined in the far field [11]. The analytical propagation expressions of Lorentz–Gauss beams beyond the paraxial approximation have been derived [12,13]. The fractional Fourier transform has been applied to treat the propagation of the Lorentz–Gauss beam [14]. The focal shift of a Lorentz–Gauss beam focused by an aperture-lens system has been numerically investigated [15]. Based on the second-order moments, the beam propagation factors of Lorentz–Gauss beams have been investigated [16]. The average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere and propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system were also investigated [17,18]. The propagation of Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis and through an apertured fractional Fourier transform optical system were also studied [19,20]. In addition, Radiation force of highly focused Lorentz–Gauss beams on a Rayleigh particle was also investigated theoretically [21].
2970
F. Rui et al. / Optik 124 (2013) 2969–2973
Fig. 1. Schematic of laser chip and rectangular coordinate system for laser diode.
More attention has been paid to the propagating properties of the Lorentz–Gauss beams. In fact, their focusing characteristics should be noticed and play an important role in many focusing optical systems, especially when diode lasers are used in these systems. In addition, researches on optical vortices have grown rapidly recently because optical vortices have some interesting properties and promising applications [21–26]. For instance, optical vortices can be used to construct highly versatile optical tweezers and arrays vortices can also assemble micro-particles into dynamically optical pumps [27]. However, to the best of our knowledge, the focusing of Lorentz–Gauss beams containing optical vortex has not been studied so far. In order to get deep insight into the properties of Lorentz–Gauss beams and extend their applications, focusing properties of linearly polarized Lorentz–Gaussian beam with one on-axis optical vortex was investigated by vector diffraction theory. The focusing principle is given in Section 2. And Section 3 shows the results and discussions. The conclusions are summarized in Section 4.
radius of optical aperture in focusing system, f is focal length of the focusing system. NA is the numerical aperture of the focusing system, defined as the multiple of refractive index of surrounding medium and sine value of aperture angle, NA = n sin(˛) =
rp f
In the focusing systems, the incident beam is Lorentz–Gauss beam, whose geometric parameters and coordinate system are shown in Fig. 1. The amplitude distributions of the electric field in the directions parallel and normal to the junction are the Lorentzian and Gaussian functions, respectively [1,9]
−
x02
ω02
02
(1)
02 + y02
where parameters ω0 and 0 are the 1/e-width of the Gaussian distribution and the half width of the Lorentzian distribution, respectively [1,9]. In order to calculate the focusing properties easily and clearly, the Eq. (1) can be rewritten as,
2 x0
E0 (x0 , y0 ) = E0 (x0 )E0 (y0 ) = exp −
ω0
1 1 + (y0 /0 )
2
(2)
(6)
n is refractive index of surrounding medium, the system we investigated is in air, so n ≈ 1, ˛ is aperture angle. By similar deviation r0 / 0 can be written as r0 r0 f rp sin() = = 0 NAy f rp 0
(7)
where y = 0 /rp is called relative beam waist in x coordinate direction, and can be called relative Lorentz parameter. Therefore, the electric field distribution can be rewritten as,
2
E0 (, ϕ) = exp −
2. Principle of focusing system
E0 (x0 , y0 ) = E0 (x0 )E0 (y0 ) = exp
Fig. 2. Intensity distributions of Lorentz–Gauss beam under condition of 0 = 0.6, and (a) ω0 = 0.6, (b) ω0 = 0.3, respectively.
cos2 (ϕ)sin ()
NA2 wx2
1 1+
sin2 (ϕ) sin2 () NA2 y2
exp(imϕ)
(8)
It is assumed that the incident Lorentz–Gauss beam is linearly polarized along the x axis. According to vector diffraction theory, the electric field in focal region can be written in the follow form [28,29] ៝ E(,
, z) =
1
2
cos + sin ϕ 1 − cos
x
˝
+ cos ϕ sin ϕ(cos − 1)y + cos ϕ sin z}
× exp −
2
cos2 (ϕ)sin ()
NA2 wx2
× exp −ik sin cos(ϕ −
1 2
2
1 + sin (ϕ) · sin ()⁄NA2 · y2
exp(imϕ)
) exp(−ikz cos ) sin ddϕ
(9)
Now the radial coordinate in z = 0 plane is introduced, therefore, x0 = r0 cos(ϕ),
y0 = r0 sin(ϕ)
(3)
where r0 is the radial coordinate, and ϕ is the azimuthal angle. When the incident Lorentz–Gauss beam contains one on-axis optical vortex, Eq. (1) can be rewritten as
r 2
E0 (r0 , ϕ) = exp − cos2 (ϕ)
1
0
ω0
2
r0 2 exp(imϕ)
1 + sin (ϕ)
0
(4)
where, m is the charge number of the optical vortex, and the r0 /ω0 can be written in the form, r0 r0 f rp sin() = = ω0 NAwx f rp ω0
(5)
where wx = ω0 /rp is called relative beam waist in y coordinate direction and also called as relative Gauss parameter. rp is the outer
where ϕ ∈ [0, 2) , ∈ [0, arcsin(NA)]. Vectors x, y, and z are the unit vectors in the x, y, and z directions, respectively. It is clear that the incident beams is depolarized and has three components (Ei , Ej and Ek ) in x, y, and z directions, respectively. The variables , , and z are the cylindrical coordinates of an observation point in focal region. The optical intensity in focal region is proportional to the modulus square of Eq. (9). 3. Results and discussions It is usually deemed that the Lorentz parameter 0 should be bigger that the Gauss parameter ω0 , which is corresponding to the condition of the laser diode. Fig. 2 illustrates the spot patterns for 0 = 0.6 and two cases of ω0 . It can be seen that the outer contour curves approaches to diamond shape. Now the focusing properties of Lorentz–Gauss beams are investigated. Without loss of validity and generality, it should be noted that the distance unit in all figures
F. Rui et al. / Optik 124 (2013) 2969–2973
2971
Fig. 3. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, and y = 1.2.
is wavelength of incident beam in vacuum. Fig. 3 gives the one focal patterns without any optical vortex under condition of NA = 0.95, ωx = 0.3, and y = 1.2, from which we can see that there is only one focal spot with shorter transverse size along y coordinate direction than that along x coordinate direction. the Now focusing patterns of linearly polarized Lorentz–Gaussian beam with one on-axis optical vortex were calculated by vector diffraction. Figs. 4 and 5 illustrate the focal patterns under condition of NA = 0.95, ωx = 0.3, y = 0.3, and different charge number m. There is one annular focal pattern with one center dark hollow focus for case of m = 1, as shown in Fig. 4(a). Optical tweezers technique has become valuable tools and accelerated many major advances in numerous areas of science since Ashkin and co-workers accomplished optical tweezers experimentally [30–35], and it is usually deemed that the forces exerted on the particle in light field consist of two kinds of forces, one is the optical gradient force, which plays a crucial role in constructing optical trap and its intensity is proportional to the optical intensity gradient; the other kind of force is scattering force, which usually has complex forms because this kind of force is related to the properties of the trapped particles, and whose intensity is proportional to the optical intensity [36]. Dark focal spot refers to those
Fig. 4. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, y = 0.3, and (a) m = 1, (b) m = 1.5, (c) m = 2, and (d) m = 2.5, respectively.
Fig. 5. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, y = 0.3, and (a) m = 3, (b) m = 3.5, (c) m = 4, and (d) m = 4.5, respectively.
focuses whose optical intensity is weaker than that around it and is stable optical trap for those particles whose refractive index is smaller than that of surrounding media, and this condition is very common, especially in life science. Therefore, the dark focus of the Lorentz–Gauss beams with one on-axis optical vortex may be used in optical tweezers technology. On increasing change number, the whole focal pattern changes very considerably. There occur two dark hollow foci along y coordinate under condition of m = 2. And when charge number changes between 1 and 2, the optical energy shifts along y coordinate and the whole focal pattern is asymmetric, as shown in Fig. 4(b). When charge number increases continuously, the focal pattern changes very regularly, the number of dark hollow foci equals to the change number of optical vortex when the charge number is integral number. There occur two big intensity focal spots along x coordinate. In addition, when charge number of optical vortex changes between the contiguous two
Fig. 6. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, y = 1.2, and (a) m = 1, (b) m = 1.5, (c) m = 2, and (d) m = 2.5, respectively.
2972
F. Rui et al. / Optik 124 (2013) 2969–2973
Fig. 7. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.95, ωx = 0.3, y = 1.2, and (a) m = 3, (b) m = 3.5, (c) m = 4, and (d) m = 4.5, respectively.
Fig. 9. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.7, ωx = 0.3, y = 0.3, and (a) m = 3, (b) m = 3.5, (c) m = 4, and (d) m = 4.5, respectively.
integral numbers, the focal pattern also changes very remarkably. So the focusing of Lorentz–Gauss beams not only can be used to construct tunable optical traps, but also may be used as one method to measure charge number of optical vortex. Figs. 6 and 7 illustrate the focal patterns under condition of NA = 0.95, ωx = 0.3, y = 1.2, and different charge number m, which is more general for commercial laser diode beams, whose output beam is usually wider in the direction perpendicular its junction. Under condition of m = 1, there are two overlapping intensity peaks along y coordinate, as shown in Fig. 6(a). On increasing m, focal pattern changes considerably. There are one center max intensity peak with two small peaks outside along y coordinate direction for case of m = 2, as shown in Fig. 6(c). On increasing charge number, the focal pattern goes on changing, as shown in Fig. 7. When charge number is 3, there is one annular focal pattern, and there are also
three hallow dark hollow foci. And for case of m = 4, one main pattern turns on one discal focal spot containing two dark hollow foci, as shown in Fig. 7(c), in addition, there are also two hallow dark hollow foci, so there are total four dark hollow foci that equals to the charge number. And under condition of non-integral charge number, the whole focal pattern evolves asymmetrically. By comparing focal patterns under condition of different relative Lorentz parameter, we can see that the effect of this parameter on the whole focal pattern is very considerably. And under all these two condition of Lorentz parameters, the focal pattern is related to the charge number, which may be used as a method to measuring the charge number of optical vortex. Now we investigate the effect of numerical aperture on focal pattern. Figs. 8 and 9 illustrate the focal patterns under condition of ωx = 0.3, y = 0.3, NA = 0.7, and different charge number of the
Fig. 8. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.7, ωx = 0.3, y = 0.3, and (a) m = 1, (b) m = 1.5, (c) m = 2, and (d) m = 2.5, respectively.
Fig. 10. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.7, ωx = 0.3, y = 1.2, and (a) m = 1, (b) m = 1.5, (c) m = 2, and (d) m = 2.5, respectively.
F. Rui et al. / Optik 124 (2013) 2969–2973
2973
References
Fig. 11. Intensity distributions of Lorentz–Gauss beam under condition of NA = 0.7, ωx = 0.3, y = 1.2, and (a) m = 3, (b) m = 3.5, (c) m = 4, and (d) m = 4.5, respectively.
optical vortex. By comparing the focal patterns under NA = 0.7 with those under NA = 0.95, we can see that focal pattern evolution on increasing charge number is very similar, the whole focal pattern under the same charge are also very similar, the considerable difference is the size of the focal pattern. Figs. 10 and 11 give the focal patterns for case of NA = 0.7, ωx = 0.3, y = 1.2, and different charge number, it was also be seen that the effect of numerical aperture on focal pattern evolution is not remarkable. 4. Conclusions Focusing of linearly polarized Lorentz–Gauss beam containing one optical vortex was investigated numerically by the vector diffraction theory. Results show that the focal pattern can be altered considerably by charge number of the optical vortex and the beam parameters. And many novel focal patterns may occur, such as dark hollow focus, multiple-peak focal pattern, and dark hollow focus chain. More interesting, under certain beam parameters, the focal pattern changes more remarkably in the direction perpendicular to the incident polarized direction than that in polarized direction, and there is one clear relation between the intensity distributions along the direction perpendicular to the polarized direction and the charge number of the optical vortex, namely, the local intensity minimum number equals charge number, which may be developed as one measurement method of charge number. Acknowledgement The authors gratefully acknowledge the financial support from the Singapore National Research Foundation (CRP Award No. NRF-G-CRP 2007-01). And this work was partly supported by the National Natural Science Foundation of China (grant no. 61176085, 60908021), the Department of Education of Guangdong Province, China (grant no. gjhz1103), the Bureau of Science and Information Technology of Guangzhou Municipality (grant no. 2010U1-D00131), and Education Commission of Zhejiang Province of China (Y201120426).
[1] A. Naqwi, F. Durst, Focusing of diode laser beams: a simple mathematical model, Appl. Opt. 29 (1990) 1780–1785. [2] W.P. Dumke, The angular beam divergence in double-heterojunction lasers with very thin active regions, IEEE J. Quantum Electron. 7 (1975) 400–402. [3] H.C. Casey Jr., M.B. Pannish, Heterostructure Lasers, Academic Press, New York, 1978. [4] W.B. Joyce, B.C. Deloach, Alignment of Gaussian Beams, Appl. Opt. 23 (1984) 4187–4196. [5] M. Saruwatari, K. Nawata, Semiconductor laser to single-mode fiber coupler, Appl. Opt. 18 (1979) 1847–1856. [6] D.D. Cook, F.R. Nash, Gain-induced guiding and astigmatic output beam of GaAs laser, J. Appl. Phys. 46 (1975) 1660–1672. [7] T. Ohtomo, S.C. Chu, K. Otsuka, Generation of vortex beams from lasers with controlled Hermite- and Ince-Gaussian modes, Opt. Express 16 (2008) 5082–5094. [8] A. Agnesi, P. Uggetti, Mesurement of thermal diffractive losses in end-pumped solid-state lasers, Opt. Commun. 212 (2002) 371–376. [9] X. Dong, A. Naqwi, Far-field distribution of double-heterostruture diode laser beams, Appl. Opt. 32 (1993) 4491–4494. [10] S. Nemoto, Experimental evaluation of a new expression for the far field of a diode laser beam, Appl. Opt. 33 (1994) 6387–6392. [11] G. Zhou, Analytical vectorial structure of a Lorentz–Gauss beam in the far field, Appl. Phys. B 93 (2008) 891–899. [12] G. Zhou, Nonparaxial propagation of a Lorentz–Gauss beam, J. Opt. Soc. Am. B 26 (2009) 141–147. [13] H. Yu, L. Xiong, B. Lü, Nonparaxial Lorentz and Lorentz–Gauss beams, Optik 121 (2010) 1455–1461. [14] G. Zhou, Fractional Fourier transform of Lorentz–Gauss beams, J. Opt. Soc. Am. A 26 (2009) 350–355. [15] G. Zhou, Focal shift of focused truncated Lorentz–Gauss beam, J. Opt. Soc. Am. A 25 (2008) 2594–2599. [16] G. Zhou, Beam propagation factors of a Lorentz–Gauss beam, Appl. Phys. B 96 (2009) 149–153. [17] G. Zhou, X. Chu, Average intensity and spreading of a Lorentz–Gauss Beam in turbulent atmosphere, Opt. Express 18 (2010) 726–731. [18] G. Zhou, Propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system, Opt. Express 18 (2010) 4637–4643. [19] C. Zhao, Y. Cai, Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis, J. Mod. Opt. 57 (2010) 375–384. [20] W. Du, C. Zhao, Y. Cai, Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system, Opt. Lasers Eng. 49 (2011) 25–31. [21] Y. Jiang, K. Huang, X. Lu, Radiation force of highly focused Lorentz–Gauss beams on a Rayleigh particle, Opt. Express 19 (2011) 9708–9713. [22] D. Rozas, Z.S. Sacks, G.A. Swartzlander Jr., Experimental observation of fluidlike motion of optical vortices, Phys. Rev. Lett. 79 (1997) 3399–3402. [23] W.M. Lee, X.C. Yuan, W.C. Cheong, Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation, Opt. Lett. 29 (2004) 1796–1798. [24] Kosta Ladavac, G. Grier David, Microoptomechanical pumps assembled and driven by holographic optical vortex arrays, Opt. Express 12 (2004) 1144–1149. [25] P.F. Chimento, G.W. Hooft, E.R. Eliel, Plasmonic tomography of optical vortices, Opt. Lett. 35 (2010) 3775–3777. [26] N. Hermosa, A. Aiello, J.P. Woerdman, Quadrant detector calculation for vortex beams, Opt. Lett. 36 (2011) 409–411. [27] D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Romamato, E. Di Fabrizio, Laser trapping and micro-manipulation using optical vortices, Microelectron. Eng. 77 (2005) 125–131. [28] M. Gu, Advanced Optical Imaging Theory, Springer, Heidelberg, 2000. [29] D. Ganic, X. Gan, M. Gu, Focusing of doughnut laser beams by a high numericalaperture objective in free space, Opt. Express 11 (2003) 2747–2752. [30] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Observation of a singlebeam gradient force optical trap for dielectric particles, Opt. Lett. 11 (1986) 288–290. [31] A. Ashkin, J.M. Dziedzic, T. Yamane, Optical trapping and manipulation of single cells using infrared laser beams, Nature 330 (1989) 769–771. [32] D.G. Grier, A revolution in optical manipulation, Nature 424 (2003) 810–816. [33] M.P. MacDonald, G.C. Spalding, K. Dholakia, Microfluidic sorting in an optical lattice, Nature 426 (2003) 421–424. [34] V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, Simultaneous micromanipulation in multiple planes using a self reconstructing light beam, Nature 419 (2002) 145–147. [35] L. Paterson, M.P. MacDonald, J. Arlt, W. Sibbett, P.E. Bryant, K. Dholakia, Controlled rotation of optical trapped microscopic particles, Science 292 (2001) 912–914. [36] K. Visscher, G.J. Brakenhoff, Theoretical study of optically induced forces on spherical particles in a single beam trap. I. Rayleigh scatterers, Optik 89 (1992) 174–180.