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Gouy phase shift of lens-generated quasi-nondiffractive beam
Optics Communications 396 (2017) 78–82
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Gouy phase shift of lens-generated quasi-nondiffractive beam a,b,⁎
Chunjie Zhai a b c
, Zhaolou Cao
MARK
c
Department of Forest Fire Protection, Nanjing Forest Police College, Nanjing 210046, China University of Science and Technology of China, State Key Laboratory of Fire Science, Hefei, Anhui 230027, China School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Bessel beam Airy beam Gouy phase Fresnel diffraction
Gouy phase shift (GPS) along geometrical rays of quasi-nondiffractive beam is studied by Fresnel diffraction to understand its phase behavior. Two typical quasi-nondiffractive beams, quasi-Bessel beam generated by axicon and quasi-Airy beam generated by cubic phase mask, were studied. Results show that those two beams have different kinds of GPS. While quasi-Airy beam follows general phenomenon of converging light wave where its phase change across focal line is -π, quasi-Bessel beam has phase change of -π/2 although it is two-dimensional.
1. Introduction Gouy phase shift (GPS), first observed by Gouy in 1892, was described as an abrupt phase change of π when converging wave crosses focal region. It reveals the phase difference of an actual beam from an ideal spherical wave [1], which accounts for several optical phenomena, such as resonant frequencies of transverse modes in laser cavities [2] and lateral trapping force of optical tweezer [3]. It is also well known that when studying the scattering of Gaussian beam by particles with geometrical optical approach, an additional phase change π needs to be added if a ray passes focal line. Many theoretical and experimental efforts have been made to investigate its properties. In Ref. [4], transverse spatial confinement was regarded as the physical origin of GPS, which was theoretically verified by Hermite-Gaussian beams and qualitatively explained that GPS exists in any other converging beams. Lots of work has been conducted so far to get precise and systematic knowledge of GPS in various kinds of beam, such as focusing beam generated by optical system with high numerical aperture where polarization cannot be ignored [5], wavefield affected by geometrical aberration (astigmatic [6] and primary spherical aberration [7]) and partially coherent focused fields [8]. At the same time, experimental work was also being conducted to observe its existence in different systems [9,10]. Here we focus on its effects on nondiffractive beam that has attracted much attention in recent years. Nondiffractive beam has invariant transverse intensity distribution along its propagation. It has been found in broad applications for the advantages of self-healing and long focal depth, such as light sheet microscopy [11,12], optical tweezer [13] and imaging with large depth of field [14]. Two typical nondiffractive beams are Bessel beam with
⁎
straight focal line and Airy beam with curved trajectory. GPS of theoretical nondiffractive beam with analytical expression has already been reported in [15,16]. Nevertheless there has been no answer to the GPS along geometrical rays in nondiffractive beam so far, which is important for studying scattering of nondiffractive beam by particles with geometrical optical method. Since theoretical nondiffractive beam has infinite energy and cannot exist in the real world, only quasinondiffractive beam can be obtained by wavefront shaping in reality. For example, Axicon is usually adopted to get quasi-Bessel beam and cubic phase mask (CPM) together with a lens can generate quasi-Airy beam. In this paper, GPS of two quasi-nondiffractive beams is studied by tracking the phase shift along geometrical rays. A numerical model based on diffraction theory is adopted to quantitatively calculate the phase. The text is organized as follows. Section 2 gives the description of the problem. Section 3 presents the results for quasi-Bessel beam and quasi-Airy beam. Conclusions are drawn in Section 4. 2. Description of problem Many methods have been developed to get quasi-nondiffractive beam, among which wavefront coding has turned out to be an effective way, as given in Fig. 1, where Fig. 1(a) and (b) show the generation of quasi-Bessel beam and quasi-Airy beam, respectively. An axicon with linear phase function in Fig. 1(a) can get a zero-order quasi-Bessel beam while a CPM with cubic phase function in Fig. 1(b) can get a quasi-Airy Beam. Unlike ideal lens where plane wave is focused into one point, the light from different parts of aperture intersects the optical axis at different positions so that it converges into a focal line in
Corresponding author at: Department of Forest Fire Protection, Nanjing Forest Police College, Nanjing 210046, China. E-mail address: [email protected] (C. Zhai).
http://dx.doi.org/10.1016/j.optcom.2017.03.040 Received 19 December 2016; Received in revised form 8 March 2017; Accepted 18 March 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
Optics Communications 396 (2017) 78–82
C. Zhai, Z. Cao
In Fig. 1(b), the phase function depends on the CPM and the lens. For two-dimensional Airy beam, ϕ(x, y) is given by
ϕ (x , y ) = − k
both beams. While zero-order quasi-Bessel beam is axisymmetrical with its focal line lying on the optical axis, quasi-Airy beam has a curved focal line that lies at the edge of the beam. Unlike Bessel beam that is always two-dimensional, Airy beam can be either one-dimensional as in Fig. 1(b) or two-dimensional if another group of cylindrical lens and CPM is added. For theoretical nondiffractive beam, analytical expression is available, which can be used to get the spatial distribution of complex amplitude so that the phase at any positions can be easily obtained. Gouy phase can thus be calculated as in [15,16]. But for quasinondiffractive beam, there is no analytical expression and therefore numerical simulation has to be adopted. Huygens-Fresnel diffraction theory is used in this paper to describe the propagation of monochromatic beam rather than vector diffraction that accounts for polarization effects since the optical system has low relative aperture. The complex amplitude is expressed as
1 iλ
∬
exp[iϕ (x, y)]
S
exp[ikl (x′, y′, x, y)] dxdy l (x′, y′, x , y)
(cos θx , cos θy ) =
(x′ − x )2 + ( y′ − y)2 + z 02
(1)
(5)
GPS(x1, y1, z1, x 0 , y0 , z 0 ) = mod[arg[U (x1, y1, z1)] − arg[U (x 0 , y0 , z 0 )] − k ( y1 − y0)2 + (x1 − x 0 )2 + (z1 − z 0 )2 , 2π ] − π
(6)
Diffracted two-dimensional converging wave are examined to validate the simulation for observing the GPS when the light passes through the focus. The phase retardation function of an ideal lens with
(2)
2π x 2 + y 2
R0=1 mm and f =50 mm is used, given by ϕ(x, y)=− λ 2f , where λ=650 nm. The amplitude of the beam is first calculated by Eq. (1) and the GPS is then evaluated by Eq. (6). Normalized on-axis intensity and GPS of the wave are given in Fig. 3. As expected, the intensity reaches its maximum value at the geometrical focus since no aberration is considered and decreases when out of focus. In agreement with the predictions of Gouy, GPS far from the focus is –π in the twodimensional case. In Ref. [4], it is believed that GPS originates from the transverse spatial confinement. Fig. 3 shows that GPS changes from 0 to –π when the intensity reaches maximum value. Therefore, the additional phase change depends on the focal depth for Gaussian beam.
Illustration of notation in Eq. (1) is shown in Fig. 2. Note that the amplitude distribution at the xy plane in Eq. (1) is assumed to be uniform. In Huygens-Fresnel diffraction theory, S usually denotes exit pupil of a system. The lens aperture plays the role of exit pupil here, since there is no extra stop in this optical layout. Therefore, in the simulation of those two systems in Fig. 1, the lens locates at the xy plane and the phase function ϕ(x, y) is defined according to the specific lens system. In Fig. 1(a), ϕ(x, y) is the phase retardation of an axicon, which linearly depends on the transverse position [17], given by
ϕ (x, y) = kA0 x 2 + y 2
1 ⎡ ∂ϕ (x, y) ∂ϕ (x, y) ⎤ , ⎢ ⎥ k ⎣ ∂x ∂y ⎦
When θx and θy are the angles between the ray and the x-axis and y-axis, respectively. When the complex amplitude of light field is obtained, we are able to get the GPS, which is also known as phase anomaly, defined to be the difference between the actual phase of the wave, arg[U (x′, y′)], and that of a nondiffracted spherical wave [1]. Unlike the previous study where the same ideal wave was employed to get the GPS at every point, here the ideal spherical wave changes for different rays in this paper. Eq. (6) gives the expression of GPS along the geometrical ray from (x0, y0, z0) to (x1, y1, z1). Therefore the origin of virtual ideal spherical wave locates at the ray all the time. Since the GPS value is wrapped between [-π, π), it is added or subtracted by 2π when the discontinuity of GPS due to phase wrapping is observed.
Where x′ and y′ are the x-coordinate and y-coordinate at the observation plane z=z0, S is the area of the aperture, k=2π/λ, λ is the wavelength, ϕ(x,y) denotes the phase function at the xy plane, l(x′,y′,x,y) is the distance, given by
l (x′, y′, x, y) =
(4)
where A0 is related to the cone angle of axicon, A1 denotes the amount of cubic phase, f is the focal length of the lens, R0 is the radius of the aperture. Different from that in Ref. [12]. where a phase mask locates at the front focal plane of a Fourier lens, the phase mask is placed near the lens for convenience in computing the geometrical ray. The position change of phase mask leads to the beam change, but the new beam still shows the curved focal line and nondiffractive beam profile [18], which are the most two important characteristics of Airy beam. When the wavefront determined by phase distribution at the exit pupil is set, the geometrical ray can be regarded to be along the normal direction of the wavefront, given by
Fig. 1. Generation of (a) quasi-Bessel beam by axicon and (b) quasi-Airy beam by cubic phase mask.
U (x′, y′, z 0 ) =
x 2 + y2 x 3 + y3 − kA1 2f R03
(3)
3. Results Here we also track a geometrical ray of quasi-nondiffractive beam
Fig. 3. Normalized on-axis intensity and GPS of diffracted two-dimensional converging wave.
Fig. 2. Illustrating the notation.
79
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C. Zhai, Z. Cao
Fig. 4. Spatial distribution of intensity of quasi-Bessel beam in the (a) yz and (b) xy plane. Where r0 denotes the radial position of the intersection of the ray and exit pupil, θr is the angle between the ray and the z-Axis. (c) Normalized intensity and (d) GPS along the optical axis.
spherical lens. It is because that the axicon steers the beam into a focal line similar to cylindrical lens, as rays with the same radius r0 converge into the same point. Therefore, when tracking a ray focused by an axicon, an additional -π/2 phase change needs to be taken into account when the ray passes the focal line. Note that as geometrical rays defined by Eq. (5) are always oblique with angle of sin−1(A0) between the rays and the z-axis, there is no on-axis geometrical ray and no -π/2 phase change is observed on the on-axis GPS in Fig. 4. Two-dimensional quasi-Airy beam is then studied. The lens parameters are the same as that in Fig. 3. Spatial distributions of intensity in the yz and xy plane are given in Fig. 6(a) and (b), respectively, which show two important properties of Airy beam: parabolic path and nondiffractive beam profile. Due to the asymmetric structure of CPM, the spatial distribution of intensity is no longer symmetric about any axis. The geometrical rays from the exit pupil with (x, y)=(0 mm, 0.6 mm) and (0 mm, −0.6 mm) are illustrated in Fig. 6(a), which have different geometrical foci. On-axis normalized intensity is given in Fig. 6(c). As expected, the increase of A1 leads to extension of focal depth. Compared to quasi-Bessel beam where the intensity linearly depends on z, the on-axis intensity at the focal line is more uniform and smoother for quasi-Airy beam. Fig. 6(d) shows the GPS along the optical axis. Similar to the ideal spherical lens, the phase change is -π/ 2, which starts and ends at the two end points of focal line and much smoother than that of quasi-Bessel beam. Comparison of the three GPSs with different amounts of cubic phase indicates that the increase of focal depth results in a more gradual π/2 phase change. Different from quasi-Bessel beam whose phase change occurs at the geometrical focus, the GPS with A1=2.0λ for quasi-Airy beam shows that the phase change mainly occurs at the two ends while it is almost constant inside the focal line. We then compared the ray along the axis and other oblique rays. A1=0.5λ is employed in the simulation. Normalized intensity is given in Fig. 7. Since the quasi-Airy beam has a curved focal line, the peak intensity at different z positions is not necessarily located at the optical axis. Therefore, the intensity of oblique rays may not be symmetric about the peak intensity as in Figs. 3 and 5. However, when x and y are of different signs, the asymmetry caused by the CPM in the x-direction and y-direction cancels each other out, as shown in the case of (x, y) =(−0.4 mm, 0.4 mm) and (x, y)=(0.4 mm, −0.4 mm) in Fig. 7. The symmetry of phase change of the ray depends on the intensity distribution, as given in Fig. 8, where the decreasing trend is symmetric when x and y are of different signs. The GPS decreases continuously and smoothly inside the focal line and it decreases faster when the corresponding intensity is higher. Although the decreasing speed is different for oblique rays, the phase change is always –π in the far field. Therefore, when tracking a geometrical ray of two-dimensional quasi-
to get the phase change along the propagation. Quasi-Bessel beam is first studied. Axicon with A0 of 0.02 is used in the simulation. Spatial distributions of intensity of the generated beam in the yz plane and xy plane are given in Fig. 4(a) and (b), respectively. As expected, the quasi-Bessel beam is nondiffractive at z < 50 mm while it gradually vanishes at z > 50 mm. In the xy plane, the beam consists of a series of concentric rings. On-axis intensity is given in Fig. 4(c), which has been normalized by the diffraction-limited intensity in Fig. 3. The intensity is linearly dependent on the propagation distance, following energy conservation from geometrical optics. Note that the on-axis GPS cannot be defined as the phase difference of actual wavefield from nondiffracted spherical wavefield as there is no geometrical ray propagating along the optical axis. Therefore, Eq. (6) cannot be adopted to describe the on-axis GPS for quasi-Bessel beam. In Ref. [16], the on-axis GPS is defined to be extra axial phase shift with respect to a reference plane wave propagating along the optical axis with new wave number k′=kcos(θ), where θ is the angle between geometrical ray and optical axis. We follow this definition in the study on on-axis GPS. Fig. 4(d) gives the on-axis GPS. Different from Fig. 3, the absolute average value of GPS is not needed here, since there is no origin for the virtual plane wave. Since there is no π phase change as in Fig. 3, the GPS can be regarded as constant along the axis if the oscillation coming from diffraction is ignored, demonstrating that the on-axis phase can be determined by the specified plane wave, in agreement with the results of theoretical Bessel-beam. GPS along geometrical rays is then calculated. Three rays from r0 =0.4 mm, 0.6 mm and 0.8 mm shown in Fig. 3(a) are studied. The angle θr between the ray and the z-axis is always 0.02, as given by Eq. (5). Normalized intensity along the ray is shown in Fig. 5. Similar to Fig. 3, the intensity is weak before the ray goes into the focal line and reaches the maximum at the geometrical focus which is z=20 mm, 30 mm and 40 mm for r0=0.4 mm, 0.6 mm and 0.8 mm. However, as the focal depth of axicon is much longer than ideal spherical lens, the intensity cannot be ignored outside the region of geometrical focus. Since GPS comes from the interaction of rays of different directions, it experiences complicated oscillation for axicon, as shown in Fig. 5. Before the ray reaches its geometrical focus, the oscillation gradually increases. Regardless of the radius r0 of the ray, a -π/2 phase change occurs when the ray goes through the focus. Although the focal depth is long, the phase change mainly occurs near the geometrical focus. The oscillation becomes much more intensive in the focal line. The discontinuity of GPS results from the sign change of the complex amplitude when the ray passes through the points where the intensity reaches zero and the phase is undefined. When the ray leaves the focal line, the oscillation gradually decreases and GPS stabilizes at -π/2 regardless of the radius of the ray, different from the -π phase change of 80
Optics Communications 396 (2017) 78–82
C. Zhai, Z. Cao
Fig. 5. Normalized intensity and GPS along the ray from r0=(a) 0.4 mm, (b) 0.6 mm and (c) 0.8 mm.
Fig. 6. Spatial distribution of intensity of two-dimensional quasi-Airy beam with A1=λ in the (a) yz and (b) xy plane. Where (x, y) is defined at the plane of exit pupil, (θx, θy) are the angles between the ray and x-axis and y-axis, respectively. (c) Normalized intensity and (d) GPS along the optical axis.
Fig. 7. Normalized intensity for rays of two-dimensional quasi-Airy beam from different parts of aperture.
Fig. 8. GPS for rays of two-dimensional quasi-Airy beam from different parts of aperture.
Airy beam, an additional phase change of –π is required if the ray get out of the focal line. But inside the focal line, the phase change depends on the inclination angle. Finally, the effects of aperture size on the GPS of both quasi-Bessel and quasi-Airy beam are examined. In the simulation, R0 is set to be
2 mm, while other lens parameters are not changed. Since A1 denotes the phase retardation at the lens edge, it is increased by 8 times and set to be 8λ to keep the same phase retardation inside the original aperture as that when R0=1 mm and A1=λ. Results are given in Fig. 9. QuasiBessel beam show little change in the GPS, while the GPS of quasi-Airy 81
Optics Communications 396 (2017) 78–82
C. Zhai, Z. Cao
Fig. 9. Normalized intensity and GPS along the ray of (a) r0=0.6 mm of quasi-Bessel beam, (b) (x, y)=(0 mm, 0 mm) of quasi-Airy beam.
References
beam becomes more oscillating, suggesting that the transverse spatial confinement comes from the rays from the ring of r=0.6 mm in the quasi-Bessel beam and the entire aperture in the quasi-Airy beam. In both cases, the GPS in the far field (far from the focal line) does not change.
[1] M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, Cambridge, 1999. [2] T. Ackemann, W. Grosse-Nobis, G.L. Lippi, The Couy phase shift, the average phase lag of Fourier components of Hermite-Gaussian modes and their application to resonance conditions in optical cavities, Opt. Commun. 189 (2001) 5–14. [3] F. Gittes, C.F. Schmidt, Interference model for back-focal-plane displacement detection in optical tweezers, Opt. Lett. 23 (1998) 7–9. [4] S. Feng, H.G. Winful, Physical origin of the Gouy phase shift, Opt. Lett. 26 (2001) 485–487. [5] X. Pang, Gouy phase and phase singularities of tightly focused, circularly polarized vortex beams, Opt. Commun. 338 (2015) 534–539. [6] T.D. Visser, E. Wolf, The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields, Opt. Commun. 283 (2010) 3371–3375. [7] X. Pang, D.G. Fischer, T.D. Visser, Wavefront spacing and Gouy phase in presence of primary spherical aberration, Opt. Lett. 39 (2014) 88–90. [8] X. Pang, D.G. Fischer, T.D. Visser, Generalized Gouy phase for focused partially coherent light and its implications for interferometry, J. Opt. Soc. Am. A 29 (2012) 989–993. [9] W. Zhu, A. Agrawal, A. Nahata, Direct measurement of the Gouy phase shift for surface plasmon-polaritons, Opt. Express 15 (2007) 9995–10001. [10] H.C. Kandpal, S. Raman, R. Mehrotra, Observation of Gouy phase anomaly with an interferometer, Opt. Lasers Eng. 45 (2007) 249–251. [11] F.O. Fahrbach, P. Simon, A. Rohrbach, Microscopy with self-reconstructing beams, Nat. Photon. 4 (2010) 780–785. [12] Z. Yang, M. Prokopas, J. Nylk, C. Coll-Lladó, F.J. Gunn-Moore, D.E.K. Ferrier, T. Vettenburg, K. Dholakia, A compact Airy beam light sheet microscope with a tilted cylindrical lens, Biomed. Opt. Express 5 (2014) 3434–3442. [13] P. Zhang, J. Prakash, Z. Zhang, M. Mills, N. Efremidis, D. Christodoulides, Z. Chen, Trapping and guiding microparticles with morphing autofocusing airy beams, Opt. Lett. 36 (2011) 2883–2885. [14] W. Chi, N. George, Electronic imaging using a logarithmic asphere, Opt. Lett. 26 (2001) 875–877. [15] X. Pang, G. Gbur, T. Visser, The Gouy phase of airy beams, Opt. Lett. 36 (2011) 2492–2494. [16] P. Martelli, M. Tacca, A. Gatto, G. Moneta, M. Martinelli, Gouy phase shift in nondiffracting Bessel beams, Opt. Express 18 (2010) 7108–7120. [17] J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, S. Bará, Nonparaxial design of generalized axicons, Appl. Opt. 31 (1992) 5326–5330. [18] Z. Cao, C. Zhai, J. Li, F. Xian, S. Pei, Light sheet based on one-dimensional Airy beam generated by single cylindrical lens, Opt. Commun. 393 (2007) 11–16.
4. Conclusion In conclusion, we studied the GPS along a geometrical ray of quasiBessel beam generated by an axicon and quasi-Airy beam generated by CPM by numerical simulation. The phase change of quasi-Bessel beam occurs mainly at the geometrical focus of the ray and oscillates intensively inside the focal line. In the beam's far field, the additional phase change is –π/2, although the beam is two-dimensional. A possible explanation is that the axicon only focuses the ray within the ring of the same radius in the aperture into one point, which is actually one-dimensional like cylindrical lens. While the two beams both have long focal depth, they show different phase behaviors. The phase change of two-dimensional quasi-Airy beam is continuous and smooth throughout the focal line although the speed of phase change depends on the inclination angle of the ray that determines the intensity distribution. In the far field, the additional phase change is –π, the same as that of Gaussian beam. Therefore, when geometrical optics is used to approximate the phase of nondiffractive beam, additional phase change has to be taken into account if the ray passes the focal line.
Acknowledgments This work was supported by the Natural Science Foundation of Jiangsu (BK20150929), Startup Foundation for Introducing Talent of NUIST (S8113099001) and National Natural Science Foundation of China (61605081).
82
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Gouy phase shift of lens-generated quasi-nondiffractive beam a,b,⁎
Chunjie Zhai a b c
, Zhaolou Cao
MARK
c
Department of Forest Fire Protection, Nanjing Forest Police College, Nanjing 210046, China University of Science and Technology of China, State Key Laboratory of Fire Science, Hefei, Anhui 230027, China School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Bessel beam Airy beam Gouy phase Fresnel diffraction
Gouy phase shift (GPS) along geometrical rays of quasi-nondiffractive beam is studied by Fresnel diffraction to understand its phase behavior. Two typical quasi-nondiffractive beams, quasi-Bessel beam generated by axicon and quasi-Airy beam generated by cubic phase mask, were studied. Results show that those two beams have different kinds of GPS. While quasi-Airy beam follows general phenomenon of converging light wave where its phase change across focal line is -π, quasi-Bessel beam has phase change of -π/2 although it is two-dimensional.
1. Introduction Gouy phase shift (GPS), first observed by Gouy in 1892, was described as an abrupt phase change of π when converging wave crosses focal region. It reveals the phase difference of an actual beam from an ideal spherical wave [1], which accounts for several optical phenomena, such as resonant frequencies of transverse modes in laser cavities [2] and lateral trapping force of optical tweezer [3]. It is also well known that when studying the scattering of Gaussian beam by particles with geometrical optical approach, an additional phase change π needs to be added if a ray passes focal line. Many theoretical and experimental efforts have been made to investigate its properties. In Ref. [4], transverse spatial confinement was regarded as the physical origin of GPS, which was theoretically verified by Hermite-Gaussian beams and qualitatively explained that GPS exists in any other converging beams. Lots of work has been conducted so far to get precise and systematic knowledge of GPS in various kinds of beam, such as focusing beam generated by optical system with high numerical aperture where polarization cannot be ignored [5], wavefield affected by geometrical aberration (astigmatic [6] and primary spherical aberration [7]) and partially coherent focused fields [8]. At the same time, experimental work was also being conducted to observe its existence in different systems [9,10]. Here we focus on its effects on nondiffractive beam that has attracted much attention in recent years. Nondiffractive beam has invariant transverse intensity distribution along its propagation. It has been found in broad applications for the advantages of self-healing and long focal depth, such as light sheet microscopy [11,12], optical tweezer [13] and imaging with large depth of field [14]. Two typical nondiffractive beams are Bessel beam with
⁎
straight focal line and Airy beam with curved trajectory. GPS of theoretical nondiffractive beam with analytical expression has already been reported in [15,16]. Nevertheless there has been no answer to the GPS along geometrical rays in nondiffractive beam so far, which is important for studying scattering of nondiffractive beam by particles with geometrical optical method. Since theoretical nondiffractive beam has infinite energy and cannot exist in the real world, only quasinondiffractive beam can be obtained by wavefront shaping in reality. For example, Axicon is usually adopted to get quasi-Bessel beam and cubic phase mask (CPM) together with a lens can generate quasi-Airy beam. In this paper, GPS of two quasi-nondiffractive beams is studied by tracking the phase shift along geometrical rays. A numerical model based on diffraction theory is adopted to quantitatively calculate the phase. The text is organized as follows. Section 2 gives the description of the problem. Section 3 presents the results for quasi-Bessel beam and quasi-Airy beam. Conclusions are drawn in Section 4. 2. Description of problem Many methods have been developed to get quasi-nondiffractive beam, among which wavefront coding has turned out to be an effective way, as given in Fig. 1, where Fig. 1(a) and (b) show the generation of quasi-Bessel beam and quasi-Airy beam, respectively. An axicon with linear phase function in Fig. 1(a) can get a zero-order quasi-Bessel beam while a CPM with cubic phase function in Fig. 1(b) can get a quasi-Airy Beam. Unlike ideal lens where plane wave is focused into one point, the light from different parts of aperture intersects the optical axis at different positions so that it converges into a focal line in
Corresponding author at: Department of Forest Fire Protection, Nanjing Forest Police College, Nanjing 210046, China. E-mail address: [email protected] (C. Zhai).
http://dx.doi.org/10.1016/j.optcom.2017.03.040 Received 19 December 2016; Received in revised form 8 March 2017; Accepted 18 March 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
Optics Communications 396 (2017) 78–82
C. Zhai, Z. Cao
In Fig. 1(b), the phase function depends on the CPM and the lens. For two-dimensional Airy beam, ϕ(x, y) is given by
ϕ (x , y ) = − k
both beams. While zero-order quasi-Bessel beam is axisymmetrical with its focal line lying on the optical axis, quasi-Airy beam has a curved focal line that lies at the edge of the beam. Unlike Bessel beam that is always two-dimensional, Airy beam can be either one-dimensional as in Fig. 1(b) or two-dimensional if another group of cylindrical lens and CPM is added. For theoretical nondiffractive beam, analytical expression is available, which can be used to get the spatial distribution of complex amplitude so that the phase at any positions can be easily obtained. Gouy phase can thus be calculated as in [15,16]. But for quasinondiffractive beam, there is no analytical expression and therefore numerical simulation has to be adopted. Huygens-Fresnel diffraction theory is used in this paper to describe the propagation of monochromatic beam rather than vector diffraction that accounts for polarization effects since the optical system has low relative aperture. The complex amplitude is expressed as
1 iλ
∬
exp[iϕ (x, y)]
S
exp[ikl (x′, y′, x, y)] dxdy l (x′, y′, x , y)
(cos θx , cos θy ) =
(x′ − x )2 + ( y′ − y)2 + z 02
(1)
(5)
GPS(x1, y1, z1, x 0 , y0 , z 0 ) = mod[arg[U (x1, y1, z1)] − arg[U (x 0 , y0 , z 0 )] − k ( y1 − y0)2 + (x1 − x 0 )2 + (z1 − z 0 )2 , 2π ] − π
(6)
Diffracted two-dimensional converging wave are examined to validate the simulation for observing the GPS when the light passes through the focus. The phase retardation function of an ideal lens with
(2)
2π x 2 + y 2
R0=1 mm and f =50 mm is used, given by ϕ(x, y)=− λ 2f , where λ=650 nm. The amplitude of the beam is first calculated by Eq. (1) and the GPS is then evaluated by Eq. (6). Normalized on-axis intensity and GPS of the wave are given in Fig. 3. As expected, the intensity reaches its maximum value at the geometrical focus since no aberration is considered and decreases when out of focus. In agreement with the predictions of Gouy, GPS far from the focus is –π in the twodimensional case. In Ref. [4], it is believed that GPS originates from the transverse spatial confinement. Fig. 3 shows that GPS changes from 0 to –π when the intensity reaches maximum value. Therefore, the additional phase change depends on the focal depth for Gaussian beam.
Illustration of notation in Eq. (1) is shown in Fig. 2. Note that the amplitude distribution at the xy plane in Eq. (1) is assumed to be uniform. In Huygens-Fresnel diffraction theory, S usually denotes exit pupil of a system. The lens aperture plays the role of exit pupil here, since there is no extra stop in this optical layout. Therefore, in the simulation of those two systems in Fig. 1, the lens locates at the xy plane and the phase function ϕ(x, y) is defined according to the specific lens system. In Fig. 1(a), ϕ(x, y) is the phase retardation of an axicon, which linearly depends on the transverse position [17], given by
ϕ (x, y) = kA0 x 2 + y 2
1 ⎡ ∂ϕ (x, y) ∂ϕ (x, y) ⎤ , ⎢ ⎥ k ⎣ ∂x ∂y ⎦
When θx and θy are the angles between the ray and the x-axis and y-axis, respectively. When the complex amplitude of light field is obtained, we are able to get the GPS, which is also known as phase anomaly, defined to be the difference between the actual phase of the wave, arg[U (x′, y′)], and that of a nondiffracted spherical wave [1]. Unlike the previous study where the same ideal wave was employed to get the GPS at every point, here the ideal spherical wave changes for different rays in this paper. Eq. (6) gives the expression of GPS along the geometrical ray from (x0, y0, z0) to (x1, y1, z1). Therefore the origin of virtual ideal spherical wave locates at the ray all the time. Since the GPS value is wrapped between [-π, π), it is added or subtracted by 2π when the discontinuity of GPS due to phase wrapping is observed.
Where x′ and y′ are the x-coordinate and y-coordinate at the observation plane z=z0, S is the area of the aperture, k=2π/λ, λ is the wavelength, ϕ(x,y) denotes the phase function at the xy plane, l(x′,y′,x,y) is the distance, given by
l (x′, y′, x, y) =
(4)
where A0 is related to the cone angle of axicon, A1 denotes the amount of cubic phase, f is the focal length of the lens, R0 is the radius of the aperture. Different from that in Ref. [12]. where a phase mask locates at the front focal plane of a Fourier lens, the phase mask is placed near the lens for convenience in computing the geometrical ray. The position change of phase mask leads to the beam change, but the new beam still shows the curved focal line and nondiffractive beam profile [18], which are the most two important characteristics of Airy beam. When the wavefront determined by phase distribution at the exit pupil is set, the geometrical ray can be regarded to be along the normal direction of the wavefront, given by
Fig. 1. Generation of (a) quasi-Bessel beam by axicon and (b) quasi-Airy beam by cubic phase mask.
U (x′, y′, z 0 ) =
x 2 + y2 x 3 + y3 − kA1 2f R03
(3)
3. Results Here we also track a geometrical ray of quasi-nondiffractive beam
Fig. 3. Normalized on-axis intensity and GPS of diffracted two-dimensional converging wave.
Fig. 2. Illustrating the notation.
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Fig. 4. Spatial distribution of intensity of quasi-Bessel beam in the (a) yz and (b) xy plane. Where r0 denotes the radial position of the intersection of the ray and exit pupil, θr is the angle between the ray and the z-Axis. (c) Normalized intensity and (d) GPS along the optical axis.
spherical lens. It is because that the axicon steers the beam into a focal line similar to cylindrical lens, as rays with the same radius r0 converge into the same point. Therefore, when tracking a ray focused by an axicon, an additional -π/2 phase change needs to be taken into account when the ray passes the focal line. Note that as geometrical rays defined by Eq. (5) are always oblique with angle of sin−1(A0) between the rays and the z-axis, there is no on-axis geometrical ray and no -π/2 phase change is observed on the on-axis GPS in Fig. 4. Two-dimensional quasi-Airy beam is then studied. The lens parameters are the same as that in Fig. 3. Spatial distributions of intensity in the yz and xy plane are given in Fig. 6(a) and (b), respectively, which show two important properties of Airy beam: parabolic path and nondiffractive beam profile. Due to the asymmetric structure of CPM, the spatial distribution of intensity is no longer symmetric about any axis. The geometrical rays from the exit pupil with (x, y)=(0 mm, 0.6 mm) and (0 mm, −0.6 mm) are illustrated in Fig. 6(a), which have different geometrical foci. On-axis normalized intensity is given in Fig. 6(c). As expected, the increase of A1 leads to extension of focal depth. Compared to quasi-Bessel beam where the intensity linearly depends on z, the on-axis intensity at the focal line is more uniform and smoother for quasi-Airy beam. Fig. 6(d) shows the GPS along the optical axis. Similar to the ideal spherical lens, the phase change is -π/ 2, which starts and ends at the two end points of focal line and much smoother than that of quasi-Bessel beam. Comparison of the three GPSs with different amounts of cubic phase indicates that the increase of focal depth results in a more gradual π/2 phase change. Different from quasi-Bessel beam whose phase change occurs at the geometrical focus, the GPS with A1=2.0λ for quasi-Airy beam shows that the phase change mainly occurs at the two ends while it is almost constant inside the focal line. We then compared the ray along the axis and other oblique rays. A1=0.5λ is employed in the simulation. Normalized intensity is given in Fig. 7. Since the quasi-Airy beam has a curved focal line, the peak intensity at different z positions is not necessarily located at the optical axis. Therefore, the intensity of oblique rays may not be symmetric about the peak intensity as in Figs. 3 and 5. However, when x and y are of different signs, the asymmetry caused by the CPM in the x-direction and y-direction cancels each other out, as shown in the case of (x, y) =(−0.4 mm, 0.4 mm) and (x, y)=(0.4 mm, −0.4 mm) in Fig. 7. The symmetry of phase change of the ray depends on the intensity distribution, as given in Fig. 8, where the decreasing trend is symmetric when x and y are of different signs. The GPS decreases continuously and smoothly inside the focal line and it decreases faster when the corresponding intensity is higher. Although the decreasing speed is different for oblique rays, the phase change is always –π in the far field. Therefore, when tracking a geometrical ray of two-dimensional quasi-
to get the phase change along the propagation. Quasi-Bessel beam is first studied. Axicon with A0 of 0.02 is used in the simulation. Spatial distributions of intensity of the generated beam in the yz plane and xy plane are given in Fig. 4(a) and (b), respectively. As expected, the quasi-Bessel beam is nondiffractive at z < 50 mm while it gradually vanishes at z > 50 mm. In the xy plane, the beam consists of a series of concentric rings. On-axis intensity is given in Fig. 4(c), which has been normalized by the diffraction-limited intensity in Fig. 3. The intensity is linearly dependent on the propagation distance, following energy conservation from geometrical optics. Note that the on-axis GPS cannot be defined as the phase difference of actual wavefield from nondiffracted spherical wavefield as there is no geometrical ray propagating along the optical axis. Therefore, Eq. (6) cannot be adopted to describe the on-axis GPS for quasi-Bessel beam. In Ref. [16], the on-axis GPS is defined to be extra axial phase shift with respect to a reference plane wave propagating along the optical axis with new wave number k′=kcos(θ), where θ is the angle between geometrical ray and optical axis. We follow this definition in the study on on-axis GPS. Fig. 4(d) gives the on-axis GPS. Different from Fig. 3, the absolute average value of GPS is not needed here, since there is no origin for the virtual plane wave. Since there is no π phase change as in Fig. 3, the GPS can be regarded as constant along the axis if the oscillation coming from diffraction is ignored, demonstrating that the on-axis phase can be determined by the specified plane wave, in agreement with the results of theoretical Bessel-beam. GPS along geometrical rays is then calculated. Three rays from r0 =0.4 mm, 0.6 mm and 0.8 mm shown in Fig. 3(a) are studied. The angle θr between the ray and the z-axis is always 0.02, as given by Eq. (5). Normalized intensity along the ray is shown in Fig. 5. Similar to Fig. 3, the intensity is weak before the ray goes into the focal line and reaches the maximum at the geometrical focus which is z=20 mm, 30 mm and 40 mm for r0=0.4 mm, 0.6 mm and 0.8 mm. However, as the focal depth of axicon is much longer than ideal spherical lens, the intensity cannot be ignored outside the region of geometrical focus. Since GPS comes from the interaction of rays of different directions, it experiences complicated oscillation for axicon, as shown in Fig. 5. Before the ray reaches its geometrical focus, the oscillation gradually increases. Regardless of the radius r0 of the ray, a -π/2 phase change occurs when the ray goes through the focus. Although the focal depth is long, the phase change mainly occurs near the geometrical focus. The oscillation becomes much more intensive in the focal line. The discontinuity of GPS results from the sign change of the complex amplitude when the ray passes through the points where the intensity reaches zero and the phase is undefined. When the ray leaves the focal line, the oscillation gradually decreases and GPS stabilizes at -π/2 regardless of the radius of the ray, different from the -π phase change of 80
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Fig. 5. Normalized intensity and GPS along the ray from r0=(a) 0.4 mm, (b) 0.6 mm and (c) 0.8 mm.
Fig. 6. Spatial distribution of intensity of two-dimensional quasi-Airy beam with A1=λ in the (a) yz and (b) xy plane. Where (x, y) is defined at the plane of exit pupil, (θx, θy) are the angles between the ray and x-axis and y-axis, respectively. (c) Normalized intensity and (d) GPS along the optical axis.
Fig. 7. Normalized intensity for rays of two-dimensional quasi-Airy beam from different parts of aperture.
Fig. 8. GPS for rays of two-dimensional quasi-Airy beam from different parts of aperture.
Airy beam, an additional phase change of –π is required if the ray get out of the focal line. But inside the focal line, the phase change depends on the inclination angle. Finally, the effects of aperture size on the GPS of both quasi-Bessel and quasi-Airy beam are examined. In the simulation, R0 is set to be
2 mm, while other lens parameters are not changed. Since A1 denotes the phase retardation at the lens edge, it is increased by 8 times and set to be 8λ to keep the same phase retardation inside the original aperture as that when R0=1 mm and A1=λ. Results are given in Fig. 9. QuasiBessel beam show little change in the GPS, while the GPS of quasi-Airy 81
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Fig. 9. Normalized intensity and GPS along the ray of (a) r0=0.6 mm of quasi-Bessel beam, (b) (x, y)=(0 mm, 0 mm) of quasi-Airy beam.
References
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4. Conclusion In conclusion, we studied the GPS along a geometrical ray of quasiBessel beam generated by an axicon and quasi-Airy beam generated by CPM by numerical simulation. The phase change of quasi-Bessel beam occurs mainly at the geometrical focus of the ray and oscillates intensively inside the focal line. In the beam's far field, the additional phase change is –π/2, although the beam is two-dimensional. A possible explanation is that the axicon only focuses the ray within the ring of the same radius in the aperture into one point, which is actually one-dimensional like cylindrical lens. While the two beams both have long focal depth, they show different phase behaviors. The phase change of two-dimensional quasi-Airy beam is continuous and smooth throughout the focal line although the speed of phase change depends on the inclination angle of the ray that determines the intensity distribution. In the far field, the additional phase change is –π, the same as that of Gaussian beam. Therefore, when geometrical optics is used to approximate the phase of nondiffractive beam, additional phase change has to be taken into account if the ray passes the focal line.
Acknowledgments This work was supported by the Natural Science Foundation of Jiangsu (BK20150929), Startup Foundation for Introducing Talent of NUIST (S8113099001) and National Natural Science Foundation of China (61605081).
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