October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
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Diffraction of orbital angular momentum carrying optical beams by a circular aperture A. Ambuj, R. Vyas, and S. Singh* Physics Department, University of Arkansas, Arkansas 72701, USA *Corresponding author: [email protected] Received May 8, 2014; revised July 15, 2014; accepted August 18, 2014; posted August 20, 2014 (Doc. ID 211639); published September 16, 2014 Far field diffraction of Laguerre–Gauss vortex (LGV) beams of different angular momentum index by a circular aperture placed at different locations with respect to incident beam waist is studied experimentally. The experiments reveal a surprisingly simple structure for the diffraction pattern and its dependence on the orbital angular momentum index of the incident beam when the aperture size is small compared to the beam radius. © 2014 Optical Society of America OCIS codes: (050.1940) Diffraction; (080.4865) Optical vortices; (260.6042) Singular optics. http://dx.doi.org/10.1364/OL.39.005475
The Airy intensity pattern produced in the far field when a plane wave is scattered by a circular aperture is a classic example of wave diffraction. [1] It is a common occurrence whenever a wave field encounters an obstacle, and numerous examples of wave diffraction can be found in textbooks on optics. [2] Most discussions of wave diffraction assume the incident field to be a plane wave or fundamental Gaussian wave as these are the fields that naturally arise in applications, for example, in astronomy or microscopy, using laser light illumination. More recently, it has become possible to produce wave fields with more complex phase profiles [3–6], and their use in achieving novel capabilities in nano-particle manipulation, microscopy, and astronomy has been demonstrated [7–12]. Of particular interest are the socalled Laguerre–Gauss vortex (LGV) beams. These beams are a subclass of Laguerre–Gauss (LG) beams, which are solutions to the paraxial scalar wave equation in circular cylindrical coordinates [13,14]. The LGV beams are characterized by a helicoidal phase front with a phase singularity (field null) at the center of the beam and are referred to as optical vortex beams. Diffraction and interference of LGV beams have been studied out to explore their phase structure [15–22]. Diffraction of LGV beams with simple apertures exhibit unusual features [15–22], which depart from the usual, textbook description. We note that the diffraction of plane waves by spiral phase plates [23, 24], despite some similarities to the problem addressed in this manuscript, leads to quite different analytical description and experimental profiles. In this Letter we wish to describe several new features that appear in the diffraction of LGV beams by a circular aperture. We find that the diffraction pattern can be described in terms of Bessel functions of order that depends on the angular momentum index of the incident LGV beam. We also investigate the effects of phase front curvature and present an analytical expression for the diffracted field, which describes the experimentally recorded diffraction patterns well. We begin with an analytical description of the diffraction of LGV beams at a circular aperture. This is followed by the description of the experimental set up to observe these features. The Letter concludes with a discussion of the experimental results and their comparison with the theoretical results. 0146-9592/14/195475-04$15.00/0
Consider a monochromatic field propagating in the z direction incident normally on a two-dimensional aperture A occupying the z 0 plane. Then the diffracted field pattern in the far zone (Fraunhofer diffraction) is proportional to the Fourier transform of the incident field in the plane of the aperture limited by the aperture. This same field pattern is obtained in the back focal plane of a lens (of focal length f ), when the aperture is placed in its front focal plane [25]. With this arrangement, referred to as the 2f -geometry, the diffracted field in the back focal plane of the lens is given by ik U f x; y 2πf
ZZ
ik
A
0
0
dAU in x0 ; y0 ; 0e− f xx yy ;
(1)
where U in x0 ; y0 ; 0 represents the spatial part of the incident wave function in the aperture plane z 0, k 2π∕λ is the wavenumber, λ being the wavelength of light, x0 ; y0 are the transverse coordinates of a point in the aperture plane, and x; y are the transverse coordinates of a point in the back focal plane of the lens. For an incident LGV beam (zero radial index) the incident field is of the form U l in x; y; z
p jlj 2ρ const × eilφikz−zo wz ρ2 × exp − e−iθz − ijlj 1θz ; wzwo (2)
p where ρ x2 y2 is the radial distance of a point from the beam axis, zR πw20 ∕λ is the Rayleigh range, beam q spot radius wz ≡ wo 1 z − zo 2 ∕z2R has its minimum
value wo at the beam waist z zo , and θz tan−1 z − zo ∕zR is the Guoy’s phase of the beam. p Using polar coordinates ρ0 x02 y02 , φ0 tan−1 y0 ∕x0 and similar relations for the unprimed coordinates, carrying out the angular integration and introducing a change of variable u ρ0 ∕a for the remaining radial integral, the diffracted field of Eq. (1) can be written as
© 2014 Optical Society of America
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Fig. 1. Side view of the experimental setup. Lens L1 forms a new waist of the incident beam. The aperture lies in the front focal plane and the imaging plane of the CCD camera in the back focal plane of lens L2 . The apparatus in the dashed box is translated as a unit along the beam axis to place the aperture at the desired location relative to the waist of the incident beam.
p jlj 2a −i eilφ−ikzo w0 Z1 a2 u2 −iθ0 kaρu jlj1 ; × duu exp − e J jlj w0wo f 0 (3)
2 U l f ρ;φ const×2πa
where J jlj kρau∕f is the Bessel function of the first kind of order l. Note that w0 and θ0 are, respectively, the spot size and Guoy’s phase of the incident beam in the aperture plane. In general, Eq. (3) must be evaluated numerically. For the special case θ0 0, which corresponds to incident beam waist at the aperture, the integral can be evaluated as an infinite series of Bessel functions [15]. However, if the incident beam spot radius is large compared to the aperture radius a, the variation of the u2 term in the exponent is small over the aperture. Then, since the integrand peaks away from the center, the leading contribution to the field in the back focal plane of the lens can be evaluated as [26] U l f ρ; φ
−
≈ Cle
a2 −iθ0 w0wo e
J jlj1 kaρ∕f : kaρ∕f
(4)
Here all nonessential factors (including the φ-dependence) have been absorbed into the constant C l . The diffracted field thus takes a simple form proportional to a Bessel function of order one more than the angular momentum index of the incident LGV beam for small a∕wo . Equation (4) can be considered as the leading term of an asymptotic series representing the integral in Eq. (3). It is valid both for LGV beams with planar or curved phase fronts and is found to be a good approximation to Eq. (3) for a∕wo < 0.2 [see Fig. 2 in the experimental section]. For the special case of the aperture at beam waist w0 wo ; θ0 0, it coincides with the leading term of the infinite series given in Ref. [15] with an intensity 2 2 distribution ∝ e−2a ∕wo J jlj1 kaρ∕f ∕kaρ∕f 2 , whereas for the aperture one Rayleigh range from the incident beam waist, it leads to an intensity distribution ∝ 2 2 e−a ∕wo J jlj1 kaρ∕f ∕kaρ∕f 2 and for the aperture in the far zone to an intensity distribution ∝ J jlj1 kaρ∕f ∕kaρ∕f 2 . Note that for l 0, we recover the result for the fundamental Gaussian beam. Using the small argument expansion of the Bessel function
in Eq. (4) or Eq. (3), we can see that the center of the diffraction pattern is always dark for all LGV beams. The apparatus for generating LGV beams has been described elsewhere [17]. Briefly, a He:Ne laser operating at 633 nm, incorporating two intra-cavity orthogonal fibers intersecting the beam axis, was used to generate Hermite–Gauss (HG) beams. By translating the fibers transverse to the beam axis, we were able to produce HG beams of different order in a controlled fashion. These HG beams were transformed into LG beams by using an astigmatic mode converter [5]. The LGV beam from the mode converter is focused using a lens L1 to produce a waist of desired beam spot size wo (Fig. 1). A circular aperture of radius a 250 μm, centered on the beam axis, is placed in the waist plane of the beam, such that the LGV beams is incident normally on it. Since the intensity profile of the incident LGV beam consists of single bright ring (intensity maxip mum), whose radius wl w0 l 1 in the aperture plane grows with l [27], the power transmitted by the aperture varied significantly from one l to another. To compensate for this effect, experiments for different l were carried out by keeping the ratio a∕wl constant. This was achieved by choosing the focal length and location of lens L1 for different l appropriately so that the ratio a∕wl had the same value. Note that this means that the ratio a∕w is not the same for different values of l in the experiment. The diffracted light was collected by the lens L2 placed one focal length f behind the aperture and was recorded by a CCD camera located in the back focal plane of L2 . Since the aperture is located in the front focal plane of the lens and the CCD camera in the back focal plane, the image captured by the camera is the Fourier transform of the incident field limited by the aperture. The diffraction of LGV beams from the circular aperture produced concentric rings around a dark center. The experimental values for the diffraction intensities were extracted by scanning the recorded images radially outward starting from the center of the central dark spot as the origin. To find the center of the diffraction pattern, the images were imported into Mathematica. A smoothing transformation was applied to the image data before using an algorithm to find the local minima. It was found that 180 pixels ≈1 mm [22]. To compare experimentally recorded intensity profiles with the theoretical calculations, one of the theoretical intensities was scaled to match the corresponding experimental intensity. No other fitting parameters were used. The top frame in Fig. 2 shows an example of experimental data (black dots) scanned from a diffraction image recorded by the CCD (inset) super-imposed on the theoretical curves (continuous and dashed lines). The dark line in the inset shows the direction of scan to obtain the experimental data in all images recorded by the CCD. The outer rings were typically much fainter than the innermost ring. To record the fainter outer rings, the intensity in the central region was deliberately allowed to saturate the camera’s sensors (the solid dashed line in the experimental graphs). Figure 2 shows the diffraction patterns for the LGV beams with l 1–4 with the input beam waist at the aperture (zo 0). The continuous 2 curve is the theoretical intensity (∝ jU l f j ) computed
October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
Fig. 2. Comparison of experimental (dots) and theoretical intensity profiles in the diffraction of LGV beams by a circular aperture for l 1–4 and a∕wl 0.28 with the input beam waist at the aperture. The continuous curve is the exact result derived from Eq. (3), and the dashed curve is derived from Eq. (4). The center of the beam in the inset in the top frame remains dark even though the central region appears overexposed to make faint outer rings visible.
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from Eq. 3 and the dots are the intensity values extracted from the radial scan of the diffraction patterns recorded by the CCD. The dashed curve represents the intensity derived from Eq. (4). Note that the ratio a∕wl 0.20 in Fig. 2 corresponds to a∕w 0.28 and 0.34, respectively, for l 1 and 2. Even for these relatively large values of a∕w, the theoretical curves obtained by using Eq. (4) are nearly indistinguishable from the exact curves in Fig. 2. For even larger values of a∕w, the difference between the predictions of Eqs. (3) and (4) is readily seen in frames for l 3 and 4. Note that the diffractionintensity profiles have the same general appearance of a dark core surrounded by concentric rings for all l, but they differ from one another in that the radii of dark rings in the diffraction of an LGV beam of order l correspond to the zeros of Bessel function J jlj1 . Experimentally recorded diffraction-intensity profiles are in good agreement with the theoretical profiles for l-values (l 1, 2, 3, and 4) accessible in this experiment. For larger values of l, the intensity distribution in the central dark region is very sensitive to the quality of the incident LGV beam, and this is reflected in the deviations seen in the graph for l 4. Even in this case, however, the intensity distribution for outer rings is well accounted for by Eq. (4). We also carried out another set of experiments, where for each l, the ratio a∕wl was varied over a decade, approximately, from 0.2 to 2, to investigate the effect of increasing aperture size on diffraction. As the ratio a∕wl increases, the diffraction pattern deviates increasingly from the predictions of Eq. (4). The first column in Fig. 3 shows diffraction intensity profiles for a∕wl 0.5 with the aperture at beam waist (zo 0) for l 1 (with a∕w 0.71) and l 3 (with a∕w 1.0). The full curves are derived from Eq. (3). A comparison of these intensity profiles with those in Fig. 2, shows that as a∕wl increases, the outer diffraction rings become less prominent, and the minima no longer correspond to the zeros of Bessel functions although their locations are still close to them. This behavior is in agreement with the expectation that as the aperture size increases compared to the beam size, diffraction effects will diminish. We also explored the effect of phase front curvature [22]. Recall that the diffraction intensity profiles of Fig. 2 correspond to the aperture at the incident beam waist, where the incident phase fronts are planar. As the aperture is moved away from the incident beam waist, phase front curvature at the aperture increases, reaching a maximum when the aperture is one Rayleigh range from the aperture (zo −zR ), so that the phase front radius of curvature at the aperture has its minimum value Rmin 2zR . From Eq. (3), we see that phase front curvature is not a significant factor as long as the ratio a∕w is small. Indeed, for small values of a∕wl , experimentally recorded diffraction intensity profiles, with the aperture one Rayleigh range away from the waist, were nearly the same as those obtained with the aperture at the beam waist. However, p for larger values of a∕wl (and, therefore, a∕w l 1a∕wl , larger still), the minima in intensity pattern begin to fill in (Fig. 3, right column). This effect of increasing aperture size relative to beam spot radius is similar to that seen with the aperture at the
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Fig. 3. Diffraction patterns for a∕wl 1.4 for l 1 and l 3 with the aperture (i) at the beam waist (left column) and (ii) one Rayleigh range away from the beam waist (right column). The continuous theoretical curves are derived from Eq. (3).
waist (left column in Fig. 3) except that the effect is more pronounced in the presence of phase front curvature (right column in Fig. 3). Finally, in agreement with Eq. (3) [and Eq. (4)], we found that the diffraction intensity pattern does not depend on the sign of the phase front curvature or the sign of the angular momentum index l. In conclusion we have studied the diffraction of orbital angular momentum carrying LGV beams from a circular aperture. The diffraction of LGV beams by a circular aperture is very different from the well known Airy diffraction pattern encountered in the diffraction of a plane wave [1]. The diffraction pattern for all nonzero l values is characterized by concentric rings around a dark center. For aperture size small compared to the incident beam spot size, diffraction intensity pattern has a simple structure proportional to J jlj1 kaρ∕f ∕kaρ∕f 2 . The minima of the diffraction pattern of LGV beam of order l are determined by the zeros of Bessel function J jlj1 . Our experiments reveal that the diffraction of LGV beams by a circular aperture has a surprisingly simple structure when compared to their diffraction by slits and other apertures [15–22]. Our observations not only expand our understanding of the properties of orbital angular momentum carrying LGV beams but also, combined with the well-known Airy pattern for the plane wave (or the fundamental Gaussian beam), present a beautifully complete picture of diffraction of LG beams with zero radial index by a circular aperture.
References 1. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), Chap. 8. 2. F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957), Chaps. 15–18. 3. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987). 4. M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993). 5. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chavez-Cerda, Opt. Lett. 29, 44 (2004) 6. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett. 29, 144 (2004). 7. R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003). 8. Q. Zhan, Opt. Express 12, 3377 (2004). 9. G. A. Swartzlander, Jr., E. L. Ford, R. S. Abdul-Malik, L. M. Close, M. A. Peters, D. M. Palacios, and D. W. Wilson, Opt. Express 16, 10200 (2008). 10. H. He, M. E. J. Friese, N. R. Heckenberg, and H. RubinszteinDunlop, Phys. Rev. Lett. 75, 826 (1995). 11. V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, Phys. Rev. Lett. 91, 093602 (2003). 12. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pasko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004). 13. H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966). 14. A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986), Chap 16. 15. G. Lenz, Opt. Commun. 123, 423 (1996). 16. H. Sztul and R. Alfano, Opt. Lett. 31, 999 (2006). 17. J. Vickers, M. Burch, R. Vyas, and S. Singh, J. Opt. Soc. Am. A 25, 823 (2008). 18. D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 47, 123 (2009). 19. A. Kumar, P. Vaity, and R. P. Singh, Opt. Commun. 283, 4141 (2010). 20. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010). 21. Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, Opt. Lett. 36, 3106 (2011). 22. A. Ambuj, H. Shiau, M. Lucini, R. Vyas, and S. Singh, J. Mod. Opt. 59, 1232 (2012). 23. V. V. Kotlyar, S. N. Khonina, A. A. Kovalev, V. A. Soifer, H. Elfstrom, and J. Turunen, Opt. Lett. 31, 1597 (2006). 24. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, J. Opt. Soc. Am. A 22, 849 (2005). 25. J. W. Goodman, Introduction to Fourier Optics (McGrawHill, 1996), Chap. 5 26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1965) p. 683. 27. R. Phillips and L. Andrews, Appl. Opt. 22, 643 (1983).
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Diffraction of orbital angular momentum carrying optical beams by a circular aperture A. Ambuj, R. Vyas, and S. Singh* Physics Department, University of Arkansas, Arkansas 72701, USA *Corresponding author: [email protected] Received May 8, 2014; revised July 15, 2014; accepted August 18, 2014; posted August 20, 2014 (Doc. ID 211639); published September 16, 2014 Far field diffraction of Laguerre–Gauss vortex (LGV) beams of different angular momentum index by a circular aperture placed at different locations with respect to incident beam waist is studied experimentally. The experiments reveal a surprisingly simple structure for the diffraction pattern and its dependence on the orbital angular momentum index of the incident beam when the aperture size is small compared to the beam radius. © 2014 Optical Society of America OCIS codes: (050.1940) Diffraction; (080.4865) Optical vortices; (260.6042) Singular optics. http://dx.doi.org/10.1364/OL.39.005475
The Airy intensity pattern produced in the far field when a plane wave is scattered by a circular aperture is a classic example of wave diffraction. [1] It is a common occurrence whenever a wave field encounters an obstacle, and numerous examples of wave diffraction can be found in textbooks on optics. [2] Most discussions of wave diffraction assume the incident field to be a plane wave or fundamental Gaussian wave as these are the fields that naturally arise in applications, for example, in astronomy or microscopy, using laser light illumination. More recently, it has become possible to produce wave fields with more complex phase profiles [3–6], and their use in achieving novel capabilities in nano-particle manipulation, microscopy, and astronomy has been demonstrated [7–12]. Of particular interest are the socalled Laguerre–Gauss vortex (LGV) beams. These beams are a subclass of Laguerre–Gauss (LG) beams, which are solutions to the paraxial scalar wave equation in circular cylindrical coordinates [13,14]. The LGV beams are characterized by a helicoidal phase front with a phase singularity (field null) at the center of the beam and are referred to as optical vortex beams. Diffraction and interference of LGV beams have been studied out to explore their phase structure [15–22]. Diffraction of LGV beams with simple apertures exhibit unusual features [15–22], which depart from the usual, textbook description. We note that the diffraction of plane waves by spiral phase plates [23, 24], despite some similarities to the problem addressed in this manuscript, leads to quite different analytical description and experimental profiles. In this Letter we wish to describe several new features that appear in the diffraction of LGV beams by a circular aperture. We find that the diffraction pattern can be described in terms of Bessel functions of order that depends on the angular momentum index of the incident LGV beam. We also investigate the effects of phase front curvature and present an analytical expression for the diffracted field, which describes the experimentally recorded diffraction patterns well. We begin with an analytical description of the diffraction of LGV beams at a circular aperture. This is followed by the description of the experimental set up to observe these features. The Letter concludes with a discussion of the experimental results and their comparison with the theoretical results. 0146-9592/14/195475-04$15.00/0
Consider a monochromatic field propagating in the z direction incident normally on a two-dimensional aperture A occupying the z 0 plane. Then the diffracted field pattern in the far zone (Fraunhofer diffraction) is proportional to the Fourier transform of the incident field in the plane of the aperture limited by the aperture. This same field pattern is obtained in the back focal plane of a lens (of focal length f ), when the aperture is placed in its front focal plane [25]. With this arrangement, referred to as the 2f -geometry, the diffracted field in the back focal plane of the lens is given by ik U f x; y 2πf
ZZ
ik
A
0
0
dAU in x0 ; y0 ; 0e− f xx yy ;
(1)
where U in x0 ; y0 ; 0 represents the spatial part of the incident wave function in the aperture plane z 0, k 2π∕λ is the wavenumber, λ being the wavelength of light, x0 ; y0 are the transverse coordinates of a point in the aperture plane, and x; y are the transverse coordinates of a point in the back focal plane of the lens. For an incident LGV beam (zero radial index) the incident field is of the form U l in x; y; z
p jlj 2ρ const × eilφikz−zo wz ρ2 × exp − e−iθz − ijlj 1θz ; wzwo (2)
p where ρ x2 y2 is the radial distance of a point from the beam axis, zR πw20 ∕λ is the Rayleigh range, beam q spot radius wz ≡ wo 1 z − zo 2 ∕z2R has its minimum
value wo at the beam waist z zo , and θz tan−1 z − zo ∕zR is the Guoy’s phase of the beam. p Using polar coordinates ρ0 x02 y02 , φ0 tan−1 y0 ∕x0 and similar relations for the unprimed coordinates, carrying out the angular integration and introducing a change of variable u ρ0 ∕a for the remaining radial integral, the diffracted field of Eq. (1) can be written as
© 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014
Fig. 1. Side view of the experimental setup. Lens L1 forms a new waist of the incident beam. The aperture lies in the front focal plane and the imaging plane of the CCD camera in the back focal plane of lens L2 . The apparatus in the dashed box is translated as a unit along the beam axis to place the aperture at the desired location relative to the waist of the incident beam.
p jlj 2a −i eilφ−ikzo w0 Z1 a2 u2 −iθ0 kaρu jlj1 ; × duu exp − e J jlj w0wo f 0 (3)
2 U l f ρ;φ const×2πa
where J jlj kρau∕f is the Bessel function of the first kind of order l. Note that w0 and θ0 are, respectively, the spot size and Guoy’s phase of the incident beam in the aperture plane. In general, Eq. (3) must be evaluated numerically. For the special case θ0 0, which corresponds to incident beam waist at the aperture, the integral can be evaluated as an infinite series of Bessel functions [15]. However, if the incident beam spot radius is large compared to the aperture radius a, the variation of the u2 term in the exponent is small over the aperture. Then, since the integrand peaks away from the center, the leading contribution to the field in the back focal plane of the lens can be evaluated as [26] U l f ρ; φ
−
≈ Cle
a2 −iθ0 w0wo e
J jlj1 kaρ∕f : kaρ∕f
(4)
Here all nonessential factors (including the φ-dependence) have been absorbed into the constant C l . The diffracted field thus takes a simple form proportional to a Bessel function of order one more than the angular momentum index of the incident LGV beam for small a∕wo . Equation (4) can be considered as the leading term of an asymptotic series representing the integral in Eq. (3). It is valid both for LGV beams with planar or curved phase fronts and is found to be a good approximation to Eq. (3) for a∕wo < 0.2 [see Fig. 2 in the experimental section]. For the special case of the aperture at beam waist w0 wo ; θ0 0, it coincides with the leading term of the infinite series given in Ref. [15] with an intensity 2 2 distribution ∝ e−2a ∕wo J jlj1 kaρ∕f ∕kaρ∕f 2 , whereas for the aperture one Rayleigh range from the incident beam waist, it leads to an intensity distribution ∝ 2 2 e−a ∕wo J jlj1 kaρ∕f ∕kaρ∕f 2 and for the aperture in the far zone to an intensity distribution ∝ J jlj1 kaρ∕f ∕kaρ∕f 2 . Note that for l 0, we recover the result for the fundamental Gaussian beam. Using the small argument expansion of the Bessel function
in Eq. (4) or Eq. (3), we can see that the center of the diffraction pattern is always dark for all LGV beams. The apparatus for generating LGV beams has been described elsewhere [17]. Briefly, a He:Ne laser operating at 633 nm, incorporating two intra-cavity orthogonal fibers intersecting the beam axis, was used to generate Hermite–Gauss (HG) beams. By translating the fibers transverse to the beam axis, we were able to produce HG beams of different order in a controlled fashion. These HG beams were transformed into LG beams by using an astigmatic mode converter [5]. The LGV beam from the mode converter is focused using a lens L1 to produce a waist of desired beam spot size wo (Fig. 1). A circular aperture of radius a 250 μm, centered on the beam axis, is placed in the waist plane of the beam, such that the LGV beams is incident normally on it. Since the intensity profile of the incident LGV beam consists of single bright ring (intensity maxip mum), whose radius wl w0 l 1 in the aperture plane grows with l [27], the power transmitted by the aperture varied significantly from one l to another. To compensate for this effect, experiments for different l were carried out by keeping the ratio a∕wl constant. This was achieved by choosing the focal length and location of lens L1 for different l appropriately so that the ratio a∕wl had the same value. Note that this means that the ratio a∕w is not the same for different values of l in the experiment. The diffracted light was collected by the lens L2 placed one focal length f behind the aperture and was recorded by a CCD camera located in the back focal plane of L2 . Since the aperture is located in the front focal plane of the lens and the CCD camera in the back focal plane, the image captured by the camera is the Fourier transform of the incident field limited by the aperture. The diffraction of LGV beams from the circular aperture produced concentric rings around a dark center. The experimental values for the diffraction intensities were extracted by scanning the recorded images radially outward starting from the center of the central dark spot as the origin. To find the center of the diffraction pattern, the images were imported into Mathematica. A smoothing transformation was applied to the image data before using an algorithm to find the local minima. It was found that 180 pixels ≈1 mm [22]. To compare experimentally recorded intensity profiles with the theoretical calculations, one of the theoretical intensities was scaled to match the corresponding experimental intensity. No other fitting parameters were used. The top frame in Fig. 2 shows an example of experimental data (black dots) scanned from a diffraction image recorded by the CCD (inset) super-imposed on the theoretical curves (continuous and dashed lines). The dark line in the inset shows the direction of scan to obtain the experimental data in all images recorded by the CCD. The outer rings were typically much fainter than the innermost ring. To record the fainter outer rings, the intensity in the central region was deliberately allowed to saturate the camera’s sensors (the solid dashed line in the experimental graphs). Figure 2 shows the diffraction patterns for the LGV beams with l 1–4 with the input beam waist at the aperture (zo 0). The continuous 2 curve is the theoretical intensity (∝ jU l f j ) computed
October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
Fig. 2. Comparison of experimental (dots) and theoretical intensity profiles in the diffraction of LGV beams by a circular aperture for l 1–4 and a∕wl 0.28 with the input beam waist at the aperture. The continuous curve is the exact result derived from Eq. (3), and the dashed curve is derived from Eq. (4). The center of the beam in the inset in the top frame remains dark even though the central region appears overexposed to make faint outer rings visible.
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from Eq. 3 and the dots are the intensity values extracted from the radial scan of the diffraction patterns recorded by the CCD. The dashed curve represents the intensity derived from Eq. (4). Note that the ratio a∕wl 0.20 in Fig. 2 corresponds to a∕w 0.28 and 0.34, respectively, for l 1 and 2. Even for these relatively large values of a∕w, the theoretical curves obtained by using Eq. (4) are nearly indistinguishable from the exact curves in Fig. 2. For even larger values of a∕w, the difference between the predictions of Eqs. (3) and (4) is readily seen in frames for l 3 and 4. Note that the diffractionintensity profiles have the same general appearance of a dark core surrounded by concentric rings for all l, but they differ from one another in that the radii of dark rings in the diffraction of an LGV beam of order l correspond to the zeros of Bessel function J jlj1 . Experimentally recorded diffraction-intensity profiles are in good agreement with the theoretical profiles for l-values (l 1, 2, 3, and 4) accessible in this experiment. For larger values of l, the intensity distribution in the central dark region is very sensitive to the quality of the incident LGV beam, and this is reflected in the deviations seen in the graph for l 4. Even in this case, however, the intensity distribution for outer rings is well accounted for by Eq. (4). We also carried out another set of experiments, where for each l, the ratio a∕wl was varied over a decade, approximately, from 0.2 to 2, to investigate the effect of increasing aperture size on diffraction. As the ratio a∕wl increases, the diffraction pattern deviates increasingly from the predictions of Eq. (4). The first column in Fig. 3 shows diffraction intensity profiles for a∕wl 0.5 with the aperture at beam waist (zo 0) for l 1 (with a∕w 0.71) and l 3 (with a∕w 1.0). The full curves are derived from Eq. (3). A comparison of these intensity profiles with those in Fig. 2, shows that as a∕wl increases, the outer diffraction rings become less prominent, and the minima no longer correspond to the zeros of Bessel functions although their locations are still close to them. This behavior is in agreement with the expectation that as the aperture size increases compared to the beam size, diffraction effects will diminish. We also explored the effect of phase front curvature [22]. Recall that the diffraction intensity profiles of Fig. 2 correspond to the aperture at the incident beam waist, where the incident phase fronts are planar. As the aperture is moved away from the incident beam waist, phase front curvature at the aperture increases, reaching a maximum when the aperture is one Rayleigh range from the aperture (zo −zR ), so that the phase front radius of curvature at the aperture has its minimum value Rmin 2zR . From Eq. (3), we see that phase front curvature is not a significant factor as long as the ratio a∕w is small. Indeed, for small values of a∕wl , experimentally recorded diffraction intensity profiles, with the aperture one Rayleigh range away from the waist, were nearly the same as those obtained with the aperture at the beam waist. However, p for larger values of a∕wl (and, therefore, a∕w l 1a∕wl , larger still), the minima in intensity pattern begin to fill in (Fig. 3, right column). This effect of increasing aperture size relative to beam spot radius is similar to that seen with the aperture at the
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Fig. 3. Diffraction patterns for a∕wl 1.4 for l 1 and l 3 with the aperture (i) at the beam waist (left column) and (ii) one Rayleigh range away from the beam waist (right column). The continuous theoretical curves are derived from Eq. (3).
waist (left column in Fig. 3) except that the effect is more pronounced in the presence of phase front curvature (right column in Fig. 3). Finally, in agreement with Eq. (3) [and Eq. (4)], we found that the diffraction intensity pattern does not depend on the sign of the phase front curvature or the sign of the angular momentum index l. In conclusion we have studied the diffraction of orbital angular momentum carrying LGV beams from a circular aperture. The diffraction of LGV beams by a circular aperture is very different from the well known Airy diffraction pattern encountered in the diffraction of a plane wave [1]. The diffraction pattern for all nonzero l values is characterized by concentric rings around a dark center. For aperture size small compared to the incident beam spot size, diffraction intensity pattern has a simple structure proportional to J jlj1 kaρ∕f ∕kaρ∕f 2 . The minima of the diffraction pattern of LGV beam of order l are determined by the zeros of Bessel function J jlj1 . Our experiments reveal that the diffraction of LGV beams by a circular aperture has a surprisingly simple structure when compared to their diffraction by slits and other apertures [15–22]. Our observations not only expand our understanding of the properties of orbital angular momentum carrying LGV beams but also, combined with the well-known Airy pattern for the plane wave (or the fundamental Gaussian beam), present a beautifully complete picture of diffraction of LG beams with zero radial index by a circular aperture.
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