December 1, 2014 / Vol. 39, No. 23 / OPTICS LETTERS
6569
Focusing by a high numerical aperture lens of distributions generated by conical diffraction Sybille Rosset,1,2 Clement Fallet,1,* and Gabriel Y. Sirat1 1
Bioaxial, Parisbiotech Sante, 24 rue du Faubourg Saint-Jacques, 75014 Paris, France 2 Ecole Polytechnique, 91128 Palaiseau Cedex, France *Corresponding author: [email protected]
Received September 12, 2014; revised October 13, 2014; accepted October 15, 2014; posted October 17, 2014 (Doc. ID 222744); published November 17, 2014 This Letter examines the diffraction of a vortex beam by a high numerical aperture of topological charge 1 generated by conical diffraction. Our research shows the behavior of the vortex beam is similar to the one already observed for Laguerre–Gauss and Bessel beams. We also highlight a similarity between the phase singularity created by a lens and the one created by conical diffraction through a thin crystal. More precisely, if the input beam is homogeneously polarized, we show the electric field in the image plane of a thin crystal caught between two orthogonal polarizers has the same expression as Ez , the component of the electric field along the propagation axis, in the focal plane of a lens. © 2014 Optical Society of America OCIS codes: (080.4865) Optical vortices; (260.1960) Diffraction theory; (260.5430) Polarization; (260.1180) Crystal optics; (260.6042) Singular optics. http://dx.doi.org/10.1364/OL.39.006569
Vortex beams have attracted considerable interest in super-resolution microscopy [1–4] among other fields. The vortex beam needs to have a neat zero of intensity at its center. For super-resolution applications in particular, the vortex must keep its zero after diffraction by a high numerical aperture (NA) lens. However, when a beam is focused through a high NA lens, singular effects take place and the vectorial diffraction theory must be used to determine the intensity in the focal plane. The focusing of a vortex beam by a high NA lens has already been studied in the case where the vortex beam is a Laguerre–Gauss or a Bessel beam [5–7]. In this Letter, we first show the phenomenon is the same for the slightly different vortex beam created by conical diffraction through a thin crystal, and then study its link with conical diffraction. Conical diffraction is the name given to the diffraction phenomenon that occurs when a beam propagates along an optic axis of a biaxial transparent crystal slab. According to [8], the incident beam is transformed by the crystal into two beams of different orbital angular momentum (OAM) and orthogonal polarizations; one beam is linked to the incident beam by a Bessel function of zero order and remains of the same OAM, while the OAM of the other output beam is different from the OAM of the input beam, differing from ℏ. It is possible to create optical vortex beams by isolating this last component [9]. More precisely, if an input beam right-circularly polarized, after having gone through the biaxial crystal passes through a left circular analyzer, the output beam is a left-circularly polarized vortex beam of topological charge 1. In the same way, an input beam left-circularly polarized may emerge as a right-circularly polarized vortex beam of topological charge −1. The amplitude distribution of these beams is nonhomogeneous, depending on the distance r from the center of the vortex, and differs from the amplitude distribution of the Laguerre–Gauss beams usually used. Richards and Wolf obtained an integral representation for the electromagnetic field in the image space of an optical system that applied beyond the usual scope of low 0146-9592/14/236569-04$15.00/0
angular systems [10]. This integral representation was expressed as a three-dimensional Fourier transform and computed using chirp z-transform (to avoid the use of zero padding and heavy costs in terms of memory and computational time) in [11], following an initial idea of McCutchen [12,13]. We used this reformulation to compute the focused electrical field. The computation was carried out for NA 0.95 and for a left-circularly polarized vortex beam of topological charge 1 generated by conical diffraction through a crystal of parameter ρ0 0.5 (where ρ0 is the parameter characterizing the width of the beam and of the crystal, as defined in [8]). The maximal intensity was scaled to 1 and the Cartesian components of the electrical field were scaled accordingly; the distance between both maxima was arbitrarily scaled to 15 pixels (equivalent to a 12 zero-padding in Fourier space, thus pixel size λ∕24). We noticed the focused vortex beam keeps its neat zero of intensity at its center, just as neatly as vortex phase beams (beams generated with a vortex phase plate, with an homogeneous amplitude and a helical phase) or Bessel beams left-circularly polarized and of topological charge 1 (Fig. 1). Phase vortices enable a better intensity confinement than Bessel beams. The vortex beam generated with conical diffraction can be regarded as an intermediate situation between these two extremes. Comparing the gradient of the distributions near the singularity, we observed the sharpest gradient is obtained for the Bessel beam (with a gradient of 0.157), while the lowest one is obtained for the vortex phase beam (with a gradient of 0.147). The vortex created using conical diffraction performs somewhere between both with a gradient of 0.149, thus possessing a slightly sharper zero of intensity than a focused vortex phase beam and slightly less sharp than a focused Bessel beam. This difference is however negligible. The same trend is observed for other polarizations of the beam. Furthermore, we observed the same phenomenon already observed for Laguerre–Gauss and Bessel beams. When focused, the vortex generated by conical diffraction of a Gaussian beam (and thus circularly polarized) © 2014 Optical Society of America
6570
OPTICS LETTERS / Vol. 39, No. 23 / December 1, 2014
0
Efm1
1 Zα 0 p @ A 0 dθaθ cos θ sin2 θ; π 0 E x iE y
(2)
where α sin−1 NA, NA being the numerical aperture of the lens. We observed that the intensity at the focal point is zero only if E y iE x ; in other words, if the vortex is left-circularly polarized. A similar reasoning gives us the electrical field Efm1 at the focal point for a topological charge of −1: 0
Fig. 1. Intensity in the focal plane along the x axis for incident vortex beams left-circularly polarized of topological charge 1, of various amplitude distributions. The plain line shows the intensity computed for a vortex phase beam, the dashed line for a Bessel beam, and the stars for a vortex beam generated by conical diffraction through a crystal of ρ0 0.5. The computations were performed for a lens of NA 0.95.
remains a vortex with a neat zero of intensity at its center, while for other kinds of polarizations, focusing a vortex beam through a high NA lens may altogether change its form, resulting in a focused intensity whose peak lies in the middle of the focal plane. This prompted us to analyze how polarization affects the zero of intensity of a vortex beam in the focal plane for a vortex beam whose phase distribution is in eimΦ (m 1 is the topological charge of the vortex) and whose amplitude distribution is circularly symmetric. It has already been investigated for a Laguerre– Gauss beam and a Bessel beam but, to the best of our knowledge, has never been done for other types of amplitude distribution. First, we verified that the conically diffracted Gaussian beam confirms these criteria. We considered an input beam left-circularly polarized, that passes after having gone through the biaxial crystal through a right circular analyzer. According to [9], the electric field after the analyzer is: 1 ; ER; Z B1 R; R0 ; Zcos θ − i sin θ −i
(1)
where R0 is a constant that only depends on the crystal used, Z describes the propagation distance, and R describes the distance to the propagation axis. The amplitude of the vortex beam is B1 R; R0 ; Z, that only depends on R and is thus circularly symmetric. In the same way, for an input beam right-circularly polarized that passes through a left circular analyzer, the amplitude of the vortex beam is also B1 R; R0 ; Z and so is circularly symmetric. Let Ei be a beam that verify these conditions. Using the same notations as in [10], we then have Ei θ; Φ eimΦ aθEExy . Since the amplitude of the field depended only on the θ parameter, we applied the results shown in [5]. The electrical field Efm1 at the focal point is given, for m 1, by:
Efm−1
1 Zα 0 p Aπ 0 @ dθaθ cos θ sin2 θ: 0 E x − iE y
(3)
Here, the intensity at the focal point is zero only if E y −iE x ; that is, if the vortex is right-circularly polarized. The presence of a zero of intensity at the focal point thus depends heavily on the polarization of the vortex beam, while it does not depend at all on the profile of the amplitude of the field in θ. As long as the field is circularly symmetric, if it is not circularly polarized with the right handedness, the intensity at the focal point will differ from zero, and the further away from the right state of polarization, the stronger the intensity. It also can be expressed in terms of orbital angular momentum (OAM) and spin angular momentum (SAM) of the beam. A vortex beam possesses an OAM of mℏ, m being the topological charge. A right-circularly polarized beam has a SAM of −ℏ, a left-circularly polarized of ℏ. The condition for the intensity at the focal point to be zero is then that the SAM and OAM of the beam must be the same, a condition that conically diffracted Gaussian beams fulfill naturally. If we look more closely at what happens, this strong dependency on the polarization of the incident beam is due to the behavior of the component of the electric field along the propagation axis zE z . If the incident beam is a vortex of topological charge 1, E z in the focal plane forms a vortex of topological charge 2 or an Airy-like pattern, depending on the handedness of the polarization. It is exactly the same phenomenon as the one observed in [9] for conical diffraction of a vortex beam. This result prompted us to examine more closely the behavior of E z . We noticed a peculiar similarity between the singularity of phase of E z and the one observed in conical diffraction. The behavior of E z is, in fact, the same as that of the intensity in the image plane of a crystal stuck between two orthogonal polarizers. We observed this similarity of behavior with homogeneously polarized input beams. If the input beam is linearly polarized, jE z j2 in the focal plane of a lens and the intensity in the image plane of a thin crystal caught between two orthogonal polarizers jE C j2 form the same pattern of two distinct lobes separated by a line of zero intensity (Fig. 2). If the input beam is circularly polarized, both jE z j2 and jE C j2 form a vortex. A closer inspection on the numerical level shows a correlation between the intensities of the distributions generated by conical diffraction and the z-component of vectorial diffraction higher than 99%.
December 1, 2014 / Vol. 39, No. 23 / OPTICS LETTERS
Ez P
−ik 2π
ZZ
6571
sin θ dsx dsy p cos θ Ω
× cos φE x s sin φE y seiks:P ;
(7)
Finally, the substitution r sin θ, r cos φ ! r sin φ 0
r
gives the following equation in a point P of the focal plane (zP 0): Fig. 2. Top line, intensity in the focal plane of a crystal between two orthogonal polarizers for input beams homogeneously (a) linearly polarized along x, (b) linearly polarized at 45°, (c) and right-circularly polarized. Bottom line, jE z j2 in the focal plane of a lens for an input beam homogeneously (d) linearly polarized along x, (e) linearly polarized along y, (f) and right-circularly polarized. The computations were performed for a lens of NA 0.95 and for a crystal of ρ0 0.5.
These similarities prompted us to investigate more closely the analogy between conical and vectorial diffraction. According to [10] and using the same notations, the electric field ex; y; z in some point x; y; z of the Fourier space of a lens is given by: −ik ex; y; z 2π
ZZ
asx ; sy dsx dsy sz Ω
× expikΦsx ; sy sx x sy y sz z;
Ez P
−ik 2π
Z
NA r0
Z
2π φ0
r dr p 1 − r2
× cos φE x r sin φE y reikr:P ;
(8)
where NA sin α is the numerical aperture of the lens (n 1). In the case of conical diffraction, consider a beam transverse and homogeneously polarized, whose polarization is EECx , such that there is an angle β between E C Cy and the x axis. The transform of this beam is Fourier then: aP aP EECx . The electric field in a point R Cy in the focal image plane of a crystal stuck between two orthogonal polarizers as expressed in [8] may then be written as EC R
(4)
Z Z k ∞ 2π E Cx dPaPeikR:P E Cy 2π r0 θ0 cos θ sin θ × coskR0 PId − i sinkR0 P ; sin θ − cos θ (9)
where !
s
sin θ cos φ sin θ sin φ ; cos θ
and Φsx ; sy 0 since we have assumed there is no spherical aberration. Furthermore, asx ; sy is the strength factor of the field, whose component along the propagation axis for an incident electric field along ex of amplitude l0 is given by [10]: p az sx ; sy −f l0 cos θ sin θ cos φ:
(5)
We deduced from this the expression of az sx ; sy for a homogeneous incident beam polarized along ex ; ey of amplitude E x ; Ey : p az sx ; sy C cos θ sin θcos φE x sin φE y ;
where P r cos θ; r sin θ, and R0 is the radius of the cylinder of refraction beyond the crystal, using the same notations as Eq. (3.3) in [8]. If the electric field is projected right after the crystal on ECy a polarization state −E (orthogonal to the polarization Cx state of the input beam), the expression of the amplitude of the output beam becomes: E C R
If we assume the beam is a Gaussian beam, ρ0 Rw0 ≪ 1 and state that EExy makes an angle −2β with the y axis, the amplitude of the electric field is then given by: E C R ∝
(6)
where C is a real constant that, combined with Eq. (4), gives us the following expression of the electric field along z in a point P of the Fourier space:
Z Z −ik ∞ 2π dPaPeikR:P sinkR0 r 2π r0 Φ0
× 2 cos ΦE Cx E Cy − sin ΦE 2Cx − E 2Cy : (10)
−ik π
Z
1∕wk r0
Z
2π Φ0
rcos ΦE x sin ΦE y eikR:P dP: (11)
For a Gaussian input beam linearly polarized, and a thin crystal verifying ρ0 ≪ 1, the expression of the electric field after having undergone conical diffraction
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OPTICS LETTERS / Vol. 39, No. 23 / December 1, 2014
and after having been projected in a state of polarization orthogonal to its input state of polarization (11) is proportional to that of the longitudinal component created by the passage through a lens of an input beam linearly polarized (8), for a small numerical aperture 1 NA ≃ wk . The expression is, in fact, approximately the same for a high NA, due to its dependence in r, while it becomes exactly the same for small NAs, while at the same time the phenomenon becomes negligible. In the same way, it is possible to show that the expression of one component of the electrical field after conical diffraction is formally the same as that of the longitudinal component created by diffraction through a lens for any homogeneous polarization state, as long as the condi1 tions ρ0 ≪ 1 and NA ≃ wk are verified. Numerically, the same trend seems to hold for nonhomogeneous amplitudes distributions, a trend we will explore in future works. In summary, the first part of this Letter focused on the specific vortex beams created by conical diffraction through a thin crystal and found they focus in the same way Laguerre–Gauss or Bessel beams do. The study of the E z component in the focal plane of a lens led us to uncover a formal similarity between E z in the focal plane of a lens and the electric field in the image plane of a crystal stuck between two orthogonal polarizers for a homogeneous input beam. On a phenomenological point of view, the inhomogeneity of thickness in the lens created the singularity observed in vectorial diffraction, while the inhomogeneity of the refractive index within the crystal created the singularity observed in conical diffraction. In both cases, an inhomogeneity of the optical path created a singularity. The first part of this Letter only examined the situation where the polarizer and analyzer placed, respectively,
before and after the crystal are circularly polarized. Note that many different beams with fractional OAM and SAM may be generated by changing the polarization of the polarizer and analyzer [14]. These beams focus just as neatly as the vortex beam created by conical diffraction. Moreover, the use of an homogeneous but anisotropic material (the biaxial crystal) to generate vortex beams allows for a simple and robust use, invariant by translation in x, y, or z. References 1. S. W. Hell and J. Wichmann, Opt. Lett. 19, 780 (1994). 2. K. I. Willig, B. Harke, R. Medda, and S. W. Hell, Nat. Methods 4, 915 (2007). 3. G. Sirat, “Procede et dispositif de mesure optique,” WO Patent App. PCT/FR2011/000,555 (September 30, 2012). 4. G. Y. Sirat, “Optical devices based on internal conical diffraction,” U.S. patent 8,514,685 (August 20, 2013). 5. Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, Opt. Lett. 32, 2357 (2007). 6. R. K. Singh, P. Senthilkumaran, and K. Singh, J. Opt. Soc. Am. A 25, 1307 (2008). 7. C. J. Sheppard, Opt. Express 22, 18128 (2014). 8. M. Berry, J. Opt. A 6, 289 (2004). 9. D. O’Dwyer, C. Phelan, Y. Rakovich, P. Eastham, J. Lunney, and J. Donegan, Opt. Express 19, 2580 (2011). 10. B. Richards and E. Wolf, Proc. R. Soc. A 253, 358 (1959). 11. J. Lin, O. Rodríguez-Herrera, and F. Kenny, Opt. Express 20, 1482 (2012). 12. C. McCutchen, J. Opt. Soc. Am. A 54, 240 (1964). 13. C. McCutchen, J. Opt. Soc. Am. A 870, 81721 (2002). 14. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, Opt. Express 18, 16480 (2010).
6569
Focusing by a high numerical aperture lens of distributions generated by conical diffraction Sybille Rosset,1,2 Clement Fallet,1,* and Gabriel Y. Sirat1 1
Bioaxial, Parisbiotech Sante, 24 rue du Faubourg Saint-Jacques, 75014 Paris, France 2 Ecole Polytechnique, 91128 Palaiseau Cedex, France *Corresponding author: [email protected]
Received September 12, 2014; revised October 13, 2014; accepted October 15, 2014; posted October 17, 2014 (Doc. ID 222744); published November 17, 2014 This Letter examines the diffraction of a vortex beam by a high numerical aperture of topological charge 1 generated by conical diffraction. Our research shows the behavior of the vortex beam is similar to the one already observed for Laguerre–Gauss and Bessel beams. We also highlight a similarity between the phase singularity created by a lens and the one created by conical diffraction through a thin crystal. More precisely, if the input beam is homogeneously polarized, we show the electric field in the image plane of a thin crystal caught between two orthogonal polarizers has the same expression as Ez , the component of the electric field along the propagation axis, in the focal plane of a lens. © 2014 Optical Society of America OCIS codes: (080.4865) Optical vortices; (260.1960) Diffraction theory; (260.5430) Polarization; (260.1180) Crystal optics; (260.6042) Singular optics. http://dx.doi.org/10.1364/OL.39.006569
Vortex beams have attracted considerable interest in super-resolution microscopy [1–4] among other fields. The vortex beam needs to have a neat zero of intensity at its center. For super-resolution applications in particular, the vortex must keep its zero after diffraction by a high numerical aperture (NA) lens. However, when a beam is focused through a high NA lens, singular effects take place and the vectorial diffraction theory must be used to determine the intensity in the focal plane. The focusing of a vortex beam by a high NA lens has already been studied in the case where the vortex beam is a Laguerre–Gauss or a Bessel beam [5–7]. In this Letter, we first show the phenomenon is the same for the slightly different vortex beam created by conical diffraction through a thin crystal, and then study its link with conical diffraction. Conical diffraction is the name given to the diffraction phenomenon that occurs when a beam propagates along an optic axis of a biaxial transparent crystal slab. According to [8], the incident beam is transformed by the crystal into two beams of different orbital angular momentum (OAM) and orthogonal polarizations; one beam is linked to the incident beam by a Bessel function of zero order and remains of the same OAM, while the OAM of the other output beam is different from the OAM of the input beam, differing from ℏ. It is possible to create optical vortex beams by isolating this last component [9]. More precisely, if an input beam right-circularly polarized, after having gone through the biaxial crystal passes through a left circular analyzer, the output beam is a left-circularly polarized vortex beam of topological charge 1. In the same way, an input beam left-circularly polarized may emerge as a right-circularly polarized vortex beam of topological charge −1. The amplitude distribution of these beams is nonhomogeneous, depending on the distance r from the center of the vortex, and differs from the amplitude distribution of the Laguerre–Gauss beams usually used. Richards and Wolf obtained an integral representation for the electromagnetic field in the image space of an optical system that applied beyond the usual scope of low 0146-9592/14/236569-04$15.00/0
angular systems [10]. This integral representation was expressed as a three-dimensional Fourier transform and computed using chirp z-transform (to avoid the use of zero padding and heavy costs in terms of memory and computational time) in [11], following an initial idea of McCutchen [12,13]. We used this reformulation to compute the focused electrical field. The computation was carried out for NA 0.95 and for a left-circularly polarized vortex beam of topological charge 1 generated by conical diffraction through a crystal of parameter ρ0 0.5 (where ρ0 is the parameter characterizing the width of the beam and of the crystal, as defined in [8]). The maximal intensity was scaled to 1 and the Cartesian components of the electrical field were scaled accordingly; the distance between both maxima was arbitrarily scaled to 15 pixels (equivalent to a 12 zero-padding in Fourier space, thus pixel size λ∕24). We noticed the focused vortex beam keeps its neat zero of intensity at its center, just as neatly as vortex phase beams (beams generated with a vortex phase plate, with an homogeneous amplitude and a helical phase) or Bessel beams left-circularly polarized and of topological charge 1 (Fig. 1). Phase vortices enable a better intensity confinement than Bessel beams. The vortex beam generated with conical diffraction can be regarded as an intermediate situation between these two extremes. Comparing the gradient of the distributions near the singularity, we observed the sharpest gradient is obtained for the Bessel beam (with a gradient of 0.157), while the lowest one is obtained for the vortex phase beam (with a gradient of 0.147). The vortex created using conical diffraction performs somewhere between both with a gradient of 0.149, thus possessing a slightly sharper zero of intensity than a focused vortex phase beam and slightly less sharp than a focused Bessel beam. This difference is however negligible. The same trend is observed for other polarizations of the beam. Furthermore, we observed the same phenomenon already observed for Laguerre–Gauss and Bessel beams. When focused, the vortex generated by conical diffraction of a Gaussian beam (and thus circularly polarized) © 2014 Optical Society of America
6570
OPTICS LETTERS / Vol. 39, No. 23 / December 1, 2014
0
Efm1
1 Zα 0 p @ A 0 dθaθ cos θ sin2 θ; π 0 E x iE y
(2)
where α sin−1 NA, NA being the numerical aperture of the lens. We observed that the intensity at the focal point is zero only if E y iE x ; in other words, if the vortex is left-circularly polarized. A similar reasoning gives us the electrical field Efm1 at the focal point for a topological charge of −1: 0
Fig. 1. Intensity in the focal plane along the x axis for incident vortex beams left-circularly polarized of topological charge 1, of various amplitude distributions. The plain line shows the intensity computed for a vortex phase beam, the dashed line for a Bessel beam, and the stars for a vortex beam generated by conical diffraction through a crystal of ρ0 0.5. The computations were performed for a lens of NA 0.95.
remains a vortex with a neat zero of intensity at its center, while for other kinds of polarizations, focusing a vortex beam through a high NA lens may altogether change its form, resulting in a focused intensity whose peak lies in the middle of the focal plane. This prompted us to analyze how polarization affects the zero of intensity of a vortex beam in the focal plane for a vortex beam whose phase distribution is in eimΦ (m 1 is the topological charge of the vortex) and whose amplitude distribution is circularly symmetric. It has already been investigated for a Laguerre– Gauss beam and a Bessel beam but, to the best of our knowledge, has never been done for other types of amplitude distribution. First, we verified that the conically diffracted Gaussian beam confirms these criteria. We considered an input beam left-circularly polarized, that passes after having gone through the biaxial crystal through a right circular analyzer. According to [9], the electric field after the analyzer is: 1 ; ER; Z B1 R; R0 ; Zcos θ − i sin θ −i
(1)
where R0 is a constant that only depends on the crystal used, Z describes the propagation distance, and R describes the distance to the propagation axis. The amplitude of the vortex beam is B1 R; R0 ; Z, that only depends on R and is thus circularly symmetric. In the same way, for an input beam right-circularly polarized that passes through a left circular analyzer, the amplitude of the vortex beam is also B1 R; R0 ; Z and so is circularly symmetric. Let Ei be a beam that verify these conditions. Using the same notations as in [10], we then have Ei θ; Φ eimΦ aθEExy . Since the amplitude of the field depended only on the θ parameter, we applied the results shown in [5]. The electrical field Efm1 at the focal point is given, for m 1, by:
Efm−1
1 Zα 0 p Aπ 0 @ dθaθ cos θ sin2 θ: 0 E x − iE y
(3)
Here, the intensity at the focal point is zero only if E y −iE x ; that is, if the vortex is right-circularly polarized. The presence of a zero of intensity at the focal point thus depends heavily on the polarization of the vortex beam, while it does not depend at all on the profile of the amplitude of the field in θ. As long as the field is circularly symmetric, if it is not circularly polarized with the right handedness, the intensity at the focal point will differ from zero, and the further away from the right state of polarization, the stronger the intensity. It also can be expressed in terms of orbital angular momentum (OAM) and spin angular momentum (SAM) of the beam. A vortex beam possesses an OAM of mℏ, m being the topological charge. A right-circularly polarized beam has a SAM of −ℏ, a left-circularly polarized of ℏ. The condition for the intensity at the focal point to be zero is then that the SAM and OAM of the beam must be the same, a condition that conically diffracted Gaussian beams fulfill naturally. If we look more closely at what happens, this strong dependency on the polarization of the incident beam is due to the behavior of the component of the electric field along the propagation axis zE z . If the incident beam is a vortex of topological charge 1, E z in the focal plane forms a vortex of topological charge 2 or an Airy-like pattern, depending on the handedness of the polarization. It is exactly the same phenomenon as the one observed in [9] for conical diffraction of a vortex beam. This result prompted us to examine more closely the behavior of E z . We noticed a peculiar similarity between the singularity of phase of E z and the one observed in conical diffraction. The behavior of E z is, in fact, the same as that of the intensity in the image plane of a crystal stuck between two orthogonal polarizers. We observed this similarity of behavior with homogeneously polarized input beams. If the input beam is linearly polarized, jE z j2 in the focal plane of a lens and the intensity in the image plane of a thin crystal caught between two orthogonal polarizers jE C j2 form the same pattern of two distinct lobes separated by a line of zero intensity (Fig. 2). If the input beam is circularly polarized, both jE z j2 and jE C j2 form a vortex. A closer inspection on the numerical level shows a correlation between the intensities of the distributions generated by conical diffraction and the z-component of vectorial diffraction higher than 99%.
December 1, 2014 / Vol. 39, No. 23 / OPTICS LETTERS
Ez P
−ik 2π
ZZ
6571
sin θ dsx dsy p cos θ Ω
× cos φE x s sin φE y seiks:P ;
(7)
Finally, the substitution r sin θ, r cos φ ! r sin φ 0
r
gives the following equation in a point P of the focal plane (zP 0): Fig. 2. Top line, intensity in the focal plane of a crystal between two orthogonal polarizers for input beams homogeneously (a) linearly polarized along x, (b) linearly polarized at 45°, (c) and right-circularly polarized. Bottom line, jE z j2 in the focal plane of a lens for an input beam homogeneously (d) linearly polarized along x, (e) linearly polarized along y, (f) and right-circularly polarized. The computations were performed for a lens of NA 0.95 and for a crystal of ρ0 0.5.
These similarities prompted us to investigate more closely the analogy between conical and vectorial diffraction. According to [10] and using the same notations, the electric field ex; y; z in some point x; y; z of the Fourier space of a lens is given by: −ik ex; y; z 2π
ZZ
asx ; sy dsx dsy sz Ω
× expikΦsx ; sy sx x sy y sz z;
Ez P
−ik 2π
Z
NA r0
Z
2π φ0
r dr p 1 − r2
× cos φE x r sin φE y reikr:P ;
(8)
where NA sin α is the numerical aperture of the lens (n 1). In the case of conical diffraction, consider a beam transverse and homogeneously polarized, whose polarization is EECx , such that there is an angle β between E C Cy and the x axis. The transform of this beam is Fourier then: aP aP EECx . The electric field in a point R Cy in the focal image plane of a crystal stuck between two orthogonal polarizers as expressed in [8] may then be written as EC R
(4)
Z Z k ∞ 2π E Cx dPaPeikR:P E Cy 2π r0 θ0 cos θ sin θ × coskR0 PId − i sinkR0 P ; sin θ − cos θ (9)
where !
s
sin θ cos φ sin θ sin φ ; cos θ
and Φsx ; sy 0 since we have assumed there is no spherical aberration. Furthermore, asx ; sy is the strength factor of the field, whose component along the propagation axis for an incident electric field along ex of amplitude l0 is given by [10]: p az sx ; sy −f l0 cos θ sin θ cos φ:
(5)
We deduced from this the expression of az sx ; sy for a homogeneous incident beam polarized along ex ; ey of amplitude E x ; Ey : p az sx ; sy C cos θ sin θcos φE x sin φE y ;
where P r cos θ; r sin θ, and R0 is the radius of the cylinder of refraction beyond the crystal, using the same notations as Eq. (3.3) in [8]. If the electric field is projected right after the crystal on ECy a polarization state −E (orthogonal to the polarization Cx state of the input beam), the expression of the amplitude of the output beam becomes: E C R
If we assume the beam is a Gaussian beam, ρ0 Rw0 ≪ 1 and state that EExy makes an angle −2β with the y axis, the amplitude of the electric field is then given by: E C R ∝
(6)
where C is a real constant that, combined with Eq. (4), gives us the following expression of the electric field along z in a point P of the Fourier space:
Z Z −ik ∞ 2π dPaPeikR:P sinkR0 r 2π r0 Φ0
× 2 cos ΦE Cx E Cy − sin ΦE 2Cx − E 2Cy : (10)
−ik π
Z
1∕wk r0
Z
2π Φ0
rcos ΦE x sin ΦE y eikR:P dP: (11)
For a Gaussian input beam linearly polarized, and a thin crystal verifying ρ0 ≪ 1, the expression of the electric field after having undergone conical diffraction
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OPTICS LETTERS / Vol. 39, No. 23 / December 1, 2014
and after having been projected in a state of polarization orthogonal to its input state of polarization (11) is proportional to that of the longitudinal component created by the passage through a lens of an input beam linearly polarized (8), for a small numerical aperture 1 NA ≃ wk . The expression is, in fact, approximately the same for a high NA, due to its dependence in r, while it becomes exactly the same for small NAs, while at the same time the phenomenon becomes negligible. In the same way, it is possible to show that the expression of one component of the electrical field after conical diffraction is formally the same as that of the longitudinal component created by diffraction through a lens for any homogeneous polarization state, as long as the condi1 tions ρ0 ≪ 1 and NA ≃ wk are verified. Numerically, the same trend seems to hold for nonhomogeneous amplitudes distributions, a trend we will explore in future works. In summary, the first part of this Letter focused on the specific vortex beams created by conical diffraction through a thin crystal and found they focus in the same way Laguerre–Gauss or Bessel beams do. The study of the E z component in the focal plane of a lens led us to uncover a formal similarity between E z in the focal plane of a lens and the electric field in the image plane of a crystal stuck between two orthogonal polarizers for a homogeneous input beam. On a phenomenological point of view, the inhomogeneity of thickness in the lens created the singularity observed in vectorial diffraction, while the inhomogeneity of the refractive index within the crystal created the singularity observed in conical diffraction. In both cases, an inhomogeneity of the optical path created a singularity. The first part of this Letter only examined the situation where the polarizer and analyzer placed, respectively,
before and after the crystal are circularly polarized. Note that many different beams with fractional OAM and SAM may be generated by changing the polarization of the polarizer and analyzer [14]. These beams focus just as neatly as the vortex beam created by conical diffraction. Moreover, the use of an homogeneous but anisotropic material (the biaxial crystal) to generate vortex beams allows for a simple and robust use, invariant by translation in x, y, or z. References 1. S. W. Hell and J. Wichmann, Opt. Lett. 19, 780 (1994). 2. K. I. Willig, B. Harke, R. Medda, and S. W. Hell, Nat. Methods 4, 915 (2007). 3. G. Sirat, “Procede et dispositif de mesure optique,” WO Patent App. PCT/FR2011/000,555 (September 30, 2012). 4. G. Y. Sirat, “Optical devices based on internal conical diffraction,” U.S. patent 8,514,685 (August 20, 2013). 5. Y. Iketaki, T. Watanabe, N. Bokor, and M. Fujii, Opt. Lett. 32, 2357 (2007). 6. R. K. Singh, P. Senthilkumaran, and K. Singh, J. Opt. Soc. Am. A 25, 1307 (2008). 7. C. J. Sheppard, Opt. Express 22, 18128 (2014). 8. M. Berry, J. Opt. A 6, 289 (2004). 9. D. O’Dwyer, C. Phelan, Y. Rakovich, P. Eastham, J. Lunney, and J. Donegan, Opt. Express 19, 2580 (2011). 10. B. Richards and E. Wolf, Proc. R. Soc. A 253, 358 (1959). 11. J. Lin, O. Rodríguez-Herrera, and F. Kenny, Opt. Express 20, 1482 (2012). 12. C. McCutchen, J. Opt. Soc. Am. A 54, 240 (1964). 13. C. McCutchen, J. Opt. Soc. Am. A 870, 81721 (2002). 14. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, Opt. Express 18, 16480 (2010).
