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3.2. GENERATION THE LONGITUDINAL COMPONENT OF ELECTRIC FIELD ON THE OPTICAL AXIS USING ASYMMETRIC BINARY AXICONS ILLUMINATED BY LINEARLY AND CIRCULARLY POLARIZED BEAMS
Материалы статьи были доложены на 21 Международной конференции по лазерным технологиям (ALT13)
Khonina Svetlana N., Doctor of Science in Physics and Mathematics, professor. Image Processing Systems Institute of the Russian Academy of Sciences, Samara State Aerospace University. E-mail: [email protected]
Abstract: It is analytically and numerically shown that by introducing the asymmetry into the axicon design it becomes possible to generate the longitudinal electric field (E-field) component on the optical axis for linearly and circularly polarized incident beams. Binary axicons with high numerical aperture (NA) in three configurations (axis-symmetric, spiral and bi-axicon) are investigated. Experimental measurements for the near-field diffraction of linearly and circularly polarized incident beams are presented.
Index terms: asymmetric binary axicon, near-field diffraction, longitudinal electrical field component
INTRODUCTION
The axicon is known to generate a zero-order Bessel beam with the central spot size for intensity FWHM = 0.36VNA [13], which is 37% smaller than the Airy disk produced with a similar NA lens. Thus, the axicons hold promise for applications where a compact transverse light spot is required.
However, if the incident beam is linearly polarized (as in most lasers), a decrease in the focal spot size generated by the total electromagnetic field intensity is hindered by the contribution of the longitudinal component, which increases with increasing NA, thus increasing the focal spot transverse dimension along the polarization axis. This problem has prompted a number of publications dealing with radially polarized incident beams. With a radially polarized incident beam, the high NA axicon produces a focal spot mainly composed of a single longitudinal component, thus enabling one to break the diffraction limit associated with a lens in compliance with the scalar theory prediction [1 - 6]. The extraction of the longitudinal component is significant for applications such as microscopy, high-resolution metrology, electron acceleration, and material processing [7, 8].
It has been demonstrated [9, 10] that using a linearly polarized beam incident on a high NA binary axicon it is possible to generate a longitudinal component on the optical axis with use of a phase jump perpendicular to the polarization axis. A similar result can be obtained with use of a bi-axicon. However, this effect is only observed at a definite mutual position of the polarization axis and the bi-axicon. Note that Ref. [10] also reported on the research of a spiral binary axicon capable of operating independently of the linearly polarized beam axis.
A linearly polarized laser beam can rather easily be transformed into a circularly polarized beam, which retains a circular symmetry even when being sharply focused. However, the resulting focal spot is also widened owing to the longitudinal component contribution. The redistribution of the longitudinal component into the focal spot central part by introducing a linear or vortex phase singularity enables the focal spot size to be reduced [9]. In Ref. [10] it was demonstrated that a similar result could be obtained using a
vortex axicon. However, for binary axicons, which are much easier to fabricate, the study has not been conducted.
In this work, an analytical and numerical study of the diffraction of linearly and circularly polarized laser beams is conducted for three types of binary diffractive axicons (axi-symmetric, spiral and bi-axicon) with the objective to strengthen the longitudinal E-field component on the optical axis. The numerical simulation was based on the Mansu-ripur's modification of the plane wave expansion (PWE) method [11] with regard for the Fresnel transmission coefficients. In this work, the experiments were conducted with high NA binary axicons fabricated by electron lithography, which enables a higher quality of fabrication when compared with Ref. [10]. The near-field diffraction pattern was recorded using a near-field microscope NT-MDT equipped with a metalized fiber probe with an aperture, enabling the measurement of both longitudinal and transverse E-field components [12]. However considering that the technique is three times as sensitive to the longitudinal component when compared with the transverse one [13], the longitudinal E-field component can be reliably recorded.
COMPUTING THE DIFFRACTION OF DIFFERENTLY POLARIZED GAUSSIAN BEAMS BY HIGH NA BINARY AXICONS
The diffractive phase axicon has the complex transmission function given by x(r) = exp (ika0 r) (1)
where a0 is the axicon parameter that defines its NA= a0.
Let us consider high NA (a0 = 0.95) binary microaxicons of the form:
T, (r) = exp {i arg [cos (ka0r )cos (m^)]} (2)
t s (r) = exp {i arg [ cos ( k a 0 r + m^)]} (3)
that operate in the visible wavelength X = 0.532 ^m and have a radius of R = 21X.
At m = 0, both formulas describe an axisymmetric axicon; at m = 1, Eq. (2) allows one to design a bi-axicon, whereas Eq. (3) describes a spiral axicon. Figure 1 depicts phases for
the binary axicons of three configurations -- axisymmetric (Fig. 1a), bi-axicon (Fig. 1b), and spiral (Fig. 1c).
Figure 1. Phases of three types of the binary axicon: (a) axisymmetric, (b) bi-axicon, and (c) spiral.
For the transmission functions in Eqs. (2) and (3), the spatial spectra of the different components are found on a circumference of the radius proportional to the axicon parameter P.(a) a S(a - a0) . In this case, the distribution
on the optical axis is very easy to analyze. In the paper [14] the distribution on the optical axis was calculated using the PWE method. In the absence of the vortex phase (m = 0), the field (9) on the optical axis (p = 0) takes the form:
ik 2
E „=„(0,0, z ) = - ^
c
v0 ;
j |j E0(r ) J 0 (kr a) r dr J exp [iW 1 -a2 ] [t, (a) + tp (a)Vl -a2 ] a da
(4)
where cx and cy are the polarization coefficients of the incident beam,
2 (w n2) 2 (yn2ni- n2) are the
ts (a)=~-r~2—^ v(a)=_n—a-;—
2yn2n - (n + n2) y (n + n2) - 2n2n1
Fresnel transmission coefficients for TE- and TM-waves, respectively, n1 is the refractive index of the optical element, and n2 is the refractive index of the medium.
From Eq. (4), it is seen that only transverse components are generated on the optical axis, with the longitudinal component being equal to zero.
If the field comprises a first-order vortex phase (| m |= 1), the field on the optical axis (p = 0) will be given by 0 0
±ic - c
E„=±1(0,0, z) = -
_ ÉL 2
0 0
c ± ic
j jE0(r)J±1 (kra)rdr a1 L 0
j jE0(r)J1(kra)rdr
exp I ikz\l 1 -a2 I tp (a) a2 da.
intensities of the longitudinal E-field compared to the transverse one the value L
= Kj/ (| Ex, maxf + |Ejmax|2) is added
into the Table 1. Size of the central light spot is defined by full with at half maximum (FWHM) of intensity in horizontal and vertical directions.
Results of Gaussian beam diffraction on the axisymmetric axicon are given in the first column of Table 1. Results for the bi-axicon are shown in the second column, and results for the spiral axicon are collected in the third column. With the bi-axicon structure containing first-order phase vortices of both signs, |m| = 1, the linearly polarized incident beam will be transformed:
{1, x - linear, 0.5, linear at 450, 0, y - linear.
(7)
Thus, for the linearly polarized beam, the proportion of energy found on the optical axis depends on the biaxicon orientation relative to the polarization axis. Hence, by rotating the optical element the intensity on the optical axis, which is entirely determined by the longitudinal E-field component, can be varied. This is illustrated in column 2 of Table 1.
Table 1.
Intensity distribution in the transverse plane at a distance of 1.5 ^m from the element for the superposition of field components (left image in the column) and for the longitudinal component (right image in the column)
expl ikz41 -a2 Itp(a)a2da =
(5)
implying that only the longitudinal component is present.
A circularly polarized beam has c = ±ic and, hence, for
y x
the longitudinal component not to vanish in Eq. (5), the circular polarization and the vortex phase need to be oppositely directed.
With the axisymmetric axicon (m = 0), the intensity distribution on the optical axis is the same for both linearly and circularly polarized beams:
|Em=0(0,0,zf = 0,25k4 [ts (a,) + tp (a0)V 1 -a02 J a02 —^0,25k4t2 (a0)a02.
(6)
The estimation (6) is received in asymptotic approach for infinite axicon. To confirm analytical expressions simulation results are collected in the Table 1. To indicate the relative
For the circularly polarized beam, Eq. (5) is dependent on the optical vortex sign, sign(m), and the polarization direction, sign (p) :
... . ! 0, sign(m) = sign(p),
cx + sign(m)i(sign(p)icx) = cx(1 -sign(m)sign(p)) = #2cx, sign(m) *Sign(p).
(8)
Considering that in the biaxicon there are optical vortices of both signs, the result is independent of the polarization
direction, because a non-zero longitudinal component can always be found on the optical axis.
The spiral axicon has a single first-order optical vortex, which means that for a linearly polarized beam, the longitudinal component will occur on the optical axis irrespective of the sign. This is illustrated numerically in column 3 of Table 1. For the circularly polarized beam, the distribution is associated with the polarization direction, as shown in column 3 of Table 1.
The phase singularity can be introduced not only into the optical element structure but also into the incident beam, say, by use of additional phase plates. Alternatively, the axisymmetric axicon can be illuminated by high-order laser modes [15].
It is noteworthy that with axisymmetric axicons, it is practically always possible to generate on the optical axis the longitudinal e-field component, with its energy being varied by either rotating the optical element or changing the beam polarization direction. In many cases, the focal spot size associated with the longitudinal E-field component is smaller than the diffraction limit at least for a single direction. Also it is possible to notice, that in the total intensity the central spot becomes more compact with increasing value of the parameter L [12-14].
EXPERIMENTAL RESULTS
The binary high NA axicons were fabricated in a SiO2 substrate using e-beam lithography and reactive ion etching of chrome. The optical elements have the size 22.68 x 22.68 ^m2, the axicon ring period d = 560 nm, and the groove depth 640 nm. For an incident wavelength 532 nm, the specified parameters correspond to NA = 0.95.
The intensity patterns produced by the fabricated optical elements were studied using the near-field scanning optical microscopy (NSOM). The near-field diffraction pattern was recorded using a near-field microscope NT-MDT with a me-talized fiber probe with aperture, enabling the measurement of both the longitudinal and transverse E-field components [12]. In [13] such a probe was shown to be selectively sensitive to different E-field components. In particular, the intensity measured was shown to be proportional to the quantity:
Jm = 2 +| 2 + 3 2 (9)
To illuminate the axicons, the laser light of wavelength 532 nm was focused with an 8x microobjective, with the axicon being placed near the focused beam waist whose size was matched with that of the axicon. The measurement was conducted so that the metalized fiber probe aperture was interacting with the element surface, allowing the electromagnetic field intensity to be measured at distances of an order of a wavelength from the surface under analysis.
Tables 2 and 3 show the near-field intensity patterns as a result of diffraction of various beams by different axicons at distances of 1 - 3 ^m from the elements. A general pattern and more detailed experimental pattern are shown. For comparison, the corresponding simulated patterns for the general intensity and the longitudinal intensity are given.
Table 2 shows the diffraction patterns for a linearly polarized beam with different polarization axes. The experimental results are in a qualitative agreement with the theoretical analysis: the diffraction pattern for a linearly polarized beam behind the high NA axicons is asymmetric and dependent on the beam rotation. Accordingly, for the axisymmetric and spiral axicons, there is a rotating diffraction pattern in the form of an oblong light spot; an entirely different pattern is produced by the bi-axicon: if the polarization and bi-axicon axes are perpendicular to each other, a small central focal spot is formed; if, however, the axes are coincident the central intensity is zero.
There is a certain discrepancy between the focal spot size evaluated quantitatively in the experiment and theoretically. The measurement problems connected with use of the NSOM were discussed in [14]. However, the discrepancy is smaller if the focal spot is mainly contributed to by the longitudinal component. This is because the microscope probe has a different sensitivity to different E-field components [13].
Table 2.
Near-field diffraction patterns for a linearly polarized beam produced by different axicons for differently oriented polarization axes.
Linear horizontal polarization Linear vertical polarization
Axy symmetric axicon 'A, * %
Exper.: FWHM = 0.22 nm = 0.4U; Theor.: FWHM=0.34J. Exper.: FWHM = 0.23 nm = 0.43?., Theor,: FWHM = 0.34J.
Bi-axicon J M* V ii
Exper.: FWHM=0.18 pm = 0.34X Theor.: FWHM = 0.52k,FWHIvL:=0.32X. Exper.: FWHM= 0.24 pm = 0.45X
Spiral axicon \ %
Exper: FWHM=0.19 nm = 0.36X Theor.: FWHM=0.4U,FWHMz=0.3X. Exper: FWHM=0.21 nm=0.39X,
Table 3 shows the results of diffraction of a circularly polarized laser beam. The experimental results are in good agreement with the theoretical analysis: for the bi-axicon, the diffraction pattern is asymmetric, whereas for the axisymmetric and spiral axicons the diffraction pattern is axially symmetric. Note that the smallest focal spot is produced by the spiral axicon. The experimental and theoretical estimates of the focal spot size are also seen to be in good agreement. Note that if the focal spot is predominantly formed by the longitudinal component the experimental estimate is closer to the theoretically predicted value specifically for this component and not for the total intensity. As was mentioned earlier, the reason is that the probe is more sensitive to the longitudinal E-field component.
In general, it should be noted that the experimental and theoretical results agree satisfactorily both in geometry and in scale. Among key factors that have led to the deviation of the experimental results from theory, one should indicate, first of all, a fairly large probe aperture (100 nm), which in a
number of cases prevented the intensity distribution from being measured with a satisfactory resolution. Some intensity peaks are not higher than 0.3A, - 0.4^, with the intervals between them being even smaller, thus resulting in a bit fuzzy experimental patterns.
Table 3.
Near-field diffraction patterns for a circularly polarized laser beam produced by different axicons.
A major quantitative discrepancy between the experimental and theoretical estimates stems from the nonlinear sensitivity of the near-field microscope NT-MDT with a meta-lized fiber probe with aperture to different E-field components [14]. For the purposes of this study, the said peculiarity is not a disadvantage as the aim is to experimentally demonstrate that when using the linearly and circularly polarized beam, the longitudinal E-field component is found in the focal spot center.
CONCLUSIONS
Based on the Mansuripur's modification of the PWE method and with regard for the Fresnel transmission coefficients the operation of various types of high NA (NA = 0.95) binary axicons: axisymmetric, bi-axicon, and spiral are analyzed and simulated.
The axisymmetric axicons (bi-axicon and a spiral axicon) are required to single out the longitudinal E-field component in the beam center when using linearly and circularly polarized incident beams.
It is shown that for a linearly polarized beam incident on the bi-axicon, the proportion of energy on the optical axis accounted for by the longitudinal E-field component can be controlled by rotating the optical element. For a circularly polarized beam, the bi-axicon allows one to form on the optical axis the longitudinal component independent of the optical element orientation or the polarization direction.
For the spiral axicon, on the contrary, the diffraction pattern has been shown to depend on the circular polarization direction - for the opposite directions of polarization and spiral the central focal spot is predominantly formed by the longitudinal E-field component, whereas the coincident direction of polarization and spiral results in the zero central intensity. With a linearly polarized beam, the spiral axicon ensures the presence of the longitudinal component in the
center of the diffraction pattern irrespective of the mutual orientation of the optical element and the polarization axis.
Thus, two types of the axisymmetric axicons discussed above are suited for operation in different modes - as controlled optical elements or as elements independent of the external influence.
The experimental and theoretical results show qualitative agreement both in geometry and in scale of the diffraction patterns. A major quantitative discrepancy between the experimental and theoretical estimates stems from the nonlinear sensitivity of the near-field microscope NT-MDT with a metalized fiber probe with aperture to different E-field components. This peculiarity has allowed us to reliably demonstrate that the longitudinal E-field component is generated at the focal spot center for linearly and circularly polarized light.
Characteristic features of the majority of the diffraction patterns, namely, the focal spot asymmetry for linearly polarized beams, dependence of the diffraction pattern on the rotation of the optical element or the incident beam, and a reduction in the focal spot size following the redistribution of the longitudinal E-field component to the central part of the spot have been confirmed experimentally. List of reference:
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