1240 Vol. 48, No.

5 / 1 March 2023 / Optics Letters Letter

Creation of pure longitudinal super-oscillatory spot

Bhavesh Pant, Hemant Kumar Meena, AND Brijesh Kumar Singh∗
Department of Physics, School of Physical Sciences, Central University of Rajasthan, Ajmer-305817, Rajasthan, India
*Corresponding author: brijeshsingh@curaj.ac.in

Received 18 November 2022; revised 19 January 2023; accepted 30 January 2023; posted 3 February 2023; published 24 February 2023

We present a method that creates a super-oscillatory focal Thus, super-oscillation provides another approach for focus-
spot of a tightly focused radially polarized beam using the ing and imaging beyond the diffraction limit in the far field
concept of a phase mask. Using vector diffraction theory, and has potential applications in super-resolution far-field light
we report a super-oscillatory focal spot that is much smaller microscopy [24,25], particle trapping [6], and optical manipula-
than the diffraction limit and the super-oscillation criterion. tion [26]. The super-oscillation can be realized by superimposing
The proposed mask works as a special polarization filter different light modes, like the Bessel [20] and Airy modes [27],
that enhances the longitudinal component and filters out the but complicated mathematics is involved in these methods. A
transverse component of radial polarization at focus, per- more suitable technique for realizing super-oscillatory beams
mitting the creation of a pure longitudinal super-oscillatory depends on designing an optical amplitude mask [28,29], a
focal spot. © 2023 Optica Publishing Group phase mask [22], and a planar binary phase lens [30]. Another
approach, super-oscillation focusing with a radially polarized
https://doi.org/10.1364/OL.481274
Laguerre–Gaussian mode, has been recently introduced for
super-resolution imaging by simply controlling the incident
It is well known that radial polarization leads to the best beam size [8]. However, a higher-order transverse mode of radial
light confinement in free space for a high numerical aperture polarization is used in this method.
(NA) focusing system [1]. The unique possibility provided by In this Letter, we numerically demonstrate a systematic
a tightly focused radially polarized (RP) beam is to create a approach for generating a super-oscillatory (SO) focal spot by
smaller focal spot with a longitudinal electric field component tightly focusing a standard radially polarized beam. For this
along the optical axis that makes no contribution to the energy purpose, we utilized the concept of an annular phase mask at
flow [2]. The smaller spot is applicable in optical data storage the pupil plane of an objective lens. This mask consists of two
devices and microscopy [3–5], while the longitudinal electric regions of opposite phases. The modulation process of the input
field component is vital for metallic particle trapping and par- beam in the focal plane is shown in Fig. 1. We identify the
ticle acceleration [6,7]. Further, the longitudinal electric field appropriate ratio between the two regions of the mask to obtain
can improve spatial resolution in laser scanning microscopy a SO focal spot. We show that this approach not only enhances
(LSM) [8]. Several applications require tailored optical beams the contribution of the longitudinal component in the central
to increase the utility of the beam. Many methods to produce hot spot but also reduces the size of the central SO spot and
tailored optical beams have been discussed, but they all need eliminates the contribution of the radial component to create a
additional elements to be aligned [9–12]. However, due to the pure longitudinal super-oscillatory spot.
diffractive nature of light, the focal spots of imaging systems According to scalar diffraction theory, the smallest possible
are restricted to a diffraction limit of 0.61λ/NA, where λ is the focal spot achieved by an annular mask of infinitely thin width
wavelength of light and NA is the numerical aperture of the lens. is 0.359λ/NA. However, due to the vector nature of the light, the
Various optical methods for focusing light beyond this diffrac- result from scalar theory is not precisely valid under high-NA
tion limit with a strong longitudinally polarized component have focusing conditions [31]. The rigorous intensity profile of an
been theoretically proposed; some of these include a negative- infinitely thin annular radially polarized beam at the focus can
index grating lens, a parabolic mirror, absorbance modulation, be written as [8]
and a 4π high-numerical-aperture focusing system [13–17]. A (︃ )︃
sub-wavelength focal spot with radially polarized light is gen- NA2 NA2
I(ρ) = 2 [J0 (kNAρ)]2 + 1 − 2 [J1 (kNAρ)]2 , (1)
erated by changing the intensity distribution of the beam cross η η
section [18].
Super-oscillation is a phenomenon where a band-limited where η is the refractive index of the medium and Jn is the Bessel
function oscillates at a local frequency higher than its high- function of the first kind of order n. The first and second terms
est Fourier component [19]. A well-designed superposition of of Eq. (1) correspond to the longitudinal and radial components
the Fourier components of a band-limited function forms a at the focus, respectively.
local hot spot whose size is much smaller than the diffraction It is well known that a tightly focused radially polarized field
limit [20–22]. The first experimental verification of super- results in a strong longitudinal component along the optical axis.
oscillation was done by using a nanohole array structure that The field in the vicinity of the focus can be analyzed using
could generate a sub-diffraction hotspot with a size of 0.36λ [23]. vector diffraction theory [31,32]. Adopting the expression from

0146-9592/23/051240-04 Journal © 2023 Optica Publishing Group

 Letter Vol. 48, No. 5 / 1 March 2023 / Optics Letters 1241

Fig. 1. Schematic sketch of the focusing of a phase-modulated

radially polarized beam by an objective lens.

vector theory, we can write the focal field of a radially polarized

beam as
⃗ z) = Eρ ⃗eρ + Ez ⃗ez ,
E(ρ, (2)
where Eρ and Ez are the amplitudes of the transverse and longitu-
dinal components of the radially polarized beam in the vicinity
of the focus, respectively (Fig. 1). These amplitudes can be
expressed as [18,32]
Eρ (ρ, z) =
A ∫0β cos 2 θ sin(2θ) p0 (θ)J1 (kρ sin θ)eikz cosθ dθ
1
(3)
and
Ez (ρ, z) = Fig. 2. Simulation results for the tightly focused RP beam. The
β 1 first row shows the results obtained without using the mask, while
2iA ∫ cos θsin2 θ p0 (θ)J0 (kρ sin θ)eikz cosθ dθ,
0
2 (4) the second, third, and fourth rows show the results obtained with a
where Jn (kρ sin θ) denotes a Bessel function of the first kind mask for ε values of 0.633, 0.72, and 0.74, respectively. The first
of order n, p0 (θ) is the pupil apodization function, k is the column represents the phase maps, the second column shows the
wavenumber, and β is the maximal angle determined by the 2D intensity profiles in the focal plane, and the third column shows
numerical aperture of the lens. In this paper, we choose p0 (θ) = the cross-sectional intensity profiles of different components in the
1. The NA of the focusing lens is chosen to be 0.8, the wavelength focal plane [the inset in (h) and (k) are zoomed SO spots with very
taken is 532 nm, and the unit of the length is normalized to the low energy, and the inset in (l) is the zoomed center].
wavelength.
Using Eqs. (3) and (4) in Eq. (2), the resultant intensity dis-
numerically calculate the modified intensity distribution in the
tribution of a tightly focused radially polarized beam in the
focal plane. We vary the value of ε and observe that for some
vicinity of the focus is shown in the first row of Fig. 2. The
specific values of ε, we can generate a radially polarized SO focal
phase map, 2D intensity profile in the focal plane, and the cross-
spot in a controlled manner followed by a sidelobe. However,
sectional intensity distributions of the different components are
the SO spot peak intensity reduces at once when we reduce the
shown in Figs. 2(a), 2(b), and 2(c), respectively. From Fig. 2(c),
focal SO spot size by increasing the value of ε. For NA = 0.8, the
one can observe that the longitudinal component (shown as a
lens diffraction limit (0.5λ/NA) and super-oscillation criterion
blue dashed line) is much stronger than the transverse (radial)
(0.38λ/NA) are 0.63λ and 0.475λ, respectively [33].
component (shown in red line), and the total intensity (shown
The numerical results for three different values of ε equal to
as a black dashed line) is the sum of these two components. The
0.633, 0.72, and 0.74 are shown in the second, third, and fourth
spot size is defined by the full width at half maxima (FWHM)
rows of Fig. 2, respectively, all of which generate the SO focal
of the total intensity profile from Fig. 2(c), which is found to be
spot of the radially polarized beam. For comparison, in the first
1.22λ. Due to the doughnut-shaped profile of the radial compo-
row of Fig. 2, we show the results when no mask is used. The
nent, there is an enlargement in the total spot size. Therefore, a
first-column figures [Figs. 2(a), 2(d), 2(g), 2(j)] represent the
smaller focal spot is expected if we can reduce the contribution
phase maps, the second-column figures [Figs. 2(b), 2(e), 2(h),
of the radial component to make the longitudinal component
2(k)] show the 2D intensity profiles of the tightly focused beam
dominant in the focal spot.
in the focal plane, and the third-column figures [Figs. 2(c), 2(f),
In our approach for realizing a SO focal spot using an inci-
2(i), 2(l)] show the cross-sectional intensity profiles of different
dent radially polarized beam, we consider the case in which an
components of the beam in the focal plane. Due to the symmetry
annular phase mask is inserted at the pupil plane of the lens and
of the beam profiles, the intensity cross-section axis is taken
consists of two regions with opposite phases (Fig. 1). This mask
along the x axis. In the second row for ε = 0.633, the measured
is expected to reduce the contribution of the radial component.
FWHM of the central spot is 0.475λ [Figs. 2(e) and 2(f)], a factor
The mask can be expressed in the form [26]
of ∼ 3.18 times smaller than the normal spot size in the case of
{︃
−1 θ ≤ ε.β no mask [Figs. 2(b) and 2(c)], and is just the super-oscillation
φmask (θ) ∝ , (5) criterion for our present study.
1 ε.β ≤ θ ≤ β
To show the level of super-oscillation of the SO spot corre-
where 0<ε.β<β, β is the maximum angle of focus defined by sponding to ε = 0.633, we compare the SO spot with the normal
the NA of the lens, and θ is the angular coordinate. ε is a variable spot (the spot in the case of no mask). A Gaussian window with
that takes a value between 0 and 1. To realize the SO focal spot, a narrow beam waist is used to filter the sidelobes of the SO spot.
we multiply Eq. (2) by the mask function defined in Eq. (5) and We apply this Gaussian window to both the normal and SO spots,
 1242 Vol. 48, No. 5 / 1 March 2023 / Optics Letters Letter

Table 1. FWHMs of Different Components in the SO

Spot
ε Value FWHM (in λ)
Longitudinal Radial Total
0.65 0.401 0.286 0.443
0.67 0.379 0.256 0.404
0.69 0.351 0.214 0.362
0.71 0.314 0.147 0.315
0.72 0.289 - 0.289
0.74 0.218 - 0.237
Fig. 3. Super-oscillation criterion of the radially polarized SO
spot.
of the first sidelobe peak intensity (I sl ) to the central SO spot
peak intensity (I so ), is smaller (∼ 1.46) when the spot size is
then calculate the local spatial frequencies of both the filtered around 0.475λ and becomes larger and larger (∼51.1 for a spot
normal and SO spots by taking their Fourier transforms. The size of 0.24λ) as the central SO spot size is further reduced.
resultant 1D plots of their intensity profiles are shown in Fig. 3 The larger peak intensity ratio indicates an enhancement in the
in green and blue, respectively. Spatial frequency is normalized peak intensity of the sidelobe. The selectivity of the SO spot
by the beam waist diameter d = (4/π)λ(f /#) of the Gaussian win- is defined as the distance between the peak intensity positions
dow, where f /# = f /D is the F-number of the lens. f and D are of the sidelobe and the central SO spot, and it is plotted as a
the focal length and the diameter of the Fourier-transforming red curve in Fig. 4. It is observed that the selectivity of the
lens. We use f = 2.87 mm and D = 4.59 mm. One can see from SO beam is larger (∼1.03λ) when the SO spot size is around
the plot that the filtered SO spot (corresponding to ε = 0.633) 0.475λ. It is reduced to ∼0.88λ (for a spot size of 0.24λ) as the
contains spatial frequencies that are higher than those of the SO spot size becomes smaller. The enhancement in the sidelobe
normal spot, thereby manifesting the super-oscillation criterion peak intensity reduces the selectivity of the SO beam. There is
at FWHM = 0.475λ [Fig. 2(e)]. Any spot with FWHM < 0.475λ a nearly linear relationship between the selectivity of the SO
is certainly a SO spot. beam and the SO spot size, as we can see from Fig. 4.
As we further increase the value of ε to 0.72, the measured As stated earlier, the doughnut shape of the radial component
FWHM of the central SO spot is reduced to 0.289λ [Fig. 2(h)]. is responsible for the broadening of the focal spot. To illustrate
The generated SO spot size is ∼ 4.22 times smaller than the the effect of our adopted method in reducing the contribution of
normal spot size [Fig. 2(b)] and a factor of ∼ 1.46 times smaller the radial component and enhancing the longitudinal component
than the super-oscillation criterion. Further, upon increasing the in the SO spot, we tabulate the FWHMs of the longitudinal
value of ε to 0.74 in the fourth row, the measured FWHM of and the radial (hollow ring) components and the total field of
the central focal spot is reduced to 0.237λ [Fig. 2(k)], a factor the SO spot in Table 1 for different values of ε. Further, the
of ∼ 5.14 times smaller than the normal spot size and a factor of corresponding peak intensity ratio of the longitudinal (I z ) and
∼ 2.0 times smaller than the super-oscillation criterion. Hence, radial (I r ) components of the central SO spot as a function of
comparing the results of the three different values of ε along the SO spot size is plotted in Fig. 5.
with the case of no mask, we notice from the third-column From Table 1, we observe that when the ε value is increased
figures [Fig. 2(c), 2(f), 2(i), 2(l)] that upon increasing the value from 0.65 to 0.71, the FWHMs of both the radial and longitudi-
of ε from 0.633 to 0.74, the spot size of the main lobe is much nal components decrease continuously, resulting in a decrease in
reduced, with a simultaneous reduction in the peak intensity of the FWHM of the total SO spot of the radially polarized beam.
the SO spot and an increase in the sidelobe peak intensity. The contribution of the radial component becomes less. In con-
Figure 4 shows the variation of the peak intensity (PI) ratio trast, the longitudinal component becomes dominant when the
of the sidelobe to the central SO spot and the selectivity of the total central field spot size is reduced from 0.443λ to 0.315λ, as
SO spot as a function of the SO spot size. As can be seen from shown by the intensity ratio of the two components in the SO
Fig. 4 (blue curve), the peak intensity ratio, which is the ratio spot (Fig. 5). Also, the spot size of 0.315λ is the smallest spot
that has both radial and longitudinal components. For ε = 0.72,
we can see from Table 1 that the FWHM of the total central

Fig. 4. Variation of the PI ratio of the sidelobe to the central

SO spot (blue color), and selectivity of the SO beam (red color)
as a function of the SO spot size. The points (red and blue) show
the calculated values, while the solid lines show the corresponding Fig. 5. Intensity ratio of the longitudinal (I z ) and radial (I r )
best-fitted curves. components in the SO spot as a function of the SO spot size.
 Letter Vol. 48, No. 5 / 1 March 2023 / Optics Letters 1243

field distribution is the same as that of the longitudinal compo- Funding. Science and Engineering Research Board (CRG/2019/001187).
nent. The radial component makes no contribution to the total Disclosures. The authors declare no conflicts of interest.
central SO field spot. Hence, the central SO spot field is purely
longitudinal for this value. Again, upon increasing the value of Data availability. Data underlying the results presented in this paper
ε up to 0.74, the central SO spot size is further reduced and may be obtained from the authors upon request.
becomes smallest for ε = 0.74. However, there is some spread
in the FWHM value of the total central SO field from that of REFERENCES
the FWHM of the longitudinal component for ε = 0.73 and 0.74,
1. T. Grosjean and D. Courjon, Opt. Commun. 272, 314 (2007).
even though no contribution of the radial component is present 2. C. C. Sun and C. K. Liu, Opt. Lett. 28, 99 (2003).
in the central SO spot. 3. W. C. Kim, N. C. Park, Y. J. Yoon, H. Choi, and Y. P. Park, Opt. Rev.
The reason behind this spread is the interaction of the side- 14, 236 (2007).
lobe with the central SO spot, which occurs due to the smaller 4. G. Terakado, K. Watanabe, and H. Kano, Appl. Opt. 48, 1114 (2009).
selectivity of the central SO spot. For ε > 0.74, we observe a 5. K. Ryosuke, K. Yoshiki, M. Hashimoto, N. Hashimoto, and T. Araki,
dark spot at the center. Opt. Lett. 32, 1680 (2007).
6. Q. Zhan, Opt. Express 12, 3377 (2004).
Thus, a SO focal spot of size 0.475λ is generated for ε = 0.633,
7. D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402
which is just the super-oscillation criterion. For ε = 0.72, an (2007).
enhanced SO focal spot of size 0.289λ is generated, which is 8. Y. Kozawa, D. Matsunaga, and S. Sato, Optica 5, 86 (2018).
longitudinally polarized, having no contribution from the radial 9. S. Quabis, R. Dorn, and G. Leuchs, Appl. Phys. B 81, 597 (2005).
component. And then, for ε = 0.74, the smallest SO focal spot, 10. G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K.
of size 0.237λ, is generated. These three characteristics (just Ohbayashi, Appl. Phys. Lett. 84, 3855 (2004).
11. K. Yonezawa, S. Sato, and Y. Kozawa, Opt. Lett. 31, 2151 (2006).
the SO criterion, totally longitudinal, and the smallest spot)
12. D. H. Ford, S. C. Tidwell, and W. D. Kimura, Appl. Opt. 29, 2234
illustrate the reason behind showing the results corresponding (1990).
to these specific values of ε in Fig. 2. 13. H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, Laser
Further, expecting a SO spot smaller than 0.237λ, we apply Photonics Rev. 6, 354 (2012).
our approach to NA > 0.8. In contrast to expectation, the 14. J. Xu, Y. Zhong, S. Wang, Y. Lu, H. Wan, J. Jiang, and J. Wang, Opt.
FWHMs of the smallest SO spots reported for NA = 0.85, 0.9, Express 23, 26978 (2015).
and 0.95 are 0.242λ, 0.25λ, and 0.254λ, respectively. All these 15. A. April, H. Dehez, and M. Piché, Opt. Express 20, 14891 (2012).
16. Q. Li, X. Zhao, B. Zhang, Y. Zheng, L. Zhou, L. Wang, Y. Wu, and Z.
spots are greater than the smallest spot size of 0.237λ for Fang, Appl. Phys. Lett. 104, 061103 (2014).
NA = 0.8 when the same method is applied. We observe that 17. D. Li, W. Ding, X. Zhang, Y. Song, Y. Wang, and Z. Nie, Opt. Express
this contradiction results from the reduced selectivity of the SO 23, 690 (2015).
spot. As we increase the NA of the system and reduce the SO spot 18. K. Kitamura, K. Sakai, and S. Noda, Opt. Express 18, 4518 (2010).
size, the selectivity of the SO spot becomes much smaller and 19. M. V. Berry, J. S. Anandan, and J. L. Safko, eds. “Quantum Coher-
the sidelobe interaction prevents it from becoming the smallest. ence and Reality: Celebration of the 60th Birthday of Yakir Aharonov,”
(World Scientific, 1994), pp. 55–65.
Hence, we can emphasize that, without using a much higher NA
20. J. Baumgartl, K. Dholakia, M. Mazilu, and S. Kosmeier, Opt. Express
system for tight focusing, smaller SO spot creation is possible 19, 933 (2011).
with this method. 21. A. M. H. Wong, G. V. Eleftheriades, M. Kim, and X. H. Dong, Optica
In conclusion, we have introduced an efficient and systematic 4, 1126 (2017).
method for generating a super-oscillatory focal spot by tightly 22. C. J. R. Sheppard, Opt. Lett. 24, 505 (1999).
focusing a doughnut-shaped radially polarized beam far beyond 23. F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia De Abajo,
Appl. Phys. Lett. 90, 091119 (2007).
the standard diffraction limit (SO criterion) of 0.475λ. This
24. E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, M.
method shows the possibility of controlled variation of the spot Segev, Y. Aharonov, D. Z. Albert, and L. Vaidman, Opt. Express 21,
size of the generated SO spot. This possibility of controlled 13425 (2013).
variation is important for utilizing the SO spot in different 25. J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev,
applications where size control is required. The study of the and K. Dholakia, Appl. Phys. Lett. 98, 181109 (2011).
selectivity and the peak intensity ratio as a function of the spot 26. B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, Light: Sci. Appl. 6,
size will be helpful for the selection of an appropriate SO spot as e17050 (2017).
27. Y. Eliezer and A. Bahabad, ACS Photonics 3, 1053 (2016).
required for the applications. The proposed mask also works like 28. G. H. Yuan, S. Vezzoli, C. Altuzarra, E. T. Rogers, C. Couteau, C.
a special polarization filter which removes the radial field from Soci, and N. I. Zheludev, Light: Sci. Appl. 5, e16127 (2016).
the center of the spot more than the longitudinal field, thus mak- 29. H. Ye, K. Huang, H. Liu, F. Wen, Z. Jin, J. Teng, and C. W. Qiu, Appl.
ing the SO spot substantially longitudinally polarized. Further, Phys. Lett. 111, 031103 (2017).
an extremely small focal SO spot size is obtained, but with low 30. A. P. Yu, G. Chen, Z. H. Zhang, Z. Q. Wen, L. R. Dai, K. Zhang, S.
peak intensity. Despite the low intensity of the SO spot, it may L. Jiang, Z. X. Wu, Y. Y. Li, C. T. Wang, and X. G. Luo, Sci. Rep. 6,
38859 (2016).
be applicable for trapping [6] and manipulating particles due
31. B. Richards and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
to its smaller diffraction-limited size. Finally, we envisage that 32. K. S. Youngworth and T. G. Brown, Opt. Express 7, 77 (2000).
the SO spot will also be useful in super-resolution microscopy, 33. K. Huang, H. Ye, J. Teng, S. P. Yeo, B. Luk’yanchuk, and C.-W. Qiu,
optical data storage, and optical tweezers. Laser Photonics Rev. 8, 152 (2014).