Synthesis of light needles with tunable length and nearly constant irradiance

We introduce a new method for producing optical needles with tunable length and almost constant irradiance based on the evaluation of the on-axis power content of the light distribution at the focal area. According to theoretical considerations, we propose an adaptive modulating continuous function that presents a large derivative and a zero value jump at the entrance pupil of the focusing system. This distribution is displayed on liquid crystal devices using holographic techniques. In this way, a polarized input beam is shaped and subsequently focused using a high numerical aperture (NA) objective lens. As a result, needles with variable length and nearly constant irradiance are produced using conventional optics components. This procedure is experimentally demonstrated obtaining a 53λ-long and 0.8λ-wide needle.

We introduce a new method for producing optical needles with tunable length and almost constant irradiance based on the evaluation of the on-axis power content of the light distribution at the focal area. According to theoretical considerations, we propose an adaptive modulating continuous function that presents a large derivative and a zero value jump at the entrance pupil of the focusing system. This distribution is displayed on liquid crystal devices using holographic techniques. In this way, a polarized input beam is shaped and subsequently focused using a high numerical aperture (NA) objective lens. As a result, needles with variable length and nearly constant irradiance are produced using conventional optics components. This procedure is experimentally demonstrated obtaining a 53λ-long and 0.8λwide needle.
About sixty years ago, McLeod 1 introduced conical lenses as a way to produce light axicons. Nowadays, optical needles are used in optical tweezers and in those techniques where long depth of focus is required. A variety of methods to achieve long needles with sub-wavelength width have been reported [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] . In general, most of this techniques are based on handling the discontinuities of the modulating distribution at the entrance pupil of the focusing system. The expected characteristics of an optical needle are small transverse width, negligible beam divergence and large longitudinal extension of the focal region. Note that polarization of the input electromagnetic field is also an important design variable. Very frequently, radial polarization is used because the produced beam displays the smallest spot size and a remarkable longitudinal polarization 17 .
The present paper aims to develop a procedure for generating needles with tunable axial extent and controlled on-axis uniformity. The incident beam is tailored by means of a special continuous modulating function designed to maximize the length of the needle according to on-axis irradiance considerations. This distribution is experimentally implemented by means of digital holography. Using the proposed technique, we produced in the laboratory a 53λ-long and 0.8λ-wide needle. Interestingly, these values are only limited by the characteristics of the electronic devices used in the optical setup. In other words, devices with improved features will be able to produce needles with better characteristics. Another remarkable characteristic of our design is that needles produced with linearly and radially polarized light have the same length. This paper is organized as follows: first, after introducing key concepts in propagation of light at the the focal area, we propose a mathematical framework for producing long needles with almost constant irradiance. Then, a modulation function that fulfils the mathematical properties of our approach is suggested. This formalism is analysed by means of computer simulations. Later on, we describe the optical system required for producing such needles and some experimental results are obtained and discussed. Finally, we present our conclusions. Moreover, in the Methods section we provide mathematical details on the design properties of the modulation function.

Results
Needle design. Let us first consider a monochromatic beam at the entrance pupil of an aplanatic high NA focusing system. The electric field E = (E x , E y , E z ) at the focal area is described by the Richards-Wolf integral 18 It can be demonstrated that the half-length of the needle L and the ratio q are related by means of the following inequality:  provides an analytic procedure to determine the size of the region around the focus where a fraction q of the energy is concentrated.
Equation (8) For instance, if Ω encloses at least the 75% of the total on-axis power content (q = 0.75), L can be calculated from Those functions h(θ) that produces distributions α  F( ) that fulfils the previous requirements are candidates to be selected for producing long light distribution along the z-axis. Provided that h(θ) is known, α  F( ) is determined using Eq. 6 and then, the needle length 2L can be theoretically estimated using Eq. (10). In this paper, we use the modulation function described by

Needle properties analysis.
To provide a better understanding on how the parameters of modulation function h(θ) (Eqs (11) and (12)) have to be selected, we have included several simulations using our design (see Fig. 2). For comparison purposes, we introduce the following parameters: z 2 • and total integrated irradiance T(z) = t(z) + l(z).   Experimental setup. In order to provide an experimental verification of our theoretical approach, we produced needles using linear and radially polarized beams. These light distributions were created with the help of an optical system able to tailor amplitude and polarization of the input beam and a high numerical aperture objective lens. This set-up is based on a modified version of a Mach-Zehnder system: each transverse component of the input beam passes through one of the arms of the interferometer where computer generated holograms displayed on liquid crystal devices modulate each component of the input beam. Then, both components are recombined and subsequently focused using a microscope objective lens. An extended description of the optical arrangement can be found in 20,21 whereas the holographic encoding method is described in 22 . The optical setup used is depicted in Fig. 3(a); insets 3(b,c) show the interferometric and recording parts of the system. The light source is a linearly polarized TEM 00 λ = 594 nm He-Ne laser with the polarization direction set at 45° with respect to the x-axis. The beam is split using a polarized beam splitter (PBS 1 on Fig. 3).
Each component of the beam E S is independently modulated by holograms h x (θ,ϕ) and h y (θ,ϕ) displayed on two inexpensive Holoeye HEO 0017 spatial light modulators (SLM 1 and SLM 2 ). The wave plates in the optical system are used to rotate the oscillating plane and set-up the modulators to the required modulation curve. These displays have been calibrated using the method described in 23 ; the corresponding modulation curves are shown in the Methods section. Holograms h x (θ,ϕ) and h y (θ,ϕ) are related with modulation function h(θ) by means of x y The polarization vector of the input beam E s are p = (1, 0) and p = (cos ϕ, sin ϕ) for the linearly and radially polarized cases, respectively. Distributions h x (θ, ϕ) and h y (θ, ϕ) are encoded on the displays using the cell-based double-pixel hologram 22 . Figure 4 shows the encoded computer generated holograms used in this paper. Note that the holographic structure can be appreciated if the image is zoomed. As explained in the previous section, Then, the resulting distributions are subsequently recombined by means of polarized beam splitter 2 (PBS 2 ). According to Eq. (4), E s reads where e x = (1, 0, 0) and e y = (0, 1, 0) (as indicated in Fig. 1); f 0 = 1.5 is the filling factor (experimentally estimated).
Finally, E s is imaged and scaled on the entrance pupil of the objective lens using relay lenses L 1 and L 2 . The spatial filter removes non-desired off-axis diffracted terms. Note that the irradiance |E S | 2 can be recorded by CCD 1 . This camera is useful to analyse the shape of |E S | 2 or to determine the Stokes images at the entrance pupil of the microscope lens. In this case, an extra polariser should be located in front of the camera.
Finally, the beam is focused using a microscope objective with NA = 0.65. Camera CCD 2 is used to image the focal area in combination with tube lens L T with a focal length f T = 400 mm. The position of the observation plane z is tuned by means of a Newport LTA-HL actuator with a minimum incremental motion δ z = 50 nm and a repeatability ε z = ±100 nm.
(, , ) 2 recorded at z = −23λ, 14λ and 23λ. Black dots and grey bands indicate the averaged values and the corresponding standard deviation. For comparative purposes, blue (transverse irradiance of the beam) and red (total irradiance) curves show numerical calculations carried out using Eq. 1. The width of the needle is obtained by calculating the Full Width at Half Maximum (FWHM) value of these curves. Accordingly, the estimated value is FWHM ~0.8λ. Figure 5(b) displays the corresponding recorded irradiances |E(r, φ, z)| 2 at z = −23λ, 14λ and 23λ. In order to provide an account of the length of the needle, a series of irradiances |E(r, φ, z)| 2 have been recorded for z ranging from −50λ to 50λ every Δz = 148 nm ≈ λ/4. This information is used to produce a visual representation I(r, z) (see Fig. 5(c)). Interestingly, the estimation of the needle length using the FWHM is ~53λ, in agreement with the theoretical prediction stated in the 'Needle properties analysis' section. It should be pointed out that axial distortion can be seen in the needle. This undesirable behaviour is due to several combined effects. First, SLM modulation curves display calibration errors (see Methods: Measurement of the modulation response of the displays) that are propagated in the holographic encoding procedure. Second, set-up misalignments, lack of flatness of the optical components (polarisers, wave plates, mirrors, et cetera) or stage drifts also deteriorates the quality of the needle. All in all, spherical aberration is present in our system. Interestingly, this aberration severely deteriorates the needle profile as it has been reported in refs 11,24 .
Second, a longitudinally polarized needle has been produced. In this case, the input beam E S is radially polarized and thus, the holograms are h x (θ, φ) = h(θ) cos ϕ and h y (θ, φ) = h(θ) sin ϕ. Again, N = m = 8. In order to test the polarization state of the input beam, the Stokes parameters S 0 , S 1 and S 2 has been measured at the entrance pupil of the focusing lens. The resulting Stokes images are shown in Fig. 6(a). Results clearly show that the beam E S is radially polarized.
As in the previous case, the irradiance |E(r, φ, z)| 2 has been recorded for z ranging from −50λ to 50λ. Nevertheless, longitudinally polarized needles cannot be visualized using conventional imaging optics. In our setup, the beam in the focal area is back-propagated trough the microscope objective and imaged on the CCD camera with the help of a long-focal length lens. In such conditions the longitudinal component is not propagated  25,26 ). Despite the needle exists, a pipe rather than a needle is detected.
Angular-averaged profiles I(r, z) at z = −23λ, 14λ, 23λ are presented in Fig. 6(b). Again, blue and red-dashed curves show the theoretical predictions of the transverse and the total irradiance of the light needle calculated using the Richards-Wolf integral [Eq. (1)]; the black dots indicate experimental values. It is apparent that instead of a bell-shaped distribution a donought-like profile is recorded: in particular, the experimental values are fitted using the theoretical estimation of the transverse part of the beam. Note that the width of the beam remains constant, around 2λ. Finally, Fig. 6(c) shows the irradiance of the imaged field |E(r, φ, z)| 2 at the above referred planes.

Concluding Remarks
In summary, we proposed a new approach for producing high quality needles. First, we developed a theoretical framework for evaluating the length of an optical needle taking into account on-axis power-content constrains. Second, we introduced a specific modulating function that depends on several parameters that are used for tuning the length of the needle. The behaviour of the needles was analysed using computer calculations. Finally, experimental needles were optically implemented by means of digital holography techniques using polarized input beams. In particular, we produced a linearly polarized 53λ-long needle with sub-wavelength width. The proposed approach was also tested with a radially polarized input beam. Despite the resulting light distribution presents a very intense longitudinal component, the full needle cannot be recorded using conventional imaging optics.

Methods
Power content of an enclosed region. In order to assess the length of the region around the focus (z = 0) where the power-content ratio of |E(0, z)| 2 is significant, it is necessary to derive a suitable criterion. This problem was extensively investigated some years ago using the so-called irradiance-moments framework 27 . The utility of this framework was confirmed because it is currently adopted as a ISO standard 28 . Nevertheless, this formalism is difficult to be implemented with fields such as the proposed one [Eq. (5)] because convergence of integrals is not guarantee 29 . The target of this section is to derive an expression to evaluate L for a given q. Equation (5) can be simplified using the change of variable α = cos θ, i.e.: ikz 1 0 with α 0 = cos θ 0 . It is important to recall that it is assumed that =  F(1) 0. For convenience, we introduce auxiliary vector q(α) defined as Note that α  F( ) 0 represents the jump discontinuity of function α  F( ) at the entrance pupil. By using q(α), Eq. (17) can be rewritten as the combination of two terms E(0, z) = E 1 (0, z) + E 2 (0, z) namely The second order intensity-moment 〈z 2 〉 1 for E 1 (0, z) is mathematically well defined and it can be calculated by means of the following expression: Note that when either E 1 or E 2 is equals to zero, prefactors 2 can be omitted obtaining a more accurate relationship between the 1 − q value and the on-axis power-content. Combining the previous inequality with Eqs (21) and (23), the following condition holds: