Birefringent masks that are optimal for generating bottle fields

ANTHONY VELLA1,2,* , HIPPOLYTE DOURDENT3,4 , LUKAS NOVOTNY3, AND MIGUEL A. ALONSO1,2 1The Institute of Optics, University of Rochester, Rochester NY 14627, USA 2Center for Coherence and Quantum Optics, University of Rochester, Rochester NY 14627, USA 3Photonics Laboratory, ETH Zürich, 8093 Zürich, Switzerland 4Institut d’Optique Graduate School, Université Paris-Saclay, 91127 Palaiseau, France *avella@optics.rochester.edu


Introduction
Trapping and manipulation of nanoparticles plays a crucial role in biology and physics.The field of laser-based trapping was pioneered by Arthur Ashkin, who developed with co-workers a singlebeam gradient-force optical trap, known as "optical tweezers" [1][2][3].Nowadays, optical tweezers have found various applications, ranging from cooling and trapping atoms to manipulating live bacteria and viruses [4][5][6].The achievement of an e cient trap relies on the optical properties of the particles and the surrounding medium, as well as the physical nature of the light-mediated trapping forces.Optical tweezers are typically implemented using a high numerical aperture (NA) objective, which generates a strong electric field gradient at the focal region.The gradient force created causes particles of higher refractive index than the surrounding medium to be attracted to the light field maximum, and it repels low index particles in a high index environment.Thus, axial trapping e ciency of low index particles can be improved through the use of "hollow" (or "donut") beams, which have intensity minima on the optical propagation axis [7,8].
In 2000, Arlt and Padgett introduced the concept of an "optical bottle" which represents a beam with a finite axial region of low (ideally null) intensity surrounded in all directions by light [9].Generating this dark region at the focus creates useful intensity gradients to trap or measure the dynamics of low index particles.For instance, the suitability of an optical bottle beam for particle trapping and manipulation has been confirmed in experiments with atoms and absorbing particles [10].Optical bottle fields also have applications in stimulated emission depletion (STED) fluorescence microscopy [11], in which the three-dimensional nature of the bottle provides enhanced resolution along the longitudinal direction [12,13].
Various techniques have been proposed to generate bottle fields [14][15][16][17].Many of these involve vortices generated by the use of phase masks, spiral phase plates, or spatial light modulators [18,19].These methods typically rely on interference between two fundamental Gaussian modes, making them sensitive to alignment errors [20].Polarization vortex beams have also been generated by propagating a circularly polarized Gaussian beam through a stressed engineered optical (SEO) window [21].The circular polarization component exiting the stressed window with the opposite handedness acquires a vortex along the axis, so a bottle can be constructed by making sure that the component emerging with the original polarization has an axial null.While SEO elements have been shown to have applications in the generation and detection of spatially varying polarization states [22][23][24][25][26], to date they have not been utilized to generate and optimize a three-dimensional bottle field, although studies on focal splitting [27] have shown fields with an intensity minimum surrounded by high intensity regions.
In this paper, we treat the problem of a general transparent birefringent mask placed in the pupil of an aplanatic high-NA focusing system.We derive the spatial variation of the mask for which a uniformly polarized incident beam is optimally converted into a high-NA bottle field.While such a mask could be fabricated using dielectric metasurfaces [28,29], we show that the optimal distribution very nearly corresponds to the simple combination of an SEO window followed by a half-wave plate.Preliminary experimental results are presented demonstrating the ability of this practical system to e ciently generate a bottle field without the need for interferometric superposition of di erent components.

System layout and notation
Consider the optical system shown in Fig. 1, in which a collimated monochromatic light source with uniform polarization is focused by an aplanatic lens with focal length f and numerical aperture NA.A thin, transparent birefringent mask (BM) is placed at the back focal (pupil) plane of the system.The radial coordinate in the pupil plane maps onto the focusing angle ⌘ after the lens as u = sin ⌘.The azimuthal pupil coordinate is denoted by , so that the spatial frequencies over the pupil may be represented using the two-dimensional vector u = (u x , u y ), which corresponds to the transverse part of the three-dimensional unit direction vector after the lens AE u = (u x , u y , u z ) = (u cos , u sin , (1 u 2 ) 1/2 ).The lens is assumed to be in air, so that u 2 [0, NA].The results presented in later sections can be modified for an immersed imaging system by changing the limits of integration to [0, NA/n], where n is the refractive index of the immersion medium.The focusing lens produces a field distribution over the spatial coordinate AE r = (x, y, z).For a bottle field, the intensity must vanish at AE r = AE 0, the focal point of the lens.The polarization e ect of the lens is assumed to be a rotation of the electric field at each point about the direction of the radial pupil coordinate [30].The lateral incident field E(u) is transformed into a focused field AE E(AE r) containing both lateral and longitudinal components.The matrix that maps the pupil coordinate to the directional variables after the lens is given by [31] M lens = p cos ⌘ . (1)

General birefringent mask
The Jones matrix of a general spatially-variant, transparent, thin BM can be written as where p 1,2 (u) are the two eigenpolarizations at each point of the BM, (u) is half the phase mismatch between these eigenpolarizations, (u) is a global phase function, and ⌦ indicates an outer product.The fact that the mask is transparent means that this Jones matrix is unitary (that is, and are real and the eigenpolarizations p j are orthonormal); the fact that the mask is "thin" means that at the pupil plane its e ect is local.Up to an arbitrary global phase, the eigenpolarizations of J may be written in the form where the functions ⇥(u) 2 [ ⇡/2, ⇡/2] and (u) 2 [0, 2⇡] are the latitude and longitude angles of p 1 (u) over the Poincaré sphere.Substituting into Eq.( 2), the Jones matrix then becomes For the special case of an SEO window with trifold symmetric stress, the retardance depends only on the radial pupil coordinate according to (u) = bu, where the stress coe cient b is proportional to the applied force [21,27].Because the birefringence results from stress, the eigenpolarizations at all points are linear (thus ⇥ = 0), and the principal stress direction rotates such that = .A half-wave plate can be inserted after the stressed window to reverse the direction of rotation to = .

Bottle field properties
In Sections 5 through 7 we find the optimal spatial variation of the BM that produces a bottle field with the sharpest possible intensity null.First we must establish the necessary quantities to optimize over, namely: (i) some measure for the width of the bottle, and (ii) the conditions under which the intensity vanishes at the focal point.

Focused intensity distribution
The focused field is the Fourier transform of the pupil distribution, so the normalized intensity is given by [31] where k = 2⇡/ is the wavenumber, A(u) is an envelope function accounting for the shape of the incident beam and the aperture (assumed to have rotational symmetry), E 0 is the polarization of the incident field, and kAE vk denotes the Euclidean norm of a vector AE v.
The normalization factor of (⇡k) 2 in this expression has been chosen for computational convenience.Without loss of generality, we may assume a right circularly polarized input field E 0 = (1, i)/ p 2, since any other polarization could be converted into this one using a wave plate, whose e ect can be absorbed into the BM.

Second derivatives of intensity
To achieve the maximum transverse and longitudinal sharpness of the null in intensity, we must maximize the second derivatives I x i x i evaluated at the origin, which are inversely proportional to the squared widths of the bottle in each dimension x i .Starting from Eq. ( 5), we find that The oscillations of the global phase factor exp(i ) contained within J will result in cancellations between the contributions to this integral from di erent parts of the pupil.Therefore, the second derivatives are as large as possible when = 0. Also, in practice it is di cult to fabricate an element with varying ⇥(u) since most real-world devices have linear eigenpolarizations.For this reason, we restrict our optimization to birefringent masks with ⇥ = 0.Moreover, we strongly suspect that ⇥ = 0 is in fact the optimal solution.Under these simplifications, .
(7) In order to avoid a discontinuity at the origin and match the periodic boundary condition on (u), we assume that (u) only depends on the radial pupil coordinate and that where m is an integer.(Note that m = 2q, where q is the topological charge of the pattern of eigenpolarization orientations of the mask.)In the paraxial limit, a bottle field can readily be produced for m = ±1.In the nonparaxial case, however, the e ects of orbital and spin angular momentum counteract each other only when m = 1 [32,33].Otherwise, the bottle becomes filled with light due to constructive interference between the various contributions over the pupil to the longitudinal component of the field.While it is possible to derive conditions guaranteeing a null in intensity for m , 1, these conditions would be di cult to satisfy and would come at the cost of broadening the spatial extent of the focused field.In the derivation that follows we assume that = , and we optimize the remaining function (u) to produce a bottle field whose minimum in intensity is as narrowly distributed as possible.For a brief discussion of the results obtained for general m, refer to Appendix A. Substituting Eqs. ( 7) and (8) into Eq.( 6) and integrating over 2 [0, 2⇡], the second derivatives of intensity become where this result may be restated as

Bottle beam constraint
The intensity at the focal point can be calculated by substituting Eqs. ( 7) and ( 8) into Eq.( 5) and integrating over .This leads to the following requirement for a bottle field:

Functional form of optimal BM retardance distribution
For the optimization problem one can consider various combinations of I xx , I yy , and I zz as a measure of the localization of the bottle field.In Sections 6.1 through 6.4 we will propose four di erent merit functions M 1 through M 4 and solve for the conditions under which each M j attains its minimum value.Using the method of Lagrange multipliers, the retardance distribution j (u) that minimizes each merit function M j is found by solving the system of equations where the Lagrange multiplier ⇤ is an arbitrary constant and @/@ denotes a functional derivative with respect to .In Section 6, we will find that each optimal solution takes the form where c j , d j , and e j are constants.Substituting where the new constants g j and h j are mutually determined by Eqs.(13a) and (13b).The function j (u) can be interpreted as the angle traced by the upper half of the elliptic curve {g j + h j (1 u 2 ) 1/2 , u} over the interval u 2 [0, NA], as illustrated in Fig. 2. In order to avoid singular behavior of the BM at the center of the pupil, we need (0) = 0 and 0 (0) , 0. The former of these two constraints requires that g j + h j > 0, while the latter is satisfied as long as g j + h j , 0. Although the constants g j and h j depend on A(u), the functional form of the optimal solution is invariant to the shape of the input beam and aperture.If A(u) can be measured experimentally, then the solution can be tailored to compensate for radial nonuniformities in the beam profile.

Lagrange multiplier constraints for optimal retardance
In this section we evaluate Eq. (13a) for each merit function M j and simplify the result to obtain a constraint on the coe cients g j and h j .In these calculations we make use of the functional derivatives @ @ ↵ nm`,c = A⇠ nm`s in , Denominator of arctan Retardance Fig. 2. Illustration of the geometrical meaning of j (u).Dashed lines are drawn to indicate a sample point u 0 on the elliptic curve (left) and the corresponding point on the retardance distribution (right).

Merit function #1: Laplacian of intensity
Let us first minimize the merit function M 1 = 1/(I xx + I yy + I zz ) 1/2 .To simplify the calculation, we solve the equivalent problem of maximizing 1/M 2  1 , which is the Laplacian of the intensity at the focal point of the bottle field.Using the Lagrange multiplier method, the condition for optimality is @ @ ⇥ Applying the results of the previous sections, this becomes Upon simplification, the solution for 1 takes the form of Eq. ( 15), subject to the constraints shown in the first row of Table 1.

Merit function #2: characteristic length
We next consider the merit function , which represents the length along the diagonal of a box "containing" the bottle.This quantity is referred to as the characteristic length.For the sake of simplicity we minimize M 2 2 , which is equivalent to minimizing M 2 .Substituting into Eq.(13a) and applying the chain rule, Upon simplification, the solution for 2 takes the form of Eq. ( 15), subject to the constraints shown in the second row of Table 1.

Merit function #3: volume of bottle
Next we consider the merit function , which is proportional to the width of the intensity profile in each dimension, thus representing in some sense the "volume" of the bottle.We will solve the equivalent problem of maximizing 1/M 2 3 .Substituting into Eq.(13a), the condition for optimality is  15), subject to the constraints shown in the third row of Table 1.

Merit function #4: cross-sectional area
Lastly, we consider the merit function M 4 = 1/(I xx I zz ) 1/2 , which is proportional to the crosssectional area of the bottle in the x-z plane.In contrast to M 3 , this merit function places equal emphasis on the transverse and longitudinal derivatives.Substitution of 1/M 2 4 into Eq.(13a) yields the condition The remainder of the derivation proceeds similarly to the previous case, leading to a solution for 4 of the form of Eq. ( 15), subject to the constraints shown in the final row of Table 1.

Numerical solutions
For each merit function, we now have a pair of constraints determining the coe cients g j and h j that minimize M j .The solutions for g j and h j can be found by numerically solving each set of constraints over a discrete set of NA values.A second solution with identical performance can then be found by flipping the signs of both g j and h j .With respect to the geometrical interpretation (recall Fig. 2), this amounts to a reflection of the elliptic curve over the vertical axis.To ensure that (0) = 0, we select the solution for which g j + h j > 0. The results obtained for each merit function are plotted as a function of NA in Fig. 3 function A(u) = 1.The solutions do not change significantly if A(u) is instead assumed to be, say, a Gaussian envelope.
For comparison, recall that the retardance distribution of an SEO window is a linear function L (u) = bu, and that the azimuthal angle when used in combination with a half-wave plate is = .In this case, the bottle field constraint given in Eq. ( 12) becomes Since the integrand contains a periodic function, this equation has infinitely many solutions for the stress parameter b.The smallest solution b 0 is most practical to fabricate (since it requires the least applied force) and produces the best results when used to generate a bottle field.The numerical solution for b 0 (assuming A(u) = 1) is plotted as a function of NA in Fig. 3(b).
The optimized retardance distributions 1 (u) through 4 (u) and the linear solution L (u) are plotted in Fig. 4 for six di erent NA values.Note that merit function M 1 is minimized by the spatially uniform retardance distribution 1 (u) = ⇡/2, which could be produced by a q-plate with half-wave retardance and topological charge q = 1/2 [34].As the NA approaches 1, the other optimized solutions 2 , 3 , 4 each become increasingly similar to L .

Evaluation of numerical solutions
The solutions found in the previous section are now evaluated in terms of the transverse and longitudinal widths of the bottle fields that they produce, which are inversely proportional to the second derivatives of intensity: As shown in Appendix B, for small numerical apertures I xx / NA 6 and I zz / NA 8 .This implies that in the paraxial limit z x , as one would expect.The transverse and longitudinal widths of the bottle field (normalized according their respective NA dependences) are shown in Fig. 5.The ratio z / x becomes closer to unity as the NA increases, as seen in Fig. 6.At NA = 1, the solution 2 produces the best balance between the widths in each dimension, with z / x = 1.596.In comparison, the linear solution produces z / x = 2.616.Again, these results assume a uniform envelope function A(u) = 1.If a Gaussian envelope A(u) = exp( u 2 ) is used instead, the width of the bottle in each dimension increases by 23% for L and by up to 30% for 1 through 4 in the worst-case scenario (NA = 1).
The performance of each solution can be visualized by plotting the cross-section of the theoretical intensity distribution in the x-z plane.The intensity profiles produced by each retardance distribution are shown in Fig. 7.These plots were simulated under the simplistic assumption that the distribution of the stressed window at the pupil gets directly mapped onto the NA disk, without any aberrations or Fresnel coe cients.Note the slight asymmetry between the upper and lower halves, due to a slight trigonal deviation from rotational symmetry of the intensity profile.In the case of 1 , the uniform retardance distribution simply writes a vortex through spin-orbit interaction, with zero intensity along the entire z-axis.This shows that a q-plate   with = ⇡/2 and q = 1/2, which has previously been used for vortex beam generation [35], produces the optimal vortex with the sharpest possible null at its center.However, it also implies that merit function M 1 is not really appropriate for qualifying bottle fields since the longitudinal confinement can be sacrificed completely in favor of the transverse one.The remaining solutions 2 , 3 , 4 , and L all produce bottles with similar appearances, although some subtle di erences can be noted.For instance, the intensity lobes along the longitudinal axis are slightly dimmer for L than for 2 and 4 .Nevertheless, these results demonstrate that a stressed window can be used to generate a bottle field with nearly optimal sharpness.

Experimental results
The generation of a bottle field using an SEO window was achieved experimentally using the system shown in Fig. 8.A linear polarizer and quarter-wave plate were used to create a circularly polarized input state.After passing through the SEO and a half-wave plate, the field was focused by a lens with a focal length of 100 mm.An input beam diameter of 3 mm was used, resulting in an e ective NA of 0.015.This low numerical aperture was required in order to create a bottle field with a large enough transverse cross-section (50 µm in diameter) to be resolved by a CMOS camera (5 µm pixel size).To characterize a high-NA bottle field focus, one could raster-scan a nanoparticle through the laser focus and detect the scattered intensity for every particle position.Similarly, a quantum emitter, such as a quantum dot or a fluorescent molecule, could be raster-scanned and the emitted fluorescence detected.These experiments are in the works and will be published elsewhere.Note that for the low-NA measurement discussed here, it is not necessary to insert a half-wave plate after the stressed window since in the paraxial limit the longitudinal component of the field vanishes at focus for both = ± .For higher numerical apertures, however, the half-wave plate is strictly necessary.
The through focus transverse intensity distribution of the experimental bottle field is shown in Fig. 9. Since the interior width of the bottle is comparable to the pixel size of the sensor, an absolute null could not be measured.However, a local minimum in intensity was observed at focus, with a ratio of 0.43 measured between the intensity at the central pixel and the peak intensity of the surrounding lobes.This demonstrates in principle that the combination of an SEO window and a half-wave plate can be used to generate a bottle field.

Concluding remarks
In summary, we have derived the optimal spatial variation of a birefringent mask for generation of a tightly distributed optical bottle field using a high-NA focusing lens.We showed that a practical real-world device, the stress engineered optical element, is theoretically capable of producing a near-optimal bottle field when used in this configuration.The current experimental results verify that an SEO window can be used to generate a focused field with a local minimum of intensity on-axis.However, at low numerical apertures the resulting intensity distribution along the propagation axis has a far greater extent than what is required in optical trapping and microscopy applications.The experimental implementation of a high-NA system comes with the added challenges of compensating for aberrations and angle-dependent polarization e ects.
If these factors can be accommodated for, the SEO element has the potential to become an e cient and reliable mechanism for bottle field generation that avoids the need of splitting and recombining the input beam.

Appendix
A. Results for m , 1 In Sections 4 through 6 we optimized the retardance distribution of the BM under the assumption that = m with m = 1.Here we reproduce the main results of Section 4 for general m, an exercise which provides additional insight as to why m = 1 is the optimal choice.We also state the form of the solutions obtained for m = 3, which are very similar to the m = 1 case.

A.1. Second derivatives of intensity
Similarly to Section 4.2, the second derivatives of intensity are obtained by substituting Eqs. ( 7) and ( 8) into Eq.( 6) and integrating over .For m = 1, this leads to where ⇠nm`= u n (1 u 2 ) m/4 (1 p 1 u 2 ) `and ij is the Kronecker delta, equal to unity when i = j and zero otherwise.Note that with the exception of the m = 1 and m = 3 cases, the first two vector components contributing to I xx and I yy vanish, i.e., the transverse derivatives consist solely of higher-order contributions from the z-component of the field.This results in a field with a very large spatial extent, which is undesirable.For reasons discussed below, the field is also widely distributed when m = 1.In the remaining m = 3 case, the second derivatives given in Eq. ( 27) become where ↵ nm`,c is defined in Eq. ( 10) and

A.2. Bottle beam constraint
As in Section 4.3, the intensity at the focal point can be calculated by substituting Eqs. ( 7) and ( 8) into Eq.( 5) and integrating over .Repeating this process for general m, we find that for a bottle field For the m = 3 case, this reduces to the condition ↵ 111,c = 0, which is identical to the constraint for m = 1.For the m = 1 case, an additional constraint ↵ 210,s = 0 is necessary to ensure that the longitudinal component of the field vanishes at the focal point.However, this leads to a wider intensity distribution since the transverse derivatives given in Eqs.(26a) and (26b) each contain an ↵ 210,s term.This suggests that among all cases where m , 1, the best performance can be achieved using m = 3.
A.3.Functional form of optimal solution and constraints for m = 3 The optimization of (u) for the m = 3 case is very similar to Section 6.The solution obtained for each merit function takes the form where the constants c j , d j , and e j are mutually determined by the constraints shown in Table 2.Note that in this case the transverse second derivatives consist of an integral with numerator 1 (1 u 2 ) 1/2 , whereas in the m = 1 case the integral depends on 1 + (1 u 2 ) 1/2 .These quantities can be interpreted as the volumes beneath the southern and northern hemispheres, respectively, of a unit sphere sitting on a plane.Therefore, the m = 1 case can be expected to yield larger second derivatives of intensity and consequently the narrowest possible bottle field.

B. NA dependence of second derivatives of intensity in the paraxial limit
The Taylor series expansions

Fig. 1 .
Fig.1.Schematic of system layout for bottle field generation, in which a thin birefringent mask is placed in the pupil plane of an aplanatic lens.The coordinate in the pupil plane is specified in terms of the focus angle ⌘.

Fig. 3 .
Fig. 3. Numerical solutions for (a) each pair of optimized coe cients g j and h j and (b) the lowest-order stress parameter b 0 required to produce a bottle field.

Fig. 4 .
Fig. 4. Comparison of optimized and linear retardance distributions for generating a bottle field, shown for six di erent NA values as indicated on the plots.From top to bottom at the right edge of each plot, the curves appear in the following order: 2 , 4 , 3 , L , 1 .

Fig. 5 .Fig. 6 .
Fig. 5. Logarithm of the transverse (left) and longitudinal (right) widths of the bottle fields generated by each solution, normalized by factors of NA 3 and NA 4 , respectively.The longitudinal width of the beam produced by 1 is not shown because it is infinite.

Fig. 7 .
Fig.7.Simulated cross-sections over the x-z plane of the focused intensity distributions generated by each solution.The results obtained using the optimized solutions 1 through 4 (a-p) and the linear solution L (q-t) are each shown for four di erent numerical apertures.The solid green curves show the intensity profiles in the x (vertical) and z (horizontal) dimensions.The dashed yellow circle is added for scale, with a radius of one wavelength.

Fig. 9 .
Fig. 9. Evolution through focus of the optical bottle beam's transverse intensity profile over a 2 cm distance along the propagation axis.The intensity distribution at focus has a diameter of 50 µm, matching simulations.

Table 1 .
Constraints for optimality of each merit function.

Table 2 .
Constraints for optimality of each merit function for the m = 3 case.