Abstract
An optical bottle field containing a three-dimensional intensity null at the focal point can be generated by placing a spatially inhomogeneous birefringent mask at the pupil of an aplanatic high-NA focusing system. We derive the optimal birefringence distribution for which a uniformly polarized input beam is converted into a bottle field with the sharpest possible null in intensity. We show that a stress engineered optical (SEO) window, which has a radially varying retardance, followed by a half-wave plate, performs nearly as well as the optimal solution. Experimental results corroborate that an SEO element can be used to generate a bottle field.
© 2017 Optical Society of America
Corrections
Anthony Vella, Hippolyte Dourdent, Lukas Novotny, and Miguel A. Alonso, "Birefringent masks that are optimal for generating bottle fields: erratum," Opt. Express 25, 19654-19654 (2017)https://opg.optica.org/oe/abstract.cfm?uri=oe-25-16-19654
24 April 2017: Corrections were made to the body text and Refs. 27 and 31.
1. 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 single-beam gradient-force optical trap, known as “optical tweezers” [1–3]. Nowadays, optical tweezers have found various applications, ranging from cooling and trapping atoms to manipulating live bacteria and viruses [4–6]. The achievement of an efficient 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 efficiency 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 [14–17]. 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 [22–26], to date they have not been utilized to 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 [27, 28], 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 efficiently generate a bottle field without the need for interferometric superposition of different components.
2. 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 η after the lens as u = sin η. The azimuthal pupil coordinate is denoted by ϕ, so that the spatial frequencies over the pupil may be represented using the two-dimensional vector u = (ux, uy), which corresponds to the transverse part of the three-dimensional unit direction vector after the lens u⃗ = (ux, uy, uz) = (u cos ϕ, u sin ϕ, (1 − u2)1/2). The lens is assumed to be in air, so that u ∈ [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 r⃗ = (x, y, z). For a bottle field, the intensity must vanish at r⃗ = 0⃗, the focal point of the lens. The polarization effect of the lens is assumed to be a rotation of the electric field at each point about the direction of the radial pupil coordinate [29]. The lateral incident field E(u) is transformed into a focused field E⃗(r⃗) containing both lateral and longitudinal components. The matrix that maps the pupil coordinate to the directional variables after the lens is given by [30]
3. General birefringent mask
The Jones matrix of a general spatially-variant, transparent, thin BM can be written as
where p1,2(u) are the two eigenpolarizations at each point of the BM, δ(u) is half the phase mismatch between these eigenpolarizations, Γ(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, Γ and δ are real and the eigenpolarizations pj are orthonormal); the fact that the mask is “thin” means that at the pupil plane its effect is local. Up to an arbitrary global phase, the eigenpolarizations of 𝕁 may be written in the form where the functions Θ(u) ∈[−π/2, π/2] and Φ(u) ∈[0, 2π] are the latitude and longitude angles of p1(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 δ(u) = bu, where the stress coefficient b is proportional to the applied force [21, 31]. Because the birefringence results from stress, the eigenpolarizations at all points are linear (thus Θ = 0), and the principal stress direction rotates such that Φ = ϕ. A half-wave plate can be inserted after the stressed window to reverse the direction of rotation to Φ = −ϕ.4. 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.
4.1. Focused intensity distribution
The focused field is the Fourier transform of the pupil distribution, so the normalized intensity is given by [30]
where k = 2π/λ 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), E0 is the polarization of the incident field, and ‖v⃗‖ denotes the Euclidean norm of a vector v⃗. (Recall that u⃗ · r⃗ = ux x + uy y + uz z.) The normalization factor of (πk)−2 in this expression has been chosen for computational convenience. Without loss of generality, we may assume a right circularly polarized input field , since any other polarization could be converted into this one using a wave plate, whose effect can be absorbed into the BM.4.2. Second derivatives of intensity
To achieve the maximum transverse and longitudinal sharpness of the null in intensity, we must maximize the second derivatives Ixixi evaluated at the origin, which are inversely proportional to the squared widths of the bottle in each dimension xi. Starting from Eq. (5), we find that
The oscillations of the global phase factor exp(iΓ) contained within 𝕁 will result in cancellations between the contributions to this integral from different parts of the pupil. Therefore, the second derivatives are as large as possible when Γ = 0. Also, in practice it is difficult to fabricate an element with varying Θ(u) since most real-world devices have linear eigenpolarizations. For this reason, we restrict our optimization to birefringent masks with Θ = 0. Moreover, we strongly suspect that Θ = 0 is in fact the optimal solution. Under these simplifications,In order to avoid a discontinuity at the origin and match the periodic boundary condition on Φ(u), we assume that δ(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 effects 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 difficult 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 Φ = −ϕ, and we optimize the remaining function δ(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 ϕ ∈ [0, 2π], the second derivatives of intensity become
where . Introducing the notation this result may be restated as4.3. Bottle beam constraint
The intensity at the focal point can be calculated by substituting Eqs. (7) and (8) into Eq. (5) and integrating over ϕ. This leads to the following requirement for a bottle field:
5. Functional form of optimal BM retardance distribution
For the optimization problem one can consider various combinations of Ixx, Iyy, and Izz as a measure of the localization of the bottle field. In Sections 6.1 through 6.4 we will propose four different merit functions M1 through M4 and solve for the conditions under which each Mj attains its minimum value. Using the method of Lagrange multipliers, the retardance distribution δj (u) that minimizes each merit function Mj is found by solving the system of equations
where the Lagrange multiplier Λ is an arbitrary constant and ∂/∂δ denotes a functional derivative with respect to δ. In Section 6, we will find that each optimal solution takes the form where cj, dj, and ej are constants. Substituting u2 = [1 + (1 − u2)1/2][1 − (1 − u2)1/2] in the dj term, this simplifies to where the new constants gj and hj are mutually determined by Eqs. (13a) and (13b). The function δj (u) can be interpreted as the angle traced by the upper half of the elliptic curve {gj + hj (1 − u2)1/2, u} over the interval u ∈ [0, NA], as illustrated in Fig. 2. In order to avoid singular behavior of the BM at the center of the pupil, we need δ(0) = 0 and δ′(0) ≠ 0. The former of these two constraints requires that gj + hj > 0, while the latter is satisfied as long as gj + hj ≠ 0. Although the constants gj and hj 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.6. Lagrange multiplier constraints for optimal retardance
In this section we evaluate Eq. (13a) for each merit function Mj and simplify the result to obtain a constraint on the coefficients gj and hj. In these calculations we make use of the functional derivatives
6.1. Merit function #1: Laplacian of intensity
Let us first minimize the merit function M1 = 1/(Ixx + Iyy + Izz)1/2. To simplify the calculation, we solve the equivalent problem of maximizing , 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 δ1 takes the form of Eq. (15), subject to the constraints shown in the first row of Table 1.6.2. Merit function #2: characteristic length
We next consider the merit function M2 = (1/Ixx + 1/Iyy + 1/Izz)1/2, 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 , which is equivalent to minimizing M2. Substituting into Eq. (13a) and applying the chain rule,
since Ixx = Iyy. Applying the results of the previous sections, this becomes Upon simplification, the solution for δ2 takes the form of Eq. (15), subject to the constraints shown in the second row of Table 1.6.3. Merit function #3: volume of bottle
Next we consider the merit function M3 = 1/(Ixx Iyy Izz)1/2, 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 . Substituting into Eq. (13a), the condition for optimality is
since Ixx = Iyy. Applying the results of the previous sections, this becomes where a factor of has been absorbed into the arbitrary constant Λ. Upon simplification, the solution for δ3 takes the form of Eq. (15), subject to the constraints shown in the third row of Table 1.6.4. Merit function #4: cross-sectional area
Lastly, we consider the merit function M4 = 1/(Ixx Izz)1/2, which is proportional to the cross-sectional area of the bottle in the x–z plane. In contrast to M3, this merit function places equal emphasis on the transverse and longitudinal derivatives. Substitution of into Eq. (13a) yields the condition
The remainder of the derivation proceeds similarly to the previous case, leading to a solution for δ4 of the form of Eq. (15), subject to the constraints shown in the final row of Table 1.7. Numerical solutions
For each merit function, we now have a pair of constraints determining the coefficients gj and hj that minimize Mj. The solutions for gj and hj 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 gj and hj. 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 δ(0) = 0, we select the solution for which gj + hj > 0. The results obtained for each merit function are plotted as a function of NA in Fig. 3(a), assuming a uniform envelope 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 δL(u) = bu, and that the azimuthal angle when used in combination with a half-wave plate is Φ = −ϕ. 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 b0 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 b0 (assuming A(u) = 1) is plotted as a function of NA in Fig. 3(b).The optimized retardance distributions δ1(u) through δ4(u) and the linear solution δL(u) are plotted in Fig. 4 for six different NA values. Note that merit function M1 is minimized by the spatially uniform retardance distribution δ1(u) = π/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 δ2, δ3, δ4 each become increasingly similar to δL.
8. 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 Ixx ∝ NA6 and Izz ∝ NA8. This implies that in the paraxial limit σz ≫ σx, 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 σz/σx becomes closer to unity as the NA increases, as seen in Fig. 6. At NA = 1, the solution δ2 produces the best balance between the widths in each dimension, with σz/σx = 1.596. In comparison, the linear solution produces σz/σx = 2.616. Again, these results assume a uniform envelope function A(u) = 1. If a Gaussian envelope A(u) = exp(−u2) is used instead, the width of the bottle in each dimension increases by 23% for δL and by up to 30% for δ1 through δ4 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 coefficients. 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 δ1, 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 δ = π/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 M1 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 δ2, δ3, δ4, and δL all produce bottles with similar appearances, although some subtle differences can be noted. For instance, the intensity lobes along the longitudinal axis are slightly dimmer for δL than for δ2 and δ4. Nevertheless, these results demonstrate that a stressed window can be used to generate a bottle field with nearly optimal sharpness.
9. 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 effective 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 Φ = ±ϕ. 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.
10. 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 effects. If these factors can be accommodated for, the SEO element has the potential to become an efficient 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 Φ = mϕ 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 ϕ. For m = 1, this leads to
where as defined in Section 4.2. For all other integers m ≠ −1, where and Δij 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 Ixx and Iyy 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 αnmℓ,c is defined in Eq. (10) andA.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 ϕ. Repeating this process for general m, we find that for a bottle field
For the m = 3 case, this reduces to the condition α111,c = 0, which is identical to the constraint for m = −1. For the m = 1 case, an additional constraint α210,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 α210,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 δ(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 cj, dj, and ej 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 − u2)1/2, whereas in the m = −1 case the integral depends on 1 + (1 − u2)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
lead to the paraxial approximations Since a bottle field is subject to the constraint α111,c = 0, the first-order component of α131,c vanishes. Therefore, α310,c and α131,c are each proportional to NA4, and α211,s is proportional to NA3. For NA ≪ 1, the transverse derivatives given in Eq. (11) are dominated by the NA6-dependent term, while the longitudinal derivative inherits the NA8 dependence of .Funding
National Science Foundation (NSF) (PHY-1507278); Swiss National Science Foundation (SNF) (200020_165841).
Acknowledgments
The authors would like to thank Thomas G. Brown for providing the SEO element and for several discussions about birefringence and polarization (and bottle beams in particular) over the years. MAA thanks Pablo Loza for useful discussions. HD and LN thank Xavier Palou for assistance with setting up the experiment.
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