Uncertainty Principle Applied to Focused Fields and the Angular Spectrum Representation

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1 Uncertainty Principle Applied to Focused Fields and the Angular Spectrum Representation Manuel Guizar, Chris Todd Abstract There are several forms by which the transverse spot size and angular spread of an optical field can be defined. In this paper we explore different properties of commonly used measures and determine whether they are useful for near field optics applications. A novel measure and uncertainty relation between direct and reciprocal space are proposed and applied to the paraxial Gaussian beam and the inhomogeneous dipole field with and without evanescent waves. Analytical and numerical results are obtained and discussed. I. PROBLEM STATEMENT ONE of the major issues in nano-optics, classical diffraction theory and Fourier optics is the relation between how tight a field, f(r), can be focused based on general characteristics of its angular spectrum, F(k x, k y, z), which is related to f(r) through a two dimensional Fourier transform. There is a well accepted notion that the minimum size of a field relates to the extent of its angular spectrum through an uncertainty relation. In other words, we wish to address the problem of how the lateral resolution of a diffraction limited system, or spot size of the field emitted by a near field optical probe, can be determined and how it is related to general properties of its angular spread. There is not only controversy over what determines a good measure for the lateral spread but also over a rigorous definition of an uncertainty relationship between the spatial and angular spreads that is suitable for non-paraxial fields, if one even exists. An appropriate measure should work well for important cases such the paraxial Gaussian beam and dipole radiation field. The measure of spatial width should be easy to compute, well defined and must correctly characterize the lateral width of the field. It is also desirable to have a relationship between the spatial and angular widths that determines how the angular and spatial spread relate to one another. In quantum mechanics the measure of the spatial and momentum spreads are given by the second moment of the field and its Fourier transform, σ k x = σ x = x f(r) dxdy f(r) dxdy, () k x F(k x, k y, z) dk x dk y F(kx, k y, z) dk x dk y, () where all integral limits are from to unless otherwise specified. The familiar result derived by Heisenberg (which holds for any Fourier transform pair) is known, in the context of quantum mechanics, as the uncertainty principle, σ x σ kx. (3) This is interpreted as the inability to know the momentum and position of a particle with infinite accuracy. To know position accurately, momentum must be uncertain and vice versa. In optics this can be interpreted as a relation between the lateral width of a field and its angular spread, which cannot both be arbitrarily small. Equations () and () make good measures for the width in many cases. However, in the context of fields with angular spectrum that extends beyond the circle of homogeneous waves, the spectrum presents a discontinuity on the first derivative upon propagation, i.e. the gradient there is not well defined. This discontinuity on the spectrum reduces the asymptotic decay of the field such that its variance diverges even when the field itself carries finite energy, which renders the variance approach useless in the context of near-field optics. An alternative way to define the resolution attainable in an optical system is through Abbe s Criterion. This definition of resolution is the distance from the maximum of the Airy function to it s first zero. Two points are said to be just resolved if the point spread functions overlap such that the maximum of one is on the first zero of the other. Therefore, the width is given by x λ 0 NA, (4) where λ 0 is the illumination wavelength and NA is the numerical aperture of the system. The problem with this measure of spread is that it only applies to a besinc field and is completely unrelated to Fourier pairs which means that there is no uncertainty relation. Since it is a phenomenological treatment of resolution for an ideal paraxial imaging system, it is hardly applicable in the context of nano-optics. Furthermore, any treatment based on function zeroes is inappropriate for near field applications since fields with evanescent components in its spectrum might not have nulls. Another example of measure can be found in a recent paper by A. Luis [], where an entropic measure is used for the spatial and angular widths. These measures are defined as and A x = A k = f(x) 4 dx, (5) F(kx ) 4 dk x. (6) Unfortunately, there is no known uncertainty relation between the two that results in an inequality such as (3). This means that we cannot tell how the widths control one another upon propagation or if they control each other at all. This result

2 (or lack thereof) renders the two measures of width useless because we have no way of telling how they are limited. Another measure that we considered was the full width half maximum of both the field and the spectrum. Aside from the obvious difficulties introduced by trying to measure the angular spectrum, the uncertainty product between these two widths has been shown to be greater than or equal to 0. Regrettably, this means that the two measures are unrestricted, e.g. both can be arbitrarily small. II. INHOMOGENEOUS AND HOMOGENEOUS DIPOLES We chose to study the inhomogeneous dipole field since it constitutes the emission of an ideal near field optical probe, or tip []. This field is a solution of the inhomogeneous Helmholtz equation with a Dirac delta dipole source oriented along the z axis at z = 0, its spatial spread is thus expected to be zero as we approach the source. The field of the inhomogeneous dipole and its angular spectrum are given by [3] f(r) = i k F(k x, k y, z) = ( ik ) e ikr z r r r, (7) π ik eiz k k ρ, z > 0, (8) where k = π/λ 0 is the light wavenumber and k ρ = k x + k y. In order to determine the effect of the evanescent terms on the resolution limit (spot size) we will also analyze what we define as the homogeneous dipole field, which has the same angular spectrum as the dipole field but where the evanescent terms are truncated. This field is thus a solution of the homogeneous Helmholtz equation and can be realized by a focused field in free-space. Although we have an analytic expression for its angular spectrum, the form of the field must be computed numerically. It should be noted at this point that, although both dipole fields are reasonably well confined transversely at any propagation distance, computation of the usual variance measure through equations ()-(3) leads to divergent results upon an infinitesimal propagation distance. This failure is an additional motivation to find a better suited estimation of spot size. III. THE PROPOSED MEASURE In this novel spread measure, the field intensity and the spectrum autocorrelation are used as the Fourier pairs for Heisenberg uncertainty relation, x = x f(r) 4 dxdy f(r) 4 dxdy (9) kx = k x F F(k x, k y, z) dk x dk y F F(kx, k y, z) dk x dk y (0) where F F(k x, k y, z) is the autocorrelation of the spectrum. The constant factors and / where introduced so that this measure agrees with the usual variance spread, equations ()- (3), for the transverse beam waist of a Gaussian beam. The initial motivation to use this measure, is that the fourth power of a function that decays as ρ 3/, where ρ = x +y, has a second moment that will converge since the radial integrand will decay as ρ. Additionally, these widths are easy to measure in an experiment, provided that the field intensity can be measured and sampled adequately. This is easy to understand if we recall that an intensity measurement provides enough information to compute the spectrum autocorrelation through a Fourier transform. IV. UNCERTAINTY OF GAUSSIAN BEAMS AND RADIATING DIPOLE In this section we will discuss and compare some of the results that were obtained with the proposed measures for the paraxial Gaussian beam and the homogeneous and inhomogeneous dipole. It is important that any new proposed measure of spread or uncertainty relation correctly characterizes the paraxial Gaussian beam, since its propagation characteristics have been widely studied. For this beam, the intensity also behaves as a Gaussian and taking its inverse Fourier transform to obtain its autocorrelation is straightforward. Since analytical expressions are available for the field and its spectrum autocorrelation, calculation of Eqs. (9) and (0) is simple x = w(z), () k x = w(z), () where w(z) = w 0 + (z/zr ) is the spot size, w 0 is the spot size at the beam waist and z R = kw0/ is the Rayleigh range. While the obtained spatial width ( x) matches that obtained with the variance approach (σ x ) at any propagation distance, the spectral width ( k x ) only matches the spectrum variance (σ kx ) at z = 0. The exhibited change in the proposed spectral spread upon propagation can be surprising if one considers that paraxial propagation does not affect the amplitude of the spectrum at any point. Although the magnitude of the spectrum remains unchanged, the changes on its phase distribution produce a variation in the amplitude of its autocorrelation, thus making the proposed measure for the spectral spread sensitive to propagation even in the paraxial regime. Notice that the paraxial Gaussian beam remains a minimum uncertainty field at any propagation distance. An analytic solution of Eq. (9) for the inhomogeneous dipole is possible by using the analytic expression for the field Eq. (7). However, the autocorrelation of the spectrum as given in Eq. (8), is not trivial to compute. Although equation (0) is written as an autocorrelation, this integral can be obtained by computing the second derivative of the field intensity. Equation (0) can be rewritten as kx = ikx F F(k x, k y, z) dk x dk y F F(kx, k y, z), (3) dk x dk y then by using F {ik x F F(k x, k y, z)} = d dx f(r), (4)

3 3 and Parseval s theorem we can rewrite k x in terms of the field intensity, d kx = dx f(r) dxdy f(r) 4. dxdy = f(r) d dx f(r) dxdy. (5) f(r) 4 dxdy The proposed spread measures for the inhomogeneous dipole are then found to be, x = a 3 + 0a + 0a 4 k + 30a + 0a4, a + 4a 4 (6) a + 70a4, (7) k x = k a where a kz. It is of interest to analyze the behavior of the spread of this beam in the limits of small and large z. When kz, the spot size approaches zero as x z/ and the angular spectrum spread diverges as k x z 5/4.035z. On the other hand, as kz the spot size grows linearly as x z/ 0.707z and the spectrum spread goes to zero as k x z 3/ z. Initially the field spot size grows at a slower rate than it does after propagation of a distance of several wavelengths. Although the spectral and spatial spreads diverge at z = 0 and kz respectively, the uncertainty product remains close to the minimum uncertainty value of 0.5. For the inhomogeneous dipole the product remains below its large z value of throughout propagation, having a value of 0.58 at z = 0. For the case of the homogeneous dipole, no analytic calculations of the spreads were possible. This is because we have neither an expression for the field intensity, nor for its autocorrelation. Instead we took a numerical approach, where we calculated both the field intensity and the angular spectrum autocorrelation using a numerical Hankel transform algorithm [4]. When calculating the spread of the autocorrelation we used 3000 points distributed in the range k ρ (0, k], where k ρ = k x + k y so that all the sampling points fell on the autocorrelation support. On the other hand, when computing the spatial spread, our frequency sampling space was distributed in the range k ρ (0, 30k], this still allowed us to radially sample the spectrum autocorrelation with 600 points while oversampling the field by a factor of 5. In both cases a trapezoidal approach was used to approximate the integrals. Figure (a) shows the computed spatial spread for both the homogeneous and inhomogeneous dipole. While the absence of evanescent components on the homogeneous field limits the minimum spatial spread that can be achieved (diffraction limit), the inhomogeneous dipole has a spread that truly goes to zero at z = 0. The computed measure for the field autocorrelation is depicted in figure (b). As expected, the homogeneous dipole Fig.. Proposed measures for (a) the spot size, (b) spectral spread and (c) uncertainty product for the homogeneous and inhomogeneous dipole (blue and red line respectively). spectral spread has a finite value for z = 0 and decreases monotonically with propagation distance. Because of the loss of evanescent components, the proposed spreads for both of the homogeneous and inhomogeneous dipoles should asymptotically tend to be equal as the propagation distance increases. From direct inspection of figure (a) and (b), it is clear that the difference introduced to the field by the evanescent components ceases to be very significant after a propagation distance of just a wavelength. An interesting feature that can be observed is that around z = 0.4λ there is a crossing of both the spatial and spectral spreads of the homogeneous and inhomogeneous dipole, resulting in a smaller spot size for the homogeneous field. This might be counterintuitive if one considers that so close to the source there is still a significant amount of evanescent waves present on the inhomogeneous field and indicates that the presence of evanescent waves is actually decreasing the attainable resolution of the near field probe. This can be understood if one considers that a tight focus requires more than a broad spectrum, it also requires the constituent plane waves to be in-phase with one another. So, as the inhomogeneous field propagates, the homogeneous waves acquire a phase distribution while the evanescent part does not. At one point the evanescent waves are out of phase from the rest of the beam spectrum and contribute to increase the spot size. This also explains why the k x is smaller while the inhomogeneous spectrum is broader. The

4 4 fact that the evanescent part is out of phase with the rest of the plane-waves increases cancelation when computing the autocorrelation integral and thus its magnitude decreases. The uncertainty product for both dipole fields is shown in figure (c) although the beams considered are fundamentally very different and have different propagation characteristics, their uncertainty products remain relatively close to the minimum uncertainty throughout propagation. In particular, the product is very similar for both dipole fields at z = 0, even when the Fourier pairs for those fields are so different. At z = 0 the inhomogeneous field and angular spectrum are a Dirac delta and a constant, while the homogeneous pair consist of a Besinc function and a circle. Having an uncertainty relation, for which the uncertainty product remains near the minimum for fields that are so different, is a nice feature. This makes the uncertainty lower limit more meaningful when attempting to estimate the spot size of a beam given certain spread of its angular spectrum. V. MAXIMUM ANGULAR SPREAD It is also of interest to calculate an upper limit for k x, given a maximum spatial frequency k max for the field angular spectrum. This can be calculated by using a spectrum given by the sum of two Dirac delta functions F(k x, k y ) = δ(k x k max, k y ) + δ(k x + k max, k y ). (8) When computing the square of the autocorrelation of this function and substituting into equation (0), we find that all the integrals diverge, but since we can assume that they diverge at the same rate, we can obtain a well behaved limit. Namely, by assuming G(k x, k y )δ 4 (k x k max, k y )dk x dk y = lim AG(k max, 0), A (9) where A is a dummy variable that governs the rate at which the integral diverges, we obtain an upper limit for the spectral spread k x 3 k max 0.87k max. (0) This result can be compared to the spread obtained for an angular spectrum that consists of a circle of maximum radius k max, 9π k = 64 (3π 6) 0.390k max, () the obtained value is about half of the upper bound of the spread. This allows us to conclude that the upper bound is one that is possible to approach by realistic fields. This upper limit and the uncertainty limit for our proposed measure can be used to estimate the improved field localization that is achieved by including evanescent fields from total internal reflection. In a typical configuration for total internal reflection microscopy, homogeneous waves traveling in a dense medium induce evanescent waves as they undergo total internal reflection. In this case k max = kn and the minimum spot size is x k x λ 0 4πn λ 0 n, () where n is the index of refraction of the medium where the homogeneous waves travel. Again if we compare this lower limit to a field with an angular spectrum of a circle of radius kn we obtain a result that is within the same order of magnitude as the lower limit, x = λ 0 πn 6 3π 6 0.λ 0 n. (3) Again showing that this limit is reasonably easy to approach with a realistic field distribution. VI. CONCLUSION Through careful consideration of several spread measures and uncertainty relations it was determined that neither of them gathered the properties required for a desirable description of the spot size for near field optical fields and a clear and meaningful relationship to its corresponding angular spectrum. The proposed measure is based on a different choice of a Fourier pair (field intensity and spectrum autocorrelation) and the application of the usual uncertainty relationship for their variances. This particular choice ensures integral convergence for any field with finite energy, even when evanescent components are present, thus overcoming one of the main limitations of the usual variance approach. For the three fields considered in this paper, the obtained spatial spreads were observed and compared against their field transverse intensity distributions at several propagation distances. Although the sample fields used behave very differently upon propagation, the predicted spot size was always in very good agreement to what could be intuitively defined as a beam waist, furthermore, it always appeared close to the half width half maximum. Results obtained for the dipole fields and the maximum spectral spread rigorously show how the spot size of a field in vacuum can be greatly reduced when evanescent components of the field are present either because of the presence of an interface or a small radiating source. Although this approach appears reasonable, there are some disadvantages associated with it. The angular spectrum spread as defined by equation (0) does not have a straightforward interpretation and furthermore, it is not descriptive of the actual angular spread of the beam, i.e. the proposed uncertainty relation is not between position and direction of rays. Unlike the usual variance spread, the proposed measure is insensitive to a change in the phase of the field, although one would expect the spectral spread to increase if the field phase is changed, both the field intensity and its autocorrelation remain unchanged. Additionally, the spectral spread k x will diverge if the field is considered at a conducting hard aperture and Kirchhoff approximation is used. This problem however might be solved by using strict boundary conditions. REFERENCES [] A. Luis, Gaussian beams and minimum diffraction, Opt. Lett. 3, (006). [] L. Novotny and B. Hecht, Principles of Nano-Optics, First Edition, Cambridge (006).

5 [3] R. Borghi, On the angular-spectrum representation of multipole wave fields, J. Opt. Soc. Am. A, (004). [4] M. Guizar-Sicairos and J. C. Gutiérrez-Vega, Computation of quasidiscrete Hankel transforms of integer order for propagating optical wave fields, J. Opt. Soc. Am. A, (004). 5