Nodal areas in coherent beams

Whereas nodal points having zero amplitude are well-known wave phenomena, destructive interference across a continuous area is rare. In fact, these strange states seem to defy nature ’ s “ abhorrence of nothingness. ” Philoso-phizing aside

Perhaps no other phenomenon distinguishes waves from particles better than total destructive interference.Interference provides the basis for an exhausting list of technologically important devices in both classical and quantum wave systems.Any new approach that provides a different means to achieve interference is therefore of great general interest.For example, the advent of optical vortices [1][2][3][4] has lead to applications in areas as diverse as astronomical and subresolution imaging [5][6][7], spinning Bose-Einstein condensates [8], and quantum entanglement [9].A vortex wave function is distinguished by a nodal point in the beam cross section, around which the scalar phase varies azimuthally by a nonzero integer multiple of 2π.Zero-valued states in a system and their asymptotic limit are intriguing manifestations of nature [10] and attract considerable experimental and theoretical attention.Vortex and other harmonic wave functions were significant in the advancement of quantum mechanics and superfluid states [11][12][13].Extensions of these topics spawned interest in fluid-like phenomena in lasers [2], nonlinear refractive media [4], and optical orbital momentum [14].The remarkable discovery of the formation of a nodal area by use of a vortex [5,6,15] or four-quadrant [16] phase mask could similarly excite widespread interest.In those cases a circular nodal area was generated when the initial beam was defined by a uniformly illuminated circular aperture.
The motivation of this report is to explore whether other types of apertures and phase masks afford the formation of nodal areas.In fact, we make the conjecture that any apertured beam of finite support may be optically transformed with a lossless phase element such that support of the final beam is complementary to the original.Loosely phrased, we assert that a beam can be "turned inside out" without loss of power by use of a phase mask in the focal plane, producing a black hole in the transverse cross section of the beam.(Diffraction of this hole is shown in Supplement 1.) An analytical transformation that turns an apertured beam inside out was first realized for a circular aperture and was the basis of a high contrast coronagraph [5,6,16].An elliptically apertured beam was later proposed to produce an elliptical nodal area [17]; experimental verification is reported here by use of a so-called q-plate [18,19].Two confirmed experimental examples, however, do not constitute a proof of the conjecture.As further evidence we provide an analytical solution for a wide variety of azimuthally invariant apertures that can be turned into nodal areas by use of a vortex phase mask.Additionally, we provide numerical examples using a point-by-point iterative phase-retrieval algorithm [20,21] to design phase elements for an aperture of arbitrary shape.Developing a proof of the conjecture for any aperture function is, as for related phase retrieval problems, complicated by the underdetermined nature of the problem.
The discovery of other closed structures admitting nodal areas may be useful in optics for high contrast imaging applications, optical patterning, or light-matter interactions.We point out that the zero-amplitude state is an idealized concept that in practice can be approached only asymptotically owing to limitations such as wave front aberrations, nonparaxial waves, the degree of coherence [22], and quantum mechanics [23].
Let us first review how a circular nodal area ("ring of fire" or corona) is produced by use of a vortex phase mask [5,6,15].A circular aperture of radius a transforms into an isomorphic circular nodal area in a 4f optical system, illustrated in Fig. 1, by centering the vortex mask with the Airy disk in the focal plane formed with lens L1.Transmission of the optical field through the vortex element is represented by the function t m x 0 ; y 0 expimθ 0 , where m is a nonzero even integer called the topological charge, and θ 0 is the azimuth in the transverse x 0 ; y 0 plane (i.e., tan θ 0 y 0 ∕x 0 ).Various methods may be used to produce the phase mask, including photoalignment of nematic liquid crystals [18,19].The transformed aperture function in the x 00 ; y 00 plane may be understood as a convolution of the circular aperture with the Fourier transform (FT) of the vortex mask transmission function [24] FTft m x 0 ; y 0 g γ m 1∕r 00 2 expimθ 00 , where r 00 and θ 00 are polar coordinates in the x 00 ; y 00 plane, γ m −i m mf ∕k, f is the focal length of the lenses, k 2π∕λ, and λ is the wavelength.Rather than re-imaging the aperture, this system remarkably diffracts all the light outside the aperture, resulting in a black nodal area where the field is zero-valued in the interior region r 00 < a.In the case where m 2, the exterior field is given by Ex 00 ; y 00 a∕r 00 2 expi2θ 00 (see Supplement 1 for derivation).
For any linear superposition of vortex transmission functions having nonzero even values of topological charge, a circular nodal area is expected to form [24].Such a superposition, however, does not change the circular size or shape of the nodal area.Therefore, there is not a unique means of transforming a circular aperture into a nodal area.This suggests that for other aperture types, there may also be families of modes that represent nodal areas.
As an important example, consider a uniformly illuminated elliptical aperture.The modes that produce a nodal area can be found by mapping the circular modes onto the ellipse [17].A focal plane phase element that accomplishes this is described by a skewed vortex mask transmission function: tm x 0 ; y 0 expimΦ, where Φ arctanby 0 ∕ax 0 , where m is a nonzero even integer, and a and b are, respectively, the major and minor axes of the aperture.
Experimental Verification: The required phase Φ was produced by means of a q-plate; i.e., a spatially variant half-wave retarder with the fast axis orientation described by αx 0 ; y 0 q arctanby 0 ∕ax 0 α 0 , where q jm∕2j and α 0 are constants (see Supplement 1 for fabrication details) [18,19].The transmitted circularly polarized components of the field may be expressed E R;L e i2α E L;R , where E R and E L are, respectively, the right and left hand circular polarization components of the incident field.Interestingly, the two output components each have a vortex phase and a polarization orthogonal to the input state [25].For m 2, the fields in the exit pupil plane may be written U R;L x 00 ; y 00 E L;R 1∕ρ 00 2 expi2Θ, where ρ 00 > 1, Θ arctanay 00 ∕bx 00 , and ρ 00 x 00 ∕a 2 y 00 ∕b 2 1∕2 .Both polarization components are zero-valued if ρ 00 < 1.
For our experiment (see Fig. 1), an expanded collimated He-Ne laser beam (λ 632.8 nm) was made circularly polarized by use of a linear polarizing filter followed by a quarterwave retarder (P1).The central nearly uniform region of the beam was then transmitted through an elliptical aperture (AP) with major and minor axes a 1.50 mm and b 0.75 mm [see Fig. 2(a)].The elliptical beam was focused by lens L1 onto the voltage-controlled q-plate, with which we achieved halfwave phase-retardation condition with a 6.6 V, 6 kHz sinusoidal signal [19].The diameter of the focal spot was 515 μm along the x 0 axis and 1030 μm along the y 0 axis.We note that the focal spot is at least 20 times larger than the 25 μm central deformation of the q-plate.Lenses L1 and L2 both have a focal length of 1 m, and diameters 25.4 and 50.8 mm, respectively.The field lens L2 imaged the entrance pupil on a CCD array in the x 00 ; y 00 plane.An image of the entrance pupil forms on the detector array when the q-plate is removed from the system, or when the center of the q-plate is displaced from the center of the beam, as shown in Fig. 2(a).However, when these centers coincide, an elliptical nodal area appears, as seen in Fig. 2(b).A circular polarizer analyzer (P2) was used to remove light that may have been transmitted in the wrong polarization state, owing to irregularities in the q-plate.A quantitative measure of the nodal area contrast is the fraction of the total beam power across the innermost half of the nodal region.We achieved an experimental value of 0.24%-a remarkable result Fig. 1.Optical configuration for producing a nodal area with a q-plate (QP).Uniform laser light that is circularly polarized by polarizing optics P1 enters an aperture AP at the x, y plane.Lens L1 focuses the transmitted light onto QP.Lens L2 forms the exit pupil at the x 00 ; y 00 plane.The second set of polarizing optics, P2, is circularly cross polarized to P1.The nodal area appears at the x 00 ; y 00 plane, where an image is captured by a CCD.given that this is the first reported measurement of an elliptical nodal area (see Supplement 1 for details).
Let us explore whether other aperture types can produce a nodal area by means of a lossless phase mask.Realizing that circular symmetry provides a path for generating circular nodal areas, we considered the important case of an annular aperture, as found in many telescopes and reflecting microscopes.The discovery of nodal areas for this system could provide a path toward extreme high-contrast astronomical imaging.For reasons that will become obvious below, we replace the "hard" central obscuration with a soft obscuration so that for r∕a <1 (i.e., inside the outer radius) Ar 1 − exp−r∕w 2 , where w is a width parameter [see Fig. 3(a)].For r∕a > 1, Ar 0. The use of pupil apodization may provide practical advantages to an imaging system [26][27][28][29].We propose that, in general, a nodal area can be found if we replace the circularly symmetric aperture function with a truncated linear superposition of zeroorder radial Zernike polynomials, R 0 n r∕a: where c n 2n 1a 2 R 1 0 AsR 0 n sds and s r∕a.Zernike polynomials are the only presently known basis functions that transform into nodal areas.The value of N may be selected based on engineering and performance metrics, such as the ease of mask fabrication and the desired darkness of the nodal area.For the Gaussian central obscuration function above, Ar, we find c 2 n rapidly decreases as exp−n 2 .It is easily shown by direct integration that a nodal area formed in the exit pupil plane if a vortex phase mask having an evenvalued topological charge jmj > N 1 is inserted into the focal plane of the telescope.A vortex phase mask having jmj 6 could then provide a perfectly black nodal area for an engineered aperture e Ar, where N 4.This result suggests that the nontruncated soft aperture function, Ar produces a dark, but nonideal nodal area, as long as the value of jmj is large.In this case the quality of the nodal area may be quantitatively expressed as the ratio of the beam power within the nodal area and the total power transmitted through the aperture, η P in ∕P tot , where P in a −2 P ∞ nm c 2 n ∕n 1.For the soft aperture function Ar shown in Fig. 3(a) (with w∕a 0.56), we find that an m 6 vortex phase element [see Fig. 3(b)] provides a power ratio of η 10 −6 .The numerically determined exit pupil amplitude shown in Fig. 3(c) exhibits a nearly ideal nodal area, as expected.Considering practical limitations such as aberrations and nonideal aperture functions, it may be difficult and unnecessary to achieve smaller values of η.
Numerical Solutions: Now that we have established additional means of achieving nodal areas (e.g., circle remappings and Zernike decompositions), we wish to explore the possibility that any aperture function can be generally transformed.To our knowledge it is currently unknown whether the desired lossless transmission functions mathematically exist.In fact, it is difficult to imagine how other apertures and phase masks could also produce a nodal area based on either a convolution or direct integral point of view.Nevertheless, the question may be formulated as a phase retrieval problem.Consider a 4f imaging system with an aperture function Ax; y and focal plane mask having a transmission function tx 0 ; y 0 expiΦx 0 ; y 0 .The field at the focal plane just before the phase element is given by the FT of the aperture function F x 0 ; y 0 FTfAx; yg.Similarly, the exit pupil field is the inverse FT of the focal plane field Gx 00 ; y 00 FT −1 fF x 0 ; y 0 expiΦx 0 ; y 0 g.We wish to calculate the phase function Φx 0 ; y 0 necessary so that Gx 00 ; y 00 0 over the support of FT −1 fF x 0 ; y 0 g.For convenience, let us again consider a real rotationally symmetric aperture.Owing to symmetry we allow for the possibility of both azimuthal and radial variations of the phase mask: Φ mθ 0 μr 0 , where μr 0 is a real-valued function.(Note that, above, we assumed only azimuthal variations of the phase mask.)The exit pupil field may be expressed in terms of an mth order Hankel transform Gr 00 ; θ 00 H m fF r 0 e iμr 0 ge imθ 0 0 .The solution for the Hankel transform may be written as the product of amplitude and phase functions: H m fF r 0 e iμr 0 g g 0 r 00 e iνr 0 0 .Being unsuccessful at finding analytical expressions for μr 0 , g 0 r 00 , and νr 00 , we instead applied a numerical phase retrieval approach, namely, a modified Gerchberg-Saxton (G-S) algorithm [20,21].Although this approach iteratively finds a nonunique solution for the phase mask that produces a nodal area, the solutions may be complicated and impractical from a fabrication point of view.The selection of a simple initial guess for the function μr 0 may, however, lessen the complexity of the final iterative solution.
The iteration scheme proceeds as follows.The field at the exit pupil plane is the FT of the product of the mask transmission function and the point spread function (FT) of the aperture.At each iterative step, if the nodal region is not zero-valued, it is forced to zero, and then the FT of the field at the exit pupil is computed.This produces new focal plane amplitude and phase functions.We replace this new amplitude function with the actual PSF of the system, and then a new exit pupil field is computed.This process is iterated until the power in the nodal area approaches zero.At that point the final phase mask profile is computed.As an example, Figs. 3(d)-3(f) show an annular aperture having a hard central obscuration, along with the computed phase mask and nodal area.The ratio of inner and outer radii was 0.38.In this case we used a vortex phase with m 2, and as a first guess in the iteration, we assumed an axicon phase, μr 0 αr 0 , where α > 2ka∕f .As designed, the nodal area in Fig. 3(f) has the same diameter as the outer diameter of the annulus.The construction of such a phase mask depicted in Fig. 3(e) may be possible with advanced fabrication techniques [25].Comparing the G-S outcome with the Zernike approach above, we find for an annular aperture having a hard central obscuration, the Zernike method would requires a vortex mask of high topological charge, whereas the G-S algorithm allows for a low vortex charge (m 2), accompanied by radial structure in the phase mask.
This, and other G-S examples not reported here, tends to confirm our conjecture that a phase-only (lossless) mask can be found to produce a nodal area having the shape of an arbitrary entrance pupil (see Supplement 1 for an additional solution).Undoubtedly, the computed structures are indicative of modal symmetries and relationships yet to be analytically discovered.Such relationships are likely related to sets of orthogonal functions in some basis that depends on the symmetries of the aperture function.
Conclusions: Nodal areas are an intriguing manifestation of destructive interference across an extended region in a cross section of a beam of light.Previously, only circular nodal areas have been observed.Here we have described what we believe to be the first experimental observation of an elliptical nodal area, produced by use of a uniformly illuminated elliptical aperture and a specially designed "elliptical vortex q-plate."After questioning whether it is possible to create nodal areas from more complicated aperture functions and phase plates, we provided both analytical and numerical approaches-each supporting (but not proving) the conjecture that nodal areas can indeed be formed.By use of Zernike mode decomposition, we have presented a method to create a nodal area from any azimuthally invariant circular aperture, such as a softly obscured annular aperture.In that case a vortex phase mask of high topological charge is typically required.This method is also suitable for elliptical apertures by mapping to a circular coordinate system.Finally, we illustrated how a G-S phase retrieval algorithm [20,21] may be used to design a focal plane phase mask that produces a nodal area.The latter result hints of yet undiscovered symmetries, which suggests to us the possibility of elegant solutions supporting nodal areas for any aperture function.We believe there is no single unique phase mask that transforms a given aperture into a nodal area.It is likely that the G-S numerical approach provides a linear combination of the underlying basis set of solutions.In optics, beams containing nodal areas may find increasing importance in applications involving structured light, such as radiation shielding, laser surfacing, radiation pressure, high contrast imaging, and dense channel line-of-sight telecommunication.We anticipate that additional experimental and theoretical investigations will help uncover general mathematical properties of nodal areas, as well as identify new applications of structured darkness in optics and other wave systems.

Fig. 2 .
Fig. 2. Experimental verification of an elliptical nodal area.Image of the elliptical aperture (a) without and (b) with the elliptical vortex q-plate.

Fig. 3 .
Fig. 3. (a) An apodized annular aperture matched with (b) an m 6 vortex phase element generates (c) a dark region with radius equal to the outer radius of the aperture.(d)-(f) Same as (a)-(c), but with (d) an annular aperture, (e) the numerically computed element, and (f) the corresponding nodal area.