We briefly recall the model used in [7] to simulate the fluid lens surface shape without considering gravity effects. The equations used are those derived by Berger [24] to determine the nonlinear, large deformation of thin isotropic elastic plates. ∇ 4 w − α 2 ∇ 2 w = q D , (1a) ∂ u ∂ x + ∂ v ∂ y + 1 2 ∂ w ∂ x 2 + 1 2 ∂ w ∂ y 2 = α 2 h 2 12 . (1b) In these equations, w x , y is the local z-displacement of the membrane, with the non-deformed state assumed to correspond to the z = 0 plane; u x , y and v x , y are the local x and y displacements, respectively; D is the membrane bending rigidity; and h is its thickness. The magnitude q x , y corresponds to the applied z-load, and α is a constant to be determined from the same equations by imposing appropriate boundary conditions.

For the case of uniform load (constant q) and elliptic aperture, analytical solutions of system 1) were obtained by the method of constant deflection contour lines derived by Mazumdar [25]. If the aperture in the plane z = 0 is an ellipse of x and y semi-axes a and b, respectively, the z-displacement of the membrane is given by w ς = Δ V π a b 2 γ γ 1 − ς 2 I 1 2 γ + I 0 2 γ ς − I 0 2 γ γ 2 + 2 I 1 2 γ − 2 γ I 0 2 γ , (2) where ΔV is the volume of the liquid. The variable ς is defined as ς 2 = x 2 / a 2 + y 2 / b 2 , (3) and the constant γ is related to ΔV by Δ V = π a b h 3 a 4 + 2 a 2 b 2 + 3 b 4 a 2 + b 2 G γ , (4) where G γ = γ 2 + 2 I 1 2 γ − 2 γ I 0 2 γ 24 γ 3 γ I 1 2 γ 2 − 2 I 2 2 γ γ I 0 2 γ + 2 I 1 2 γ . (5)

We now consider the effect of gravity when the (x, y) plane of the lens aperture is vertical. In Eq. 1in the study by Berger load q has the expression q = q 0 − ρ g x ⁡ sin ⁡ θ + y ⁡ cos ⁡ θ , where q ₀ is the load at the lens center (x = y = 0), ρ is the mass density of the filling fluid, g is the acceleration of gravity, and θ is the angle between the vertical direction and the y-axis (see Figure 1).

In the case that ρgL/q ₀ ≪ 1, with L as the characteristic length of the lens pupil, we can treat gravity effects as a perturbation to the case with uniform load q ₀, as previously obtained, and write w = w 0 + w 1 , α = α 0 + α 1 , with w ₀ and α ₀ corresponding to the solution of the case with q = q ₀. Linearization of Eq.1a in the perturbations w ₁ and α ₁ yields ∇ 4 w 1 − α 0 2 ∇ 2 w 1 − 2 α 0 α 1 ∇ 2 w 0 = − ρ g D x ⁡ sin ⁡ θ + y ⁡ cos ⁡ θ . (6)

For a clamped membrane with no pre-stretching, u = v = 0 at ψ = 0, and so the integral of the linearized version of Eq.1b over the area S ₀ of the lens aperture gives ∫ S 0 ∂ w 0 ∂ x ∂ w 1 ∂ x + ∂ w 0 ∂ y ∂ w 1 ∂ y d x d y = α 0 α 1 h 2 6 S 0 . (7)

We now consider the solution to Eq. 6 in the usual case in which the membrane forces dominate: α 0 2 L 2 ≫ 1 . In this case, except extremely close to the membrane border, one has ∇ 4 w 1 ≪ α 0 2 ∇ 2 w 1 , and so it is easy to check that the solution to Eq. 6 satisfying the boundary condition w ₁ = 0 at the border of the membrane is given by w 1 = − 2 α 1 α 0 w 0 − ρ g a 2 b 2 2 D α 0 2 1 − x 2 a 2 − y 2 b 2 x ⁡ sin ⁡ θ a 2 + 3 b 2 + y ⁡ cos ⁡ θ 3 a 2 + b 2 . (8)

Using expression 8) in Eq. 7, one readily obtains α ₁ = 0 so that the complete solution with the inclusion of gravity effects is w = w 0 − ρ g 2 D γ 2 a 4 b 4 a 2 + b 2 3 a 2 + 2 a 2 b 2 + 3 b 4 1 − x 2 a 2 − y 2 b 2 x ⁡ sin ⁡ θ a 2 + 3 b 2 + y ⁡ cos ⁡ θ 3 a 2 + b 2 , (9) where w ₀ is the solution previously obtained for uniform load, Eq. 2, and the constant γ is the one determined in that solution using Eq. 4.

In order to determine the aberrations of a fluid lens with one plane surface, we consider a plane wavefront with normal incidence on the plane side of the membrane, taken at z = 0 (see Figure 2). The corresponding rays, parallel to the z-axis of unit vector e _z , are then refracted according to Snell’s law when they cross the membrane curved surface at z = w x , y . The external normal unit vector at that surface is given by (in Cartesian components) n = − w x , − w y , 1 1 + w x 2 + w y 2 , (10) where the subscripts x and y indicate derivatives with respect to the corresponding coordinate.

The angle θ _i of the rays incident from inside the lens, relative to the external normal direction at the corresponding point of the membrane curved surface, is thus given by cos θ i = n ⋅ e z = 1 1 + w x 2 + w y 2 . (11)

The Snell law then determines the angle of the refracted ray emerging from the lens, also relative to the normal direction, as sin θ r = n f w x 2 + w y 2 1 + w x 2 + w y 2 , (12) where n _f is the index of refraction of the filling fluid, relative to that of air.

The refracted ray is contained in the plane determined by the normal unit vector n and the unit vector tangent to the surface t = e z − n ⋅ e z n e z − n ⋅ e z n = w x , w y , w x 2 + w y 2 1 + w x 2 + w y 2 w x 2 + w y 2 , (13) so that the ray direction is given by the unit vector k r = n ⁡ cos θ r + t ⁡ sin θ r . (14) The explicit expressions of the Cartesian components of k _r are k r x , y = n f − 1 + 1 − n f 2 w x 2 + w y 2 1 + w x 2 + w y 2 w x , y , (15a) k r z = n f w x 2 + w y 2 + 1 + 1 − n f 2 w x 2 + w y 2 1 + w x 2 + w y 2 , (15b) which are functions of the point (x, y) in the plane z = 0, at which the ray originated.

In this way, a generic ray starting at the point (x, y) in the plane z = 0 inside the lens is refracted at the point X 0 = x , y , w x , y on the membrane surface, and after traversing in air a distance L reaches the point X _L = X ₀ + k _r L so that (from now on, we do not write the explicit dependence on (x, y) of w and of k _r ) z L = w + k r z L , (16) and x L = x + k r x k r z z L − w , (17a) y L = y + k r y k r z z L − w . (17b) The phase at (x _L , y _L , z _L ) is thus ϕ L = ϕ 0 + 2 π λ n f w + z L − w k r z , (18) where ϕ ₀ is the phase of the front at z = 0 and λ is the wavelength in air.

From (18), we can determine the z _W position of a wavefront of given phase ϕ _W as z W = w 1 − n f k r z + k r z λ 2 π ϕ W − ϕ 0 , (19) to which correspond the (x _W , y _W ) coordinates: x W = x + k r x λ 2 π ϕ W − ϕ 0 − n f w , (20a) y W = y + k r y λ 2 π ϕ W − ϕ 0 − n f w . (20b) These two relations can, in principle, be solved to give x = x x W , y W , (21a) y = y x W , y W , (21b) which, if replaced in (19), give the wavefront geometry z W = z W x W , y W .

If this wavefront is analyzed at a position z _A , we can, without loss of generality, take this position as that of the image of the origin, x = y = 0: z A = w 0 1 − n f k r z 0 + k r z 0 λ 2 π ϕ W − ϕ 0 , where w 0 = w 0,0 and k r z 0 = k r z 0,0 so that λ 2 π ϕ W − ϕ 0 = z A − w 0 1 − n f k r z 0 k r z 0 , (22) and analyze the deviation from a plane front: Δz _W = z _W − z _A , which is conveniently written as Δ z W = w 1 − n f k r z − k r z k r z 0 w 0 1 − n f k r z 0 + k r z k r z 0 − 1 z A . (23)

The corresponding (x _W , y _W ) coordinates are written as x W = x + k r x k r z 0 z A − w 0 − n f k r z 0 w − w 0 , (24a) y W = y + k r y k r z 0 z A − w 0 − n f k r z 0 w − w 0 . (24b) In this way, Eqs 23, 24a, 24b give the wavefront geometry, analyzed at z = z _A , parameterized in terms of the (x, y) coordinates on the plane at z = 0.

We further model the wavefront analyzer at z = z _A as having a circular aperture of radius r _A so that the section of the wavefront Δ z W x W , y W to be studied is expressed as Δ z W r A ξ ⁡ cos ⁡ ϕ , r A ξ ⁡ sin ⁡ ϕ , with 0 ≤ ξ ≤ 1, and decomposed in Zernike polynomials in polar coordinates Z n ξ , ϕ , Δ z W = ∑ n a n Z n ξ , ϕ , (25) with a n = 1 π ∫ 0 2 π ∫ 0 1 Δ z W r A ξ ⁡ cos ⁡ ϕ , r A ξ ⁡ sin ⁡ ϕ Z n ξ , ϕ ξ d ξ d ϕ . (26)

We have followed the standard OSA/ANSI indexing and normalization scheme, used in the Shack–Hartmann wavefront sensor, for which the first 15-term orthonormal Zernike circle polynomials are as follows: Z 0 = 1 , Z 1 = 2 ξ ⁡ sin ⁡ ϕ , Z 2 = 2 ξ ⁡ cos ⁡ ϕ , Z 3 = 6 ξ 2 ⁡ sin ⁡ 2 ⁡ ϕ , Z 4 = 3 2 ξ 2 − 1 , Z 5 = 6 ξ 2 ⁡ cos ⁡ 2 ⁡ ϕ , Z 6 = 8 ξ 3 ⁡ sin ⁡ 3 ⁡ ϕ , Z 7 = 8 3 ξ 3 − 2 ξ sin ⁡ ϕ , Z 8 = 8 3 ξ 3 − 2 ξ cos ⁡ ϕ , Z 9 = 8 ξ 3 ⁡ cos ⁡ 3 ⁡ ϕ , Z 10 = 10 ξ 4 ⁡ sin ⁡ 4 ⁡ ϕ , Z 11 = 10 4 ξ 4 − 3 ξ 2 sin ⁡ 2 ⁡ ϕ , Z 12 = 5 6 ξ 4 − 6 ξ 2 + 1 , Z 13 = 10 4 ξ 4 − 3 ξ 2 cos ⁡ 2 ⁡ ϕ , Z 14 = 10 ξ 4 ⁡ cos ⁡ 4 ⁡ ϕ .

In order to characterize numerically the spectral response of optical aberrations and compare directly with experimental data, we expanded the wavefront aberrations of a single elastic membrane in terms of Zernike polynomials up to order 14, for a beam with wavelength λ) in the range 400–1,500 nm, thus numerically characterizing the spectral response in the visible and infrared domains. Even though we analyze wavefront aberrations in terms of Zernike polynomials up to order 14, we only display those coefficients for Zernike polynomials which are not negligible, namely, P ₀, P ₂, P ₃, P ₆ (for circular apertures) and P ₀, P ₂, P ₃, P ₅ (for elliptic apertures). The remaining Zernike coefficients are all below 10^–14, so they are not displayed in the figures.

Numerical results of wavefront aberrations for fluidic lenses with circular aperture (axes a = b = 1.7 cm) are displayed in Figure 5. Figures 5A–D, correspond to normalized Zernike coefficients for polynomials of orders P ₀, P ₂, P ₃, and P ₆, respectively. The remaining polynomials are not reported because their coefficients are negligible (≪ 10^–14). Numerical results for the chromatic response of fluidic lenses with elliptic apertures characterized by ellipse axes (a = 1.5 cm, b = 1.7 cm) and (a = 1.3 cm, b = 1.7 cm) are displayed in Figures 6, 7, respectively. Figures 6A–D and Figures 7A–D correspond to Zernike coefficients for polynomials of orders P ₀, P ₂, P ₃, and P ₅, respectively. The remaining polynomials are not displayed because their coefficients are negligible. As it is apparent from numerical results, the dependence of wavefront aberrations is of the general form |1/λ|. Moreover, spectral fluctuations in wavefront aberrations are within a fraction of λ.

As readily reported in a previous publication [7], the fluidic lens consists of two layers of the elastic membrane of the polydimethylsiloxane (PDMS-type). The two elastic films are held together by an aluminum frame, sealed with the elastic membrane. An optical fluid of refractive index matched to the polymer, such as glycerol or distilled water, is injected between the elastic layers. By increasing or decreasing the fluid volume mechanically injected, it is possible to tune the focal distance across several centimeters and adjust the optical power of the lens. Furthermore, we tune one additional degree of freedom, given by the shape of the aperture. By modifying the aperture shape from circular (Figure 8B ) to elliptical (Figure 8C ), we can introduce different optical corrections. The typical size for the circular lens is given by a diameter d = 17 mm, and the elliptic lenses have a major axis b = 17 mm and minor axes a = 15 mm and a = 13 mm. Please note the axes labeling used in the theoretical model is not necessarily the same as the axes labelling used in the experimental setup. Further details regarding the fabrication of the fluidic lens prototype are reported in a previous article [7].

In order to characterize the chromatic response of the light field transmitted by the fluidic lenses, we reconstructed the wavefront transmitted through the lenses using a Shack–Hartmann wavefront sensor Figure 8A (Model Thorlabs WFS150-5C, raw experimental data can be found at our GitHub repository [27]). To this end, we used a collimated incoherent RGB LED source (RGB: red–green–blue). Perfect collimation of a polychromatic beam can never be achieved. Here, by collimation, we refer to the fact that we verified the beam size did not diverge significantly over large distances (3 m or more), and we also confirmed that the wavefront impinging on the fluidic lenses was nearly a plane wavefront so that we could use the internal calibration of the Shack–Hartmann wavefront sensor, and all measured wavefront aberrations could be ascribed to the fluidic lenses themselves.

Shack–Hartmann wavefront sensors (SHWSs) enable analyzing the shape of an incident beam’s wavefront by dividing the beam into an array of discrete intensity points using a micro-lens array. These data are then used to reconstruct and analyze the shape of the wavefront using Zernike polynomials. In addition to analyzing classical optics phenomena, they are increasingly employed in applications where real-time monitoring of the wavefront is required to control adaptive optics with the intent of removing the wavefront distortion before creating an image. In particular, SHWSs enable two types of wavefront characterizations. I) Direct measurement (not displayed in Figure 9): shows the wavefront that is directly calculated from the measured spot deviations using a 2-dimensional integration procedure. II) Zernike reconstruction (left column in Figure 9): displays the wavefront that is reconstructed using a selected set of the determined Zernike coefficients. The advantages of Zernike reconstruction are as follows: i) selecting only a few lower-order Zernike modes for reconstruction smooths the wavefront surface (noise canceling), ii) the lowest-order Zernike modes (for instance, Z0 piston, Z1 tip, and Z2 tilt) are always present, but they are of less interest. Using an appropriate reconstruction (e.g., starting from Z3) can omit the Z0, Z1, and Z3 Zernike modes in order to see only the higher-order modes. iii) If selecting particular Zernike modes, they can be displayed and analyzed separately. Difference (right column in Figure 9): displays the difference between the I) directly measured wavefront and II) reconstructed wavefront and is therefore an indicator of the fit error.

The incident field had a residual field curvature below λ/6. The sensor was placed 10 cm apart from the fluidic lens, with an aperture limited by the pupil size of the sensor itself, typically 3 mm in diameter. We reconstructed the wavefront produced by a circular fluidic lens filled with V _max = 6 mL corresponding to an optical power (OP) = 50 D (Figure 9A) and by an elliptical fluidic lens filled with V _min = 4 mL corresponding to OP = 36 D (Figure 9C). The qualitative difference in the wavefront due to the shape of the aperture is apparent. Residual differences between the measured wavefront and the reconstructed wavefront are displayed in Figures 9B, D). Further details on the Shack–Hartmann wavefront sensor are provided in Ref. [7].

In order to experimentally characterize the spectral response of optical aberrations in the central region of the fluidic lens prototype, we use the experimental setup described in Figure 8A. A collimated incoherent beam, produced by a programmable LED source (ALIC Smart Life, 14W, Luminous Flux 1400 lm, λ = 400–1,045 nm) propagates through the fluidic lens and is imaged by a Shack–Hartmann wavefront sensor (Model Thorlabs WFS150-5C), located at a distance of 2 cm from the fluidic lens, in order to image the near-field wave produced by the lens. The area of the beam to be characterized is determined by the aperture of the sensor (typically 3 mm). We verified that the transverse profile of the beam did not change significantly when tuning the wavelength of the source across the entire spectral range. Spectral characterizations in the visible range are mostly qualitative due to the broad spectrum produced by the incoherent LED source.

Measured aberrations in μm, in terms of the coefficients associated with Zernike polynomials of order 0 to 14, for red, green, and blue LED illumination are displayed in Figures 10A-I. First column: blue LED source, (A) circular aperture (a = b = 1.7 cm), (D) elliptic aperture 1 (a = 1.5 cm, b = 1.7 cm), and (G) elliptic aperture 2 (a = 1.3 cm, b = 1.7 cm). Second column: green LED source, (B) circular aperture (a = b = 1.7 cm), (E) elliptic aperture 1 (a = 1.5 cm, b = 1.7 cm), and (H) elliptic aperture 2 (a = 1.3 cm, b = 1.7 cm). Third column: red LED source, (C) circular aperture (a = b = 1.7 cm), (F) elliptic aperture 1 (a = 1.5 cm, b = 1.7 cm), and (I) elliptic aperture 2 (a = 1.3 cm, b = 1.7 cm).

In order to quantify the agreement between experimental results and numerical simulations, we calculated the distance between measured Zernike coefficients for red–green–blue (RGB) wavelengths. We considered three different distance measures defined for two sets of data {a, b, c} and {x, y, z} in the following form: 1) Euclidean distance D 1 = | a − x | 2 + | b − y | 2 + | c − z | 2 , 2) Canberra distance D ₂ = |a − x|/(|a| + |x|) + |b − y|/(|b| + |y|) + |c − z|/(|c| + |z|), and 3) Bray–Curtis distance D 3 = ( | a − x | + | b − y | + | c − z | ) / ( | a + x | + | b + y | + | c + z | ) .

A brief comparison between the different distance measures is in order: D ₁ corresponds to the “Pythagorean distance and is the only measure that can be subject to a direct geometrical interpretation; therefore, in this sense, it is the most intuitive one. D ₂ and D ₃ distance measures are similar in essence as they are both based on the algebraic concept of norm of a vector. They differ in the normalization factor. While D ₂ normalizes each element of the vector independently, D ₃ introduces a global normalization factor, and is therefore less sensitive (larger in modulus), as can be verified in Figures 11G, J. Note that D ₁ is not normalized, and for this reason, it is typically larger in modulus than D ₂ and D ₃. We did not include the Manhattan distance in this analysis because it returned practically identical results to the Euclidean distance. The usefulness of the Manhattan measure was clearly revealed when employed in clustering techniques (see Section 2.4; Figure 12).

Comparison between spectral distances for measured Zernike coefficients in μm, for RGB wavelengths are displayed in Figure 10. Blue markers, magenta markers, and brown markers correspond to B–R distance, B–G distance, and R–G distance, respectively. Figures 10A–C depict spectral distances D ₁, D ₂, and D ₃ for fluidic lenses with circular apertures, respectively. Figures 10D–F depict spectral distances D ₁, D ₂, and D ₃ for fluidic lenses with elliptic 1 aperture, and Figures 10G–I depict spectral distances D ₁, D ₂, and D ₃ for fluidic lenses with elliptic 2 aperture. Distances are within a fraction of the wavelength, in agreement with numerical simulations.

In order to further classify the spectral response of optical aberrations, we partitioned the data into a predetermined number of clusters (N) across the visible range (λ = 400–650 nm). More specifically, Zernike coefficients were partitioned into subgroups (or clusters) representing proximate collections of elements based on a distance or dissimilarity function. In particular, we consider the Manhattan distance, given by the sum of the absolute difference between the elements. Identical element pairs have zero distance or dissimilarity and are grouped into a given cluster, and all others have positive distance or dissimilarity.

Clustering techniques provide for a robust quantitative tool to classify large sets of data according to the distance between the elements in the clusters. This, in turn, can enable to identify emerging trends in experimental data. In addition, they can enable quantitative comparisons between experiments and numerical/theoretical predictions. Moreover, in our case, we performed clustering techniques based on an alternative distance measure, e.g., the Manhattan distance, which provided for further insights into the way in which averaged experimental data are grouped and distributed, according to the input wavelength. For instance, from the clustering analysis, one can infer that average positive and negative Zernike coefficients are typically distributed with similar probabilities, for all input wavelengths. Note that the insights provided by clustering techniques are complementary to the direct calculations of distances between elements (Figure 11).

Experimental clusters for average Zernike coefficients (⟨P _N ⟩) in the range λ = 400–650 nm, partitioned into a predetermined set of N = 2, 4, and 6 clusters, are presented in Figures 12A–I. First column: N = 2 clusters. (A) Circular aperture (a = b = 1.7 cm), b) elliptic aperture 1 (a = 1.5 cm, b = 1.7 cm), and (C) elliptic aperture 2 (a = 1.3 cm, b = 1.7 cm). Second column N = 4 clusters. (D) Circular aperture (a = b = 1.7 cm), (E) elliptic aperture 1 (a = 1.5 cm, b = 1.7 cm), and (F) elliptic aperture 2 (a = 1.3 cm, b = 1.7 cm). Third column: N = 6 clusters. (G) Circular aperture (a = b = 1.7 cm), (H) elliptic aperture 1 (a = 1.5 cm, b = 1.7 cm), and (I) elliptic aperture 2 (a = 1.3 cm, b = 1.7 cm). For N = 2, Zernike coefficients can be classified into two main clusters, corresponding to either an average positive amplitude ⟨P ₂⟩ = +10 (blue dots) or an average negative amplitude ⟨P ₂⟩ = −2 (orange dots). Next, for N = 4, Zernike coefficients can be classified into four clusters, one with an average positive amplitude ⟨P ₄⟩ = +15 (blue dots), one with an average negative amplitude ⟨P ₄⟩ = −15 (magenta dots), and the remaining two with nearly vanishing amplitudes ⟨P ₄⟩ ≈ 0 (green and orange dots). Finally, for N = 6, Zernike coefficients are classified into six clusters, one with an average positive amplitude ⟨P ₆⟩ = +20 (blue dots), one with an average negative amplitude ⟨P ₆⟩ = −20 (purple dots), and the remaining four clusters with nearly vanishing amplitudes ⟨P ₆⟩ ≈ 0 (green, orange, magenta, and red dots). The decreasing amplitude of the average Zernike coefficients for decreasing the number of clusters (N) can be ascribed to averaging over a broader range of amplitudes since reducing N increases the diversity of the elements.

Convolution (CN) and correlation (CR) measurements are robust analytical tools that enable quantitative analyses of the interrelation between two experimental magnitudes, in this case Zernike coefficients vs wavelength. Specifically, CR/CN = +1 (-1) represents a maximal positive (negative) interrelation, while CR/CN = 0 represents no interrelation at all. Moreover, these methods enable to identify emerging trends or salient features for specific values of the measured quantities. In addition, they enable direct contrast and comparison with other characterization methods, such as clustering techniques, and with theoretical/numerical predictions. From the CR and CN data, we can conclude that the correlation between measured Zernike coefficients and wavelength is typically medium CR/CN = +0.5 (-0.5), uniformly distributed between positive and negative values for all wavelengths, with no specific wavelength-dependent salient features. These results are in agreement with the conclusions obtained from clustering techniques, and from numerical predictions.

In both CN and CR, the basic idea is to combine a kernel list with successive sub-lists of a list of data. The convolution of a kernel K _r with a list u _s has the general form ∑ _r K _r u _s−r , while the correlation has the general form ∑ _r K _r u _s+r . In particular, for a kernel list K _r = [x, y] and list of data u _s = [a, b, c, d, e], the convolution (CN) results in the combined list C N = b x + a y , c x + b y , d x + c y , e x + d y , (30)

while the correlation (CR) results in the combined list C R = a x + b y , b x + c y , c x + d y , d x + e y . (31)

We calculated the convolution (CN) and correlation (CR) between the wavelength (λ) and the measured Zernike coefficients (P _s ), where s = 0, … , 14 labels the polynomial order in each sub-list. We consider a kernel specified by the wavelength range K _r = [400, 450, 500, 550, 600, 650] in nm and a list of measured Zernike coefficients (P _s (λ)) in nm for each different input wavelength λ) of the form u _s = [P _s (400), P _s (450), P _s (500), P _s (550), P _s (600), P _s (650)]. A plot of the normalized correlation (CR) and convolution (CN) is presented in Figures 13A–F. Left column: CN [nm ²]. (A) Circular aperture (a = b = 1.7 cm), (B) elliptic aperture 1 (a = 1.5 cm, b = 1.7 cm), and (C) elliptic aperture 2 (a = 1.3 cm, b = 1.7 cm). Right column: CR [nm ²]. (D) Circular aperture (a = b = 1.7 cm), (E) elliptic aperture 1 (a = 1.5 cm, b = 1.7 cm), and (F) elliptic aperture 2 (a = 1.3 cm, b = 1.7 cm). As a general trend, Zernike polynomials display either a positive correlation with λ (CR/CN = +0.5) or a negative correlation with λ (CR/CN = −0.5). Fluctuations on this trend increase as the asymmetry in the ellipse axes increases. The significant color spread indicates that there is no particular correlation, neither positive nor negative, between the Zernike order (s) and wavelength (λ).