A ray-trace analysis of x-ray multilayer Laue lenses for nanometer focusing

X-ray MLLs can be considered as thick diffractive optical elements, in which diffraction occurs as a volume effect. Efficient diffraction is obtained by satisfying the Bragg condition as mentioned above, which requires the layers to be tilted by θ, half the deflection angle. MLLs are fabricated by the alternating deposition of two (or more) materials onto a substrate followed by slicing to produce a structure similar to that illustrated in the inset of figure 1(a). The variation of period d(r) throughout the lens is controlled by varying the amount of material deposited in each layer (e.g. by changing the time the substrate is exposed to each sputtering target) and the required tilt can be obtained by placing a mask edge between the sputtering target and the substrate [25]. MLLs provide high efficiencies for x-ray energies above about 10 keV (wavelengths less than about 0.1 nm) with periods typically in the range of 1 to 100 nm. The optimum thickness τ of the lens that the x-rays transmit through is half the so-called Pendellösung period Λ_L , which can vary from several micrometers at lower x-ray energies to many tens of micrometers at harder energies, depending on the materials and (weakly) on the period.

X-ray MLLs are usually fabricated as flat lenses with bi-layers (periods) positioned according to the Fresnel zone plate condition of (2) with z = 0. Given the large thickness τ of an MLL, rays can be thought to reflect not only on the front surface of the structure but at points some distance z into the interior of the MLL. In order that these rays are directed to a common focal point the layers should follow the paraboloids given by (2). The approximation of the paraboloids to cones (or tilted wedged layers for one-dimensional focusing) can be found from a Taylor series of r_n (z), showing the gradient of each layer is approximately r_n /(2f), which is indeed equivalent to the angle θ(r) of layers mentioned above and shown in figure 1. Also, given a bi-layer thickness $d_n = r_{n+1}-r_n$ it can be shown that $d_n^2 \approx f \lambda/(4(n+1/2))$, and from (2) it follows that $d_n \approx f\lambda/r_n$. For the purpose of ray tracing, the phase imparted to the ray that intersects the lens at a position r, equal to 2πn, is given by φ(r) of (3).

We first consider the ray-trace analysis of perfect MLLs that are designed to deflect incident parallel rays to the focal point. In section 6 we consider lenses with imperfections (due to manufacturing processes, for example) where φ(r) must be modified due to deviations in d(r) from the ideal. Generally, ray tracing tracks a ray from a particular point in an object, via straight-line propagation through homogeneous space to an interface where it is refracted (according to Snell's law) or diffracted (according to a grating equation or Bragg's law) before propagating to the next interface, and so on, until the plane of interest is reached [18]. Unlike a more complete calculation that may utilise Huygens wavelets formed at each scattering point of the structure (such as used in a multislice calculation [26]), a ray described by the unit vector $\hat{\boldsymbol{r}}$ incident upon a diffractive optic is taken to follow a single trajectory only. As mentioned above, the direction of the diffracted ray $\hat{\boldsymbol{r}}'$ is set by Bragg's law, cast here as $(2\pi/\lambda_m)\hat{\boldsymbol{r}}' = \boldsymbol{k}_\mathrm{out} = \boldsymbol{k}_\mathrm{in}+\boldsymbol{q}$ for an incident wave-vector $\boldsymbol{k}_\mathrm{in} = (2 \pi/\lambda_m)\hat{\boldsymbol{r}}$ (see figure 2(a)). That is, the ray has a 'measurement' wavelength λ_m which may differ from the parameter λ used in the design of the diffracting structure that modulates with a period $d = 2\pi /|\boldsymbol{q}|$ in the direction of $\boldsymbol{q}$. This presumes that the diffracting structure will indeed be oriented in the Bragg condition. For thin structures, or to account for diffraction of a ray in an off-Bragg condition (whether that be due to an angular deviation or a change in wavelength), it can be assumed that the termination of the periodic structure by the surface leads to truncation rods in reciprocal space that extend from $\boldsymbol{q}$ along the direction of the surface normal $\hat{\boldsymbol{n}}$, in which case the 'refraction' of the single ray can be described as

$$\begin{equation} \hat{\boldsymbol{r}}' = \hat{\boldsymbol{r}}+\frac{\lambda_m}{2 \pi} \boldsymbol{q}+\epsilon \hat{\boldsymbol{n}}, \end{equation} \tag{ 7 }$$

where the deviation parameter is chosen to ensure that $|\hat{\boldsymbol{r}}'| = 1$, and $\boldsymbol{q}$ points towards the optical axis (rather than away from it) for a focussing optic. Although (7) was derived in the context of the geometric theory of diffraction (see chapter 3.3 of [27]), the expression also holds in the framework of dynamical diffraction theory (see chapter 4.8.5 of [27]). A similar approach used in the ray-tracing of thick HOEs is referred to as the blurred grating vector approach [28].

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As the ray is traced sequentially from interface to interface, the optical path length (OPL) of the ray is accumulated. In a perfect imaging system based on refractive optics, rays originating from a point in the object plane should all accumulate the same optical path (proportional to the flight time) upon their intersection of the image plane at the image point. A map of the differences of the OPLs of the rays to that of a reference ray (optical path difference, OPD), as a function of the pupil coordinates of the rays, gives the map of the wavefront aberration. This aberration can be related to the PSF of the lens through a wave optic calculation, treating the aberration as a phase error [29]. In a diffractive optic such as an MLL, however, rays reflecting from each subsequent bi-layer accrue an extra wavelength of path-length, for a total path difference of n  λ_m for n periods, or a phase of 2πn. For a wavelength equal to the design wavelength, λ_m  = λ, this phase is exactly −φ(r) as provided by the diffracting structure. The formation of the image is dictated by the constructive interference of waves at the measurement plane rather than the principle of least time. The lag of rays by a wavelength per period of the structure is thus not apparent and it is accounted for by forming the OPD as [18]

$$\begin{equation} \mathrm{OPD}(r) = \mathrm{OPL}(r)-\mathrm{OPL}(r_0) + \frac{\lambda_m}{2\pi}\phi(r), \end{equation} \tag{ 8 }$$

where r₀ is the coordinate of the reference ray (such as the chief ray). It is not necessary to discretise φ(r) into the effects of individual layers since the coordinates r in the pupil need not be determined with a precision better than the period of the diffracting structure. Chromatic aberrations can be computed when the measurement wavelength λ_m is different to the design wavelength λ.

Equations (7) and (8) can be used to trace rays through the optical system and analyse its wavefront aberrations. These equations do not account for any variation in diffraction efficiency, which drops precipitously for rays outside of the so-called Darwin width of the reflection profile (also referred to as the rocking-curve width), which will occur for rays originating from a source point off the optic axis, for example, as well as for rays incident at the Bragg angle but which have the wrong wavelength. No matter whether a deviation from the Bragg condition is caused by a change in wavelength λ_m or a change in incident ray direction $\hat{\boldsymbol{r}}$, or a combination of both, solving $|\hat{\boldsymbol{r}}'| = 1$ using (7) will generate a non-zero error term . If the normal $\hat{\boldsymbol{n}}$ is perpendicular to $\boldsymbol{q}$ (which is the case for symmetric Laue diffraction) then the equivalent angular deviation from the Bragg condition can be estimated as

$$\begin{equation} \Delta \theta \approx \frac{2 \pi \epsilon}{\lambda_m |\boldsymbol{q}|} = \frac{\epsilon d}{\lambda_m}. \end{equation} \tag{ 9 }$$

The diffraction efficiency of a ray as a function of Δθ can be calculated by dynamical diffraction theory [27] or, equivalently, coupled waved theory [30]. Bajt et al [31] and Yan et al [7] give expressions for dynamical diffraction of multilayer structures. In particular, for a thick multilayer grating constructed of equal layer heights of the two materials per period, a thickness of the lens τ, and ignoring absorption, the normalised rocking curve of the symmetric Laue reflection is given by

$$\begin{equation} I_L(\eta) = {\sin}^2 \left(\frac{\pi\,\tau}{\Lambda_L}\sqrt{1+\eta^2} \right)\,\frac{1}{1+\eta^2} \end{equation} \tag{ 10 }$$

for a Pendellösung period Λ_L and the normalised deviation parameter η equal to

$$\begin{equation} \eta = \frac{\Delta\theta}{w_\theta} = \frac{\Delta\theta \, \pi\,\sin 2\theta}{2|\delta_1-\delta_2|} \approx \frac{\epsilon\,\pi}{2|\delta_1-\delta_2|} = \frac{\epsilon}{w_\epsilon} \end{equation} \tag{ 11 }$$

for a Bragg angle θ, and where δ₁ and δ₂ are the real parts of the optical constants of the layer materials at the particular photon energy used. The expression of η in terms of was obtained from sin 2θ ≈ 2θ ≈ λ_m /d. In the following we assume that the MLL is cut at a thickness τ = Λ_L /2 to give the maximum diffraction efficiency such that I_L (0) = 1. The width of the Laue rocking curve is given by the Darwin width $w_\theta$, at which the efficiency drops to 0.317. The efficiency $I_L(\epsilon/w_\epsilon)$ is plotted in figure 2(b) and can be included in the ray tracing procedure [20] by multiplying the ray intensity (initially unity) by the diffraction efficiency at each interface. As seen from equations (10) and (11), this calculation does not require any further specification of the d spacing of the multilayer or the wavelength, which are implicitly accounted for in the prescription of $\boldsymbol{q}$, as demonstrated in the next section.

Since all rays from the on-axis field point exactly match the Bragg condition in the lens, they will be fully transmitted (under the ideal assumptions discussed in section 3.3) and the transmission of the lens will be uniform across the pupil. Rays from off-axis field points will deviate from the Bragg condition with an efficiency I_L () where is the deviation parameter. An off-axis field angle leads to an angular deviation Δθ ≈ α_y , so we expect from (9) that $|\epsilon| \approx \lambda \alpha_y/d$. Since the smallest periods d occur at the edge of the lens, the lens will be apodised—it will lose transmission at the outermost regions of the pupil as the field angle is increased from zero. Off-axis field points therefore become vignetted (have lower transmission) [33] with reduced NA and degraded resolution. The expression for is obtained in the analysis that leads to (20) by solving for $|\hat{\boldsymbol{r}}'| = 1$. The series expansion of this expression to first-order in α_y is given by

$$\begin{equation} \epsilon^{(1)}(\rho) = -\alpha_y \rho \cos\psi + \frac{f}{2~{\rm R}}\left(1-\frac{f}{R} \right)\,\alpha_y\,\rho^3 \cos\psi. \end{equation} \tag{ 21 }$$

That is, the magnitude of the deviation parameter does indeed increase linearly with field angle and also depends linearly on the pupil coordinate in the direction of the field angle, ρ cos ψ. In this direction, lens apodisation occurs when the deviation parameter at the edge of the lens ($\rho = \mathrm{NA}$) exceeds the rocking-curve width, or

$$\begin{equation} \alpha_y \lt \frac{w_\epsilon}{\mathrm{NA}} = \frac{2 |\delta_1-\delta_2|}{\pi\,\mathrm{NA}} = w_\theta. \end{equation} \tag{ 22 }$$

This restriction is independent of the focal length of the lens.

The paraxial focal length of a diffractive lens scales inversely with the wavelength, and so such lenses exhibit strong chromatic aberrations dominated by defocus. The design of the layers according to (16) will give zero on-axis aberrations (such as spherical aberration) only for the wavelength at which it was designed. Off-axis aberrations will also be modified with a change in wavelength, since the image of an off-axis source point at infinity will be focused to a plane a distance $f_m = f\,\lambda/\lambda_m$ from the lens with a transverse displacement of f_m   α from the optic axis instead of f  α. The ray tracing procedure can be carried out using the governing equations of (7) and (8) and solving the path length $l_2 = (f_m-s)/(\hat{\boldsymbol{r}}'\cdot \hat{\boldsymbol{z}})$ to determine the wavefront aberrations encountered when utilising the lens at a different wavelength to its design. In this case it is found that the spherical aberration term is given by

$$\begin{equation} \mathrm{OPD}_\mathrm{SA}(\rho;\lambda_m) = \frac{3 f}{8} \frac{\lambda_m\, \Delta \lambda}{\lambda^2}\left( 2 -\frac{2f}{R}+\frac{\Delta \lambda}{\lambda}\right)\rho^4, \end{equation} \tag{ 23 }$$

where Δλ = λ_m −λ. This expression indicates that in addition to the design wavelength (Δλ = 0), spherical aberration is nulled at the wavelength $\lambda_m = \lambda (2 f/R -1)$. That is, by setting the radius to $R = 2f/(1+\lambda_m/\lambda)$, the spherical aberration is zero for both λ and λ_m . Of course, images at each of these wavelengths will be located at different planes, so such a design could have utility in scanning transmission or fluorescence microscopy of thick objects at two discrete wavelengths (above and below an absorption edge, for example) with a detector able to discriminate photon energies. Equation (23) also shows that in the limit $\lambda_m \rightarrow \lambda$, the optimum radius for minimising the spherical aberration is R = f. In this case $\mathrm{OPD}_\mathrm{SA}$ varies quadratically with Δλ instead of linearly.

As with a change in the field angle, rays will no longer obey the Bragg condition as the wavelength is moved from the design. The series expansion of the deviation parameter for the on-axis field point, as determined in the analysis used to derive (23) is given by

$$\begin{equation} \epsilon^{(2)} (\rho) = \frac{\lambda_m \,\Delta \lambda}{2 \lambda^2} \rho^2. \end{equation} \tag{ 24 }$$

For this on-axis field point, apodisation will therefore occur radially with a tolerance given by

$$\begin{equation} \frac{\Delta \lambda}{\lambda} \lt \frac{2 w_\epsilon}{\mathrm{NA}^2} = \frac{4|\delta_1-\delta_2|}{\pi\, \mathrm{NA}^2}. \end{equation} \tag{ 25 }$$

Both apodisation and spherical aberration may scale approximately linearly with Δλ (as is the case for a flat lens, for example), but the degree of apodisation increases with the square of the NA whereas spherical aberration grows with the fourth power of NA and linearly with the focal length. Thus for focal lengths shorter than about $0.15 \lambda / (\mathrm{NA}^2 w_\epsilon)$, the limitation to wavelength changes will be the reduction of the active pupil diameter. (Here the spherical aberration tolerance from the Marechal condition is $\mathrm{OPD}_\mathrm{SA} (\mathrm{NA}) /\sqrt{10} \lt \lambda/14$.)

As an example, we consider a lens designed to achieve a resolution of about 1 nm at a wavelength λ = 0.075 nm (17 keV photon energy) and $\mathrm{NA} = 0.0375$. The Rayleigh resolution of this lens for incoherent imaging (such as the case in a scanning transmission microscope) is $0.61 \lambda / \mathrm{NA} = 1.22~{\rm nm}$. The focal length of the lens is chosen to be f = 1 mm, and thus the diameter of the lens is 75 µm. The ray-tracing procedure detailed above was used to numerically compute the wavefront aberrations exactly—that is, without approximating to a fourth-order series expansion as for (20). The procedure, as formulated above, applies to the case of field points at infinity, displaced from the optical axis by a field angle α_y as set by the ray direction $\hat{\boldsymbol{r}}$. This situation represents, for example, the formation of a focused beam for a scanning microscope. In this case the field angle also represents the angular misalignment of the optic axis of the lens relative to say a 'beam axis' set by the direction of a far-off source. These calculations also correspond to the aberrations expected in full-field imaging at high magnification (where the image plane is far from the lenses and the sample is near to the focal plane). Finite conjugates can be simulated too, with an appropriate re-definition of $\hat{\boldsymbol{r}}$ and l₁.

At the design wavelength the calculated map of the OPD as a function of pupil coordinates for the on-axis field point (α_y  = 0) is zero to numerical precision, for both the flat MLL ($R = \infty$) and for R = f. Thus, the prescription of the layer periods and tilts according to (16) avoid spherical aberration and all other aberrations as desired. The wavefront map for an off-axis field point at α_y  = 1 mrad for a flat axisymmetric MLL is given in figure 4(a). This wavefront error, with an RMS value of 0.0062 nm (0.082 wavelengths), is dominated by coma and astigmatism in agreement with (20). Further calculations show that as the field angle increases, coma increases linearly and becomes the dominant term. However, by curving the lens surface such that R = f, this term is essentially eliminated, leaving astigmatism as the dominant aberration. For the same field angle of 1 mrad the RMS wavefront for the curved lens is only 0.0001 nm and so the OPD map is not visible on the same scale as for figure 4(a). Instead, the map is shown for R = f at a field angle of 7.5 mrad in figure 4(b), after subtracting the best-fit tilt and focus, with an RMS error of 0.0080 nm.

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According to the Marechal condition [29], diffraction-limited imaging with a Strehl ratio above 80 % requires an RMS wavefront aberration less than λ/14, or 0.005 nm in this example. The quarter-wave rule of Rayleigh corresponds to an absolute deviation of the OPD by 0.019 nm. Thus it is seen that the field angles chosen for the calculations of figures 4(a) and (b) just slightly exceed the aplanatic region that provides diffraction-limited imaging. The RMS error of both flat and curved lenses is plotted as a function of the field angle in figure 4(c). It can be seen that by curving the surface of the lens to match the focal length, the radius of the aplanatic field increases from 0.8 to 6.0 mrad. The quadratic dependence of the RMS on field angle for the curved lens indicates this is dominated by the Seidel term for astigmatism in (20). Indeed, after subtracting defocus and tilt, $\mathrm{OPD}^{(4)} (\rho_x,\rho_y) = (f/4) \alpha_y^2 (\rho_y^2 - \rho_x^2)$ for R = f. The RMS of this aberration over the circular pupil is equal to $f\,\alpha_y^2\,\mathrm{NA}^2/(4\sqrt{6})$. A plot of this curve cannot be distinguished from that shown in figure 4(c), and the residual RMS aberration after subtracting this term from the numeric computation of OPD is less than 1 × 10⁻³ nm. Indeed, the contribution of other aberrations besides astigmatism only reach an RMS value equal to λ/14 at a field angle of 100 mrad. The RMS error is also plotted in figure 4(c) for the zero-astigmatism condition of R = (2/3)f. As mentioned above, coma is still present with $\mathrm{OPD}^{(4)} = (f/2) \alpha_y \,\rho_y^3$, giving rise to the linear dependence of the RMS wavefront error on field angle for lenses with this surface radius.

The aplanatic field of view was calculated here for the angular positions of objects at infinity, but this also corresponds to forming a magnified image in the limit of high magnification. In this case the field of view at the object plane is the field angle multiplied by f = 1 mm, or a diameter of 12 µm for the curved lens, which would be increased to 200 µm if the astigmatism were corrected. A well-known approach to do this in lens design is to position the stop at a different plane. For a single refractive surface of radius R, astigmatism can be eliminated by placing the stop a distance R/2 from that surface. Unfortunately, this remedy does not work well here since the reduction of astigmatism requires a stop to be placed close to the focal plane. It may be possible, instead, to use a refractive meniscus lens near to the focal plane to change the curvature of the wave-field in the direction of the field displacement and by an amount that increases with that displacement.

As discussed in section 4.1, the field of view of a MLL may be limited not by the field dependence of the aberrations, but rather by the Darwin width of the Laue reflection, proportional to the difference of the optical constants of the materials that make up the layers of the MLL, as given by (22). For SiC and WC layers with densities of 2.47 and 14.6 g cm⁻³, respectively, $|\delta_1-\delta_2| = 6.7 \times 10^{-6}$ at a wavelength of 0.075 nm, giving a half width of $w_\theta = 2|\delta_1-\delta_2|/(\pi\,\mathrm{NA}) = 0.11~{\rm mrad}$ for the layers at the boundary of the lens considered here. A plot of the total lens transmission obtained from the numerical analysis of this lens is given in figure 5 as a function of the field angle. Maps of the transmission of the lens pupil are also shown for particular field angles, showing the loss of transmission in the thinner layers first. As seen in the maps, this transmission loss depends only on the projection of the layer period in the direction of the transverse displacement of the field, giving a band of the lens that transmits. The reduction of the aperture leads to a corresponding reduction in resolution in the direction of the object or image point (for example, vertical lines will be less resolved than horizontal lines when located at a horizontal position in the field). The lens transmission does not depend noticeably on the radius R.

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As seen in section 4.1, the largest field angle that can be tolerated by apodisation due to a loss of diffraction efficiency depends only on the multilayer materials and inversely on numerical aperture, whereas the magnitude of the wavefront aberrations are proportional to the focal length and inversely on powers of the numerical aperture (depending on whether coma or astigmatism dominate). The competition of the effects of the wavefront aberration and apodisation of the pupil on the imaging properties of the lens is illustrated in figure 6 where the combined complex-valued pupil function is shown for various field angles. The square modulus of the Fourier transform of this yields the PSF [28, 29, 34], also shown in the figure. The radius of the aplanatic field for the f = 1 mm, $\mathrm{NA} = 0.0375$ lens considered here is much larger than the limits due to diffraction efficiency, as can be seen in the top row of figure 6. When the field angle exceeds 0.2 mrad (or a image field radius of 0.2 µm) efficiency is lost from two edges of the lens, leading to a broadening of the PSF in the direction of the displacement of the object point. Increasing the focal length of the lens to 10 mm for the same NA (implying a 10 times increase in the lens diameter) leads to a corresponding 10-times increase in the OPD. As seen in the second pair of rows of figure 6, coma causes a degradation of the PSF at lower field angles, before apodisation becomes a problem. Here, the field is limited by the Marechal condition to 0.1 mrad or a field radius of 1 µm. The aberrations can be removed in principle by adopting an aplanatic design by cutting the lens on a spherical surface with a radius R = f, giving a performance similar to the 1 mm lens (bottom pair of rows) and an image field radius of about 2 µm (or 2000 resolution elements).

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Away from atomic resonances, the optical constants of materials, and hence their difference $|\delta_1-\delta_2|$, vary with λ². For lenses of a given resolution (i.e. made with a minimum d spacing) the NA increases in proportion with λ so that the angular width $w_\theta$ is linearly proportional to wavelength. Thus the largest ratio of field of view to resolution is achieved at longest wavelength.

The dominant aberrations induced by a change in wavelength are tilt and defocus due to the change in focal length (which varies inversely with wavelength) and the corresponding change in the paraxial image position. For an on-axis field point the OPD due to defocus is given by (f/2)(Δλ/λ)ρ² with a RMS value of $(f/4) (\Delta\lambda/\lambda) \mathrm{NA}^2$. Thus, applying the Marechal condition (equivalent to a defocus of less than $(2/7)\lambda/\mathrm{NA}^2$), a relative bandwidth of $2\Delta\lambda/\lambda \lt (8/14) \lambda /(f \,\mathrm{NA}^2) = 3.0 \times 10^{-5}$ can be tolerated for 1 nm imaging for the lens considered here. A shorter focal length allows a broader bandwidth since this gives fewer periods in the lens.

A MLL manufactured for a particular wavelength might be used at another, for example in spectro-microscopy applications where the sample must be repositioned at the focal distance of $f_m = f\,\lambda/\lambda_m$. The range of wavelengths that can be used depends both on the spherical aberration and the apodisation of the lens caused by the incorrect tilt of the layers for the particular wavelength. The RMS wavefront error is plotted in figure 4(d) for the on-axis field point as a function of wavelength. Here, the ray-tracing was performed at the paraxial image plane at f_m . As expected from (23), the best compensation is achieved for R = f, where the dependence on Δλ is quadratic. However, as seen from figure 5(b), and expected from (25), the limitation on using the lens at different wavelengths arises from apodisation, not spherical aberration.

To determine the performance of crossed 1D MLLs we trace rays through an optical system of a 1D MLL focusing in the y direction with a focal length f₁ followed by another focusing in the x direction with a focal length f₂. The distance between the two lenses is set to $f_1-f_2$ to ensure they both focus to a common point, and for generality the lenses are polished to cylindrical surfaces with radii R₁ and R₂, respectively (see figure 7). The functions describing the geometry and optical properties of the lenses are given in table 2. The calculation of the OPD proceeds as before, but now with a source point located at a position given by the angles α_x and α_y such that incident rays on the first MLL have a direction $\hat{\boldsymbol{r}}_1 = (\sin \alpha_x, \sin \alpha_y, \{1-{\sin}^2\alpha_x -{\sin}^2\alpha_y\}^{1/2})$. These rays are traced a distance l₁ from the plane perpendicular to their direction to the first lens at the position $\boldsymbol{s}_1 = (x_1,y_1,s_1)$, and then deflected to directions $\hat{\boldsymbol{r}}'_1$ as described in sections 4 and 5 (solving for the deviation parameter ₁). A ray intersects the second lens at the point $\boldsymbol{s}_2 = \boldsymbol{s}_1 + l_{12}\,\hat{\boldsymbol{r}}'_1$ after propagating a distance l₁₂. The x component of this expression yields $x_2 = x_1+l_{12}\,r'_{1,x}$, which can be substituted into the expression for s₂ given in table 2 to obtain, from the z component, a quadratic equation that can be solved for l₁₂. The ray at $\boldsymbol{s}_2$ incident at an angle $\hat{\boldsymbol{r}}_2 = \hat{\boldsymbol{r}}'_1$ can then be deflected into the direction $\hat{\boldsymbol{r}}'_2$ that is calculated using (7) and the expressions for $\hat{\boldsymbol{n}}_2$ and $\boldsymbol{q}_2$ in table 2, solving for the off-Bragg deviation parameter ₂ by requiring that $|\hat{\boldsymbol{r}}'_2| = 1$. The distance l₂ from $\boldsymbol{s}_2$ in the second lens to the image plane can be then obtained from $l_2\,\hat{\boldsymbol{r}}'_2 \cdot \hat{\boldsymbol{z}} = f_2-s_2$.

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As stated above, two crossed cylindrical lenses do not have a well-defined pupil (that is, an aperture in a single plane that limits the angular extent of all rays) and therefore this optical system does not have a well-defined chief ray. Nevertheless, we can express the OPD as a function of coordinates $(x_1,y_1)$ of the first lens, as the difference between the OPL of a ray that passes through those coordinates and the vertex of that lens, as

$$\begin{align} \mathrm{OPD}(x_1,y_1) & = \mathrm{OPL}(x_1,y_1) - \mathrm{OPL}(0,0)\nonumber\\ & \quad-\frac{\lambda_m}{2\pi}\left[\phi_1(y_1)+\phi_2(x_1+l_{12}\,r'_{1,x})\right] \end{align} \tag{ 28 }$$

with $\mathrm{OPL} = l_1 + l_{12}+l_2$ and λ_m is the measurement wavelength. The phase lags φ₁ and φ₂ are proportional to the number of diffracting periods between the ray of interest and the reference ray, and is thus determined by the coordinate y₁ or x₂ of the ray on the particular lens. Retaining terms to fourth order in pupil and field coordinates, when λ_m  = λ the OPD is evaluated to

$$\begin{align} & \mathrm{OPD}^{(4)}(\rho_x,\rho_y) = \frac{3}{4} f_2 \,\rho_x^2\,\rho_y^2 \nonumber\\ & \quad +f_2\left(\frac{f_2}{R_2}-1\right) \alpha_x \, \rho_x^3 + f_1\left(\frac{f_1}{R_1}-1\right) \alpha_y \, \rho_y^3 \nonumber\\ & \quad - \frac{3}{2} f_2 \,\alpha_y \, \rho_x^2 \, \rho_y + \frac{3}{2} (f_1-2f_2)\, \alpha_x \, \rho_x \, \rho_y^2 \nonumber\\ & \quad + \frac{3}{4} \left[ 4f_1\left(\frac{f_2}{R_2}-1\right) +f_2\left(7-\frac{6f_2}{R_2} \right)\right]\alpha_x^2 \,\rho_x^2 \nonumber\\ & \quad +\frac{1}{4} f_1\left (3-\frac{2f_1}{R_1}\right)\alpha_y^2\,\rho_y^2 + \frac{3}{4}f_2\,\alpha_y^2\,\rho_x^2 \nonumber\\ & \quad +\frac{1}{4}\left[11(f_2-f_1)+\frac{3f_1^2}{f_2}\right] \alpha_x^2 \,\rho_y^2 -3(f_1-2f_2)\,\alpha_x\,\alpha_y\,\rho_x\,\rho_y \nonumber\\ & \quad +\frac{3}{2}(f_1-2f_2)\,\alpha_x\,\alpha_y^2 \,\rho_x -\frac{1}{2}\left [11(f_2-f_1)-\frac{3 f_1^2}{f_2}\right]\alpha_x^2\,\alpha_y\,\rho_y \nonumber\\ & \quad +3\left(\frac{f_1}{f_2}-2\right)\left[f_1\left(\frac{f_2}{R_2}-1\right)+ f_2\left(\frac{3}{2}-\frac{f_2}{R_2}\right)\right] \alpha_x^3 \,\rho_x \nonumber\\ & \quad - \frac{3f_1}{2}\,\alpha_y^3\,\rho_y . \end{align} \tag{ 29 }$$

Here, $\rho_x = x_1/f_2$ and $\rho_y = y_1/f_1$. That is, $(\rho_x,\rho_y)$ are equal to the tangents of the angles of rays converging onto the focus, ignoring deflections of rays in the x direction by the first lens.

There are 14 terms in (29), out of a possible 16 allowed by the symmetry of an anamorphic system [35]. The two absent fourth-order aberrations are the spherical aberration terms of the individual lenses, proportional to $\rho_x^4$ and $\rho_y^4$, which are zero by design (as given by the forms of φ and $\boldsymbol{q}$). However there is a term in (29) proportional to $\rho_x^2\,\rho_y^2$ which thus varies with the fourth power of the pupil coordinate along the diagonal and is the only aberration for the on-axis field points $\alpha_x = \alpha_y = 0$. It is seen from (29) that the coma terms (proportional to $\alpha_y\,\rho_x^3$ and $\alpha_x\,\rho_y^3$) are zero under the same conditions as for the individual lenses, namely that the curvatures of the lenses are set equal to the focal lengths, $R_1 = f_1$ and $R_2 = f_2$. However, even then, there are some coma-like cross terms (proportional to $\alpha_y \, \rho_x^2 \, \rho_y$ and $\alpha_x \, \rho_x \, \rho_y^2$) that are independent of R₁ and R₂. Indeed, the only dependence of the OPD on the radii of curvature occur in terms that are proportional to only x or only y coordinates, and we find through computational ray tracing that, in general, errors in or modifications to the wavefronts of the individual lenses only affect the wavefront in the corresponding coordinate of the system wavefront. Thus, a radius of curvature of the lenses equal to their focal lengths does reduce coma, and does minimise off-axis aberrations, but it does not produce an aplanatic system as is achievable for an axi-symmetric lens. The remaining terms of (29) are either quadratic in the pupil coordinates, representing curvature of field and astigmatism, or linear in pupil coordinates, describing image distortion. The field-dependent 45^° astigmatism (or more correctly, oblique astigmatism), proportional to $\rho_x\,\rho_y$, is zero for $f_2 = f_1/2$, as are some of the other terms. When $f_1 = f_2 = R_1 = R_2$ the expression for OPD is invariant in swapping the x and y coordinates ($\rho_x \leftrightarrow \rho_y$ and $\alpha_x \leftrightarrow \alpha_y$) as expected.

The minimum aberration occurs for a field point on axis, and unlike the case of the individual lenses this aberration is non-zero, with $\mathrm{OPD} = (3/4)f_2\,\rho_x^2\,\rho_y^2$. This non-zero aberration exists even though each individual lens focuses an incident collimated beam without aberration. It can be termed 'oblique spherical aberration' since it varies with the fourth order of the pupil radial coordinate along the diagonal (ψ = 45^° for a square pupil), and is zero in the x and y directions. For on-axis cylindrical lenses where ρ_x and ρ_y range from $-\mathrm{NA}$ to $+\mathrm{NA}$, the RMS wavefront error is evaluated to $(3/20) f_2 \, \mathrm{NA}^4$. This can be compensated slightly by introducing defocus proportional to $\rho_x^2+\rho_y^2$, reducing the RMS wavefront error to a minimum of $\sqrt{2/7}(3/20) f_2 \,\mathrm{NA}^4 = 0.11 f_2 \,\mathrm{NA}^4$. This error is significant for high-NA lenses for imaging at 1 nm resolution. For example, for $\mathrm{NA} = 0.0375$, λ = 0.075 nm, and f₂ = 1 mm, the RMS error is equal to $0.16~{\rm nm} = 2.1 \,\mathrm{waves}$, excluding tilt, defocus, and astigmatism. The Marechal condition at this wavelength (i.e an RMS wavefront less than λ/14 = 0.005 nm) can be satisfied for lenses with a 1 mm focal length only for $\mathrm{NA} \lt 0.016$, or a resolution of 2.3 nm. Alternatively, a resolution of 1 nm at this wavelength requires a focal length $f \lt 34\,\mu$m. Such a lens would have a very small aperture, so a more practical approach would be to compensate the on-axis aberration with an appropriately shaped refractive phase plate [36], akin to a Schmidt corrector plate [37].

The $\rho_x^2\,\rho_y^2$ error is not dependent on the curvatures of the surfaces of the individual lenses, nor the distance between them, and is independent of the field angle. The origin of this error can be traced to the curvature of field of the second MLL (here focusing in the x direction) due to rays incident out of the x–z plane. This is the situation addressed by (27) (but with the x and y directions reversed). That is, consider rays from the on-axis field point with $\hat{\boldsymbol{r}}_1 = (0,0,1)$ intersecting the first MLL at a particular height y₁ and a range of values x₁. These will be all be deflected in the y direction by the same angle, $\alpha_y \approx y_1/f_1$, and will impinge on the second lens with this angle, leading to a defocus term that according to (27) is proportional to $\alpha_y^2\,x_1^2$ and thus to $x_1^2 \,y_1^2$. One may expect that this could be addressed by polishing the surface of the second lens to a cylinder of radius f₂ that curves in the y direction so that the rays deflected by the first lens always impinge normal to the surface (and hence seem to always be on axis). Failing that, setting $R_2 = (2/3)f_2$ may eliminate the curvature of field as suggested by (27). The ray-trace analysis shows, however, that the on-axis aberration remains unchanged for any choice of curvature, as given by the term in (29). The wavefront is dependent upon the tilts of the diffracting layers in the MLL, which follow the cylindrical symmetry dictated by their deposition onto flat substrates. The normals of the layers of the second lens are in the x–z plane as described by $\boldsymbol{q}_2$ in table 2.

The transmission of the lens system is also affected by the violation of the Bragg condition in the second lens. For the on-axis field point, the analysis leading to (29) gives the solution of the deviation parameter of the first lens as $\epsilon_1(\rho_x,\rho_y) = 0$, showing that the Bragg condition is satisfied across the entire pupil of the first lens, but in the second lens

$$\begin{equation} \epsilon_2^{(4)}(\rho_x,\rho_y) = \rho_x^2 \rho_y^2/4. \end{equation} \tag{ 30 }$$

This equation suggests that the loss of transmission of the lens starts to occur at the corners of the pupil ($\rho_x = \rho_y = \mathrm{NA}$), even for the on-axis field point, when $\mathrm{NA}^4 > 4 w_\epsilon$, independent of focal length. For the SiC/WC multilayer system considered above for a wavelength of 0.075 nm where $|\delta_1-\delta_2| = 6.7 \times 10^{-4}$, (30) sets a limit of the NA of crossed 1D lenses to 0.064, corresponding in this case to a resolution of 0.59 nm.

Equation (30) confirms that the on-axis aberration stems from the fact that the convergent beam incident on the second lens does not mirror the reference wavefront that conceptually would produce the design of the second MLL as a computer-generated hologram. An elimination of the on-axis aberration (and the apodisation) would thus require that the layers of the second lens are fabricated to conform to the cylindrical symmetry introduced by the first lens and the desired spherical convergent beam created by the second. The use of a phase plate, as suggested above, would address the aberration but not the lens transmission.

From the cylindrical symmetries of the two lenses, any transverse displacement of one lens relative to the other either leaves the system invariant or simply shifts the optic axis, defined by the y coordinate of the first lens and the x coordinate of the second lens (see figure 7). A tilt of one lens about either the x or y axes will cause an aberration of similar magnitude as that from a field angle α_y or α_x . However, in that case there no longer exists an axis of symmetry of the system, and instead one can consider the coordinate system to be fixed to one of the lenses, such that a tilt of that lens is equivalent to an opposite change in the field angle. Alignment tolerances are investigated in section 5.5 for lenses with 1 nm resolution.

As known from the analysis of crossed cylindrical lenses, an error in the orthogonality of the two lenses, caused by a rotation of one relative to the other about the z axis, leads to oblique astigmatism with a wavefront error proportional to $\rho_x\,\rho_y$ [9]. This can be seen in the simplifying approximation of the system wavefront as the sum of the phase contributions of each lens, scaled to to the coordinates of the second lens, given by $\lambda/(2\pi)\,\phi_1 = -y^2/(2f_2)$ and $\lambda/(2\pi)\,\phi_2 = -x^2/(2f_2)$. The combined wavefront of $-f_2 (\rho_y^2 + \rho_x^2)/2$, represents (in the paraxial approximation) a spherical wave converging onto the focus at f₂. However, if the second lens is rotated about the optic axis by an angle χ_z then the second lens produces a phase proportional to $-f_2(\rho_x \cos\chi_z + \rho_y \sin\chi_z)^2$. For small χ_z the wavefront error found by subtracting the converging spherical wave is then equal to $f_2 \,\chi_z\,\rho_x\,\rho_y$. This is oblique (45^°) astigmatism, and this contribution is constant across the field. This term is orthogonal to the 0–90^° astigmatism and to defocus, and gives an RMS error of $f_2 \,\chi_z\, \mathrm{NA}^2 /3$. Given a tolerable degree of astigmatism of half a wave, the requirement on the alignment of the two lenses is then $\chi_z \lt 3 \lambda/(2 f_2 \,\mathrm{NA}^2)$. For the example above of λ = 0.075 nm, f₂ = 1 mm, and $\mathrm{NA} = 0.0375$ we find a requirement of $\chi_z \lt 80\, \mu {\rm rad}$. This tolerance can in some cases be relaxed by finding the field location with an oblique astigmatism that cancels this term due to the non-orthogonality. As seen in (29), the coefficient of the field-dependent oblique astigmatism is $-3(f_1-2f_2)\,\alpha_x\,\alpha_y$. However, choosing the appropriate α_x and α_y to zero the oblique astigmatism will introduce other off-axis aberrations.

Due to the anamorphism of crossed MLLs, and the fact that focal length scales inversely with λ_m , a change in wavelength causes a different amount of defocus in each focused direction. That is, besides defocus, crossed MLLs exhibit a chromatic astigmatism that is proportional to the difference in focal lengths $f_1-f_2$. The magnitude of off-axis aberrations (including distortion) will also depend on the field direction. For the on-axis field point, the chromatic defocus and astigmatism can be compensated by repositioning the lenses so that their distances to the image plane are $f_1\,\lambda/\lambda_m$ and $f_2\,\lambda/\lambda_m$ (respectively for the first and second lenses). As in section 4.2, the ray tracing procedure can be carried out for the repositioned lenses for λ ≠ λ_m , in which case the on-axis wavefront aberration is evaluated to

$$\begin{align} & \mathrm{OPD}_\mathrm{SA}(\rho;\lambda_m) = \frac{3f_2}{4}\frac{\lambda_m^3}{\lambda^3}\,\rho_x^2\,\rho_y^2 + \frac{3 f_1}{8} \frac{\lambda_m\, \Delta \lambda}{\lambda^2}\nonumber\\ & \quad \left( 2 -\frac{2f_1}{R_1}+\frac{\Delta \lambda}{\lambda}\right)\rho_y^4 + \frac{3 f_2}{8} \frac{\lambda_m\, \Delta \lambda}{\lambda^2}\left( 2 -\frac{2f_2}{R_2}+\frac{\Delta \lambda}{\lambda}\right)\rho_x^4. \end{align} \tag{ 31 }$$

The coefficients of the $\rho_y^4$ and $\rho_x^4$ terms of (31) are equivalent to the spherical aberration term for the axisymmetric lens of (23). These terms represent a distorted spherical aberration, which can be zeroed independently in each direction, for a given λ_m , with the appropriate radius of curvature R₁ and R₂. As for the axisymmetric lens, setting $R_1 = f_1$ and $R_2 = f_2$ gives a quadratic dependence of the spherical aberration on the wavelength deviation Δλ and therefore the lowest chromatic aberration for positive and negative wavelength deviations from the design wavelength. However, even though these terms can be minimised, the oblique spherical aberration (proportional to $\rho_x^2\,\rho_y^2$) scales with $(\lambda_m/\lambda)^3 = (1 +\Delta \lambda/\lambda)^3$ and therefore dominates.

As an example, we consider the above-mentioned lens system with a resolution of 1 nm at a wavelength λ = 0.075 nm (17 keV photon energy) with a square exit pupil of $\mathrm{NA} = 0.0375$ in the x and y directions, (and $\mathrm{NA} = 0.053$ along the diagonal). Here we define the resolution for a square-pupil optic to be $0.5 \lambda / \mathrm{NA}$ in the x and y directions. The focal lengths of the lenses are taken to be f₁ = 1.25 mm and f₂ = 1 mm, and thus the heights of the lenses are 93 and 75 µm, respectively. The ray-tracing procedure detailed above was used to numerically compute the wavefront aberrations exactly—that is, without approximating to a fourth-order series expansion as for (29). The particular procedure applies to the case of field points at infinity, displaced from the optical axis by field angles α_x and α_y as set by the ray direction $\hat{\boldsymbol{r}}_1$. This situation represents, for example, the formation of a focused beam for a scanning microscope. In this case the field angles also represent the tilt of the optic axis of the lenses relative to say a 'beam axis' set by the direction of a far-off source. The calculations also correspond to the aberrations expected in full-field imaging at high magnification (where the image plane is far from the lenses and the sample is near to the focal plane).

The aberrations are computed in terms of the points of intersection of rays on the two lenses. A map of the OPD as a function of the coordinates of the second lens is shown in figure 8(a), for flat lenses ($R_1 = R_2 = \infty$). The extent of the plot is limited to the same numerical aperture of 0.0375 in the two orthogonal directions. The best-fit defocus has been subtracted and this map represents the phase error of the exit wave converging onto the plane of best focus. This map is dominated by the fourth-order term $\rho_x^2\,\rho_y^2$, and the residual after subtracting that term from the calculated OPD is proportional to higher order terms with the same symmetry as displayed in figure 8(a). The RMS wavefront error of the OPD shown in figure 8(a) is 0.134 nm, compared with the contribution of the fourth-order term $0.08 f_2 \,\mathrm{NA}^4 = 0.158~{\rm nm}$. As previously, the PSF can be computed from a Fourier transform of the complex-valued pupil function of the lens system, constructed from the OPD and lens transmission. The PSF for the on-axis field point of the lens system is shown in figure 8(b) for a wavelength of 0.075 nm, compared with the PSF for a perfect system of the same NA and wavelength in figure 8(c). In addition to the increased width of the PSF with a full-width at half maximum of 1.64 nm, compared with 0.85 nm for the perfect system, the Strehl ratio of the PSF is 0.076.

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The calculated OPD for the on-axis field point far exceeds both the Marechal condition of the RMS error being less than λ/14 or the Rayleigh condition of an absolute deviation of the OPD by more than λ/4. Further computations show that the OPD at field angles less than about 1 mrad appear similar to the on-axis aberration map, so we would expect to achieve aplanatic imaging over a field of view of about this extent by compensating the wavefront equally at all field points. Such a compensation could be achieved by locating a phase plate close to either of the lenses.

The aberrations of such a phase-compensated system were simulated simply by subtracting the OPD computed for $\alpha_x = \alpha_y = 0$ from wavefront maps at other field angles. This should be a valid assumption when the phase plate is close enough to the second lens that its correction is the same for all field angles. The wavefront map for the field located at α_x  = 1.25 mrad and α_y  = 1 mrad is given in figure 9(a). The RMS error is 0.024 nm, which is certainly an improvement over the non-compensated lens system, but still does not meet the Marechal condition. It is apparent from this figure that the wavefront is dominated by coma oriented along the same direction as the field point (ψ = 45^°, accounting for the anamorphism). That is, the wavefront is proportional to $\rho_x^3 + \rho_y^3$. These are the terms in the second line of (29), which can be eliminated by setting the radii of curvature of the lenses equal to the focal lengths. Figure 9(b) shows the calculated wavefront for that case. The wavefront aberration is reduced slightly (especially along the $\rho_y = -\rho_x$ diagonal), and the RMS error is also slightly improved with a value of 0.019 nm. The aberration is still dominated by coma, now given by the terms on the third line of (29). Equation (29) suggests one more optimisation to reduce the wavefront aberration, which is to set the focal length of the second lens to half that of the first, $f_2 = f_1/2$, to eliminate one of the remaining two coma-like terms. With this, and cylindrical curvatures of the two lenses equal to the focal lengths, we find the lowest RMS error of 0.014 nm and the map of the wavefront given by figure 9(c). Here there is a remaining coma-like term that only depends on the position of the field in the y direction, which is the focusing direction of the first lens.

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Given a linear dependence on the field angle for coma, achieving the Marechal condition of 0.005 nm RMS aberration error therefore requires reducing the magnitude of the field angle to about a quarter to a third of the value simulated in figure 9 and thus we expect the aplanatic region to have a radius of about 0.3 mrad. Plots of the RMS error as a function of the field position $(\alpha_x,\alpha_y)$ are given in figure 10 for the same lens configurations as for figure 9, with the additional case of flat lenses with $f_1 = 2 f_2$. The aplanatic area is indicated by a dashed line, which gives the contour for an RMS of λ/14 ≈ 0.005 nm. Comparing figures 10(a) and (b), it is seen that setting the radii of the lenses to the focal lengths only provides a slight enlargement of the aplanatic area over the case of flat lenses. The RMS error increases linearly with the magnitude of the field angle in these cases, indicating coma in accordance with the terms of (29). The largest aplanatic field (and hence the largest tolerance for misalignment of lenses) is achieved by setting the focal length of the second lens to half that of the first, as in figure 10(c) and additionally curving each lens to a radius equal to the focal length as depicted in figure 10(d).

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Plots of the RMS wavefront error as a function of the relative wavelength deviation $\Delta\lambda / \lambda = \lambda_m/\lambda-1$ for the on-axis field point are given in figure 11, for the cases of curved lenses ($R_1 = f_1$ and $R_2 = f_2$) and flat lenses ($R_1 = \infty$ and $R_2 = \infty$), as computed by ray tracing. It is assumed in these calculations that both lenses are refocused for each wavelength change. As compared with the aberrations for the axisymmetric lens, plotted in figure 4(d), the RMS wavefront error is larger for the crossed lenses. It is found that the dominant aberration is proportional to $\rho_x^2\,\rho_y^2$, as expected from (31). The calculations of the RMS errors given by the solid lines in figure 11 furthermore assume that this error is corrected at Δλ = 0, for example by a phase plate, but that the correction is independent of wavelength. However, the refractive index decrement δ of a phase plate constructed from a material of low atomic number such as carbon (diamond) varies as λ² at wavelengths comparable to the design wavelength of 0.075 nm. In this case, given that the phase of the corrector is much less than unity, the correction scales as $(\lambda_m/\lambda)^2$. This somewhat compensates for the $(\lambda_m/\lambda)^3$ dependence of the $\rho_x^2\,\rho_y^2$ aberration of the crossed lenses, giving lower RMS errors plotted as dashed lines in figure 11.

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As with the case for the axisymmetric lens, the Darwin width of the Bragg reflections of these particular crossed lenses limits the field angles more stringently than the wavefront aberrations. A map of the total transmission of the lens pair is given in figure 12(a) as a function of field angle, for the same multilayer properties as in the calculations of figure 5: $|\delta_1-\delta_2| = 6.7 \times 10^{-6}$. For λ = 0.075 nm the half width $w_\theta = 0.11 ~{\rm mrad}$, as for the axisymmetric lens. The transmission is found to be independent of the radii of the lenses, as expected from (30). Likewise, the apodisation caused by detuning the wavelength has a similar behaviour to that found for the axisymmetric lens. As seen in figure 12(b), a deviation of the wavelength causes the effective numerical aperture of the lens to decrease, and a transmission of at least 0.5 requires $\Delta\lambda/\lambda \lt 0.9 \%$.

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The two lenses must be oriented with respect to each other by angles that are comparable to the allowable field angles. The tolerance can be explored by considering one lens to be fixed (which sets the coordinate system) and setting the tilt of the other lens and the field coordinates as free parameters. Ignoring the orthogonality of the lenses (which was discussed in section 5.2) this four-dimensional space is illustrated in figure 13. Here ray tracing calculations are presented for two flat lenses ($R_1 = R_2 = \infty$). The second lens was held fixed and the first tilted by χ_x about an axis passing through its vertex parallel to the x axis or by an angle χ_y about the y axis. The tilt by χ_x is given in the left-hand column of the figure. This tips the lens in the direction that it focuses (refer to figure 7) and it is seen that rays incident on the lens at that tilt angle are efficiently transmitted through it and the second lens, as indicated by the false-colour maps of the transmission as a function of field angle. That the region of high transmission tracks the tilt angle of the lens shows that the transmission of the second lens (which focuses in the x direction) is not sensitive to misalignment in the y direction. The contours in the maps of figure 13 show the RMS wavefront aberration. For the tilts of χ_x (the left column) the aplanatic area shrinks and only tracks the tilt at about a third of the rate of the lens tilt. Thus, the region of the field with low aberrations moves out of the region of high transmission and the tolerable magnitude of the lens tilt χ_x (where the system gives high transmission and low aberrations) is about 0.2 mrad.

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The situation is somewhat more relaxed for tilts of the first lens by χ_y about the y axis, as displayed in the right-hand column of figure 13. Just as the χ_x tilt had little effect on the transmission of the second lens, the tilt of χ_y has little effect on the transmission of the first lens. Therefore, the map of the transmission of the lens system remains unchanged. However, the aplanatic region of the field tracks with about half the rate of the lens tilt and does not diminish in area as rapidly as for χ_x tilts. The tolerable magnitude of the χ_y tilt is about 0.4 mrad. This tilt of the first lens by χ_y is equivalent to a rotation by −χ_y of the second lens about the axis that passes through its vertex parallel to y, combined with a rotation of the field angle. Although the dependence of the aberrations on field angle differs for different values of R₁, R₂, and the ratio of focal lengths, it is found that the tolerances of the lens misalignment remain approximately the same for the various cases that were investigated in figure 10.

The imaging performance depends both on the wavefront aberration and the apodisation, as is visualised in figure 14 for a particular misalignment of two flat lenses with χ_x  = 0.4 mrad and χ_y  = 0.4 mrad, compensated by a phase plate. The phase plate is assumed to conjugate the wavefront of perfectly-aligned lenses. The complex-valued pupil functions were constructed from the phase as given by the OPD and the amplitude equal to the square root of the transmission, plotted in the figure over a range of field angles α_x and α_y . As expected from figure 13 the field position that gives maximum transmission, at $(\alpha_x, \alpha_y) = (0, 0.2)\, \mathrm{mrad}$ does not correspond to the location of lowest aberration at $(\alpha_x, \alpha_y) = (0.1, 0.1)\, \mathrm{mrad}$. The wavefront at the location with maximum transmission is dominated by a 45^° coma. The width of the apodised PSF at $(\alpha_x, \alpha_y) = (0.1, 0.1)\, \mathrm{mrad}$ is similar to that of the aberrated PSF at $(\alpha_x, \alpha_y) = (0, 0.2)\, \mathrm{mrad}$. Further from these field points, the PSF is dominated by the severe loss of diffraction efficiency at the edges of the pupil.

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An overall scaling of the MLL structure by a factor h may occur due to an error in the deposition rate used to make it. The dilation of the structure gives rise to a phase φ_m (r) = φ(r/h) (of the 'manufactured' lens in terms of the designed lens) which, from a Taylor expansion based on (3), yields a lowest-order term equal to $r^2/(2 f h^2)$. The scaled MLL therefore has a focal length f  h². The scale error places the $n^\mathrm{th}$ layer at hr_n , or at the position r_n(m) given by $r_{n(m)}/h = r_n$. Each layer provides an extra wavelength of path, with $\phi_m(h r_n) = \phi(r_n) = n\lambda$. The scaling changes the spacings of the layers and hence also the deflection angles of rays. Since $\partial \phi(r/h)/\partial r = \phi'(r/h)/h$, it follows from (5) that the periods of the scaled MLL are given by hd(r/h) (for the case of the flat lens). That is, the period of the relocated $n\mathrm{th}$ layer is scaled by h. From (6), this again shows that the focal length becomes h² f. The scaling also has an effect on higher-order aberrations. One way to conceptualise this case is to consider that scaling the lens and the wavelength by h similarly scales the focal length, as well as the OPD (in normalised pupil coordinates), by the same factor. That is, the spherical aberration of the scaled lens will be zero for a wavelength hλ. Reverting to the original wavelength will require scaling that by 1/h (giving a focal length h times larger again), and will be akin to the discussion of chromatic aberrations described in section 4.2, where now $\Delta\lambda/\lambda = 1/h -1$. That is, for an axi-symmetric lens originally designed with focal length f and radius R, the spherical aberration of the scaled lens at the original design wavelength will be

$$\begin{equation} \mathrm{OPD}_\mathrm{SA}(\rho; h) = \frac{3 f}{8}\frac{1-h}{h}\left(1+\frac{1}{h}-\frac{2f}{R}\right)\rho^4. \end{equation} \tag{ 32 }$$

As with the chromatic aberration, the highest tolerance to scaling errors is achieved for R = f, but the allowable scaling error is limited primarily by violating the Bragg condition in the outermost layers, assuming that the layer tilts are also scaled (that is, that the entire lens structure is dilated uniformly in all directions). For the lens parameters and wavelength considered in section 4.3, a scaling error of less than about 1% can be tolerated, as indicated by figure 5(b).

A drift in the rate of deposition of materials in the fabrication of an MLL will lead to an error in the d spacing and the position of those layers. Consider a deposition recipe that assumes a constant rate p of the accumulation of material which requires the material pairs to be alternated at deposition times T_n such that $r_n = p T_n$. However, if that rate changes in time then those layers will occur at positions

$$\begin{equation} r_{n(m)} = \int_0^{T_n} p (1 + c(t)) \,{\rm d} t. \end{equation} \tag{ 33 }$$

A linear drift, c(t) = c₁ t, deposits layers according to $r_{n(m)} = p T_n + (c_1/2) p T_n^2 = r_n + (\beta/2) r_n^2$, where β = c₁/p is the relative change in deposited thickness per thickness of material deposited (with units of inverse length). This coordinate error can be compared with the simpler scale error of section 6.1, showing that

$$\begin{align} \frac{\lambda}{2 \pi} \phi_m(r) & = \frac{\lambda}{2 \pi} \phi(r-\beta r^2/2) = s(r-\beta r^2/2) \nonumber\\ & \quad + \sqrt{(r-\beta r^2/2)^2 + (f-s(r-\beta r^2/2))^2} - f \nonumber\\ & \approx \frac{r^2}{2 f} - \frac{\beta r^3}{2 f} + \left(\frac{2}{R}-\frac{1}{f} \right) \frac{r^4}{8 f^{\,2}}. \end{align} \tag{ 34 }$$

Here the inverse of the coordinate error was determined by solving $r_n + \beta r_n^2 /2 = r_{n(m)}$ for r_n . When β is positive, the number of periods deposited over a height r is reduced, and so $\phi_m(r) \lt \phi(r)$ in agreement with (34).

As with off-axis aberrations of ideal (non-distorted) lenses, the incident beam impinging on the structure described by (34) no longer matches a reference beam that would give rise to a hologram described by φ_m , and accordingly the Bragg diffraction is modified due to the change in d spacing. The OPD must therefore be calculated according to (19), with a modified form of $\boldsymbol{q}$. In particular, since from (34) $\partial \phi_m(r) / \partial r = (1-\beta r) \,\phi'(r-\beta r^2/2)$, (16) must be replaced by $(1-\beta r)\,\boldsymbol{q}(r-\beta r^2/2)$. Analogously to the scaling error above, the period of the layer at r is modified by a multiplicative factor $1/(1-\beta r) \approx 1+\beta r$ and repositioned to $r_n+\beta r_n^2/2$. We find for the on-axis field point of a curved axi-symmetric lens,

$$\begin{equation} \mathrm{OPD}(\rho) \approx - \beta f^{\,2} \rho^3, \end{equation} \tag{ 35 }$$

for the normalised pupil coordinate ρ = r/f. For a 1D lens, a similar scaling as for $\boldsymbol{q}$ must be applied to $\boldsymbol{q}_1$ in table 2. In this case we obtain the same result as (35) but with ρ replaced by the linear coordinate ρ_x or ρ_y .

Equation (35) shows that a linear drift of the deposition leads to a third-order error. For an axi-symmetric lens this is not coma (which has a direction and is not radial) and is a phase error that is approximately constant across the field. For a 1D lens, this does vary in a single direction as per the the coma terms proportional to $\rho_x^3$ and $\rho_y^3$ in (29). Thus, in that case, a small degree of deposition drift β can be compensated by tilting the lens, such that $(f_1/R_1-1) \alpha_y = \beta f_1$ (for $f_1 \neq R_1$). This compensation is only practical for field angles α_y that are smaller than the Darwin-width acceptance $w_\theta$ of the finest layers, which considerably limits the range of β that can be compensated. The RMS OPD error for a 1D lens with coma given by (35) is equal to $\beta f^{\,2} \mathrm{NA}^3 / (6 \sqrt{7})$, minimised here by a defocus. The Marechal condition is satisfied when the RMS OPD is less than λ/14, and so without any compensation, we require

$$\begin{equation} \beta \lt 1.13 \lambda / (f^{\,2} \mathrm{NA}^3) \approx 9 \delta / D^2 \end{equation} \tag{ 36 }$$

where $\delta = \lambda/(2 \mathrm{NA})$ is the lens resolution and $D = 2\mathrm{NA} f$ is the lens diameter. For f = 1 mm, $\mathrm{NA} = 0.037$, and λ = 0.075 nm, $\beta \lt 1.6 \times 10^{-6} \mu{\rm m}^{-1}$ or a relative error of 1.5 × 10⁻⁴ over the course of the entire deposition. This corresponds to an absolute error of 7.1 nm in the height of the lens.

Another example of a deposition error is a constant offset of the period, d_m (r) = d(r)+Δd. This can occur if the materials used to form the layers interdiffuse and create a thin interface with a different density to either of the unmixed materials [38]. It can be seen that this is in fact equivalent to the example just considered, since we have from (6) that $d(r) + \Delta d \approx \lambda f/r \,(1+ \Delta d \,r/(\lambda f))$, so that β = Δd/(λf). The magnitude of the coma generated by a given contraction error Δd increases as the wavelength is reduced, but from (36) a reduction in the focal length and the size of the lens relaxes the tolerance of Δd.

MLLs are usually sliced to the desired thickness τ from an extended structure formed by material deposition. The angle at which a flat lens is cut might not be exactly normal to the optic axis, or the centre of curvature of a spherical surface might not exactly lie on that optic axis. We consider here the case of a flat lens cut at an angle that differs from the intended normal face by an angle γ. If the structure from which the lens is cut replicates the family of paraboloids that are described by (2), then the resulting optical element would be a tilted holographic lens, exhibiting zero OPD for the field point on the optical axis. Indeed, this will be the case for any arbitrary surface shape z = s(r). Thus, in this case, cutting a lens at an incline will only change the off-axis aberrations. However, if the deposited layers only match the paraboloid forms on a particular surface (for example, z = 0) and differ in form away from that surface, then a mis-cut will cause an aberrated image of the on-axis field point. The aberration depends on the form of the layers at the positions cut by the surface, and we consider here the case of a flat 1D lens constructed from layers that are tilted planes.

We consider a lens that focuses in the y direction with a focal length f. The tilted planes of the structure from which the lens is sliced all intersect the optic axis a distance 2f from the lens vertex, as shown schematically in figure 15. The lens is cut on a plane that is tilted about the x axis by an angle γ, with the optic axis along z. A ray from the on-axis field angle intersects the lens at a height y and a longitudinal distance z = ytanγ from the vertex. The layer of the MLL intersected by this ray, which we presume to be the $n\mathrm{th}$ period, passes through the z = 0 plane at a height $y_n = 2fy/(2f-y\tan\gamma)$. Since the lens was intended to be cut normal to the optic axis, the position y_n follows the prescription of (2) for z = 0. We see from figure 15 that for positive γ the layers the ray intersects are thinner and placed closer to the optic axis than intended, but have almost the correct tilt to satisfy the Bragg condition. With analogy to the analysis in section 6.1 the lens can be considered to be locally scaled by the factor $y/y_n = 1-y\tan \gamma/(2f)$ so that

$$\begin{align} \frac{\lambda}{2 \pi} \phi_m(y) & = \frac{\lambda}{2 \pi} \phi\left(\frac{y}{1-y \tan\gamma/(2f)}\right) \nonumber\\ & = \sqrt{f^{\,2}+\frac{y^2}{(1-y\tan\gamma/(2f))^2})} - f. \end{align} \tag{ 37 }$$

Even though the surface of the MLL follows a tilted plane, we take s(y) = 0 in the derivation of φ_m (y) from (3) since the scaling accounts for this. Following the analyses of the previous sections,

$$\begin{equation} \frac{\partial \phi_m(y)}{\partial y} = \frac{1}{\left(1-y\tan \gamma/(2f)\right)^2} \phi'\left(\frac{y}{1-y\tan \gamma/(2f)} \right), \end{equation} \tag{ 38 }$$

and the expression for $\boldsymbol{q}(y)$ must be scaled in a similar way. The normal vector is

$$\begin{equation} \hat{\boldsymbol{n}}(y) = \left(0,\,\sin \gamma,-\cos \gamma \right). \end{equation} \tag{ 39 }$$

For the on-axis field angle, we follow the ray-tracing procedure established in section 4 with l₁ = y tan γ and $l_2\, \hat{\boldsymbol{r}}'\cdot \hat{\boldsymbol{z}} = f-y\ \tan\ \gamma$. Using a symbolic mathematics program the Taylor expansion of the wavefront aberration is evaluated to

$$\begin{equation} \mathrm{OPD}^{(4)}(\rho_y) = \frac{f}{2}\tan \gamma \,\rho_y^3+\frac{f}{4}{\tan^2}\gamma \,\rho_y^4. \end{equation} \tag{ 40 }$$

The dominating term is coma, proportional to $\rho_y^3$. This term depends linearly on the tangent of the mis-cut angle γ. To maintain the Marechal condition of an RMS wavefront error below λ/14 requires a mis-cut angle

$$\begin{equation} \gamma \lt 2.3 \frac{\lambda}{f\,\mathrm{NA}^3}, \end{equation} \tag{ 41 }$$

or less than 3.3 mrad for the 0.0375 NA lens considered in section 5.4.

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Perhaps more importantly, the linear dependence of coma on the mis-cut angle gives a way to easily compensate for the main component of drift in the deposition rate as explored in section 6.2. The coma error of (35) can be removed by cutting the lens at an angle given by tanγ = 2βf. Setting a limit of $\tan \gamma \lt 1$ to ensure that the additional spherical aberration term of (40) is kept manageable, the largest drift in the deposition rate that can be tolerated is raised to β = 1/(2f). For a lens with f = 1 mm, this corresponds to 5 × 10⁻⁴ µm^-1 or a relative error of 3.75% over the height of a 75 µm diameter lens. The range of angles at which the lens can be cut may be limited by satisfying the Bragg condition. The series expansion of the expression of the deviation parameter determined by solving $|\boldsymbol{r}'| = 1$ in the ray-trace procedure is given by

$$\begin{equation} \epsilon = \frac{\tan \gamma}{2 \cos \gamma} \rho^3 + \frac{7 {\tan^2} \gamma}{8 \cos\gamma}\rho^4. \end{equation} \tag{ 42 }$$

Maintaining Bragg efficiency at the edge of the lens ($\rho = \mathrm{NA}$) therefore requires

$$\begin{equation} \gamma \lt \frac{4 |\delta_1-\delta_2|}{\pi}\frac{1}{\mathrm{NA}^3}. \end{equation} \tag{ 43 }$$

For the materials and parameters of the lens considered in section 5, the largest tolerable cut angle is thus about 160 mrad, which then limits the correctable deposition drift to no more than 0.8 × 10⁻⁴  µm⁻¹.