1. Introduction

A common design problem is to split one laser beam into multiple beams with equal energy. For example, binary phase beam-splitting gratings utilize a pattern consisting of two phase levels [1–3]. Since these gratings are limited to only two phases, efficiency is relatively low. Dammann and Görtler were the first to address the problem of beam-splitting phase gratings [3], their class of binary phase grating solutions are now referred to as Dammann gratings. Continuous phase gratings were developed to further improve efficiency [3–5]. Using calculus of variations, Romero and Dickey found analytical expressions for the optimal continuous phase functions of one- and two-dimensional gratings that maximize energy into N desired output beams [6,7]. While these gratings theoretically provide optimal efficiency into N outgoing beams, the presence of fabrication errors can significantly degrade their performance. However, consideration of fabrication errors in the design can significantly improve tolerances [8]. Applications that benefit from beam-splitting gratings are parallel processing in laser machining and material processing, sensor systems, interferometry, communication systems, and image processing and gathering system [9].

One disadvantage of single layer beam-splitting gratings is that they are only optimized for a single wavelength. If the wavelength deviates from the designed wavelength, uniformity of the output beams degrades. This degradation may be unacceptable for certain applications. An achromatic beam-splitting grating has wavelength-independent diffraction efficiency. By using multiple layers, a diffractive optical element (DOE) can be achromatized for multiple wavelengths [10,11]. The sensitivity of multilayer DOEs has been studied by analyzing the change in the diffraction efficiency of multilayer DOEs with respect to some common fabrication errors [12,13]. However, only DOEs that are designed to diffract all energy into one diffraction order were analyzed. By optimizing the levels of a single relatively thin multi-level phase grating, a grating can be achromatized at multiple wavelengths [14]. However, these designs typically have several closely spaced large discontinuities in the height profiles, which may make fabrication difficult. In this paper, a generalized achromatic design method is used to design multilayer achromatic beam-splitting grating doublets with equal energy output modes at two wavelengths. By studying the sensitivity of these achromatic grating doublets, it is shown that grating profiles with constant spatial derivatives perform significantly better than continuous grating profiles with respect to fabrication errors.

Section 2 details the theory of achromatic beam-splitting grating doublet design. Section 3 analyzes the sensitivity of grating doublets to lateral shifts between grating layers for an achromatic continuous phase grating doublet and an achromatic Dammann grating doublet. Section 4 analyzes the sensitivity of grating doublets to fabrication errors in the heights of each layer for an achromatic Dammann grating doublet.

2. Achromatic beam-splitting grating doublet design

A lossless, one-dimensional grating is characterized by periodic height function h(x). In the absence of fabrication errors and using a thin grating approximation, the transmitted phase in air of a laser beam at position x is changed by an amount

The efficiency of each beam at wavelength λ is defined as $|a_p(\lambda ){|}^2$. A 1xN vector $\eta \left[\varphi (x,\lambda )\right]$ is defined that contains efficiencies of each desired output beam at wavelength λ,

The efficiency of each output beam depends on the amount of fabrication error and the phase function of the grating. Total efficiency E of the grating at wavelength λ is

In the design algorithm, Fourier coefficients of an error-free uniform beam-splitting grating are subject to the constraint

To demonstrate the performance change due to wavelength, performance of a 1x5 beam-splitting phase grating is simulated from 400 nm – 700 nm. The form of the phase function used to design beam-splitting continuous phase gratings into odd numbers of beams is described analytically by [6]:

 

Fig. 1 a) Transmitted phase in radians of achromatic 1x5 continuous phase beam-splitting grating, b) Standard deviation σ and c) total efficiency E of the 5 desired output modes as a function of wavelength. At 700 nm, σ_max = 0.165. d) The efficiency of each output mode $|a_p{|}^2$ as a function of wavelength. At 700 nm, $\Delta |a_p{|}_{\max }^2$ = 0.403.

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Analogous to how a refractive achromatic doublet lens is designed to have the same focus for two wavelengths, an achromatic beam-splitting grating doublet is designed to produce σ = 0 at two wavelengths. In order for a beam-splitting grating to have equal energy output modes at two different wavelengths, two grating layers with different refractive indices are placed in series, as shown in Fig. 2 (a). To design achromatic grating doublets, start with the desired phase profile for a beam-splitting grating that is represented by ϕ(x) in radians, like the phase function in Fig. 1(a). Achromatization with two grating layers sets ϕ(x) to be the same at two wavelengths. The optical path lengths in units of microns for two desired wavelengths is given by:

 

Fig. 2 (a) Diagram of the OPL for an achromatic grating. Index of refraction for (b) BK7 and (c) SF5 as a function of wavelength. (d) Height profile in microns for grating layer 1. (e) Height profile in microns for grating layer 2. Peak-to-valley heights for layer 1 and layer 2 are 17.24 μm and 12.58 μm, respectively.

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An example of an achromatized continuous phase 1x5 beam-splitting phase grating doublet is shown in Fig. 2 that is optimized for λ₁ = 486 nm and λ₂ = 656 nm. The desired phase profile ϕ(x) is shown in Fig. 1(a). Grating layer 1 is made from BK7 glass, and grating layer 2 is made from SF5 glass. At these wavelengths, n₁(486 nm) = 1.5224, n₁(656 nm) = 1.5143, n₂(486 nm) = 1.6875, and n₂(656 nm) = 1.6667. Figures 2(b) and 2(c) show the index of refraction as a function of wavelength for BK7 and SF5, respectively. Height profiles for the two grating layers are shown in Fig. 2(d) and 2(e). Grating layer 1 has a peak-to-valley height of H₁ of 17.24 μm and grating layer 2 has a peak-to-valley height H₂ of 12.58 μm. Note that h₂(x) is an inverted copy of h₁(x), and is scaled by a different height. When light is transmitted through both grating layers using illumination at λ₁ or λ₂, the transmitted phase of the light is given by the desired phase profile ϕ(x). For all other wavelengths, the transmitted phase deviates from ϕ(x), and the 5 desired output modes are no longer equal energy.

A plot of the standard deviation σ as a function of wavelength is plotted in Fig. 3(a), where σ = 0 for λ₁ and λ₂. For all other wavelengths, the modes are not equal energy. The efficiency for each mode $|a_p{|}^2$ is shown as a function of wavelength in Fig. 3(b). Note the transmitted phase through both grating layers deviates significantly from the desired phase profile at λ = 400 nm, which results in σ = 0.2. If this is unacceptable for a certain application, the achromatic grating doublet can be redesigned in order to minimize σ at λ = 400 nm by using a combination of different materials for the grating layers, different design wavelengths λ₁ and λ₂, and different phase profiles. Between 486 nm and 700 nm_, the maximum difference in efficiencies between modes is $\Delta |a_p{|}_{\max }^2$ = 0.047, and the maximum standard deviation σ_max = 0.020 at λ = 554 nm. Comparing this to $\Delta |a_p{|}_{\max }^2$ and σ_max at 700 nm for the grating optimized for a single wavelength, there is an improvement by a factor of at least 8 in both $\Delta |a_p{|}_{\max }^2$ and σ_max between 486 nm and 700 nm. Using this method, beam-splitting achromatic grating doublets can be designed for any ϕ(x). The ability to fabricate an achromatic grating doublet is explored in the following sections.

 

Fig. 3 a) Standard deviation σ and b) efficiency E of the 5 desired output modes as a function of wavelength for achromatic 1x5 continuous phase beam-splitting grating doublet. σ_max = 0.020 between 486 nm and 700 nm. d) The efficiency of each output mode $|a_p{|}^2$ as a function of wavelength. The max difference between 486 nm and 700 nm is $\Delta |a_p{|}_{{}^{\max }}^2$ = 0.047.

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3. Sensitivity to lateral shifts between grating layers

3.1 Theory

This section studies how a lateral shift between two grating layers of an achromatic design affects performance. First, assume the achromatic solution is known, where ϕ₁(x,λ) is the transmitted phase through grating layer 1 and ϕ₂(x,λ) is the transmitted phase through grating layer 2. When added together, ϕ₁(x,λ) and ϕ₂(x,λ) have a transmitted phase ϕ(x,λ) that produces equal energy output beams at two wavelengths λ₁ and λ₂. The electric field amplitude of the beam in the far field with uniform illumination is proportional to:

To satisfy Eq. (18), ${{\varphi }^{\prime }}_2(x,\lambda )$ must be constant. The types of gratings that approximate this property are blazed gratings, Dammann gratings and multi-level phase gratings. Blazed gratings are ideally a sawtooth pattern, which has a constant non-zero first spatial derivative, except for the transitions at the end of every period. Dammann gratings and multi-level phase gratings have profiles with either binary or multi-level phases. Therefore, the grating profiles are a series of steps, where each step has a first spatial derivative equal to zero.

It makes intuitive sense that constant derivative grating profiles are the most resistant to lateral shifts between gratings. For example, as light is transmitted through two binary phase gratings shifted relative to each other, the only portion of the transmitted phase that deviates from the designed transmitted phase is near the transitions of the grating. All other portions of the transmitted phase have the designed phase. In comparison, if a continuous phase grating is shifted, the transmitted phase at every position x deviates from the designed transmitted phase after passing through both gratings. Therefore, the far field pattern from a shifted continuous phase grating deviates significantly from its designed performance.

3.2 Simulation of performance of a laterally shifted continuous phase grating doublet

A continuous phase grating doublet is analyzed to demonstrate problems associated with a lateral shift between the two continuous grating layers that are designed in Sec. 2. Figure 4(a) shows the transmitted phase ϕ₂(x,λ) of the profile from Fig. 2(e) at λ₁. Figure 4(b) shows ${{\varphi }^{\prime }}_2(x,{\lambda }_1)$. Figure 4(c) shows |U_deriv| assuming Δx = 0.01, where units of Δx are fractional period. Note that |U_deriv| exhibits significant energy in almost all modes. When U_deriv is convolved with U_equal, a low-amplitude copy of U_equal is centered at each of the modes shown in Fig. 4(c). This solution yields extremely poor performance, even with the small shift of Δx = 0.01. Figure 4(d) shows the total efficiency E and Fig. 4(e) shows the standard deviation σ of the 5 desired output beams plotted as a function of the relative shift between gratings at the two design wavelengths. For |Δx| > 0.008, efficiency E drops to below half its designed value, which indicates the majority of the energy in these modes are distributed to higher, undesired modes. The difference in E between Δx = 0 and Δx = 0.1 is 0.86. The maximum value of σ is σ_max ≈0.1 at Δx ≈0.004, which is a significant departure from the design value, while σ_max = 0.020 in Fig. 3(a). Figures 5(a) and 5(b) show $|a_p{|}^2$ for the continuous phase grating for λ₁ and λ₂, respectively. After |Δx| > 0.01, the efficiency of each output mode is below half its designed value. The performance of this grating deteriorates significantly with small lateral shifts. Due to the high sensitivity to lateral shifts between the grating layers, continuous phase gratings are ill-suited for achromatic grating doublets. Continuous phase achromatic grating doublets require nearly perfect alignment, which may be extremely difficult to achieve during fabrication.

 

Fig. 4 (a) Transmitted phase ϕ₂(x,λ) in radians through the grating layer 2 of 1x5 achromatic continuous phase beam-splitting grating doublet. (b) The spatial derivative ${{\varphi }^{\prime }}_2(x,{\lambda }_1)$ in units of radians/μm. (c) |U_deriv|. Note nearly all modes are non-zero. (d) Efficiency E and (e) standard deviation σ of desired 5 modes as a function of the lateral shift of grating layer 2.

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Fig. 5 Efficiency of each output mode $|a_p{|}^2$ at (a) λ = 486 nm and (b) λ = 656 nm as a function of lateral shift of grating layer 2. At Δx = 0, $|a_0{|}^2\approx |a_{\pm 1}{|}^2\approx |a_{\pm 2}{|}^2$.

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3.3 Simulation of performance of laterally shifted Dammann grating

Binary phase gratings have a first derivative of zero over nearly the entire period, which make them excellent candidates for achromatic beam-splitting grating doublets. The solution for an achromatic binary phase Dammann grating doublet that splits an incident beam into 5 equal energy output beams is shown in Fig. 6. The desired transmitted phase ϕ(x) has transitions between the high and low phases at x = (−0.471, −0.133, 0.133, 0.480)T, where T is the period of the grating, with a maximum phase of 2.993 radians [9]. This grating is optimized to be achromatic at λ₁ = 486 nm and λ₂ = 656 nm. Grating layer 1 is made from BK7 glass and grating layer 2 is made from SF5 glass. Figure 6(a) shows the transmitted phase through the ideal combined grating layers, and Figs. 6(b) and 6(c) show height profiles of grating layers 1 and 2, respectively. The peak-to-valley heights are 11.29 μm and 8.24 μm for grating layers 1 and 2, respectively. Once again, note that h₂(x) is an inverted copy of h₁(x), and is scaled by a different height. Figure 6(d) shows the efficiency E, which is nearly constant after λ = 486 nm. Figure 6(e) shows the efficiency of each of the 5 desired modes $|a_p{|}^2$ as a function of wavelength. After λ = 460 nm, the efficiency of each mode remains relatively constant. A magnified portion of this wavelength range is shown in Fig. 6(f). This grating doublet has 5 nearly equal energy output beams from 486 nm – 700 nm, which makes it an excellent achromatic grating over this wavelength range with $\Delta |a_p{|}_{\max }^2$ = 0.010 and σ_max = 0.004 at λ = 700 nm. Note $\Delta |a_p{|}_{\max }^2$ and σ_max are approximately 5 times lower than the achromatic continuous phase beam-splitting grating doublet from 486 nm – 700 nm.

 

Fig. 6 Transmitted phase ϕ(x) in radians of an achromatic 1x5 beam-splitting Dammann grating doublet. (b) Height profile in microns for grating layer 1. (c) Height profile in microns for grating layer 2. Peak-to-valley heights are 11.29 μm and 8.24 μm, respectively. (d) Efficiency E of each of the desired 5 output modes from λ = 400 nm – 700 nm. (e) The efficiency of each output mode $|a_p{|}^2$ as a function of wavelength. (f) Magnified portion of $|a_p{|}^2$ from λ = 460 nm – 700 nm.

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The transmitted phase ϕ₂(x, λ₁) is shown in Fig. 7(a) and ${{\varphi }^{\prime }}_2(x,{\lambda }_1)$ is shown in Fig. 7(b). ${{\varphi }^{\prime }}_2(x,{\lambda }_1)$ is zero, except at the transitions between the high and low phase of the profile. Figure 7(c) shows |U_deriv| assuming Δx = 0.01, which is approximately a delta function. Therefore, this grating is resistant to shifts between grating layers. The efficiency E and standard deviation σ are plotted as a function of the lateral shift in terms of the fraction of the grating period in Figs. 7(d) and 7(e), respectively. The difference in efficiency E between Δx = 0 and Δx = 0.1 is 0.05, which is 17 times smaller than the shifted continuous phase grating design. Additionally, σ_max ≈0.04 at Δx = 0.1 for λ = 486 nm, which is nearly 2.5 times less than with the shifted continuous phase grating design. Figures 8(a) and 8(b) show efficiencies of each desired output mode as a function of the shift of the grating layers for λ₁ and λ₂, respectively. For small shifts there is very little change in the efficiencies of the desired modes, so the performance of this achromatic grating doublet is maintained, even in the presence of a lateral shift between the two grating layers, whereas efficiency for each mode in the continuous phase grating doublet decreased dramatically after a small shift. The Dammann grating doublet performs better by a factor of 17 over the continuous phase grating doublet in the presence of a shift between grating layers.

 

Fig. 7 (a) Transmitted phase in radians through ϕ₂(x,λ₁) of achromatic Damman phase grating doublet. (b) The spatial derivative ${{\varphi }^{\prime }}_2(x,{\lambda }_1)$ in units of radians/μm. (c) |U_deriv|. Note it is nearly a delta function. (f) Efficiency E and (e) standard deviation σ of desired 5 modes as a function of the lateral shift of grating 2.

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Fig. 8 (a) The efficiency of each output mode $|a_p{|}^2$ at (a) λ = 486 nm and (b) λ = 656 nm as a function of lateral shift of grating layer 2. At Δx = 0, $|a_0{|}^2\approx |a_{\pm 1}{|}^2\approx |a_{\pm 2}{|}^2$.

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Greisukh et al demonstrated scalar diffraction theory reliably models two-layer phase-relief diffraction structures when the ratio of the structure period to the height of the diffraction structures is greater than 2.5 [16]. Based off this result, scalar diffraction theory is reliably accurate for periods greater than 28 μm and 43 μm for the achromatic Damman grating doublet and the continuous phase achromatic grating doublet, respectively. To maintain σ<0.04 for the achromatic continuous phase grating doublet at λ = 486 nm with a 43 μm period, the grating layers have to be aligned with Δx < 0.002, which corresponds to a lateral shift less than 0.086 μm. In contrast, the achromatic Dammann grating doublet with a 43 μm period has to be aligned within 4.3 μm to maintain σ<0.04 at λ = 486 nm. For this example, the tolerance for alignment is 50 times smaller for the achromatic continuous phase grating doublet, which makes alignment extremely difficult.

A summary of the results for the achromatic continuous phase grating doublet and the achromatic Dammann grating doublet is shown in Table 1. Note the achromatic Dammann grating doublet significantly outperforms the achromatic continuous phase grating doublet in all cases. Therefore, gratings that have a height profile where the spatial derivative is constant over approximately the entire period like Dammann gratings, multi-level gratings, and blazed gratings minimize sensitivity to relative shifts between the two grating layers. While continuous phase beam-splitting gratings are superior for single-wavelength designs, since they can yield higher efficiencies and better uniformity than Dammann and multi-level gratings, their sensitivity to lateral shifts between grating layers makes it extremely difficult to fabricate an achromatic continuous phase grating doublet with good performance. Dammann gratings, multi-level gratings, and blazed gratings are superior for fabrication of achromatic beam-splitting grating doublets.

Table 1. Summary of 1x5 achromatic grating doublet results

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4. Sensitivity to height fabrication errors

4.1 Theory

In addition to lateral shifts between the two grating layers, each grating layer may be fabricated with an error in its height. Start with the solution for an achromatic grating doublet, where ϕ₁(x, λ) is the transmitted phase through grating layer 1 and ϕ₂(x, λ) is the transmitted phase through grating layer 2. Together, ϕ₁(x, λ) and ϕ₂(x, λ) have a transmitted phase ϕ(x, λ) that creates equal energy output beams at two wavelengths λ₁ and λ₂. Assume there is a constant multiplicative fabrication error in the height, so the total phase transmission is (1 + Δ)ϕ(x, λ). The electric field amplitude in the far field in the presence of fabrication errors in the height is proportional to:

The transmitted phase is rewritten in terms of phase amplitude ϕ_q0 and normalized base profile b_q(x), where the subscript q indicates the layer of the doublet,

It is instructive to observe the conditions on Δ₁ and Δ₂ for which Eq. (25) is satisfied with different b₁(x). For an arbitrary base function, the condition for which Eq. (25) is satisfied is given by

4.2 Simulation of height fabrication errors for achromatic Dammann grating doublets

The performance change due to height errors in both layers of the achromatic Dammann grating doublet described in Sec. 3.3 is simulated in this section. Figure 9(a) shows σ as a function of Δ₁ and Δ₂ for λ = 486 nm. Note that σ is periodic, as predicted by Eq. (27). Furthermore, the minima of σ are located along the lines given by Eq. (28), shown as solid lines in Fig. 9(a). Assume grating layer 1 is fabricated with the incorrect height after fabrication and is measured to have multiplicative height error Δ₁. Instead of trying to fabricate grating layer 2 at the designed value Δ₂ = 0, σ is minimized if the height of grating layer 2 is calculated using the line given by Eq. (28).

 

Fig. 9 (a) Standard deviation σ of 5 desired output modes of achromatic Dammann grating doublet as a function of height fabrication errors Δ₁ and Δ₂ at λ = 486 nm. The lines given by Eq. (28) for λ = 486 nm are plotted as solid lines. The line perpendicular to the solid lines is plotted as a dashed line. (b) σ along dashed line in (a) for λ = 486 nm (solid line) and λ = 656 nm (dashed line).

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As predicted by Eqs. (27) and (28), σ is also minimized when m is non-zero. To demonstrate the periodic nature of σ, the line perpendicular to the lines given by Eq. (28) is shown as a dashed line in Fig. 9(a). For λ = 486 nm, σ is periodic along the dashed line with a period T₄₈₆ as shown in Fig. 9(b). However, this period is wavelength dependent. Figure 9(b) also shows σ perpendicular to the lines given by Eq. (28) for λ = 656 nm, which has a larger period T₆₅₆. When m is zero in Eq. (27), the minima of σ for different wavelengths overlap. However, when m is non-zero, the minima of σ for different wavelengths do not overlap. Thus, when averaged over multiple wavelengths, the performance of the doublet is reduced when m is non-zero.

Since the achromatic Dammann grating doublet has excellent performance from λ = 486 nm – 700 nm, Fig. 10(a) shows σ averaged over this wavelength range as a function of Δ₁ and Δ₂. The multiplicative height error combinations that minimize σ_avg are shown along the solid line in Fig. 10(a). Even with errors from Δ₁ = [-0.1,0.1], σ_avg only changes by a maximum of 5 × 10⁻⁵ along this line. In comparison, suppose Δ₂ is fabricated and has a value along the dashed line shown in Fig. 10(a). Figure 10(b) shows σ_avg along the dashed line as a function of Δ₁. Along this line, σ_avg oscillates, but does not return to the minimum value at Δ₁ = 0 since Eq. (27) cannot be satisfied for multiple wavelengths when m is non-zero.

 

Fig. 10 (a) Standard deviation σ of 5 desired output modes of achromatic Dammann grating doublet as a function of height fabrication errors Δ₁ and Δ₂ averaged over λ = 486 nm – 700 nm. Solid line is height combinations that minimize σ_avg. Dashed line is the perpendicular to the solid line. At red square (0.05,0), σ_avg = 0.429. At yellow circle (0.05,0.049), σ_avg = 0.002. (b) Average σ from 486 nm – 700 nm along dashed line in (a).

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As an example, suppose grating layer 1 is fabricated with Δ₁ = 0.05. If grating layer 2 is fabricated with Δ₂ = 0, σ_avg = 0.429, which is indicated as the red square in Fig. 10(a). However, if grating layer 2 is calculated using Eq. (28) it is fabricated with Δ₂ = 0.049, which is indicated as the yellow circle in Fig. 10(a). At this Δ₂, σ_avg reduces to 0.002, which is a 215 times improvement. Therefore, the performance of the achromatic Dammann grating doublet is improved significantly over a large wavelength range if the height of grating layer 2 is calculated taking Δ₁ into account.

5. Conclusion

Simulations for an achromatic continuous phase grating doublet and an achromatic Dammann grating doublet are shown in this paper. Sensitivities of achromatic beam-splitting grating doublets to lateral shifts between grating layers are studied in detail. It is shown that continuous phase grating doublets are extremely sensitive to lateral shifts between the grating layers, which makes fabrication extremely difficult. Grating profiles with a constant first spatial derivative decrease the performance change caused by lateral shifts of the grating layers. Grating profiles with a nearly constant first spatial derivative over a full period include blazed gratings, Dammann gratings, and multi-level phase gratings.

An achromatic Dammann grating doublet and an achromatic continuous phase grating doublet that split an incident beam into 5 equal energy outgoing beams at λ₁ = 486 nm and λ₂ = 656 nm are designed. A simulation of an achromatic Dammann grating doublet with a lateral shift between grating layers of one-hundredth of the period has 17 times better performance than the achromatic continuous phase grating doublet with the same lateral shift. Grating profiles with a constant first spatial derivative are superior designs for achromatic grating doublets, since they are significantly more resistant to lateral shifts between grating layers.

By studying the sensitivities to fabrication errors in the height, it is shown that certain height error combinations yield no performance change for a single wavelength. Additionally, for binary gratings it shown that if the height fabrication errors between the two grating layers combine to produce a phase error that is an integer multiple of 2π, there is no change in the far-field pattern at a particular wavelength. When averaging over a large wavelength range, the performance of achromatic grating doublets can be preserved even with significant fabrication errors in the height of each grating layer. It is shown that the performance of an achromatic Dammann grating doublet can be improved by a factor of 215 if the height of the grating layers are chosen in order to minimize the performance change in the presence of height fabrication errors.

In addition to lateral shifts between grating layers and errors in the fabricated height of each layer, a grating layer may also be rotated or tilted with respect to the other grating layer. Both of these alignment errors further degrade the performance of a grating doublet. The diffraction pattern from a grating doublet is convolution of the diffraction pattern from grating layer 1 with the diffraction pattern from grating layer 2. A rotation between grating layers rotates the diffraction patterns from each layer with respect to each other, which may degrade the performance of the grating doublet. A tilt in one grating layer changes the optical path length through that grating layer, which may further degrade the performance of the grating doublet. Future work may quantify the impact of rotation and tilt errors on achromatic grating doublets.