Strategy for design of achromatic diffractive optical elements with minimized etch depths Tobias Gühne* and Jochen Barth EADS Deutschland GmbH, Cassidian, Landshuter Str. 26, D-85716 Unterschleissheim, Germany *Corresponding author: [email protected] Received 9 August 2013; revised 22 October 2013; accepted 2 November 2013; posted 6 November 2013 (Doc. ID 195479); published 25 November 2013
Achromatization of a diffractive optical element (DOE) is achieved by a doublet of two holograms etched into different substrates. We show that for optimal results, it is not sufficient to just maximize the diffraction index difference, but the dispersion must also be regarded. A simple but effective strategy is presented to select substrates which allow minimum etch depths for the holograms. We apply this strategy to design an achromatic DOE for the near-infrared range. The introduction of a corridor in which the DOE phase shift is allowed to vary increases the spectral range for which the figures of merit (diffraction efficiency, nonuniformity error) indicate excellent performance. © 2013 Optical Society of America OCIS codes: (050.1970) Diffractive optics; (220.1000) Aberration compensation. http://dx.doi.org/10.1364/AO.52.008419
1. Introductions
Diffractive optical elements (DOEs) provide many advantages for the design of optics and open new applications in sensing and illumination. In this paper, we address the correction of the chromatic error of DOEs with particular emphasis on the minimization of the required etch depth in order to limit the high aspect ratios. DOEs contain a pixilated surface structure etched into the optical material which introduces phase lags between waves transmitted through the DOE along different paths. By designing the surface structure we can control the phase lags and thus the interference conditions of the transmitted waves in the image plane. Because the phase lag results as a measure of the glass thickness in units of the wavelength, it is a linear function of the wavelength of the transmitted light. Thus, a single DOE is designed for a particular wavelength, and when the DOE is used for wavelengths different from the design wavelength a chromatic error occurs. 1559-128X/13/348419-05$15.00/0 © 2013 Optical Society of America
Arieli et al. [1,2] have established a model to calculate the DOE phase lag in a scalar approach, and investigated the reduction of the chromatic error by combining two DOEs made from glasses of different refractive indices and dispersions, with equal surface structures but different etch depths. They also suggest a criterion for the selection of an optimum glass combination for the achromatization. In a recent paper Kleeberg et al. [3] extend the same approach to correct the chromatic error for gradient–index, subwavelength, and cut-and-paste DOE designs and emphasize the importance of low etch depths. This present paper is based on the same scalar approach which applies to DOEs with lateral structures that are large compared to the wavelength of light. We present a strategy to find an optimum glass combination for achromatic DOEs with minimum etch depths which goes beyond the previous suggestion [1,2], and apply this strategy to a case study in the NIR spectral region where dispersion coefficients of glasses are smaller than in the visible regime and achromatization is harder to achieve. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
8419
2. Achromatization of Diffractive Optical Elements
Figure 1 illustrates the DOE achromatization with two etched steps of depths d1 and d2 placed adjacent to each other. The refractive indices are n1 and n2 . Light transmitted along paths A and B emerges with the phase difference φA−B according to φA−B d d n1 − 1 · 1 − n2 − 1 · 2 ; 2π λ λ
(1)
where the etch depth d is entered as a positive value, and λ is the wavelength. We can thus calculate pairs of d1 and d2 , which will solve the equation for a desired phase difference. Of this multitude of pairs of d1 and d2 satisfying Eq. (1), we select the achromatic solution by taking the derivative of the phase difference with respect to the wavelength and setting it equal to zero dn1 dn2 · λ − n1 − 1 d φA−B dλ · λ − n2 − 1 dλ · d − · d2 0: 1 2 dλ 2π λ λ2 (2) Combining this with Eq. (1) we obtain an expression of the etch depths d1 and d2 as functions of the refractive indices and the dispersions of the hologram materials
d1;2
3. Selecting Optical Materials for DOE Achromatization
If we consider that the lateral, pixilated, structures of a DOE are of the order of a few micrometers and that the etch depth typically amounts to several wavelengths, we encounter aspect ratios close to 1, or even larger. The etch depth of a DOE should thus be chosen as small as possible, not only for manufacturing reasons but also to reduce shadowing effects if the DOE is designed as part of imaging optics. This case normally implies that rays are incident
n2;1 − 1 − λ0 · n02;1 λ λ0 φA−B · λ0 · : n1 − 1 · n2 − 1 − λ0 · n02 λ λ0 − n2 − 1 · n1 − 1 − λ0 · n01 λ λ0 2π
Here, λ0 denotes the design wavelength. Observe that d1 is calculated by inserting the associated refractive index n1 in the numerator of Eq. (3) and d2 by inserting n2 while the denominator remains the same in both cases. While the basic equations are already contained in earlier publications [1,2] our conclusions and the derived strategy to minimize etch depths are beyond the previous discussion.
n1
path A
n2
path B
d1
d2
Fig. 1. Etched step in an achromatic DOE consisting of two holograms. 8420
Before proceeding on to discuss the optimum choice of refractive materials, one remark should be made. Achromatization is achieved by correcting the chromatic error of the phase lag using the dispersion of the glass materials. In fact, if we set the dispersion in Eq. (3) equal to zero, i.e., n0 λ0 0, there is no solution for the etch depth. As a more quantitative observation, we note that we correct an error of the order of the difference between the wavelength of regard and the design wavelength by the dispersion which varies with wavelength to a much lesser extent. Although Fig. 1 implies that etch depths of the order of the design wavelength will introduce any desired phase shift, we must apply much larger etch depths in order to compensate the wavelength dependence of the phase shift with the effect of the dispersion.
APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013
(3)
onto the DOE under angles different from the surface normal. From Eq. (3) we observe that the denominator is the same for both DOE materials. Thus, we conclude that the relative magnitude of d for each material depends mainly on the refractive index because, in general, n − 1 ≪ λ0 · n0 in the transparent region of optical materials. Furthermore, we find that the low-index material has a higher etch depth because its numerator depends on the optical data of the high-index material. Nevertheless, selecting a material with the highest available index for the regarded wavelength range for one DOE is a good starting point because the phase lag results as a measure of the glass thickness in units of the internal wavelength which gives high-index materials a significant advantage. The above arguments provide a strategy to find the optimum combination of materials for an achromatic DOE doublet with minimum etch depths for a given phase lag. Starting with the highest index material which is suitable for the application will define two variables in Eq. (3). Selecting an arbitrarily lowindex material for the first DOE and setting the
phase lag equal to π we can express the etch depth d1 as a function of λ0, n2 , and dn2 ∕dλ. Treating d1 as parameter we find that the two optical variables which characterize the desired low-index material are connected by a linear relationship according to n1
A λ A · λ0 · n01 0 · 1 − 1; B B 2 · d1
(4)
with A n2 − 1 and B λ0 · n02 . This suggests a visualization in an index– dispersion diagram for which Eq (4) defines contour plots of constant etch depths as straight lines, and which contains the available optical glasses as points according to their values of refractive index and dispersion, quite similar to the well-known Abbé diagram. From the index–dispersion diagram we obtain the material for the second DOE which provides the intended achromatization with a minimum required etch depth. The required data of optical materials are obtained from standard glass catalogs, e.g., [4], or online services [5]. We note in passing that the Abbé diagram itself may be taken as an approximate index–dispersion diagram because the Abbé number v, which is used as the abscissa in the Abbé diagram, is approximately proportional to the inverse of the refractive index derivative for a design wavelength of the Fraunhofer spectral line λD v ≈
nD − 1 nD − 1 C nF − nC λF − λC · nλFF −n −λC nD − 1 n −1 · n0 jλD −1 . D λF − λC · n0 jλD 170 nm
(5)
However, if we substitute the index derivative in the above equations using this approximation the resulting etch depths are calculated with a significant error, and the Abbé diagram is readily available only for the design wavelength of λD, i.e., for the visible spectral range. Thus, the index–dispersion diagram provides more accuracy and better design flexibility. Here, we illustrate our suggested strategy by an NIR case study. While the index and dispersion of optical glasses are lower than in the visible region, we have the advantage of using low bandgap crystals such as ZnSe or ZnS with refractive indices around 2.5. Defining ZnSe as the high-index material (DOE with index n2 ), we establish the index– dispersion diagram of Fig. 2. The bold solid line intersecting the ZnSe point at n 2.52 and −dn∕dλ 0.254 is the contour plot for infinite etch depth for which the denominator of Eq. (4) is equal to 0, and the shaded area to the right of this line marks the region for which optical materials would become the high-index partners to ZnSe for achromatization. To the left we show a series of contour plots for various etch depths converging toward the bold line, and the material with the largest distance to
Fig. 2. Index–dispersion diagram for NIR achromatization of a DOE made of ZnSe.
this line on the left side requires the minimum etch depth for achromatization. The best result is obtained for a DOE pair of ZnS and ZnSe with etch depths of 4.3 and 4.0 μm, respectively. If the selection of the partner material is constrained to conventional glasses, the best choice is N-LASF31A with etch depths of 5.7 and 3.0 μm (N-LASF31A/ZnSe). If the plastic material of PMMA is favored because of its low price and ease of manufacturing, the resulting larger etch depth is 8.5 μm, which is about 50% larger than for the combination of N-LASF31A and ZnSe. At the same time, the lower etch depth decreases but if the achromatic DOE is also used for rays incident at larger angles this does not provide an advantage. The inset in Fig. 2 summarizes the etch depths achieved for different combinations of materials at a 800 nm design wavelength. If the high-index material ZnSe is replaced by a conventional glass, the etch depths increase significantly. The dashed line in Fig. 2 shows a contour line for the achromatization of -NSF66 with a 15 μm larger etch depth, and even the best partner material exceeds this value. In the visual range, etch depths for the combination of N-LAK34 and N-SF66 are 5.5 μm∕4.0 μm (λ0 588 nm), for comparison. 4. Design and Modeling of an Achromatic DOE
We investigate the achromatized DOE consisting of N-LASF31A glass and crystalline ZnSe. Using the field tracing software VirtualLab by Lighttrans [6], the DOE is modeled in a 2f-setup and built up virtually from unit cells which represent the desired pixilated surface profile. When designing the phase transmission, care is taken to keep the feature size of the DOE surface structures large enough as shown in Fig. 3. Within the 12 μm × 12 μm large unit cell all features are at least five times larger than the design wavelength thus supporting the validity of our simulation approach. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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Fig. 3. Unit cell of the investigated DOEs. The unetched substrate surface is shown in black and the etched regions in gray.
Combining two holograms with complementary height profiles as in Fig. 1 permits us to investigate the properties of the achromatic DOE doublet as part of the optical design. We obtain five dots in the image plane of the optics from parallel incident light, arranged as on dice: four dots associated with first-order diffraction in two dimensions, plus a zeroorder spot attenuated to match the intensity of the first-order spots. As an independent check, we verified the transmission function by calculating the diffraction pattern based on Raleigh–Sommerfeld equations for a planar wave which samples the unit cell of Fig. 3. We obtain the expected output pattern in the image plane with five dots of homogenous intensity distribution thus validating the simulation results of VitualLab. The phase lag for the achromatic DOE is calculated according to Eqs. (1)–(4) with the maximum phase lag intentionally set to π 5% at 800 nm. The resulting phase variation is plotted in the upper part of Fig. 4 yielding a useful spectral range between 667 and 1050 nm, where the phase lag is constrained to π − 5% < φ < π 5%. The resulting etch depths are 6 μm for N-LASF31A and 3.14 μm for ZnSe, slightly larger than calculated above due to the increased phase lag. The benefit of the achromatization becomes evident from a comparison with a single DOE without correction of the chromatic error which is modeled using the same unit cell. The single DOE is modeled with its surface profile etched into fused silica. For better comparability we choose a design wavelength of 950 nm for the chromatic DOE because at this wavelength the achromatic DOE produces a phase lag of π as shown in the upper part of Fig. 4. Thus, at this reference wavelength the phase lag of both corrected and uncorrected DOEs agree. 8422
APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013
Fig. 4. (Top) Phase shift of the achromatic DOE versus wavelength. (Bottom) Efficiency, nonuniformity error, and overall quality of the achromatic DOE doublet compared to a single, uncorrected DOE.
The DOE efficiency η and nonuniformity error ε are calculated, and in addition we define a figure of merit Q combining η and ε to assess the overall DOE quality η
I pattern ; I total
ε
I max − I min ; I max I min
Q ‖η · 1 − ε‖; (6)
where I total , I pattern , I max , and I min denote the total intensity which is incident onto the DOE, the integral intensity obtained within the design output pattern, and the maximum and minimum intensities within this pattern, respectively. A high-performance DOE has a low nonuniformity error and a high diffraction efficiency. We compare the figures of merit of both DOEs in Fig. 4, where η, ε, and Q for the two types of DOEs are plotted versus wavelengths. All values are obtained by simulating the DOE output pattern for different incident wavelengths using a step width of 25 nm. The DOE with uncorrected chromatic error shows a relatively stable diffraction efficiency but the nonuniformity error increases even within a rather narrow spectral range, and the overall DOE quality decreases accordingly. This is mainly due to an increasing intensity in the zero-order spot which is very sensitive to changes in the phase lag. In contrast, the achromatized DOE has a nonuniformity close to zero increasing to less than 8% at 1100 nm, and the DOE quality exceeds 92% between 700 nm and about 1100 nm. Hence, the VirtualLab model matches well with our phase calculations: within the intended useful spectral range the achieved quality (Q) is a direct result of the very small variation of the phase lag. Note that our calculation also regards light incident under different angles by modeling the DOE combined with an objective with a 40° field of view.
5. Conclusion
Our case study demonstrates the effectivity of DOE achromatization by using a doublet of holograms etched into suitably selected optical substrates. This selection is optimized by a simple, yet effective, strategy which uses contour lines of constant etch depth in a diagram containing index and dispersion values for the substrates of interest, at the design wavelength (index–dispersion diagram). This yields minimum required etch depths for DOEs forming an achromatized doublet. This allows us to also use the doublet for light incident under oblique angles. In addition, the introduction of a corridor in which the phase shift may vary leads to a maximized spectral range. As a result, even in the difficult NIR spectral range and including partial constructive interference for the zero-order
diffraction spot which is highly sensitive to the phase condition, we obtain a DOE which exhibits excellent performance over a wide spectral range in our simulation. References 1. Y. Arieli, S. Ozeri, N. Eisenberg, and S. Noach, “Design of a diffractive optical element for wide spectral bandwidth,” Opt. Lett. 23, 823–824 (1998). 2. Y. Arieli, S. Noach, S. Ozeri, and N. Eisenberg, “Design of diffractive optical elements for multiple wavelength,” Appl. Opt. 37, 6174–6177 (1998). 3. B. H. Kleeberg, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high-efficiency DOEs,” J. Europ. Opt. Soc. 3, 08015 (2008). 4. Schott AG, Optisches Glas, Version 1.8d (2009). 5. http://refractiveindex.info. 6. F. Wyrowski and H. Schimmel, “Electromagnetic optical engineering—an introduction,” Photonik Int. 1, 1–4 (2006).
1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
8423
Achromatization of a diffractive optical element (DOE) is achieved by a doublet of two holograms etched into different substrates. We show that for optimal results, it is not sufficient to just maximize the diffraction index difference, but the dispersion must also be regarded. A simple but effective strategy is presented to select substrates which allow minimum etch depths for the holograms. We apply this strategy to design an achromatic DOE for the near-infrared range. The introduction of a corridor in which the DOE phase shift is allowed to vary increases the spectral range for which the figures of merit (diffraction efficiency, nonuniformity error) indicate excellent performance. © 2013 Optical Society of America OCIS codes: (050.1970) Diffractive optics; (220.1000) Aberration compensation. http://dx.doi.org/10.1364/AO.52.008419
1. Introductions
Diffractive optical elements (DOEs) provide many advantages for the design of optics and open new applications in sensing and illumination. In this paper, we address the correction of the chromatic error of DOEs with particular emphasis on the minimization of the required etch depth in order to limit the high aspect ratios. DOEs contain a pixilated surface structure etched into the optical material which introduces phase lags between waves transmitted through the DOE along different paths. By designing the surface structure we can control the phase lags and thus the interference conditions of the transmitted waves in the image plane. Because the phase lag results as a measure of the glass thickness in units of the wavelength, it is a linear function of the wavelength of the transmitted light. Thus, a single DOE is designed for a particular wavelength, and when the DOE is used for wavelengths different from the design wavelength a chromatic error occurs. 1559-128X/13/348419-05$15.00/0 © 2013 Optical Society of America
Arieli et al. [1,2] have established a model to calculate the DOE phase lag in a scalar approach, and investigated the reduction of the chromatic error by combining two DOEs made from glasses of different refractive indices and dispersions, with equal surface structures but different etch depths. They also suggest a criterion for the selection of an optimum glass combination for the achromatization. In a recent paper Kleeberg et al. [3] extend the same approach to correct the chromatic error for gradient–index, subwavelength, and cut-and-paste DOE designs and emphasize the importance of low etch depths. This present paper is based on the same scalar approach which applies to DOEs with lateral structures that are large compared to the wavelength of light. We present a strategy to find an optimum glass combination for achromatic DOEs with minimum etch depths which goes beyond the previous suggestion [1,2], and apply this strategy to a case study in the NIR spectral region where dispersion coefficients of glasses are smaller than in the visible regime and achromatization is harder to achieve. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
8419
2. Achromatization of Diffractive Optical Elements
Figure 1 illustrates the DOE achromatization with two etched steps of depths d1 and d2 placed adjacent to each other. The refractive indices are n1 and n2 . Light transmitted along paths A and B emerges with the phase difference φA−B according to φA−B d d n1 − 1 · 1 − n2 − 1 · 2 ; 2π λ λ
(1)
where the etch depth d is entered as a positive value, and λ is the wavelength. We can thus calculate pairs of d1 and d2 , which will solve the equation for a desired phase difference. Of this multitude of pairs of d1 and d2 satisfying Eq. (1), we select the achromatic solution by taking the derivative of the phase difference with respect to the wavelength and setting it equal to zero dn1 dn2 · λ − n1 − 1 d φA−B dλ · λ − n2 − 1 dλ · d − · d2 0: 1 2 dλ 2π λ λ2 (2) Combining this with Eq. (1) we obtain an expression of the etch depths d1 and d2 as functions of the refractive indices and the dispersions of the hologram materials
d1;2
3. Selecting Optical Materials for DOE Achromatization
If we consider that the lateral, pixilated, structures of a DOE are of the order of a few micrometers and that the etch depth typically amounts to several wavelengths, we encounter aspect ratios close to 1, or even larger. The etch depth of a DOE should thus be chosen as small as possible, not only for manufacturing reasons but also to reduce shadowing effects if the DOE is designed as part of imaging optics. This case normally implies that rays are incident
n2;1 − 1 − λ0 · n02;1 λ λ0 φA−B · λ0 · : n1 − 1 · n2 − 1 − λ0 · n02 λ λ0 − n2 − 1 · n1 − 1 − λ0 · n01 λ λ0 2π
Here, λ0 denotes the design wavelength. Observe that d1 is calculated by inserting the associated refractive index n1 in the numerator of Eq. (3) and d2 by inserting n2 while the denominator remains the same in both cases. While the basic equations are already contained in earlier publications [1,2] our conclusions and the derived strategy to minimize etch depths are beyond the previous discussion.
n1
path A
n2
path B
d1
d2
Fig. 1. Etched step in an achromatic DOE consisting of two holograms. 8420
Before proceeding on to discuss the optimum choice of refractive materials, one remark should be made. Achromatization is achieved by correcting the chromatic error of the phase lag using the dispersion of the glass materials. In fact, if we set the dispersion in Eq. (3) equal to zero, i.e., n0 λ0 0, there is no solution for the etch depth. As a more quantitative observation, we note that we correct an error of the order of the difference between the wavelength of regard and the design wavelength by the dispersion which varies with wavelength to a much lesser extent. Although Fig. 1 implies that etch depths of the order of the design wavelength will introduce any desired phase shift, we must apply much larger etch depths in order to compensate the wavelength dependence of the phase shift with the effect of the dispersion.
APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013
(3)
onto the DOE under angles different from the surface normal. From Eq. (3) we observe that the denominator is the same for both DOE materials. Thus, we conclude that the relative magnitude of d for each material depends mainly on the refractive index because, in general, n − 1 ≪ λ0 · n0 in the transparent region of optical materials. Furthermore, we find that the low-index material has a higher etch depth because its numerator depends on the optical data of the high-index material. Nevertheless, selecting a material with the highest available index for the regarded wavelength range for one DOE is a good starting point because the phase lag results as a measure of the glass thickness in units of the internal wavelength which gives high-index materials a significant advantage. The above arguments provide a strategy to find the optimum combination of materials for an achromatic DOE doublet with minimum etch depths for a given phase lag. Starting with the highest index material which is suitable for the application will define two variables in Eq. (3). Selecting an arbitrarily lowindex material for the first DOE and setting the
phase lag equal to π we can express the etch depth d1 as a function of λ0, n2 , and dn2 ∕dλ. Treating d1 as parameter we find that the two optical variables which characterize the desired low-index material are connected by a linear relationship according to n1
A λ A · λ0 · n01 0 · 1 − 1; B B 2 · d1
(4)
with A n2 − 1 and B λ0 · n02 . This suggests a visualization in an index– dispersion diagram for which Eq (4) defines contour plots of constant etch depths as straight lines, and which contains the available optical glasses as points according to their values of refractive index and dispersion, quite similar to the well-known Abbé diagram. From the index–dispersion diagram we obtain the material for the second DOE which provides the intended achromatization with a minimum required etch depth. The required data of optical materials are obtained from standard glass catalogs, e.g., [4], or online services [5]. We note in passing that the Abbé diagram itself may be taken as an approximate index–dispersion diagram because the Abbé number v, which is used as the abscissa in the Abbé diagram, is approximately proportional to the inverse of the refractive index derivative for a design wavelength of the Fraunhofer spectral line λD v ≈
nD − 1 nD − 1 C nF − nC λF − λC · nλFF −n −λC nD − 1 n −1 · n0 jλD −1 . D λF − λC · n0 jλD 170 nm
(5)
However, if we substitute the index derivative in the above equations using this approximation the resulting etch depths are calculated with a significant error, and the Abbé diagram is readily available only for the design wavelength of λD, i.e., for the visible spectral range. Thus, the index–dispersion diagram provides more accuracy and better design flexibility. Here, we illustrate our suggested strategy by an NIR case study. While the index and dispersion of optical glasses are lower than in the visible region, we have the advantage of using low bandgap crystals such as ZnSe or ZnS with refractive indices around 2.5. Defining ZnSe as the high-index material (DOE with index n2 ), we establish the index– dispersion diagram of Fig. 2. The bold solid line intersecting the ZnSe point at n 2.52 and −dn∕dλ 0.254 is the contour plot for infinite etch depth for which the denominator of Eq. (4) is equal to 0, and the shaded area to the right of this line marks the region for which optical materials would become the high-index partners to ZnSe for achromatization. To the left we show a series of contour plots for various etch depths converging toward the bold line, and the material with the largest distance to
Fig. 2. Index–dispersion diagram for NIR achromatization of a DOE made of ZnSe.
this line on the left side requires the minimum etch depth for achromatization. The best result is obtained for a DOE pair of ZnS and ZnSe with etch depths of 4.3 and 4.0 μm, respectively. If the selection of the partner material is constrained to conventional glasses, the best choice is N-LASF31A with etch depths of 5.7 and 3.0 μm (N-LASF31A/ZnSe). If the plastic material of PMMA is favored because of its low price and ease of manufacturing, the resulting larger etch depth is 8.5 μm, which is about 50% larger than for the combination of N-LASF31A and ZnSe. At the same time, the lower etch depth decreases but if the achromatic DOE is also used for rays incident at larger angles this does not provide an advantage. The inset in Fig. 2 summarizes the etch depths achieved for different combinations of materials at a 800 nm design wavelength. If the high-index material ZnSe is replaced by a conventional glass, the etch depths increase significantly. The dashed line in Fig. 2 shows a contour line for the achromatization of -NSF66 with a 15 μm larger etch depth, and even the best partner material exceeds this value. In the visual range, etch depths for the combination of N-LAK34 and N-SF66 are 5.5 μm∕4.0 μm (λ0 588 nm), for comparison. 4. Design and Modeling of an Achromatic DOE
We investigate the achromatized DOE consisting of N-LASF31A glass and crystalline ZnSe. Using the field tracing software VirtualLab by Lighttrans [6], the DOE is modeled in a 2f-setup and built up virtually from unit cells which represent the desired pixilated surface profile. When designing the phase transmission, care is taken to keep the feature size of the DOE surface structures large enough as shown in Fig. 3. Within the 12 μm × 12 μm large unit cell all features are at least five times larger than the design wavelength thus supporting the validity of our simulation approach. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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Fig. 3. Unit cell of the investigated DOEs. The unetched substrate surface is shown in black and the etched regions in gray.
Combining two holograms with complementary height profiles as in Fig. 1 permits us to investigate the properties of the achromatic DOE doublet as part of the optical design. We obtain five dots in the image plane of the optics from parallel incident light, arranged as on dice: four dots associated with first-order diffraction in two dimensions, plus a zeroorder spot attenuated to match the intensity of the first-order spots. As an independent check, we verified the transmission function by calculating the diffraction pattern based on Raleigh–Sommerfeld equations for a planar wave which samples the unit cell of Fig. 3. We obtain the expected output pattern in the image plane with five dots of homogenous intensity distribution thus validating the simulation results of VitualLab. The phase lag for the achromatic DOE is calculated according to Eqs. (1)–(4) with the maximum phase lag intentionally set to π 5% at 800 nm. The resulting phase variation is plotted in the upper part of Fig. 4 yielding a useful spectral range between 667 and 1050 nm, where the phase lag is constrained to π − 5% < φ < π 5%. The resulting etch depths are 6 μm for N-LASF31A and 3.14 μm for ZnSe, slightly larger than calculated above due to the increased phase lag. The benefit of the achromatization becomes evident from a comparison with a single DOE without correction of the chromatic error which is modeled using the same unit cell. The single DOE is modeled with its surface profile etched into fused silica. For better comparability we choose a design wavelength of 950 nm for the chromatic DOE because at this wavelength the achromatic DOE produces a phase lag of π as shown in the upper part of Fig. 4. Thus, at this reference wavelength the phase lag of both corrected and uncorrected DOEs agree. 8422
APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013
Fig. 4. (Top) Phase shift of the achromatic DOE versus wavelength. (Bottom) Efficiency, nonuniformity error, and overall quality of the achromatic DOE doublet compared to a single, uncorrected DOE.
The DOE efficiency η and nonuniformity error ε are calculated, and in addition we define a figure of merit Q combining η and ε to assess the overall DOE quality η
I pattern ; I total
ε
I max − I min ; I max I min
Q ‖η · 1 − ε‖; (6)
where I total , I pattern , I max , and I min denote the total intensity which is incident onto the DOE, the integral intensity obtained within the design output pattern, and the maximum and minimum intensities within this pattern, respectively. A high-performance DOE has a low nonuniformity error and a high diffraction efficiency. We compare the figures of merit of both DOEs in Fig. 4, where η, ε, and Q for the two types of DOEs are plotted versus wavelengths. All values are obtained by simulating the DOE output pattern for different incident wavelengths using a step width of 25 nm. The DOE with uncorrected chromatic error shows a relatively stable diffraction efficiency but the nonuniformity error increases even within a rather narrow spectral range, and the overall DOE quality decreases accordingly. This is mainly due to an increasing intensity in the zero-order spot which is very sensitive to changes in the phase lag. In contrast, the achromatized DOE has a nonuniformity close to zero increasing to less than 8% at 1100 nm, and the DOE quality exceeds 92% between 700 nm and about 1100 nm. Hence, the VirtualLab model matches well with our phase calculations: within the intended useful spectral range the achieved quality (Q) is a direct result of the very small variation of the phase lag. Note that our calculation also regards light incident under different angles by modeling the DOE combined with an objective with a 40° field of view.
5. Conclusion
Our case study demonstrates the effectivity of DOE achromatization by using a doublet of holograms etched into suitably selected optical substrates. This selection is optimized by a simple, yet effective, strategy which uses contour lines of constant etch depth in a diagram containing index and dispersion values for the substrates of interest, at the design wavelength (index–dispersion diagram). This yields minimum required etch depths for DOEs forming an achromatized doublet. This allows us to also use the doublet for light incident under oblique angles. In addition, the introduction of a corridor in which the phase shift may vary leads to a maximized spectral range. As a result, even in the difficult NIR spectral range and including partial constructive interference for the zero-order
diffraction spot which is highly sensitive to the phase condition, we obtain a DOE which exhibits excellent performance over a wide spectral range in our simulation. References 1. Y. Arieli, S. Ozeri, N. Eisenberg, and S. Noach, “Design of a diffractive optical element for wide spectral bandwidth,” Opt. Lett. 23, 823–824 (1998). 2. Y. Arieli, S. Noach, S. Ozeri, and N. Eisenberg, “Design of diffractive optical elements for multiple wavelength,” Appl. Opt. 37, 6174–6177 (1998). 3. B. H. Kleeberg, M. Seeßelberg, and J. Ruoff, “Design concepts for broadband high-efficiency DOEs,” J. Europ. Opt. Soc. 3, 08015 (2008). 4. Schott AG, Optisches Glas, Version 1.8d (2009). 5. http://refractiveindex.info. 6. F. Wyrowski and H. Schimmel, “Electromagnetic optical engineering—an introduction,” Photonik Int. 1, 1–4 (2006).
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