1. Introduction

A diffractive optical element (DOE) can perform several key functions in optical instruments such as beam splitting, focusing and filtering. DOEs can replace conventional optical components that perform these functions, promising several advantages such as small footprint, low weight and low cost. Several micro-spectrometer designs with DOEs have been suggested recently, such as [1] and [2], taking advantage of these possibilities. Furthermore, silicon processing facilitates integrated optical microspectrometers, as reviewed in [3]. In an earlier report from our research group, the design of focusing diffractive optical elements for spectroscopy applications in the near-infrared was described [4]. Quantitative spectroscopy is a very demanding application for a DOE, requiring the fabricated DOE surface to match the design very closely, with submicron accuracy.

The DOE is used in a (low-cost) spectrometer where specific wavelength components of the incident light are analyzed in order to determine the sample material (see Fig. 1(a)). In addition to defining the spectral bandwidth, the spectrometer entrance aperture also has the side-effect of a pin-hole camera. The object (sample) is imaged on the DOE surface, resulting in a non-uniform illumination that depends on the type of object and its position. Thus an important property of the spectrometer is a spectral response that is uniform over the DOE, in addition to being reproducible and environmentally stable. Industrial silicon processing today is very highly developed, and should be well suited to meet the stringent fabrication requirements presented by DOEs for spectroscopy.

Ideally the spectral response should be the same, regardless of which part of the DOE is illuminated. We call this property the spectral uniformity of the DOE. A setup was built to study this property experimentally, as sketched in Fig. 1(b). The incidence and diffraction angles vary with position on the DOE surface, in general giving rise to variations in the spectral response over the DOE surface. Thus one would not expect a perfect spectral uniformity in any case, and some variations must be acceptable. However, severe anomalies that are wavelength dependent can cause a significant change in the response for one wavelength relative to the others. For instance a lowered diffraction efficiency for one wavelength may be misinterpreted as absorption in the sample material. Figure 2 illustrates this problem.

Furthermore, if the incident or diffracted waves do not propagate perpendicular to the plane of the DOE, the spectral response can be strongly dependent on the polarization of the incoming light, and the DOE may cause polarization conversion. In many cases it is necessary to study fully vectorial (electromagnetic) models in order to determine the diffraction efficiency before producing such an element. Despite these challenges we show here an example of a successful DOE design in the focusing four-level DOE fabricated in silicon.

The studied DOE has a complicated, non-periodic surface profile, and thus scalar Fourier-optical methods are the only ones feasible for direct simulation of the diffracted waves from the DOEs that we are interested in. However the scalar approximation does not describe polarization, and its validity is limited to small angles of incidence and diffraction, and structures where the feature sizes are typically much larger than the wavelength of the incident wave. Our design does not fulfill these conditions, so more rigorous methods must be used. To this end, we show that some important qualitative features of the optical filter response can be explained with the help of a local linear grating (LLG) [5] model of the DOE surface.

 

Fig. 1. (a) A schematic drawing of the spectrometer system. The tilt angle of the DOE, and the distance between source and detector are exaggerated. (b) The characterization setup consists mainly of a reflective DOE, an input fiber from the source, and a detector. The source position in the figure represents the input fiber end. The input fiber is excited by a halogen lamp emitting white light. The DOE focuses certain wavelength components in the near-infrared on the detector. During characterization, the DOE is partly covered by a movable opaque screen with an aperture which allows us to study the response at different positions on the DOE. In order to switch between operating wavelengths the DOE is rotated around a scan axis, focusing light of one wavelength at a time on the detector at predefined scan angles.

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Fig. 2. To the right, the spectral response for two wavelength bands, labeled λ ₁ and λ ₂, is plotted for two areas of the DOE shown on the left. Typically the diffraction efficiency is not spectrally uniform over the DOE. The figure illustrates a severe case where the resulting response is highly dependent on whether area A or area B is illuminated. The intensity difference between the case where wavelength λ ₂ is incident on the detector, compared to λ ₁, is larger when area B (bottom) is illuminated than when area A (top) is illuminated. If a sample was present, this could be misinterpreted as absorption of the λ ₂ component in a spectroscopic measurement.

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In the following we will describe the DOE designs studied, and present the simulation model. We have studied two binary designs for the DOE, with two and four levels, as shown in Fig. 3. The two-level profile is important because it represents the simplest profile in terms of fabrication, requiring no mask alignment. The fabrication of the four-level DOE on the other hand requires precise relative alignment of two masks in order to yield the desired surface pattern. Thus for a low-cost spectrometer the two-level DOE would be a natural choice if the spectral performance was acceptable, even if it is expected to give a lower diffraction efficiency than the four-level profile. The simulation model gives estimates of the variations in the spectral response between different areas of the DOE, in reasonable agreement with measurements for the two-level DOE design as well as for the four-level design. Our simulations show clear evidence that the physical mechanism responsible for the variations is closely related to Wood’s anomalies, well-known from the theory of grating diffraction, see for instance [6]. The Wood’s anomalies are rapid variations of diffraction efficiency with either wavelength or angle of incidence. These variations are typically either due to a change in the number of propagating diffraction orders, or a resonance phenomenon, such as the excitation of a surface wave in the grating surface.

 

Fig. 3. Two periods of the cross-section of a grating with perfectly straight grooves that run perpendicular to the figure. (a): Two-level binary approximation to the sawtooth profile denoted by the dotted line. (b): Four-level binary approximation.

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For our geometrical set-up and choice of wavelengths we thus show that, in addition to the expected lower diffraction efficiency, the two-level DOE yields a much poorer spectral uniformity than the equivalent four-level DOE. The measured spectral uniformity of the fabricated DOEs is improved by a factor of three for the four-level DOE compared to the two-level DOE.

2. DOE function and surface profile

The diffractive optical elements were designed and fabricated for spectroscopy applications in the near-infrared. Our DOEs are planar, with an area of 14×15 mm². They are designed so that the incident light is split into five focal lines each containing a spectrum centered on one out of five design wavelengths, as shown in Fig. 4. The spatial positions of the focal lines are such that a simple rotation of the DOE moves the lines across the detector. The separation into several focal lines allows one to easily know which band is on the detector without knowing the exact scan angle. The separation comes at the expense of light intensity, since the incoming optical power is divided between the five focal lines.

When the focal lines are scanned across the detector, a wavelength band (i.e. wavelength interval) of the corresponding spectrum is detected. The bandwidth is determined by the area of the detector. Hence the DOE is used as a dispersive element, and also performs focusing just like an off-axis reflective Fresnel lens, for each of the five desired wavelength bands. Thus when the DOE is rotated, the different bands are sequentially focused on the detector one at a time.

 

Fig. 4. Sketch of the DOE principle for diffracting the incident light into five focal lines that run in the x direction. Different wavelengths come to a focus at different x positions, indicated with different colors in the figure, and the grooves on the DOE surface hence run predominantly in the y direction. As the DOE is rotated, the focal lines L ₁ to L ₅ are scanned across the detector, and the desired wavelength bands from the different spectra are detected. In the figure, the wavelength band of L ₃ is detected. After the DOE is rotated through an angle α, the wavelength band of L ₄ is detected.

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Several DOEs are fabricated on each silicon wafer, and the four-level process requires two photolithography steps, and two anisotropic etch steps. The two-level process only requires one photolithography step and one anisotropic etch step. In this work we focus on the modeling and characterization of the DOEs, but we shall first briefly describe the surface profile and how this is generated. The description of the fabrication itself will be presented elsewhere.

The design of the DOE surface involves specification of the reflection coefficient for the optical field in every position of the plane of the DOE. We use the design algorithm described in [4], an algorithm that produces an amplitude and a phase for the reflection coefficient. The computed amplitude is disregarded, and a perfectly reflecting surface is assumed in the design algorithm. The effect of the DOE surface depth profile is then simply to modify the delay between source and detector. The delay is converted to a phase delay with the help of an average wavelength λ_avg, so that if the desired reflection coefficient has a phase ϕ (x,y), the corresponding depth profile f (x,y) for the DOE surface is

where k_avg=2π/λ_avg. For a DOE element reflecting and focusing a single wavelength, the phase function ϕ (x,y) is given by the formula

where k is the wavenumber, and r_s and r_d are the distances from (x,y) to the source and detector, and the mod function ensures that ϕ (x,y) is between 0 and 2π. The reflection coefficient of a single wavelength DOE corresponds to that of a Fresnel lens, with a constant amplitude A_n and a phase ϕ_n(x,y). For an n-wavelength DOE, the total complex reflection coefficient is found by adding the reflection coefficients of each Fresnel lens:

The amplitude A_tot (x,y) is approximated to unity. The phase of the resulting total reflection coefficient is converted to a surface depth using the average of the n wavelengths. This procedure for incorporating the five Fresnel lenses into a single surface yields a complex pattern, including the bifurcations seen in Fig. 5 below. Note that the design algorithm in [4] is based on the scalar-optical approximation.

The optimum groove depth of such a DOE varies with period and wavelength [7]. The standard silicon processing procedure produces DOEs with a groove depth that is the same everywhere on the DOE. Thus the design is not optimized (with regards to height) for maximum diffraction efficiency. However, as we shall see, this design nevertheless yields a DOE surface with very good spectral uniformity given the right choice of binary profile.

The design algorithm described above yields a surface pattern where the individual grooves are sawtooth-like in shape. The silicon DOEs studied here were fabricated as binary approximations (we refer here to the usage of the term “binary optics” as described in for instance [8], where the binary profile has 2^N levels, fabricated through N etch steps) to the sawtooth profile as shown in Fig. 3. The simplest binary approximation is the two-level profile (Fig. 3(a)) with one etch step in the DOE fabrication process, etching half the depth of the corresponding sawtooth profile. As given in [8] and [7], the maximum efficiency is achieved when the peak to peak phase variation introduced by the grating depth is exactly 2π. For normal incidence this is achieved by a maximal depth of λ_avg/2 for the sawtooth profile. Thus, as in scalar theory the optimum depth varies slowly with incidence angle (see [7]), we chose this depth for the grating even if we do not have normal incidence. However, it is clear that the design height could be further optimized, for instance using numerical optimization algorithms based on rigorous diffraction theory. For our two-level profile (Fig. 3(a)) the depth is then

The four-level profile (Fig. 3(b)) has the depth

Figure 5 shows a SEM micrograph of the resulting height profile in a small section of a two-level and a four-level DOE. The grating grooves of the two-level DOE were not smooth, but rough. This will be commented below when studying the results from measurements, and it will be evident that this groove roughness may also explain some of the differences between measurements and simulations for the two-level DOE.

3. Simulation model

The diffraction efficiency of small parts of the DOE is estimated by finding the corresponding diffraction efficiency of plane, linear diffraction gratings. The model is a local linear grating model, first introduced in [5] for the treatment of diffractive-lens arrays. In our case, the model relies upon the assumption that when a small spot is illuminated, the diffraction efficiency of the composite Fresnel lens can be approximated by an ideal linear diffraction grating. A plot of the surface modulation in a small area can be seen in Fig. 6, showing that it resembles a perturbed diffraction grating with straight grooves. This approach to studying the diffraction efficiency is also similar to that outlined for diffractive Fresnel lenses in [7], but there it is suggested as a method to determine the total diffraction efficiency of the Fresnel lens as a whole.

For each position on the DOE we determine the incidence angle and the diffraction angle from the measurement setup. These range from approximately 14° to 33° with respect to the DOE surface normal (see Fig. 7(a)). We then use the grating equation

to determine the period of a local linear grating corresponding to this DOE position. Here P is the period of the grating, θ_i is the incidence angle, θ _d,m is the diffraction angle of diffraction order m, and λ is the wavelength of the incident wave. For each DOE position the simulation is run for a linear grating with parameters determined as described here. In the following sections we present results for three wavelengths, λ ₁=1.736 µm, λ ₂=1.685 µm, and λ ₃=1.649 µm.

 

Fig. 5. SEM micrograph of a small part of the two-level (a) and four-level (b) DOE surface. The surface is designed to focus five different wavelength components, and thus incorporates five different Fresnel lenses. In (b) the wafer is cut in half, showing a cross-section of the DOE surface profile with the gold coating on top. The bifurcation apparent in (b) is due to the five separate Fresnel lenses incorporated into the surface, giving a complex surface pattern.

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Fig. 6. A plot of the height profile of a small part of the DOE area of approximately 40×20 µm. The grooves form a pattern resembling a perturbed linear diffraction grating. A typical period for the grooves on the DOE surface is 2 µm. The surface profile in this figure is representative of the DOE surface, although the typical groove width varies with DOE position.

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The incidence angle increases in the positive x-direction. The source is offset from the center of the DOE and the DOE itself is rotated around the scan axis (parallel to the x-axis) as illustrated in Fig. 1(b). Thus the incidence angle is not symmetrical about the DOE center (y-position 5) in the y-direction. The grating period decreases with incidence angle, and ranges from approximately 1.5 µm to 3.3 µm (see Fig. 7(b)). The ideal grating operates in a classical (in-plane) mount in the simulation model. The local period is determined such that the m=-1 diffraction order has the same diffraction angle as the angle between the point on the DOE and the detector in the actual measurement setup (θ_d in Fig. 8(a)). At each aperture position we compute the diffraction efficiency at four positions within the aperture and average over the results to take into account the finite aperture.

Figure 8(a) illustrates how the incidence and diffraction angles at a specific point on the DOE are determined, and Fig. 8(b) how these are employed in the simulation model. The dotted line in the DOE plane (xy-plane) is assumed to be approximately perpendicular to the grating grooves at the specific DOE position. Even though a conically mounted diffraction grating is needed to model the diffraction from most of the spots on the DOE surface accurately, a classically mounted grating is a good approximation over much of the DOE surface. In this classically mounted grating model the incidence and diffraction angles lie in the same plane. This plane makes a right angle with the DOE plane along the dotted line in the DOE plane shown in Fig. 8(b).

 

Fig. 7. (a): Incidence angle on the DOE at 9×9 positions. Each triplet of curves belong to the three wavelengths at 9 y-positions (y ₁…y ₉) for one x-coordinate. The wavelengths are λ ₁=1.649 µm, λ ₂=1.685 µm, and λ ₃=1.736 µm, drawn in solid blue, dashed red, and dotted black, respectively. The triplets corresponding to x ₁ and x ₉ are denoted at the top of the figure. (b): Modeled grating period on the DOE determined from the grating equation, Eq. (6). Note that the curve triplet corresponding to position x ₁ is to the right (longer periods) in this figure, so the subsequent positions x ₂…x ₉ are found from right to left.

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We compute the diffraction efficiency for an ideal linear grating, employing full vectorial representation of the optical field. The simulations were carried out using the GD-Calc [9] software, an implementation of rigorously coupled wave analysis (RCWA), see [10] and [11]. The results to be presented here are for simulations with plane waves incident on grating profiles in bulk gold. We have also done simulations of silicon gratings with a thin coating of gold (as is the case for the real DOE), using the complex refractive index data of gold and silicon. We found that when the thickness of the metal coating exceeds the skin depth of the metal by a factor of approximately 3 (the skin depth was calculated from the complex refractive index data [12], and is approximately 25 nm for λ=1.67 µm), the material beneath the coating does not influence the diffraction efficiency. The thickness of the metal coating on the measured DOEs was much greater than 25 nm, so we present only the results from the simulations on DOEs in bulk gold.

A more accurate model of our experimental setup is to have a conically mounted planar periodic grating at each point, where the incidence and diffraction planes are not perpendicular to the grooves. Diffraction efficiency calculations were also carried out for this model, and the results were found to be similar to those of the simpler model outlined above. Thus we present only the results using the classically mounted grating model.

4. Results from simulations

Due to a minor adjustment in the wavelengths used in the spectroscopic application, the set of wavelengths used in the two-level and four level design were not the exact same. The simulation results are presented for three wavelengths λ ₁=1.736 µm, λ ₂=1.685 µm, and λ ₃=1.649 µm for the two-level DOE. For the simulations of the four-level DOE the wavelengths λ ₁=1.733 µm, λ ₂=1.685 µm, and λ ₃=1.649 µm were used.

Two more wavelengths were used when designing the two-level DOEs: λ ₄=1.694 µm and λ ₅=1.712 µm. The height of the two-level DOE surface profile was determined taking into account all the five wavelengths, giving a maximal height of h ₂=λ_avg/2≈424 nm. For the four-level DOE, we again present the results for λ ₁, λ ₂ and λ ₃. However, as mentioned the wavelength λ ₁ was changed to λ ₁=1.733 µm, and also λ ₄=1.58 µm and λ ₅=1.715 µm. Thus the maximal height of the four-level DOE is h ₄=3λ_avg/8≈314 nm. This minor change in operating wavelengths has a very small effect on the diffraction efficiency, and does not affect the discussion in this work.

 

Fig. 8. (a): The incidence angle θ_i and the diffraction angle θ_d determined for a point on the DOE. The source is denoted by S and the detector by D. (b): The incidence and diffraction angles are used in the simulations for a classically mounted grating with linear grooves.

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For each wavelength the diffraction efficiency is plotted for 9×9 positions (x_i,y_j) on the DOE. The indices i and j increase in the direction of the x- and y-axis respectively as shown in Fig. 8(a). When discussing results for the different wavelengths, one must remember that with a change in wavelength there is also a small change in the scan angle of the DOE. The scan angles α_k for the three wavelengths are α ₁=1.1°, α ₂=0° and α ₃=-1.65°. The diffraction efficiency presented here is given for “unpolarized” light, i.e. the average of the results from TE and TM illumination. All the presented simulation results are for the order m=-1. To compute the diffraction efficiency of order m=-1 at a position (x_i,y_j) on the DOE, the incidence and diffraction angles from the experimental setup (the incidence angles for each position are shown in Fig. 7(a)) are used in the grating equation Eq. (6) to determine the period of the corresponding representative straight grating. Figure 9 shows the absolute diffraction efficiency for the order m=-1, found in simulations for the DOEs with two-level profile. The corresponding diffraction efficiencies for the four-level profile is given in Fig. 10. We immediately see that the two-level profile gives much larger variations in diffraction efficiency, both for each wavelength when comparing between the profiles, and when comparing the variations with wavelength for each profile.

 

Fig. 9. Diffraction efficiency for order m=-1 from simulations for the two-level profile, for the three wavelengths λ ₁=1.736 µm, λ ₂=1.685 µm, and λ ₃=1.649 µm in figure (a), (b) and (c), respectively. The diffraction efficiency varies dramatically depending on DOE position.

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Fig. 10. Diffraction efficiency from simulations for the four-level profile, for the three wavelengths λ ₁=1.733 µm, λ ₂=1.685 µm, and λ ₃=1.649 µm in figure (a), (b) and (c), respectively.

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4.1. Normalization of data

Below we will study the spectral uniformity, i.e. the wavelength dependence of the diffraction efficiency at different positions on the DOE. The results of our simulations were normalized as follows.

The diffraction efficiency determined for one wavelength component λ_k at one aperture position (x_i,y_j) is denoted γ_ijk. The coordinates take values

and

where Δ_x and Δ_y are the distances in the x- and y-directions between two adjacent DOE positions. The wavelengths are labeled λ_k, where k=(1, …,n_λ).

The mean global diffraction efficiency for one wavelength λ_k is the mean of the efficiency for all the positions for this wavelength:

We define the local normalized efficiency

This normalization is used because the measurements only provide the local normalized diffraction efficiency Γ_ijk, and not absolute diffraction efficiencies. Taking the mean of Γ_ijk over the wavelengths, gives

To study the spectral uniformity at the different positions we study how much the local normalized efficiency (Eq. (10)) deviates with respect to the mean of those of all the wavelength components (Eq. (11)). Thus the quantity of interest is the ratio of the normalized local diffraction efficiency to the mean of the same quantity taken over the wavelengths:

If the diffraction efficiency of the various wavelength components at each point gives the same ratio with respect to their mean global diffraction efficiencies, η_ijk is equal to one for the whole DOE. This is the unrealistic ideal situation, because then at each point the relationship between the components is the same. In reality η_ijk is different from one because the diffraction efficiency varies with λ_k. This variation is a result of the actual change in wavelength, and also due to the rotation of the DOE through scan angles as illustrated in Fig. 1(b). So a change in wavelength λ_k is also accompanied with a change in DOE positioning. The scan angles α_k for the three wavelengths are α ₁=1.1°, α ₂=0°, and α ₁=-1.65°. The goal of the design is then to find a profile which minimizes |η_ijk-1| over the DOE surface.

In the next sections we present simulations and measurements of spectral uniformity using the normalization described in Eq. (12) above.

4.2. Spectral uniformity

We present here the maximal deviations in spectral uniformity |η_ijk-1| (see Eq. (12)) across the DOE at a given x-position number i. So for instance for wavelength λ_k at x-position x_i we find the maximum of |η_ijk-1| for the positions (x_i,y ₁), (x_i,y ₂)…(x_i,y ₉). This does not give information about which y-position gives the maximum deviation, but ultimately the most interesting figure for this discussion is the worst case scenario, i.e. the maximal deviation in spectral uniformity. The results from simulations are given in Fig. 11(a) for the two-level profile and Fig. 11(b) for the four-level profile. In order to take into account the finite aperture being used in the measurement setup, the simulation results are averaged over four positions at distances of 0.25mm from the (x_i,y_j) coordinate on the DOE.

 

Fig. 11. The maximal normalized deviation |η-1| in diffraction efficiency across the DOE at each of the 9 x-coordinates for the three wavelengths. (a) Simulations for the two-level profile. (b) Simulations for the four-level profile.

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4.3. Discussion of simulation results

In the simulation results for the two-level profile we see in Fig. 11(a) two large deviations around position x ₃ and x ₆. These correspond to Rayleigh and resonance type anomalies respectively. The fact that the anomalies have an impact on the spectral uniformity is due to the wavelength dependence of the excitation of the anomalies. For the four-level profile in Fig. 11(b), we see that the resulting grating groove profile yields much better spectral uniformity than the two-level profile. The anomaly around position x ₃ is also seen in the simulations for the four-level profile, but it is not as pronounced as for the two-level profile. The resonance anomaly is not excited for this type of profile. The two-level profile is clearly a less desirable choice due to the large spectral variations. However it is an important design example because it is much simpler to fabricate than the four-level DOE. Not only does the four-level DOE have groove profiles with smaller line-widths, but the fabrication also requires very precise alignment of two masks. Furthermore, our simulation results for the two-level profile happen to show clearly some vectorial optical field effects that must be taken into account in reflective DOE design in general. Below we will discuss in detail the two different anomalies.

4.3.1. Rayleigh’s anomaly

The x-positions x ₂ and x ₃ correspond to incidence angles approximately between 18° and 20°, as seen in Fig. 7 above. In Fig. 12, the absolute value of the diffraction angle for the beams of order m=1 and m=-2 is shown as a function of incidence angle for three different wavelengths, and a grating period of P=2.5 µm. We see that the increase in diffraction efficiency coincides with the transition of the diffracted beam of order m=1 (Fig. 12(a)) from a propagating to an evanescent wave. The studied diffraction efficiency of the beam of order m=-1 decreases at higher incidence angles when the diffracted beam of order m=-2 (Fig. 12(b)) appears at a slightly higher angle of incidence. The anomaly around x-position x ₂ and x ₃ is hence

a result of the redistribution of energy carried by the diffraction order m=1 into the remaining propagating orders. This sudden change in the diffraction efficiency is wavelength dependent, and this leads to poor spectral uniformity for the positions where this occurs.

 

Fig. 12. The diffraction angle of the orders m=1 (a), and m=-2 (b), calculated from the grating equation, Eq. (6), for a grating with period P=2.5 µm and plotted against the incidence angle for the three wavelengths. We see that the limiting incidence angle for diffraction order m=1 is between 18° to 20°.

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The simulations for the four-level profile show that the effect of the Rayleigh anomaly on the spectral uniformity is less severe in this case. This is because the four-level profile resembles the blazed grating for the order m=-1 more closely, and thus less of the diffracted energy is in the order m=1 in this case. When the energy is redistributed from the m=1 order to the propagating orders it has less impact on the m=-1 order which is already carrying most of the energy. One could avoid this type of anomaly by designing the setup such that the involved incidence angles and grating periods are such that they lie outside the parameter space where the number of propagating diffraction orders vary. However, there are other requirements to take into account, such as separating the focal regions of the wavelength components, and other instrument specifications. Rayleigh’s anomaly represents an undesirable additional design consideration.

4.3.2. Surface shape resonance

The second anomaly around x-position x6 (corresponding to approximately 25° incidence angle, and 2 µm grating period) in Fig. 9, is due to the excitation of a standing wave in the grooves of the two-level profile. This excitation occurs only for the TM polarization defined such that the magnetic field is perpendicular to the plane of incidence and parallel to the grating grooves. Figure 13(a) shows the shape of the total intensity of the TM polarized electric field for the two-level profile, contrasting that of the four-level profile in 13(b). The groove acts as an open resonator sustaining a standing wave with a maximum amplitude in the bottom of the groove.

 

Fig. 13. E-field intensity for a grating with a two-level (a) and a four-level (b) profile at resonance conditions (for the two-level profile). The period is P=2 µm, incidence angle is 25° and the wavelength is 1.685 µm. Both the incident and reflected fields are plotted.

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In Figs. 14(a) (Media 1) and 14(b) (Media 2) we show animations of the contrasting scenarios seen upon excitation of the two-level and four-level gratings respectively. The two-level grating supports the standing wave, thus leading to the anomaly in the diffraction efficiency which we have seen above. The corresponding four-level profile yields a resonator with a completely different shape, and the simulations show that the Q factor of this resonator is much smaller than for the two-level case. A reason for this may be that a potential standing wave is forced to have its maximum higher up in the groove, and thus the resonator is effectively more open. The FDTD simulations were carried out using the simulation software in [13] for an ideal periodic grating with linear grooves, which is an approximation to the real DOE as described in Section 3. The parameters of the grating and incidence angle were the same as for the RCWA simulation results presented in Fig. 13.

Such surface shape resonances in lamellar metallic gratings have been studied extensively, see for instance [14] and [15]. For some choices of parameters one can achieve total absorption of the incoming wave due to the excitation of a surface plasmon wave propagating along the surface. This type of effect is naturally undesirable in our context, and must be taken into account when designing a DOE for spectroscopy. The solution to the problem of surface shape resonances in our case is to find a groove profile which does not support such standing waves. The four-level profile removes this problem entirely.

5. Experimental results

5.1. Experimental setup

A measurement setup was built to characterize the DOEs. The experiments are carried out by sending white light from a halogen lamp into an optical fiber that ends in the position labeled “Source” in Fig. 1(b), pointing towards the DOE. The DOE is designed to focus and reflect the light back at a detector. The DOE is scanned stepwise through the n_λ predefined angles α_k at which light of n_λ wavelengths is successively focused on the detector. When collecting data from the experiments, we assume that only light of the correct wavelength is focused on the detector at the predefined scan positions. The DOE is covered by a movable screen, with a circular aperture with a diameter of 1 mm that was centered on 9×9 positions on the DOE as described above. In the analysis of measurement data we disregarded the fact that the projected area of the aperture (seen from the source or the detector) depend on the incidence and diffraction angles. Given an aperture position, our main concern is variations in diffraction efficiency between the five wavelength bands, and the difference in the illumination would only come from the small scan angles α_k that separate the wavelength bands, when the aperture position is given.

 

Fig. 14. The intensity of the electric field upon excitation of the two grating profiles, determined by FDTD simulation. (a) (Media 1) Shows the resonance for the two-level profile, where the incident field excites a standing wave in the grating grooves. This creates a strong, localized field, and results in a dip in the diffraction efficiency for these parameters as discussed above. (b) (Media 2) The corresponding four-level profile does not support the standing wave resonance, so the field is reflected back into the propagating orders.

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As mentioned in Section 4.1, the measurements are not of absolute diffraction efficiency. The illumination may vary slightly over the DOE and the DOE is also rotated through a small angle when changing between the detection of different wavelength bands. Thus we discuss only the diffraction efficiency at a specific position relative to the other wavelength channels for one band center relative to the other four band centers at this position. This gives sufficient information about the spectral uniformity as ultimately the important issue is the variation of the diffraction efficiency of the wavelength bands relative to one another.

5.2. Results from measurements

Figures 15(a) and 15(b) show the results from the measurements. The plotted quantity is the maximal deviations in spectral uniformity |η_ijk-1| across the DOE at a given x-position the same way as for the simulation results above.

5.3. Discussion

The real DOEs have a more complicated surface pattern than our simulation model, and from the measurements it seems that the anomalies predicted for the perfect, linear gratings are somewhat suppressed by the irregularities of the real DOE surface. Thus it is important to determine the cause of an anomaly in order to know whether or not it should be expected in the measurements. We do however expect anomalies of both the Rayleigh and resonance type to be excited in the measurements, but the model does not accurately predict where these will occur.

Also in the measurements we see that the spectral uniformity is much better for the four-level profile, Fig. 15(b), than for the two level profile in Fig. 15(a). This is due to the profile dependence of the amplitude of the Rayleigh’s anomaly, and the fact that the resonance anomaly is only excited for the two-level profile. The anomalies were discussed in detail in Section 4.3. For the two-level profile in Fig. 15(a), we see a broad peak around x-positions x ₃ and x ₄, but no large defined peak around x-position x ₆, as was seen in the simulations in Fig. 11(a). This broad peak resembles the result from the simulations where the diffraction efficiency from gratings of three different groove widths were averaged over, Fig. 16(a). Thus we believe the reason we measure a broad peak and not the more sharply defined high-amplitude peak as predicted by the simulations in Fig. 11(b), is because of variations in grating period and groove profile in the real DOE. Furthermore, the roughness of the grating lines of the fabricated two-level DOE would also lead to scattering and may also to some extent prevent the excitation of the resonance anomaly predicted in simulations. We emphasize that for our application the spectral uniformity of the four-level DOE (see Fig. 5(a)) is found to be very good both in the simulations and measurements. The uniformity is naturally even better if larger areas of the DOE are compared, with the help of a larger aperture in the measurements. To further improve the spectral uniformity, one could perform an optimization with regards to the height and width of the steps in the profile, as reported in [16] for transmission gratings.

 

Fig. 15. (a) Maximal normalized deviation from measurements on a DOE with two-level groove profile. (b) Maximal normalized deviation from measurements on a DOE with four-level groove profile.

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5.3.1. Variations from ideal linear gratings

As mentioned, small variations in the groove profile will shift the position of the resonance anomaly. For instance a change in the groove width will make the grating support a standing wave at different incidence angles. Thus the conditions for exciting this resonance are highly dependent on the grating groove profile, and the approximate simulation model can not be used to determine the exact DOE position for which it is excited. Due to this sensitivity to the grating profile parameters, such sharply located anomalies do not show up in the diffraction efficiency of the real DOE, which is not an ideal linear grating, and where the fabricated groove walls are not as perfectly vertical and smooth as in the simulations. In Fig. 16, the diffraction efficiency is averaged over three simulations for two- and four-level gratings with groove width of 45%, 50% and 55% of the period.

Comparing Fig. 16 with the simulations in Fig. 11, we see that the result for the two-level profile is a lower variation, while the four-level result is virtually the same. This shows that if at a certain position we see such variations in groove width, it is likely that the resonance anomaly is not excited in the whole illuminated area as is the case in the simulations with a single groove width. This variation would cause the anomaly to have a weaker effect on the diffraction efficiency determined within an aperture, and thus on the spectral uniformity. Indeed this is what we also see in the results from measurements in Fig 15(a) for the two-level profile.

For the four-level DOE Figs. 16(b) and 11(b) show that the Rayleigh’s anomaly for x-position x ₃ is almost unchanged by the averaging over different groove widths. This illustrates the fact that the conditions for exciting this anomaly are not dependent on the detailed groove profile. The variations for the four-level profile are generally larger in the measurements, Fig. 15(b), than the simulations in Fig. 11(b). However, for the four-level profile, the measurements and simulations agree very well, to the extent that they both predict very small variations compared to the two-level results. On a relative scale, however, the agreement is quite poor, since the measured maximal variations are many times greater than the simulated variations, except in x-position 3. For the two-level profile, measurements and simulations agree much better on a relative scale, within a factor of 2 for most x-positions.

 

Fig. 16. The maximal normalized deviation |η-1| (see Eq. (12)) across the DOE at each of the 9 x-coordinates for the three wavelengths. The data is the mean of three simulations where the groove width was 45%, 50% and 55% of the period, and averaged over 4 positions 0.25 mm from the (x_n, y_m) coordinate on the DOE in order to take into account the finite aperture. (a) Simulations for the two-level DOE. (b) Simulations for the four-level DOE.

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6. Conclusion

Diffractive optical elements used to separate and focus light in five different wavelength bands were fabricated for use in an application of near-infrared spectroscopy. The DOEs were fabricated in silicon using standard silicon processing technology, and were coated with gold for improved reflectivity. The design procedure for the DOE yields a surface of total area 14×15 mm² with a complex pattern of grooves with sawtooth-like profiles. We studied two binary approximations to this profile, with two and four levels, respectively.

Simulations and measurements were presented for the two- and four-level profiles. In the measurements a screen with a 1 mm diameter aperture was used in 81 (9×9) positions on the DOE, and the resulting response was analyzed in order to study the spectral variations over the DOE. The simulation model is based on a local linear grating model, representing small parts of the DOE surface corresponding to the area underneath the aperture for each aperture position in the measurement setup. The diffraction problem for such a small part of the DOE was approximated as classically mounted linear diffraction gratings, with incidence and diffraction angles taken from the actual measurement setup. The simulations were then carried out with a vectorial representation of the optical field.

We found that both the simulations and measurements yield a much larger variation in the spectral response for the two-level DOE than for the four-level DOE. Furthermore, the simulations give qualitative and to some extent quantitative information on the variations for the two-level profile, whereas for the four-level profile the agreement can only be said to be qualitative. The two-level DOE had such large variations in the spectral response that it was found to be an unsuitable design for our application. One would have to alter the design to avoid the problems with anomalies in the response. However, this problem is avoided by using the four-level profile, which suppresses these anomalies. The absolute maximum deviation of normalized diffraction efficiency, relative to the other wavelengths at a given DOE position, is only 5% for the four-level DOE. This meets our requirements, and the four-level DOE was found to be a good candidate for fabrication of DOEs for near-infrared spectroscopy, using standard silicon processing techniques.