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Abstract
The generation of wide-angle diffraction patterns can be done in different ways using either thin diffractive optical elements with small features sizes or arrays of microoptics with large optical paths that are thick diffractive optical elements. Our aim is to create as many high contrast diffraction-limited dots in the far-field as possible with a uniform intensity distribution. As a model system, we use a sinusoidal phase grating and as a peculiarity, we introduce non-uniform illumination using a Gaussian beam illumination. By making use of the self-imaging phenomenon, a large number of peaks with uniform distribution are generated for a defined range of the phase grating thicknesses due to the sinusoidal curvature. For very high structures, the pattern distribution is not uniform and it demonstrates that very thick sinusoidal phase gratings are not suitable pattern generators. For simulation, we compare thin element approximation, fast Fourier transform beam propagation method, and the rigorous finite difference time domain method. The large-angle diffraction is considered using a high numerical aperture propagator for far-field simulation. We demonstrate that the beam propagation and the Fraunhofer approximation are not accurate enough. Also, our rigorous near-field calculation versus phase grating thickness confirms the significant influence of reflection of thick structures on the far-field distribution, especially on pattern uniformity. Finally, experiments were carried out to confirm our findings and a good agreement between the simulation and experimental far-field distributions confirms our approach.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Arbitrary high-contrast light pattern generation has various applications including imaging, microscopy or sensing [1–3]. For some sensing applications, generating large numbers of dots with a wide field of view and equal intensity peaks in the far-field is essential and also challenging to do. For this purpose, using micro-optical and diffractive optical elements with different aspect ratios from low to high is a practical method especially in terms of pattern field of view [4–7]. Figure 1 shows the two concepts.
Fig. 1. (a) Dot pattern generation for a binary diffractive optical element. A series of well-defined dot positions are designed and dots are optimized for intensity. (b) Dot pattern generation for a periodic refractive-diffractive optical element. Dots appear by diffraction and the intensity distribution is designed by the surface profile of the micro-optical components. (c) Ray tracing for one period of a sinusoidal phase grating which shows two different focal points in one period
Using binary diffractive optical elements which are designed using optimization techniques produce a defined functionality for structured pattern generation [8–10]. Using binary diffractive optical elements, a uniform distribution of spots is designed in the far-field, as seen in Fig. 1(a). However, using this method is limited by some effects: First, the design is monochromatic. Second, there can be unwanted intensity in zero-order and outside the field of view due to fabrication errors [11]. Not all diffraction orders are addressed and it is quite challenging to design very high angle pattern distributions using this technique due to the very small feature size [12].
Using small period refractive optical elements that combine refraction and diffraction is another method for spot array generation in which the pattern distribution very much depends on the surface profile of the optical element [8,13–15]. In this case, as sketched in Fig. 1(b), all diffraction orders can be used and the intensity distribution is managed by the form of the structure. There is no very small feature but the precision of the surface profile needs to be high.
We investigate here micro-optical elements with a period of 75$\lambda $ which allows us to control both diffractive and refractive effects with the height of the structure [8]. Using a regular micro-optical element such as lens array under plane wave illumination, one obtains a periodic pattern in the far-field with the period of ${\sim} \frac{\lambda }{P}$ according to the diffraction theory. To increase the numbers of diffraction orders, one solution is to increase the lens array period P. By increasing the period P, the refraction becomes predominant and the effect of diffraction is reduced which is not desirable. Alternatively, introducing a point source instead of plane wave, the periodic element field distribution would reproduce in the far-field including a magnification factor for certain distances between the source and the periodic element based on the known self-imaging phenomenon [16,17]. Numbers of publications are devoted to using this phenomenon for pattern generation using a lens array in which the point source is modeled by a single-mode Gaussian beam [4,15]. In this paper, we make use of the self-imaging phenomenon under point source illumination for a sinusoidal phase grating. We choose the sinusoidal shape as it is a combination of concave and convex lenses in one period according to Fig. 1(c) and its phase distribution would reproduce in the far-field. In such a situation one generates two peaks in one period instead of one. This phenomenon can be generalized for the generation of more numbers of points by designing the surface curvature in one period of the phase grating. Using a thick phase grating, one may obtain a bigger field of view. In [18], we briefly demonstrated the importance of using a rigorous method as the most accurate near-field simulation tool for thick phase gratings under the Gaussian beam illumination. Near-field and far-field definitions in this paper are referred to Fresnel and Fraunhofer regions as in classical optics [19].
In many design schemes for diffractive optics thin element approach TEA is used to specify the basic parameters of a configuration. In the first part of the paper, we analytically compare the number of diffraction orders generated by the point source and the plane wave illumination using TEA and demonstrate that more numbers of points can be generated with a point-like source in comparison to the plane wave. We model the point source as a single mode Gaussian beam which is the most practical to compare results with measurements. According to the analytical calculations, the two most important parameters are the source beam waist w0 and the thickness of the phase grating. We start by comparing the validity of the TEA and the Fast Fourier beam propagation method (FFT-BPM) [20] for a thin phase grating considering different beam waists. FFT-BPM is a scalar approach which is more accurate than TEA and less time consuming than the rigorous FDTD method. We will demonstrate that the pattern field of view is wider for smaller Gaussian source beam waists and we show that TEA is not accurate enough even for thin phase gratings with the aspect ratio (thickness over the grating period) of 0.24. In the next step, we compare the near-field and far-field for thin and thick phase gratings for a certain source beam waist w0. We confirm that for thick sinusoidal phase gratings, only the near field rigorous FDTD and far-field high NA propagator [21] are valid. The rigorous near-field simulation results show a complicated behavior of amplitude and phase due to several effects such as total internal reflection which significantly influence field distribution. The result gives important insights into the optimization of the structure. We conclude that for very high structures, the far field pattern distribution is non-uniform and such a configuration is not a practical point generator. The best point generator for our case of 50 micron period was obtained for an aspect ratio of 0.5, having a uniform pattern distribution with a wide field of view. In the last part of the paper, we briefly introduce a possibility to fabricate sinusoidal phase gratings and compare the design of point generators with experiments for selected cases of which generate the most uniform patterns.
2. Problem definition
2.1. Analytical investigation of point source versus plane wave illumination
In this section, we compare the far-field distribution when a sinusoidal phase grating is illuminated by a point source and a plane wave. We use TEA for a principle design and show that more numbers of points in the far field are generated for point source. The effect is based on self-imaging [15,16]. Figure 2 gives a diagram of the point source propagating to the far-field observation plane. For a clear understanding of the self-imaging phenomenon for point source illumination, we use the notation and formulas as found in [16].
The far-field is given by the following integral
Using the TEA, the complex field at $z{\; = \; }{D^\textrm{ + }}$ is described by
Fig. 3. (a) Phase modulation generated by the sinusoidal phase grating using TEA, (b) The retrieved phase in the far-field according to Eq. (7).
As seen, the phase modulations are comparable except that one graph is flipped around x axis and the scale in x has changed. The sinusoidal phase grating modulation is imaged in the far-field. Interestingly this leads to the appearance of two peaks in one period of the far-field pattern. We will investigate this effect in more detail in the simulation part below.
Now, we consider the sinusoidal phase grating under the plane wave illumination. For the plane wave illumination, the far-field pattern is simply the Fourier transform of the sinusoidal phase modulation based on TEA
2.2. Simulation tools
Consider a Gaussian beam with a finite spatial dimension that illuminates a sinusoidal phase grating of period p, as shown in Fig. 4. The observation plane is in the far-field and we aim to generate high contrast patterns. As we discussed in the previous section, for particular values of the distance D, high contrast intensity peaks are observed because of interference effects of the curved wavefront and the grating [15,16]. We choose D to fulfill the self-imaging condition to be 3.84 mm for a period P = 50 um in all simulations. As a source, a single-mode Gaussian beam with $\lambda $ = 650 nm is chosen.
Fig. 4. Far field light pattern generation for sinusoidal phase grating under the Gaussian beam illumination.
2.2.1 Source definition
The source is a single-mode Gaussian beam with the following complex amplitude [1]:
2.2.2. Field propagators
According to our configuration in Fig. 4, the simulation is approached in three steps. The first step is the Gaussian beam propagation for the distance D from the source to the phase grating. Immediately before the phase grating, one finds an amplitude and phase distribution which can be calculated using Eq. (10). The second step is the propagation inside the phase grating. There are different means depending on the complexity and dimensions of the structure. We use three methods: thin element approach TEA, beam propagation method FFT-BPM and the rigorous FDTD method. In the thin element approach TEA, phase profiles are used that are calculated from the local phase difference a structure will produce. In our case, there is no amplitude variation as we consider a pure phase grating. The thin element approach is known to deliver false results for high spatial frequency diffractive optical elements or thick elements but is useful to obtain basic specifications of the systems.
The beam propagation method FFT-BPM is a scalar approach for which the phase grating is divided into sub-sections along the propagation direction z and the angular spectrum of plane waves (ASPW) [19] method in combination with the TEA for each subsection is implemented. This method is more accurate than TEA but it is still based on paraxial approximation. It leads to both near-field amplitude and phase modulations for thick diffractive optical elements but the effects of reflection are not included [5]. We use the VirtualLab Fusion software [22] to perform the FFT-BPM simulations. In rigorous simulations, all effects resulting from interference and scattering (forward and backward) are included and it represents the most advanced but also the slowest method to analyze thick diffractive optical elements. Rigorous simulations are based on the software package of Lumerical FDTD [23].
As seen in Fig. 4, the phase grating is invariant in y direction and the near-field 2-D simulation box in the x-z plane needs to be chosen large enough to cover the entire Gaussian beam intensity distribution. The 1-D near-field is monitored on a length of 2 mm in the ${x_\textrm{1}}$ plane for both the FFT-BPM and FDTD simulation methods. For the rigorous simulation, we choose the perfectly matched layer (PML) boundary conditions along x and z directions and the mesh size is 50 nm. In the third step, the near-field is extracted a few microns behind the phase grating and the far-field irradiance pattern is obtained using the Fraunhofer approximation for thin sinusoidal phase gratings and with the high numerical aperture (NA) method for thick sinusoidal phase gratings [21].
For the Fraunhofer approximation, the complex field in the observation plane (${x_\textrm{2}}$, z) is the Fourier transform of the phase gratings complex near field. It is obtained by the following formula, which is based on the paraxial approximation:
For the high diverging light pattern generation, the paraxial approximation is not valid in the observation plane ${x_\textrm{2}}$ and a fast and more accurate method is required. To obtain the far-field for a high diverging beam, we use the high NA propagator as proposed by Engelberg [21]. It is defined by the following formula assuming that the paraxial approximation is lifted in the observation plane.
Using the high NA method, the coordinate mapping between the near-field plane ${x_1}$ and the observation plane ${x_\textrm{2}}$ is not linear anymore. This is the major difference in comparison with the Fraunhofer approximation for which the mapping between ${x_1}$ and ${\textrm{x}_\textrm{2}}$ plane is linear. Also, the electric field is proportional to $z\textrm{/}R_\textrm{2}^\textrm{2}$ which may significantly change the pattern envelope in comparison to Fraunhofer propagator if the paraxial approximation is not valid in the observation plane ${x_\textrm{2}}$.
3. Simulations
Next, we investigate the effect of the source beam waist w0 and the thickness of the phase grating h on the far-field irradiance using different approximations.
3.1. Variable source beam waist ${w_\textrm{0}}$
In this section, we explore the dot pattern generation in the far field under the diverging Gaussian beam in comparison with the plane wave illumination. We investigate the effect of the source divergence angle on the irradiance pattern and compare TEA, FFT-BPM for modeling the passage of the beam through the thin phase grating of 12 um. The Fraunhofer approximation is applied for the far-field calculation of this thin periodic phase grating. For the plane wave illumination, the simulation dimension along the x-axis is 3 mm.
3.1.1 Thin element approach (TEA) calculations
In Fig. 5(a) and (b), the irradiance patterns for a plane wave in comparison to the Gaussian beam with the beam waist of 2 um are given. The pattern period matches in both cases the diffraction theory: a period of 50 um for $\lambda $ = 650 nm delivers 7.4° diffraction angle. Under plane wave illumination, the energy is not uniformly distributed over the diffraction orders. When comparing the inset of Fig. 5(a), a number of peaks have almost zero intensity and the field of view is small in comparison to the Gaussian beam illumination in Fig. 5(b). However, as we discussed in the analytical part for the Gaussian beam, two non-equal peaks with a period of 7.4° are repeated in the pattern as it is seen in the inset of Fig. 5(b). The two series of peaks have different intensities and diameters. They are caused by the convex-concave curvature form of the sinusoidal phase grating which can be interpreted as a combination of concave and convex lenses. Assuming the Gaussian beam as a spectrum of plane waves with different incident angles allows interpreting the increase of the field of view when compared to the plane wave illumination.
Fig. 5. Irradiance pattern using TEA for (a) plane wave illumination and (b) Gaussian beam with the beam waist of 2 um.
In Fig. 6 the irradiance patterns are illustrated for the Gaussian beam waists ${w_{\textrm{0}}} = \; 1,\; 2\; \textrm{and}\; 3\; um$ using the TEA. The pattern field of view increases by decreasing the source beam waist down to 1 um. Enlarging the pattern field of view is more pronounced from 2 to 1 um beam waist in comparison with 3 to 2 um beam waist. The reason is that the Gaussian beam divergence angle is proportional with $\textrm{1/}{w_\textrm{0}}$ and it indicates that the divergence angle increment is higher from 2 to 1 um in comparison with 3 to 2 um beam waist which results in a more pronounced enlargement in the pattern field of view.
Fig. 6. Irradiance pattern using TEA for the Gaussian beam with the beam waist of (a) 3 um, (b) 2 um and (c) 1 um.
To explore the effect of the Gaussian beam waist ${w_{\textrm{0}}}$ on the irradiance pattern in more detail, we calculate the pattern envelope in the far field which is the Fourier transform of one single period of the phase grating near field. Our configuration is sketched in Fig. 7(a) and we illuminate only one single period of the phase grating with a Gaussian beam with 1 um beam waist. If the element is centered the result is a symmetric distribution of peaks as visible in Fig. 5(b). Next, we move the element along x-axis to probe the influence of the tilted local phase and compare the pattern envelope for different positions at -400 um and -800 um. As seen in Fig. 5(b), the pattern envelope moves in the observation plane by moving the optical element along x-axis because of the tilted source phase profile. This clearly shows the reason for an increased field of view: the different illumination angles that are present in the Gaussian beam.
Fig. 7. (a) Moving one single period of the phase grating along the x-axis and (b) taking the Fourier transform from near field to obtain the pattern envelope for x = 0, -400 and -800 um.
3.1.2. TEA versus FFT-BPM
For the thin phase grating of 12 um thick, using a rigorous method is not required but using a propagator like FFT-BPM which includes the beam propagation inside the structure including some approximations is necessary for evaluating the TEA results. In Fig. 8, we compare the irradiance patterns using TEA and FFT-BPM for the Gaussian source beam waists ${w_{\textrm{0}}} = \; 1,\; 2\; \textrm{and}\; 3\; um$ and a plane wave. The major difference for FFT-BPM is that the effects that arise because of the propagation through a certain thickness of the structure are now considered. As seen, for the plane wave illumination, FFT-BPM predicts a wider field of view in comparison with the TEA although the pattern envelope is still similar. For the Gaussian beam illumination using the FFT-BPM method, both the pattern field of view and the pattern envelope differ significantly from the TEA. It demonstrates that the TEA is not accurate even for a shallow surface profile of 12 um thickness and the field of view calculated with FFT-BPM is about 20% wider.
Fig. 8. Normalized far-field irradiance using TEA and FFT-BPM under (a) plane wave and Gaussian beam with the beam waist of (b) 3 um, (c) 2 um and (d) 1 um.
To explore the effect of the near-field behind the structure on the far-field distribution we consider again one single period of the phase grating under the Gaussian beam illumination for the beam waist of 1 um and move the optical element along x-axis, as sketched in Fig. 9(a). In Fig. 9(b), the pattern envelope is obtained by taking the Fourier transform from the near field which is calculated using FFT-BPM. As seen, the calculated pattern envelope is already bigger using FFT-BPM for x = 0 compared with TEA in Fig. 7(b). Also, the obtained pattern envelope using FFT-BPM at x = -400 and -800 um is completely non-symmetric however for TEA, the pattern envelope non-symmetry is weak as only the phase near-field is modulated using TEA.
Fig. 9. (a) moving one period of phase grating along the x-axis. (b)The calculated pattern envelope for x = 0,-400 and -800 um using FFT-BPM.
3.1.3. Transmission
One important practical criterion is the total transmission of the phase grating. We define the transmission as the normalized transmitted power immediately after the phase grating with respect to the source power immediately before the phase grating. As mentioned in the beginning, different effects influence transmission such as total internal reflection. Figure 10 shows the transmission versus the source beam waist ${w_\textrm{0}}$ using the FDTD simulation. As seen, by increasing the source beam waist ${w_\textrm{0}}$ the transmission increases until it reaches a constant value which is identical with the transmission for the Plane-wave illumination. As the Gaussian beam divergence angle is proportional to $\textrm{1/}{w_\textrm{0}}$, small changes in ${w_\textrm{0}}$ for ${w_\textrm{0}} \to \textrm{0}$ correspond to big changes in divergence angle and as a result large changes in transmission. Thus for ${w_\textrm{0}}\;<\;{3\; }um$, the smaller the beam waist the lower the transmission (maximum 4% reduction in transmission). On the other side according to Fig. 8, the smaller the beam waist the higher the field of view. It demonstrates a trade-off between obtaining higher transmission and higher field of view for small source beam waist ${w_\textrm{0}}$.
3.2. Variable phase profile thickness h
In the previous section, we have shown that even for the small height of the sinusoidal phase grating, one obtains large numbers of peaks with a uniform distribution. The thickness of the structure also plays an important role in the design and may generate a bigger field of view. In this section, we, therefore, compare the irradiance pattern for different structure heights and three aspect ratios of h/P = 0.24, 0.5 and 1 for P = 50 um that represents the transition between thin to thick gratings. For thick structures, a careful analysis needs to be done as the non-uniform illumination condition modifies the angular spectrum of the arriving wavefront and hence influences the far-field pattern. As we discussed in the previous section, TEA is not enough accurate even for thin phase gratings and we concentrate here on comparing the FFT-BPM and FDTD and discuss their accuracies for different thicknesses. The source beam waist is fixed at ${w_\textrm{0}}\; = \; {2\; }um$ for all the simulations.
3.2.1. Near field simulations, FFT-BPM versus FDTD
In Fig. 11, the field amplitude is shown around the structure for different h/P = 0.24, 0.5 and 1 ratios. As one can see heavy amplitude modulations can be noticed and two hot spots are observed behind the structure. The structure seems to behave like lenses and the hotspots move toward the phase grating surface by increasing the thickness h. For thin height h = 12 um, the near field is the same using the FFT-BPM and FDTD as the effect of reflection is negligible. As seen in Fig. 11(b) and (c) by increasing h, near field distribution is more complicated due to the effect of reflections. Especially for the height of 25 um, we observe that some of the beams that are transmitted using FFT-BPM, are reflected back using FDTD and it significantly changes the near-field distribution.
The near-field simulations demonstrate that both the phase grating thickness and surface profile remarkably influence the near field distribution for thick sinusoidal phase gratings. The near field simulation done by FFT-BPM is not enough accurate for thick sinusoidal phase gratings as it is still based on some approximations and does not take into account the effect of reflection. In the following section, we will determine the limits of using FFT-BPM by comparing it with FDTD.
3.2.2 Far-field simulations, Fraunhofer versus a high NA propagator
One of the problems in determining the correct field of view of a diffractive structure is the validity of the far-field propagator. For large fields, a simple Fourier transform might not be valid anymore to describe the situation correctly. In this section, we investigate the validity of using the Fraunhofer approximation in comparison with the high NA propagator for thin and thick sinusoidal phase gratings. We start by showing in Fig. 12 the near field and far field simulations for a grating height of h = 12 um. As seen, the near field amplitude and phase distributions are the same using FFT-BPM and FDTD. The far-field patterns are similar using Fraunhofer and high NA approximations because the pattern field of view does not go far beyond the paraxial approximation and most of the energy is distributed in +/- 25$^\circ $ angles which $sin{\; (}\theta {)\; } \cong {\; }\theta $ with less than 5% error. Nevertheless, the pattern field of view is slightly bigger using the high NA propagator. The pattern envelope is also calculated by taking the Fourier transform from one single period of phase grating near-field at x = 0 which is basically the pattern envelope under the plane wave. By comparing the pattern envelope and the irradiance pattern, we can observe how the Gaussian beam modifies and smoothens the energy distribution over the peaks.
Fig. 12. Near field and far field for h = 12 um. (a) Near field distribution using FFT-BPM and FDTD and near field line plot of the amplitude and phase distribution. The corresponding far-field distributions and pattern envelopes using (b) Fraunhofer approximation and (c) high NA approximation.
In Fig. 13, the far-field patterns are compared using the Fraunhofer approximation and high NA propagator for both the FFT-BPM and FDTD near field simulations for h = 25 um. As seen in Fig. 13(a), the amplitude and phase near field distributions are different using FFT-BPM and FDTD. According to Fig. 13(b), the far-field distributions using Fraunhofer approximation for FFT-BPM and FDTD are similar although their near-field distributions are different. In Fig. 13(c) the far-field pattern using the high NA propagator is totally different for the FDTD near-field simulation in comparison with the FFT-BPM. Especially the peaks in the far-field pattern at around ${ \pm \; 45^\circ }$ vanish for the FDTD calculations using the high NA propagator.
Fig. 13. Near field and far field for h = 25 um. (a) Near field distribution using FFT-BPM and FDTD and near field line plot of the amplitude and phase distribution. The corresponding far-field distributions and pattern envelopes using (b) Fraunhofer approximation and (c) high NA approximation.
The far-field distributions for the grating height of h = 25 um indicate that the Fraunhofer propagator is not valid because the pattern field of view goes beyond the paraxial approximation and using a non-paraxial far-field propagator is necessary to obtain the correct field of view. Also, it is demonstrated that obtaining the same far-field distribution for the FFT-BPM and FDTD near field simulations (as shown in Fig. 13(b)) does not guarantee the validity of the used near field and far-field approximations.
In Fig. 14, the near field and far-field patterns are compared for h = 50 um. As seen in Fig. 14(a), the near field phase distribution is different using the FFT-BPM and FDTD. According to Fig. 14(b) for the Fraunhofer approximation, the null in the center of the pattern is deeper using the FDTD in comparison with the FFT-BPM. In Fig. 14(c) using the high NA propagator, the null in the center of the pattern is much wider using the FDTD in comparison with the FFT-BPM.
Fig. 14. Near field and far field for h = 50 um. (a) Near field distribution using FFT-BPM and FDTD and near field line plot of the amplitude and phase distribution. The corresponding far-field distributions and pattern envelopes using (b) Fraunhofer approximation and (c) high NA approximation.
Finally, Fig. 13 and 14 demonstrate that both the amplitude and phase near field modulations determine the far-field distribution. Also, a rigorous method (eg. FDTD) should be used to find the correct near-field distribution for thick sinusoidal phase grating and FFT-BPM is not valid for thick phase gratings of more than 0.5 aspect ratio. Besides, the paraxial approximation is not valid in the far-field observation plane for thick sinusoidal phase gratings.
3.2.3 Evaluation of the spot generators
In this section, we compare the performance of our gratings of different heights (12, 25 and 50 u) as spot generators. The distribution for 50 um thickness is very non-uniform and not a practical point generator. To compare the 12 and 25 um height, the pattern uniformity and the number of peaks are applied, as the criterion. we calculate two critical parameters for the patterns. The first parameter is the pattern intensity standard deviation which measures uniformity. The second parameter is the number of points in the pattern. We consider only dots intensities higher than 13% of the maximum intensity. Table 1 summarizes the findings.
Table 1. Pattern standard deviation and numbers of points for 12 and 25 um thicknesses
Comparing the 12 and 25 um height for point pattern generation, more numbers of points are generated with a higher uniform distribution for 25 um thickness as seen in Table 1. In the experimental evaluation section, we will compare the pattern distribution generated by 12 and 25 um thick sinusoidal phase grating point generators.
4. Experimental evaluation
4.1. Sample fabrication
We fabricated sinusoidal phase gratings for h = 12 and 25 um by means of a direct laser writing technique using the commercial Nanoscribe Photonic Professional GT which is a femtosecond laser lithography system [24–26]. We apply a 25x immersion objective and the light intensity is optimized to be sufficient at the focal point to initiate the photo-polymerization. The polymerization and structure build-up is done layer by layer. The distance between planes (resolution) is 100 nm and 200 nm for 12 and 25 um thickness, respectively. Structures are written on a glass substrate that is coated with ITO to obtain sufficient refractive index contrast (of more than 0.1) between writing material IP-S and the substrate. After the writing process, the sample is developed in a bath of PGMEA for 10 min to remove the non-polymerized photoresists and rinsed in isopropanol for 2 min. The photograph of the 12 um thick sample is shown in Fig. 15(a) and Fig. 15(b) shows scanning electron microscopy (SEM) of the sample.
Fig. 15. (a) Photograph from one of the fabricated samples. (b) Scanning electron microscopy (SEM) of the sample, showing a small gap between the stitched areas. (c) Zoom at a small area demonstrating the surface roughness.
There are small gaps between the stitching areas [27] which is a result of the lateral movement of the sample during the exposure using the Galvano actuator which is not as accurate as piezo-actuators. The lateral structure dimension of 0.5 mm x 1.2 mm is needed in the experiment and there is a comprise between quality and wring time. In Fig. 15(c) the zoom-in SEM image of the structure illustrates the surface roughness for the 12 um thick sample.
4.2. Comparison of the simulation and the experiment
The schematic of the measurement setup is given in Fig. 16(a). A laser at 660 nm wavelength is coupled into an optical monomode fiber (P1-630Y-FC Thorlabs) with an NA between 0.1 and 0.14. Using the subsequent 1:1 aspheric pair lenses (Thorlabs, C110M-B, mounted match pair), the beam coming out of the fiber is imaged 8 mm away from the second lens flat front and a Gaussian beam source is created in very good approximation. In such a way, the setup is very versatile even for sample substrate of 800 um thick and avoids small distance to phase gratings surface. Distance between Gaussian beam waist and grating surface D is chosen to be 3.8 mm, the same as the value for simulations. The pattern is projected on a screen 24 cm away from the sample and the image of the far-field distribution on the screen is captured by a camera (IDS CMOS camera) using an objective (JC10M Kowa lens series) with a big field of view of 80 degrees along the x-axis. The 24 cm distance between the sample and the screen is not in the far-field region under plane wave illumination but for the Gaussian beam, the far-field region is at smaller distances from the sample [28] and for this reason, we are able to observe the far-field already at 24 cm. In Fig. 16(b) and (c), the experimental far-field pattern is compared with the simulation result for h = 12 and 25 um, respectively. All the simulations are done using FDTD and high NA propagator. The measurements verify the simulation results especially the pattern distributions match very well for 25 um thick sample. For 12 um thick configuration, the experiment pattern envelope is not exactly the same as simulation although the number of peaks clearly matches with simulation. Also, the measurement verifies that more numbers of peaks are generated for 25 um in comparison to 12 um thick phase grating. The quality of the experimental result is limited by two constraints: First, the fabrication of high quality micro-optical periodic structures is challenging with a period of 50 um and a big lateral dimension of 0.5 mm x 1.2 mm. As we discussed in the fabrication part, there are small gaps between the stitched areas which may influence the pattern quality although we have optimized the fabrication parameters for a high-quality sample. The second and most important limitation is the low resolution (1280 × 1024) and limited dynamical range of the camera which lowers the contrast in measurements and the lines are not resolved with high quality in the picture.
Fig. 16. (a) the schematic of the optical setup. (b) and (c) Experimental versus simulation far-field pattern for h = 12 and 25um, respectively.
The measurements for 12 and 25 um thick sinusoidal phase grating confirm the high accuracy simulation results. It demonstrates that choosing accurate near-field and far-field simulation tools is essential to find the correct pattern distribution. The accurate simulation tool is defined according to the phase grating thickness and surface curvature as well as the generated pattern field of view in the far-field. Especially in the thick case of 25 um, only the high precision FDTD simulation tool leads to the correct pattern distribution. To calculate the far-field for high diverging patterns in which the paraxial approximation is not valid, but the high NA propagator is enough accurate [21] Also, the measurements confirm that 25 um thick phase grating generates the most number of points with uniform distribution as a dot pattern generator.
4. Conclusion
Structured dot pattern generation under diverging Gaussian beam illumination for sinusoidal phase grating is investigated using the self-imaging phenomenon. Under the Gaussian beam illumination, large numbers of peaks with a bigger field of view are generated in the far-field for sinusoidal curvature in comparison to the plane wave illumination. The influence of the source beam waist and the sinusoidal phase grating thickness on the far-field pattern is studied. We compared using TEA, FTT-BPM and the rigorous FDTD method for the near field simulations. TEA is not accurate for the simulation of thin phase gratings for different source beam waists, especially in terms of far-field pattern envelope and field of view. According to the rigorous FDTD simulations, the reflection from sinusoidal phase grating increases by 4% for the smallest source beam waist in comparison with the plane wave illumination due to the source wave-front curvature.
We studied the effect of changing the sinusoidal phase grating thickness on the far-field pattern using the FFT-BPM in comparison with the FDTD for near field simulation and Fraunhofer approximation compared with the high NA propagator for far-field simulation. The far-field intensity distribution is very sensitive to the phase grating thickness and completely different pattern distributions are obtained for different thicknesses. As a result of the FDTD simulations, we found that the total internal reflections inside the thick phase grating due to both the thickness and also the sinusoidal curvature of phase grating influence the near field distribution considerably. Consequently, using the scalar approach of FFT-BPM is not valid for thick sinusoidal phase gratings with more than 0.5 aspect ratio. Moreover, we demonstrated that the far-field Fraunhofer approximation cannot be used for thick sinusoidal phase gratings as the pattern field of view is large and using the high NA far-field propagator is obligatory. On the other hand, the comparison between thick and thin sinusoidal phase gratings demonstrates that very high structures with the unity aspect ratio generate non-uniform pattern distribution which is not suitable for point pattern generation. The 0.24 and 0.5 phase grating aspect ratios generate large numbers of peaks with uniform distribution which makes them suitable as point pattern generators. Finally, we demonstrated that the experimental results confirm the FDTD and high NA propagator simulations.
Funding
Horizon 2020 Framework Programme.
Acknowledgments
The authors thank J. Dorsaz (CMI-EPFL) for useful discussions during the fabrication process using the Nanoscribe Photonic Professional GT at Center of Micro Nano Technology (CMi) at EPFL. Also, the authors thank Jeonghyeon Kim (NAM-EPFL) for coating the sample with carbon. We needed one carbon coated sample for taking SEM images.
Disclosures
The authors declare no conflicts of interest.
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