2.1

Design and Simulation Methods

To use the DOEs to replace high-NA objectives for fast scanning with high resolution, they need to be able to produce dense spot arrays with high-NA. This is achieved by the concept of overlapping aperture, which is explained by the design procedures in figure 2.

Figure 2.

Design procedures for a DOE spot generator with overlapping apertures.

First, the target spot field distribution u_spot is defined. It propagates back and forms a spherical-wave-like field distribution u_r. Rayleigh-Sommerfeld integral^7–9 is used as the simulation method for the diffraction field propagation, which is shown as

where Σ denotes the surface on the boundary, i.e. the plane which u_spot lies in and the semi-infinite sphere behind it, r = (x, y, z) is the coordinate of u_r, r′ = (x′, y′, z′) is the coordinate on Σ, and k is the wave number.

Afterwards, in order to construct a spot array, u_r is duplicated and overlapped with a certain pitch to form u_D. Then the phase of the overlapping field u_D is extracted and binarized with a binarization factor B³ in the following equation,

Finally the binarized field propagates back by simulation to verify the spot array which will be actually produced. In this case, the spot is not solely produced by the small unit cell above it, but it will be produced by the original spherical-wave-like field which is already overlapped with the adjacent ones. Every single spot in the array can receive contributions from other unit cells around it. Thus, the NA of the spot is not limited by the pitch anymore. A dense spot array with high NA can be generated in this way.

However, as described in Section 1, such DOEs are not capable of measuring opaque objects, in which case the DOEs and the imaging system must be placed on the same side. To improve that, the idea of superposition is used to add more flexibility to the current DOEs. By simply adding different fields generated from different target light distributions, all the target patterns can be generated with one single DOE¹⁰. Such kind of DOEs is known as the multi-functional DOEs.

In the following sections, two DOE concepts are designed with this method to overcome the shortcomings of the DOEs in current research for confocal microscopy^3–5. The new designs have different functions and are both able to measure opaque surfaces.

2.2

See-through DOE design

In order to illuminate and image the sample on the same side for confocal surface measurements, the spots need to be seen through the DOEs without severe disturbances.

Figure 3 shows a DOE-based multi-spot scanning confocal microscope for surface measurements. The DOE generates high-NA spots which are imaged by a low-NA objective. In this way, although the low-NA objective will produce larger spots on the image plane, the lateral resolution is still governed by the high-NA illumination spots, which is similar to the principle of super-resolution microscopy like STED¹¹ and PALM¹². Meanwhile, a low-NA lens can offer a large field of view. Thus high resolution and large-area scanning can be achieved at the same time.

Figure 3.

See-through configuration of a DOE-based confocal surface measurement system.

However, in the setup above, the original DOEs with overlapping apertures in figure 2 will introduce disturbances during the imaging process. Simulation shows that the spots become irregular or even invisible when they are imaged by the objective and camera¹³. In order to reduce the disturbances added by the DOEs, a plane-wave component is added to form a new field distribution,

where u_D is the original overlapped field distribution in figure 2 and W is a constant.

In this way, the zero-order diffraction of the DOEs becomes higher. In other words, the DOEs become more transparent. The parameters of the DOEs are optimized iteratively to achieve the highest intensities in the spot centers in the image with the working distance z = 1.095 mm, the plane-wave weight factor W = 13 and the binarization factor B = 0.41π in Eq. (3) for a 11 × 11 spot array. The pitch is 100 μm and the designed wavelength is 785 nm.

The see-through DOEs can improve the lateral resolution by the high-NA spots produced by the DOE. However, according the the theory of confocal microscopy, such configuration in figure 3 cannot significantly improve the axial resolution when measuring a plane object¹⁴. The axial resolution will still be strongly restricted by the low-NA objective for surface measurement.

2.3

Direct-imaging DOE design

In order to increase the axial resolution and make the DOE suitable for 3D surface measurements, the optical configuration for the direct-imaging DOE design is proposed as figure 4 shows. In this case, the DOE is the superposition of two kinds of lenses with different focal lengths and overlapping apertures. One acts as an illumination lens which converts an incident plane wave into a spot array. The other one is the imaging lenses which acts as part of the imaging optical system. Alternatively, it can also directly image the spots onto a imaging sensor. The two lenses are overlapped as figure 2 shows. In this case, the field distribution for the new DOE is calculated by

Figure 4.

Direct-imaging configuration of a DOE-based confocal surface measurement system.

In this way the illumination and the imaging are both high-NA and the system can have the same performance as a high-NA confocal microscope.

The direct-imaging DOEs are also optimized iteratively to achieve the highest intensities in the image spot centers with the working distance z = 1.11 mm, the distance from the DOE to the intermediate image spots d = 21.262 mm, the weight factor W = 0.045 and the binarization factor B = 0.97π in Eq. (3) for a 5 × 5 spot array. The pitch is 100 μm and the designed wavelength is 785 nm.

The direct-imaging DOEs provide both high lateral and axial resolution. However, simulation shows that irregular interference will be introduced when the number of spots in the array increases. An 11 × 11 spot array will already make the spots indistinguishable from the background¹³. As mentioned previously, the DOE is composed of two components which are generally two kinds of lenses with different focal lengths. They act as the illumination lenses and imaging lenses respectively. However, they will not work separately as we want for illumination and imaging. They will always take effect at the same time. When projecting the illumination spots, the imaging lenses will also produce a blurred spot around the original one, which is shown in figure 5a. Besides, on the imaging side, due to the overlapping apertures, one spot will not only pass the designed lens to form a spot on the image sensor, but also it will go through the adjacent lenses to form other blurred spots, which is shown in figure 5b. As the number of spots increases, the disturbances will accumulate and the image quality will be reduced.

Figure 5.

Side effects which cause irregular interference patterns for the direct-imaging DOE.

One possible solution is using a line of spots instead of a 2D grid to reduce the overlapping. In this case, there is no longer interference from all the directions. Furthermore, one spot will receive less disturbances from the spots farther away. Thus the overall disturbances for a single spot can be controlled to an acceptable level and the length of the line can be extended to any desired measurement range.

3.1

See-through DOE Experiments

Figure 6c shows the real experiment setup of figure 3. In the setup, a diode laser is coupled into a single mode fiber. It connects to a collimator and produces plane-wave illumination. The light is reflected by the beam splitter and illuminates the DOEs. Underneath there is a CMOS sensor to measure the illumination spots and a mirror to reflect the light. At the top, a camera and an objective is used to image the spots through the DOEs.

Figure 6.

Experiments to measure the illumination spots.

In the first experiment, the 11 × 11 spot arrays produced by both the original and see-through DOEs are measured by the CMOS sensor from Raspberry Pi Camera V2. Because the 1.12 μm pixel size of the sensor is not small enough to resolve the spot laterally, the sensor is moved up and down in z direction to measure the axial point spread function (PSF) of the spots in order to determine the actual NA which has been achieved. The axial PSF measurements for the original and see-through DOEs are shown in figure 6a and 6b respectively.

For the original DOE, the simulated axial full-width at half maximum (FWHM) is 2.3 μm and the measured FWHM is 2.67μm, which corresponds to an NA = 0.77^{15, 16}. For the see-through DOE, the simulated FWHM is 2.5 μm and the measured FWHM is 2.96 μm, which corresponds to an NA = 0.73. The experiment results are close to the simulation. The slight deviation can be attributed to that the real diffraction efficiency of the DOEs might be lower and the acceptance angle of the light for the CMOS sensor is limited.

In the second experiment, the illumination spots are reflected by a mirror and they are imaged by the objective and the camera through the DOE. Then the mirror is also moved axially in z direction to measure the confocal axial response. The central spot in the 11 × 11 spot array is selected and the axial intensity response in the image by moving the mirror is recorded. The objective has an NA = 0.15.

Figure 7a and 7b show the results of the confocal axial responses for the original DOE and the see-through DOE separately. Obviously, the intensity response of the original DOE is very noisy and its peak is hard to define. Contrarily, the see-through DOE provides a much smoother signal, which proves that it can be fit into the configuration in figure 3 without introducing severe disturbances. It shows a FWHM of 46.2 μm which corresponds to an NA = 0.18. As predicted by the theory and the simulation, the axial resolution for a plane is only slightly improved and is still highly restricted by the low-NA objective for imaging.

Figure 7.

Experiments to measure the confocal axial response.

Finally a rough surface on a coin is randomly selected and used as a measurement object. The confocal axial responses of all the spots in the array are recorded. The FWHM of the spots within the measurement range are calculated. The distribution is shown in the histogram in figure 7c. The results show that although still limited by the low-NA objective, a large portion of spots can produce significantly smaller axial FWHM, which increase the measurement sensitivity. The reason is that when the surface is not a perfect horizontal plane, it will be more “point-like”. And for a point object, the axial resolution is no longer governed by the low-NA imaging objective in the configuration of figure 3. Detailed mathematical derivation can be found in Ref. 13, 14.

3.2

Direct-imaging DOE Experiments

For the direct-imaging DOE, the axial PSF of the produced illumination spot is also measured as figure 8a shows. The measured FWHM is 3.54 μm which corresponds to an NA = 0.67. The produced spots are larger compared to the ones produced by the see-through DOEs because of the contribution from the additional superimposed field as shown in figure 5a.

Figure 8.

Experiments to measure the confocal axial response.

After that, a mirror is placed underneath to measure the confocal axial response as figure 4 shows. The results for the central spot of the direct-imaging DOEs in the 5 × 5 and 9 × 9 spot arrays are shown in figure 8b and 8c respectively. The confocal axial response of the central spot in the 5 × 5 array shows a FWHM of 4.57 μm, which provides much better resolution than the imaging objective with NA = 0.15. figure5 increase with the number of spots, the central spot in the 9 × 9 array is completely missing in the irregular interference. Thus a line should be used instead of a grid to avoid such disturbance in real applications.