This paper presents a simple method based on the measurement of the 3D intensity point spread function for the quality evaluation of high numerical aperture micro-optical components. The different slices of the focal volume are imaged thanks to a microscope objective and a standard camera. Depending on the optical architecture, it allows characterizing both transmissive and reflective components, for which either the imaging part or the component itself are moved along the optical axis, respectively. This method can be used to measure focal length, Strehl ratio, resolution and overall wavefront RMS and to estimate optical aberrations. The measurement setup and its implementation are detailed and its advantages are demonstrated with micro-ball lenses and micro-mirrors. This intuitive method is adapted for optimization of micro-optical components fabrication processes, especially because heavy equipments and/or data analysis are not required.
© 2014 Optical Society of America
The fabrication processes optimization of micro-optical components requires correct characterization of their performances in order to tend to good reproducibility and high optical quality. Different optical parameters can be used to determine these optical performances  such as focal length, shape of the wavefront and aberrations .
Basic evaluation of micro-optical elements quality is usually first based on the measurement of their topography since their optical properties are largely defined by their shape. Surface 2D topography along with surface roughness measurements are then performed by mechanical  or optical devices. The latter, so-called non-contact profilometers, derive the sample altitude from an optical signal reflected from the sample surface. Although various non-contact profilometry methods have been developed, such as white light interferometry [4–6] and digital holography , they are usually geometrically limited to rather low numerical aperture (NA) of the measured components. Furthermore, surface topography is not sufficient and optical characteristics have to be derived from the measured geometry using optical propagation algorithms (ray-tracing or diffraction theory).
An alternative method is the direct measurement of sample optical properties. The advantage of this approach is its ability, unlike topography measurements, to characterize higher NA components. Moreover, it can be applied not only to single elements but also to more complex optical systems composed of several optical components. The most popular direct optical measurement method is the wavefront measurement  which can be performed with interferometric systems [7,8] or by discrete wavefront sampling, usually achieved with Shack-Hartmann sensors . Nevertheless, wavefront detection requires high quality probing wavefronts. In case of micro-optical elements, this task can be difficult because the probe has to be scaled down to the size of the measured element and later rescaled to the size of the detector. This multi-scale optical measuring system has to maintain high optical quality and needs accurate alignment of measured micro-elements .
Consequently, we propose a simple method based on the measurement of the 3D Intensity Point Spread Function (IPSF) generated by the focusing optical elements. This method is intuitive and does not require heavy equipments so that it can be used for quality evaluation during, e.g., optimization of micro-optical component fabrication processes. It can be noted that many papers in the literature dealing with, e.g. fabrication of microlenses, often rely only on topography measurements without further derivation of optical features [10–14].
The direct characterization of focusing response through the measurement of IPSF allows estimation of the optical quality, leading to identification of aberrations (without quantitative wavefront mapping) and asymmetry, as well as measurements of parameters such as spot size and radius of curvature. The IPSF  has been already employed as a characterization tool for complex transmissive optical systems like confocal microscopes [16, 17] or objectives [18, 19]. In here, we adapt this approach to build a simple setup for micro-optics characterization.
In particular, our system can be easily adapted for two different imaging cases: the first one consists in the investigation of focusing by a transmissive element of a collimated probing beam which corresponds to imaging a point located at infinity. The second one, adapted to the characterization of reflective elements, relies on imaging a point (focused probing beam) located at the center of curvature of the investigated element. For both cases, the fixed imaging configuration allows rapid estimation of quality and repeatability of fabricated focusing optical components. Moreover, the optical resolution of the micro-optical elements can be directly derived from the recorded 3D shape of the focal spot.
2. 3D IPSF of a focusing element
The 3D intensity distribution of light focused by an optical component can be described by the classical diffraction analysis  and is given by:
This intensity distribution corresponds to the optical response of the system, i.e. to the 3D IPSF. Figure 1 displays lateral and longitudinal slices of focus volumes with different types of aberrations calculated from Eq. (1). It can noticed that the images of the focal plane in Fig. 1(a) do not differ so much. However, the evolution of the focused beam along its propagation (Figs. 1(b)–(e)) is significantly and specifically affected by each type of aberration. Consequently, imaging beam propagation around the focus can allow easy estimation of aberrations, more efficiently than from focus plane images.
3. Measuring setup
The characterization of such beam transformation depends on the nature of the micro-optical element to be analyzed, whether it is transmissive or reflective. The proposed measurement station architecture has then two arrangements for both transmissive (Fig. 2(a)) and reflective (Fig. 2(b)) measurements of micro-optical elements. In order to switch from one configuration to another, the two beam splitter cubes (BS1 and BS2) are introduced or not (slid in or out in our setup). Moreover, the imaging part is translated in transmission mode whereas in reflection mode, the sample is displaced. In our case, the beam emitted from a He-Ne laser source (λ =632.8nm) is spatially filtered and expanded in order to get a single source point, i.e. a flat wavefront incident onto the sample. The diameter of the output beam is in the order of few millimeters and is larger than the pupil of the measurement system (objective pupil in reflection mode and microlens size in transmissive mode) assuring its uniform illumination. A half wave plate and a polarizer are added to adjust the incident light intensity.
In the transmissive configuration (Fig. 2(a)), the expanded probing beam is reflected by two mirrors to be easily directed onto the micro-optical element. The focused spot is collected by the microscope objective MO (x50 and NA=0.45) and projected by the tube lens TL (f’=125mm) onto the CMOS camera (μEye UI-1240SE from IDS). The microscope objective used as a collection lens provides a high NA to the imaging system and a large magnification. In this case, the pupil function from Eq. (1) can be written as:
The three components MO, TL and camera are part of the imaging system, which is translated in order to image the different slices of the focal volume. This translation is achieved by a servo motor (Ealing 37-1104) whose minimum incremental motion and travel range are 50nm and 25mm, respectively. The step size is then adjusted depending on the NA of the investigated element in order to record typically 100 to 200 slices around the focal plane, i.e. within ±5λ/NA2. The camera and the motor are controlled by a PC, for which system control and data processing have been programmed in Python .
In reflective configuration (Fig. 2(b)), the probing beam is reflected by the first beam splitter cube BS1 and is directed towards the MO by the second beam splitter cube BS2. The latter can be placed between the MO and the TL without deterioration of the optical beam quality thanks to the infinite conjugate configuration of the imaging objective. Then, the micro optical component to be characterized reflects the probing beam onto the imaging system. The 3D intensity distribution is slightly different since light encounters a round-trip. On the one hand, this round-trip leads to a multiplication by a factor 2 of the aberration term in Eq. (1). It is important to note that the aberration function Φ(θ, ρ) is in here directly defined by the geometry of the measured object, literally how it geometrically deviates from a sphere. This is not true for measurements of transmissive focusing elements such as microlenses for which basic geometrical analysis demonstrates that perfectly spherical lenses do not generate spherical wavefronts . On the second hand, the round-trip leads to a factor 2 multiplying the defocus term. In consequence, the measured 3D intensity does not correspond directly to the object IPSF but to its axial compression. In addition, in reflection mode, the microscope objective used in the imaging system encounters also a light round-trip. Then, its quality is twice more important than in the transmissive system, since any aberration generated by the objective is also doubled. The pupil function P(ρ, θ) in Eq. (1) has then to be written as:
The image of each focus slice is then recorded by the CMOS camera. Since the dynamic range of the CMOS sensor (8 bits) is not sufficient to register the rapidly changing structure along Z of the observed focal spot, the dynamic range is improved by recording multiple frames of each focus slices with different exposure times . In here, 6 consecutive frames (with exposures varying from 0.05ms to 100ms) are recorded and the final image is reconstructed from multi-exposure data. The reconstruction involves images normalization (according to exposure time) of the different frames, followed by frames averaging with exclusion of saturated or under-threshold pixels. This method allows increasing significantly the dynamic range while lowering the noise level as it can be seen on Fig. 3 that displays an example based on 3 frames.
To obtain data about the whole investigated micro-optical element, the numerical aperture of the microscope objective NAobj has to be higher than the numerical aperture of the investigated sample NAm. Otherwise, only a part of the sample is characterized. In this case, the pupil function in Eq. (1) is written as:
In other terms, by setting the numerical aperture of the collection system NAobj < NAm, we can choose to probe a determined zone of the sample using a known NA (defined by NAobj). In addition, this allows the characterization of elements in similar conditions than in the targeted applications.
4. Data analysis
Recorded 3D IPSF can provide different information about the characterized optical component. The qualitative (visual) assessment of the PSF shape can simply indicate the presence of aberrations as well as the asymmetry of optical components. When a quantitative analysis is required, the same data can be used to derive the optical resolution by analyzing the IPSF geometrical spreading. The optical resolution can then be defined in different ways such as FWHM (full width at half maximum), RMS (root mean square) spot radius or radius encircling a certain amount of energy. Another parameter that can be extracted from the measured IPSF data is the Strehl Ratio that quantifies contrast (peak intensity value of the IPSF) in comparison to a perfect optical component having the same NA.
Optical performance of components is usually defined at the focal plane. However, in practice, locating precisely the focal plane can be difficult. When 3D focal volume data is recorded, desired figures of merit like spot radius or Strehl ratio can be calculated for all collected slices and the real position of focal plane can be easily identified.
4.1. Measurement example of a 3D IPSF
Recorded data correspond to the 3D intensity variation around the focal plane of the micro-optical element. Figure 4 shows the IPSF measurement in transmission from a ball microlens (Edmunds Optics). It can be noted that this component is usually difficult to characterize although it is widely employed in micro-optical systems.
The focal length and the diameter of the N-BK7 glass microlens (nd = 1.57) are equal to 367μm and 500μm, respectively. In this configuration (whole lens illuminated), the numerical aperture of the ball microlens (NAm=0.57) is higher than the numerical aperture of the microscope objective NAobj=0.45, limiting the observation area to about 75% of the lens diameter.
The image of focal plane (Fig. 4(a)) shows the well defined spot similar to an Airy disk, however, with stronger outer rings than in the case of aberration free imaging systems. It can be noted that the geometrical deviation from the ideal case is difficult to notice simply by investigating the image of the focal plane. This is not true when observing XZ slice of the focal volume (Fig. 4(b)) which is very different from the ideal case (Fig. 1(b)). The visible lack of symmetry with respect to the focal plane is attributed to the wavefront aberrations introduced by the microlens. When comparing the XZ slice to the one from the Fig. 1(c), strong outer rings before the focal plane are the signature of a significant spherical aberration. In general, the level and type of asymmetry in the XZ focus slice, with respect to the focal plane, is an easy qualitative indication of aberrations generated by the characterized element.
As mentioned earlier, NAobj has to be higher than NAm in order to get the full picture of the component under investigation. Thus, Fig. 5(a) shows a measurement of the same ball microlens performed with NAobj = 0.80. However, if IPSF XZ slices can be easily employed for comparison of different components when using the same NAobj, it is more difficult when employing different NAobj. For this case, the same data should be plotted in normalized (u,v) coordinates as shown in Figs. 5(b) and (c). These figures demonstrate that the ball microlens performs significantly better in a lower NA configuration than with full accessible numerical aperture (NA=0.57).
The lateral resolution of optical systems can be estimated by simple analysis of their 3D IPSF (Fig. 4). The lateral resolution (FWHM based criterion) of an aberration free micro-optical element (Φ=0) can be defined as:
Therefore, the spatial spread of the focus spot can be analyzed by means of root mean square (RMS) spot size or an encircled energy plot. RMS spot size is defined as a squared mean:Figure 6(a) shows the evolution of the RMS spot size during the propagation of the beam. The minimum spot size is equal to 2.5μm and does not correspond to the maximum peak intensity.
An additional way to quantify the performance of the optical components is to derive the power focusing efficiency. This can be analyzed via encircled energy plot, i.e. a plot illustrating the power distribution in the focal plane. Figure 6(b) presents the power analysis for the considered ball-lens. The numerical analysis (performed with ray tracing ZEMAX software where a 500μm diameter ball lens is considered) is compared with data calculated from the measurement. The curve for a diffraction limited system is also plotted to illustrate the large difference between aberration free and aberrated cases. Although a slight difference between measured and simulated curves for small radii exists, 75% of energy for both cases is contained in a circle of 6μm which can be considered as the effective resolution of the component.
4.3. Strehl ratio
Another often used measurement of the optical aberrations is the Strehl ratio (SR). It compares the maximal values of IPSF of an aberrated system with an ideal one. The Strehl ratio is given by:18].
Total power is calculated by integrating the intensity over the whole focal plane. To do so, total power can be approximated by the sum of the intensity values recorded by the camera, i.e. Ptot = ∑i,j I(i, j)δx2, where I(i, j) is the intensity level in the focal plane at the grid coordinate system defined by pixel indexes i, j, and δx2 represents the surface of the pixel (δx = 0.168μm at object space coordinates). Peak intensity is taken as the maximal value in the measured data. SR can be calculated from 3D IPSF data obtained from both transmissive and reflective configurations.
In the following, the example measurements are performed in reflection. This choice is governed by the possibility to calibrate concurrently the measuring setup. Since the component at cat’s eye position does not generate aberrations, the measured ones can be attributed to the imaging system only. Figure 7 displays the measured focal spots of a ball microlens at cat’s eye (objective focal plane located at the apex of the microlens) and confocal positions (positions are indicated on Fig. 2). The calculated Strehl ratio are SRCE = 0.85 for cat’s eye position and SRCF = 0.5 for confocal position. SRCE is high and defines the precision of the measuring system (optical system characterized by SR > 0.8 can be considered as diffraction limited).
In the low aberration systems, the SR is defined by a statistical measure of the aberration function, i.e. the wavefront RMS related by Marechal approximation:Eq. (10), it can be noted that determination of SR allows the estimation of wavefront deviation σΦ that is the most common aberration quantification. The strong relation SR(σΦ) allows precise estimation of the optical quality of the analysed optical system through the experimental measurement of SR. Thus, wavefront aberration of employed imaging system can be estimated using Eq. (10) as . This value corresponds to the round-trip, thus, for a single pass, the performance of the imaging system can be estimated in the transmission mode as , that corresponds to SR = 0.96. This value demonstrates the high quality of our imaging system based on a microscope objective (Nikon CF Plan, 50x/0.45 SLWD).
Then, the measurement performed at the confocal position contains information about the investigated microlens. Here, wavefront deviation reads as , i.e. a geometry RMS deviation equal to . This value corresponds to a RMS figure error of the microlens reflective surface . In this measuring system, contribution of the surface roughness of the measured object is not expected, since light scattering due to the roughness is weakly coupled by the imaging system. The manufacturer specification on the lens geometry accuracy mentions only the sphericity precision, defined as the maximal expected diameter difference of lens sections. In consequence, the used ball lens exhibits some level of spheroidal form. From maximal expected diameter difference (specified as 1μm), the RMS geometrical deviation on an observed zone (45% of lens diameter defined by NAobj = 0.45), is calculated as 27nm, which is consistent with the measured value.
It is worth to notice that Eq. (10) connecting SR and σRMS is only valid for system having low aberrations (σΦ¡ 0.15λ). For systems characterized by higher aberrations, the peak intensity value is not directly related to σΦ but depends on specific form of the system aberrations . In consequence, the SR is meaningful only for the elements of relatively high optical quality.
4.4. Radius of curvature measurement
Although the primary target of our system is the measurement of micro-optical components quality via the 3D IPSF analysis, this system can also be used (without any modification) to precisely measure the radius of curvature (ROC) and then the focal length of the investigated micro-optical element. The ROC measurement principle is sketched in Fig. 8(a). It is based on the axial scanning of the sample between the confocal and the cat’s eye positions. They correspond to locations of the object for which the probing beam does not undergo changes and is completely collected by the imaging system. The distance between the confocal and the cat’s eye positions is then equal to the ROC of the object (Fig. 8(b)).
In practice, since the used detector is a pixelized camera, the real intensity information is not accessible. Moreover, since perfect alignment of the optical system is never possible (measurement setup - sample), the optical axis of the system is not well defined. To address this problem, the axial dependence of power encircled in a small virtual pinhole located in the beam center instead of axial intensity is investigated. The beam center is found for each z-frame separately, to make the method independent of possible beam misalignment. The power is calculated by numerically integrating the intensity located in a virtual circular pinhole of a given size (usually pinhole size is chosen in the order of the lateral size of the focal spot).
In order to validate the ROC measurement configuration, tests were also performed with a ball microlens. The measurement data used for ROC determination are presented on Figs. 8(b)–(c). The resulting ROCmeas = (247.5 ± 0.7)μm is consistent with specifications given by the manufacturer (deduced from the diameter precision: ROCspec = (250 ± 1.5)μm). It can be noted that the lens was used without any additional reflective coatings. Reflections observed by the system originate from the Fresnel reflections on the dielectric boundary only.
We propose a simple setup for characterization of micro-optical focusing elements. The method is based on the direct measurement of the probing beam focused by the micro-optical elements and allows characterizing transmissive and reflective components. The setup, described extensively, is based only on a camera, a microscope objective, a tube lens and two single axis servo motors. Moreover, it does not require complicated analysis to interpret the results. In this paper, a ball microlens is tested as an example. In addition to qualitative straightforward evaluation of the component quality, the paper presents the simple analyses of the recorded data corresponding to 3D IPSF. These analyses allow deriving resolution, Strehl ratio, wavefront deviation and radius of curvature. This simple and intuitive method is adapted for micro-optical components, for which only topography measurements are usually reported. It can be also noted that more sophisticated data analysis  can allow calculating wavefront shape from the 3D IPSF.
This work was supported in part by the Agence Nationale de la Recherche and the Federal Ministry of Education and Research Joint-Programme Inter Carnot Fraunhofer under Project DWST-DIS and in part by the collaborative project VIAMOS of the European Commission (FP7, ICT program, grant no. 318542). The authors would like to thank Olivier Gaiffe for fruitful discussions.
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