So called progressive addition lenses (PALs) are produced as ophthalmic devices, which compensate for presbyopia. They are aspheric in the most general sense and made up of surfaces without any spatial symmetry. Lenses made up by one or two of such surfaces are called freeform lenses. Freeform essentially means that a surface is described and manufactured by pointwise information on a given grid, according to a given optical design. Typically, the design comprises a near zone and far zone, which show a more or less constant power distribution and a corridor, the progression zone, where the power increases smoothly from the far zone to the near zone. Various designs might differ in the way the progression zone is designed. Because of the theorem of Minkwitz,1 the design parameters of the progression zone have direct consequences to the unwanted astigmatism beside the progression zone. Thus, the performance of the lens depends crucially on the changes in the progression zone, which have to be validated in a finished lens. Since, globally, such lenses produce highly deformed wavefronts, which challenge the dynamic range of any metrology tool, a closer look to the progression zone requires special approaches.
For many years, PALs were only produced with one standard progressive surface. The individual spherocylindrical power, changing from lens to lens, was included by the second surface, which might be of toric shape. Compared with the freeform surface, this shape is geometrically simple. The most complex surface might have changed with certain classes of dioptric power but has to be controlled, say, only once for several batches in production. At present, there are modern lenses, where one or both surfaces are of freeform type, and, additionally, according to individual parameters of the patient, these surfaces change from lens to lens. These lenses are generated by highly flexible multiaxis spindle machines. The obvious problem during production is the validation of the finished lenses and the comparison with the optical design data.
Various approaches are used to measure PALs. The main distinction stems from the mode of how the lens is tested. Either the entire lens is measured in transmission mode or the two surfaces are serially controlled in reflection mode.
Phase measuring deflectometry is a way to investigate surfaces in the reflection mode.2 Although of high accuracy this method up to now has the disadvantage that the results of both surfaces have to be combined, say numerically, to describe the final lens. Relative centration problems of the two surfaces must be carefully controlled. Additionally, until now, the back surface has to be blackened during the measurement. Clearly, this prohibits the use of the lens after the measurement.
Moiré-deflectometry, Hartmann-sensors, or Hartmann-Shack-sensors3–7 are used to measure the complete lens in transmission. Also, shape of surfaces has been characterized by physical height measurement and hence the second-order optical properties.8 Interferometric tests have the advantage of high accuracy, but generally, the dynamic range is not enough for ophthalmic lenses (OLs) and the use of a reference lens or non-null interferometers is often mandatory.9 This approach is not useful if individualized lenses have to be measured. Additionally, in an industrial environment, interferometric measurements are not robust enough because of their sensitivity for vibrations. However, as we will show here the point diffraction interferometer (PDI)10 can be tailored to measure PALs. As a common path interferometer, it is less sensitive to environmental disturbances.11–13
We applied this type of interferometer to characterize PALs by spatially resolved measurements of the local values for the sphere, cylinder, and axis in a serial way. Although the spatial resolution of Hartmann-Shack sensors is limited by the microlens array's pitch, the resolution of this device is only limited by the pixel density of the captured imaged. It goes without saying that noise is the ultimate limit in every metrology setup.
The PDI consists basically in a semitransparent plate with a clear pinhole as shown in Fig. 1. When a beam reaches the plate, a spherical reference wave is produced by diffraction at the clear pinhole, whereas the rest of the beam passes through the plate without any change in its phase. Only its amplitude is attenuated by the transmittance of the plate (t). If the size of the pinhole and the transmittance of the plate are chosen in such a way that both beams have similar amplitude then well-contrasted fringes will be seen in any observation plane, Σ, placed after the plate.
An extended development of the theoretical basis of the interferometer can be found in Acosta et al.10; here, we will only review those features that form the basis to PAL measurement.
On the one hand, it is easy to understand that if an incoming spherical wave focuses exactly on the pinhole, a bright field illuminates the observation plane, as both transmitted and reference beam are essentially the same beam. If the incoming wave focuses after or before the pinhole, two different spherical waves are generated, and the classical interference pattern of concentric spherical fringes will be observed in Σ. The density of the interference fringes will increase with the axial distance between the focus of the wave and the pinhole plate while contrast will decrease (from here on we name defocus, and it will be represented by ε). Fringe patterns will show a contrast inversion depending on which side of the PDI plane the wave focuses.10
On the other hand, for big pinholes, the contrast of interference fringes is not constant across the observation plane because of the fact that the transmitted wave is modulated by a Bessel function.10 This translates in three effects as follows:
For very small defocus, i.e., for a few interference fringes in the observed region, no true interference pattern can be detected.
As defocus increases, the contrast of the periphery fringes decreases and some fringes disappear, nevertheless the central fringes remains and correspond to those of the interference between the reference spherical beam and the incoming beam.
As defocus increases, the radius and the visibility of the central fringes also decrease. The image acquisition system will determine the upper limit of measurable defocus.
Fig. 2 summarizes these characteristics.
If the PDI plate is illuminated by an astigmatic wave, the shape of the fringes turns into general conics depending on the relative position of the focal lines with respect to the pinhole. The shape of the interference maxima and minima is defined by
where D is the distance from the PDI plate to the observation plane, λ the wavelength, and εx and εy the distance from the pinhole to the focal lines. If both focal lines lay on one side of the pinhole plane, εx. εy > 0, the interference patterns become ellipses with a central maximum or minimum depending on the sign of εx and εy. On the contrary if εx × εy < 0 the fringes become hyperbolas.
In case of ellipses (or circles), it can be straightforwardly deduced that from the measurement of the major and minor axes (or radius) of any interference fringe both the distance and the sign of the focal lines (or focus) related to the pinhole plane can be easily evaluated. Moreover, the center of the interference fringes in the observation plane will change with the relative lateral shifts between the incoming beam (or equivalently a tilted wavefront14) and the pinhole ([xs, ys] and [xp, yp]) with respect to a given optical axis, being the position of the center given by:
Being the angular tilt of the wavefront given by14
For illustrative purposes see Fig. 3.
The interferograms will be rotated if focal lines are rotated, i.e., the angle of astigmatism, θ, can be also obtained from the orientation of the fringes (Fig. 4).
Finally, as this will be the key point for the working principle of the interferometer for PALs, it is worthwhile to stress the fact that for a pure spherical or astigmatic incoming beam, the measurement of the radius or axis and orientation of a single interference fringe (maxima or minima) will provide the values of ε and εx, εy, and θ, respectively.
The experimental setup is shown in Fig. 5. A monochromatic source, S (He-Ne laser @633nm) is collimated by means of a microscope objective and positive doublet, the plane wave illuminates the OL placed 100 mm before a converging lens (CL) with a focal length of f = 50 mm. The observation plane, where the interferograms are going to be recorded, is placed at 100 mm from CL; in this way, the OL and the observing plane are conjugated with magnification −1. The PDI plate is made on a glass substrate coated with a chromium oxide layer with an optical density of 2.3. The clear pinhole has a diameter of 15 μm. The position of the PDI plate will define what we call absolute configuration, i.e., when the plate is placed in the focal plane of the CL, δ = 0, and differential configuration, when the plate is not in the focal plane, δ ≠ 0. The change from one configuration to another only implies an axial movement of the PDI plate what will allow us, as we will explain below, to increase the dynamic range of the apparatus. Fig. 5 shows a scheme of the device and the laboratory setup.
CALIBRATION AND DYNAMIC RANGE
To calibrate the device and establish the dynamic range and accuracy, we used a set of trial lenses with dioptric powers ranging from −10.00D to +10.00D at steps of 0.25D. First, we placed the PDI plate at the focal plane of CL (absolute configuration, δ = 0) and we measured the radius of the first minima; hence, we evaluate the defocus, ε, with respect to the focal plane with eq. 1 (m = 2 for positive powers and m = 1 for negative one). Taking into account that the position of the trial lens is known (2f from CL), a very simple geometrical calculation provides the dioptric power of the lens from the measured values of defocus, ε.
In Fig. 6A, a comparison between measured results and the nominal power of the trial lenses is shown. Results are fitted to a straight line passing through the origin of coordinates. The slope of the line, a, as well as the regression coefficient, r, shows a good agreement between both values. Due to the fact that commercial trial lenses are produced under some tolerances, additional small deviations from a perfect linear behavior may show up in the results. The radii of the first minimum fringe vs. the power of the trial lenses are plotted in Fig. 6B. Since the observation plane and trial lenses are placed at conjugate planes with magnification −1 what implies that measurements correspond to the dioptric power within circles with radii ranging from 0.4 to 1.8 mm.
From Fig. 6A it can be deduced that small powers within the interval of −0.5D to +0.5D cannot be evaluated in this configuration because the image spots fall near the focal plane and because the size of the pinhole interference fringes do not correspond to the interference pattern between two spherical beams (as in Fig. 2B, C). It can be also deduced that only a limited range of dioptric powers from −3D to +6D can be measured in this configuration.
The so-called differential configuration overcomes these two problems; in other words, it increases the dynamic range in both directions: larger powers and smaller ones as well. The reason why the absolute configuration cannot go beyond the aforementioned range is because the image spots lie so far away from the pinhole or the focal plane that fringes are too small. With such a low contrast, the CCD device cannot detect them (as in Fig. 2H, I). The obvious way to increase the size of the fringes as well as the contrast is to translate axially the PDI plate toward the focused spot to get fringes as in Fig. 2D to G.
Displacing the pinhole by an amount δ, is equivalent to compensate for some amount of vergence of the OL, and in this way, the dynamic range can be extended. Besides by displacing the pinhole far enough from the focal plane for those lenses with small dioptric power makes that contrasted central fringes appear and therefore they can be measured.
Fig. 7 shows the measured powers of trial lenses (in steps of 1.00D) for the range of δ values where fringes can be accurately detected. For dioptric powers equal to or bigger than +7.00D, the lower limit in δ values is not because of the low contrast and small size of the fringes but to the mechanical constraints of the experimental setup (delimited with a vertical arrow).
Table 1 shows the nominal power and the averaged measured power (for all δ values) of the trial lenses (in steps of 1.00D) as well as the standard deviation of the measurements from the nominal value what allow us to establish the dynamic range of the setup between −10.00D and +10.00D and an accuracy of about 0.1D (as an upper limit). In all cases, the radius of the first minimum fringe ranged from 0.4 mm to 1.8 mm.
As explained earlier in the text, the shape of the interference fringes turns into conics for toric lenses. Ellipses are obtained if the pinhole is placed right after or right before both focal lines and hyperbolas if in between. In order to apply the same image processing for the fringes and avoid the mechanical constraints of experimental setup in what follows we are going to choose only differential configurations, where elliptical or circular fringes with a central minimum are obtained.
All these calibrations have been performed when both the trial lens and the pinhole are centered on the axis of the optical system. We have already shown that the dioptric power of the lens can be evaluated within a small region. The size of this patch can be chosen just by controlling the axial position of the pinhole.
When the pinhole plate is translated in its own plane, the fringes also displace transversally in the observation plane. Because Σ and the OL are placed at conjugate planes, the fringes provide the local power of that portion of the lens whose image falls on the first minimum fringe's region.
In this way, a simple translational movement of the pinhole can scan the whole lens without any prism system for both absolute and differential configuration as shown in Fig. 8.
In Fig. 9, a superposition of interferograms for three different OLs and different configuration of the setup are shown. Dioptric powers have been evaluated from the first minimum fringe in all cases. From Fig. 9A, B, it can be deduced that both absolute and differential configuration provides the same results for the same lens.
It can be also deduced that the local dioptric power for a low power spherical lens is the same at the different points on the lens surface. In Fig. 9C, the superposition of interferograms in differential configuration is shown for a spherical lens with a dioptric power of +9.00D. It can be seen how aberrations translate in elliptical fringes at the periphery of the lens. In Fig. 9D, we show the local power in both meridians of a cylindrical lens. It can be also observed how the dioptric power is slightly changed as the measurement points are displaced toward the periphery of the lens.
Now, we will show how the PDI performs for a set of PALs.
There are two aims in our measurements of PALs. First, we want to demonstrate that the PDI-setup produces the global information on the performance all over the lens. To this end, we compare our results with a commercial system. Second, we like to emphasize the possibility to concentrate on regions of interest, like the progression zone, where the spatial resolution can be adapted to the properties under consideration. This renders the option to compare PAL-designs according to selected preferences.
We have found that for a PAL elliptic or circular fringes about a point on the lens can be also observed by means of the proper axial and transverse displacement of the pinhole. Depending on the spherocylindrical power and the addition of the lens, the total mapping can be performed with the pinhole placed in a unique fixed axial position (whether absolute or differential configuration) or for several axial positions changing from one to another when first minimum fringes become too small to be detected or too big and because of the presence of high order terms, the second order approach within this region is not valid, and therefore fringes are no longer good approaches to conic sections. Six PALs were measured with the interferometer and the corresponding power maps compared with those obtained with a Rotlex Class Plus lens mapper. The axial position of the pinhole was selected in such a way that the radius (or axis) of the first minimum circular (or elliptical) fringe ranges from 0.4 mm to 1.5 mm in order to keep the second-order approach accurate. Table 2 shows the parameters of the 6 lenses.
Fig. 10 shows a superposition of interferograms for lens L3 (without cylindrical power) and for only one position of the pinhole for the whole mapping.
Fig. 11A to F show the corresponding addition (first row) and cylinder (second row) maps obtained from the two different devices (first column for the PDI and second one for the Rotlex Class Plus lens mapper). Here, we must point out that the laboratory setup is not yet automatic; therefore, each point is selected by hand. Therefore, maps are drawn from approximately 300 measurement points in a square grid. For an automatic device, this number and position can be chosen by the user, and therefore spatial resolution can be dramatically improved as well speed up the process but this goes beyond the scope of this work.
The mapping of lenses with addition smaller than 2.5D could be done with only one differential configuration. The other lenses needed two different positions of the pinhole to ensure the size of the major axis of the ellipses smaller than 1.5 mm. Lenses L3 and L4 have the same dioptric power, but they are from different manufactures. It is clear that they have different addition and cylinder distributions.
The slight differences come from the smaller number used here by the PDI and the consequent differences in the interpolation of points to find the isolevel lines by the software.
Finally, another advantage of the device is the possibility of isolating a point or a set of points where the local dioptric values want to be evaluated. Therefore, we will show how we can find slight differences in the designs between two lenses with the same dioptric power (S = 0.25D, Add = 2.25D) and the same mounting height manufactured by different lens makers by analyzing the differences along the corridor.
For a given height of the lens, we perform a horizontal scan of the local dioptric powers and we choose the point where the cylindrical power minus the prescribed cylindrical power reaches a minimum. In Fig. 12, we show a scan with the corresponding measured cylinder values for lens L4 at a height −11 mm having as origin of coordinates the fitting cross. The point with coordinates (−1mm, −11mm) is chosen as a corridor point.
After choosing all the points of the corridor in both lenses, we plotted in Fig. 13 addition profiles and the differences between makers arise like slight differences in the slope of the addition.
We propose here a modified PDI as a lens mapper of PALs. The experimental setup has only one movable part that allows to measure dioptric powers ranging from −10.00D to +10.00D, choose the point or the set of points on the lens where the local spherocylindrical power is going to be measured and select the radius of the measurement region between 0.4 mm and 1.5 mm. This ensures that the local approximation of a second order wavefront works with an accuracy better than 0.1D.
The possibility to measure the local dioptric matrix in one point or a set of desired points all over the lens allows to analyze or compare lens designs in the regions of interest.
Finally, it is worthwhile to say that the axial movement of the pinhole allows considering bigger regions of interest within which higher order aberrations of the wavefront can be measured, but this will be dealt with in a future work.
Facultad de Física
Universidade de Santiago de Compostela
Campus Vida, 15782
Santiago de Compostela
We thank INDO LENS GROUP S.L.U for let us measure the lenses with Rotlex Class Plus. This work was supported by the Spanish Ministerio de Educacion y Ciencia grant FIS2010–16753, the FEDER and AiF, grant FKZ: 1782X07.