The Effect of Adding Aspheric Surfaces to a Fixed Focal Length Lens System

THE EFFECT OF ADDING ASPHERIC SURFACES TO A FIXED FOCAL LENGTH LENS SYSTEM

Rhiannon Katarina Jenkins

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A Report Submitted to the Faculty of OPTICAL SCIENCES (GRADUATE) In Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCES In the Graduate College UNIVERSITY OF ARIZONA 2016

ACKNOWLEDGEMENTS

I would like to thank Jeremy Govier for his help with designing these lenses and being a great support throughout the duration of this project, Jose Sasian for his help in structuring the written project and discussing the project with me and to Jim Schwiegerling for being a part of the final committee. I am greatly indebted to Edmund Optics Inc. for funding my Masters degree at the University of Arizona. I would especially like to thank Robert Edmund for first suggesting that I could move out to the US to undertake this course.

TABLE OF CONTENTS

LIST OF FIGURES………………………………………………………………………………………….…..3 ABSTRACT…………………………………………………………………………………………………….….5 1. THE OPTICAL TRANSFER FUNCTION………………………………………………………….…6 History of the OTF…………………………………………………………………………………….…6 Concepts of the OTF…………………………………………………………………………………...9 Linear Shift Invariant Approach to the OTF……………………………………………….11 Tolerancing Using the MTF……………………………………………………………………….12 2. ASPHERIC LENSES…………………………………………………………………………………..…14 History of Aspheric Lenses………………………………………………………………………..14 Describing Aspheric Surfaces…………………………………………………………………….17 Manufacturing Aspheres……………………………………………………………….………….18 3. RESULTS……………………………………………………………………………………………………22 The Original Lens with all Spherical Surfaces…………………………………………….22 Re-Design Using a Single Aspheric Surface………………………………………..………23 Re-Design Using Two or More Aspheric Surfaces………………………………………31 4. CONCLUSION…………………………………………………………………………………………….33 APPENDIX 1……………………………………………………………………………………………………34 REFERENCES…………………………………………………………………………………………………..41

LIST OF FIGURES

Figure 1.1 1951 USAF resolution test target imaged with progressively worsening resolution. (6)

Figure 1.2 Representation showing an imaging system forming a sinusoidal image. (6)

Figure 1.3 Spread of OTF data measured for a 50mm PCX lens by 13 separate laboratories. (7)

Figure 1.4 Schematic of the linear shift invariant system analysis of an optical system. (8)

Figure 1.5 Illustration of the difference between square and sinusoidal spatial frequencies. (9)

Figure 1.6 The MTF for a 12.5mm focal length micro video lens calculated in Zemax. (10)

Figure 1.7 A defocused image of a star pattern with visible spurious resolution. (10)

Figure 1.8 Diagram depicting how lower resolution results in spurious resolution. (11)

Figure 1.9 The MTF for a defocused 12.5mm focal length micro video lens calculated in Zemax, with a discontinuity at around 275cycles/mm. It would display spurious resolution in the approximate range 275-500cycles/mm. (11)

Figure 1.10 Convolution theorem. (12)

Figure 2.1 Depiction of a lens displaying spherical aberration, on the left and a corrected asphere on the right. (14)

Figure 2.2 Example of a Cartesian oval. (15)

Figure 2.3 Lens design #4 from Sloan and Hopkins 1967 paper. (16)

Figure 2.4 Comparison of Double Gauss lens performance with variation in asphere placement. (17)

Figure 2.5 Table of conics, Cyclopaedia 1728. (17)

Figure 2.6 Descartes’ aspheric grinding machine (1638). (19)

Figure 2.7 Active fluid jet polishing by OptoTech. (19)

Figure 2.8 Hybrid moulding technique as utilised by Edmund Optics Inc. (20)

Figure 3.1 The original #59-870 with all spherical surfaces. (22)

Figure 3.2 Lens data for the #59-870. (22)

Figure 3.3 MTF for #59-870. (23)

Figure 3.4 Design #1 with the highlighted surface aspherized. (23)

Figure 3.5 MTF data for Design #1. (24)

Figure 3.6 Design #2 with the highlighted surface aspherized. (24)

Figure 3.7 MTF data for Design #2. (24)

Figure 3.8 Design #3 with the highlighted surface aspherized. (25)

Figure 3.9 MTF data for Design #3. (25)

Figure 3.10 Design #4 with the highlighted surface aspherized. (25)

Figure 3.11 MTF data for Design #4. (26)

Figure 3.12 Design #5 with the highlighted surface aspherized. (26)

Figure 3.13 MTF data for Design #5. (26)

Figure 3.14 Design #6 with the highlighted surface aspherized. (27)

Figure 3.15 MTF data for Design #6. (27)

Figure 3.16 Design #7 with the highlighted surface aspherized, the original design is on the left and the re-sized version on the right. (27)

Figure 3.17 MTF data for Design #7. (28)

Figure 3.18 Design #8 with the highlighted surface aspherized. (28)

Figure 3.19 MTF data for Design #8. (28)

Figure 3.20 Design #9 with the highlighted surface aspherized. (29)

Figure 3.21 MTF data for Design #9. (29)

Figure 3.22 Design #10 with the highlighted surface aspherized. (29)

Figure 3.23 MTF data for Design #10. (30)

Figure 3.24 Design #11 with the highlighted surface aspherized. (30)

Figure 3.25 MTF data for Design #11. (30)

Figure 3.26 Design #12 with the highlighted surface aspherized. (31)

Figure 3.27 MTF data for Design #12. (31)

Figure 3.28 Table showing the MTF performance of all designs, listing the aspheric deviation in µm for all aspherized surfaces. (32)

ABSTRACT

In imaging applications there is a growing demand for higher performance imaging lenses that are able to meet the demands of high resolution sensors. MTF is an important performance metric used to describe the performance of an imaging lens. MTF is especially important in the design phase, since optical design programs such as Zemax can quickly calculate the theoretical MTF in real time as the design is progressing and many designs will use MTF as a criteria for specifying the final design. Aspheric surfaces are used to reduce optical aberrations in optical systems and are, if carefully designed, able to correct the optical performance of an optical system, pushing the performance closer to the diffraction limit.

This project is concerned with the redesign of a 16mm fixed focal length lens that uses all spherical surfaces, to accommodate one or more aspheric surfaces and investigating the effect that this has on the MTF performance of the lens with the aim of producing a design that has an improved optical design and is suitable for manufacture using industry standard machinery.

1. THE OPTICAL TRANSFER FUNCTION

‘The Optical Transfer Function (OTF) is the frequency response, in terms of spatial frequency, of an optical system to sinusoidal distributions of light intensity in the object plane; the OTF is the amplitude and phase in the image relative to the amplitude and phase in the object as a function of frequency, when the system is assumed to respond linearly and to be space invariant.’ [1]

History of the OTF

Before the introduction of the OTF as a standardized way to define the resolution and contrast of an imaging system, resolution was thought of and tested in terms of blur circles and bar charts, the most famous example of which is the 1951 USAF resolution test chart.

Figure 1.1 1951 USAF resolution test target imaged with progressively worsening resolution [2].

Optical design at this time depended solely on geometrical ray tracing and even after the development of wave theory, it was not immediately implemented into design techniques. In fact modern computational ray-tracing programs still use geometrical calculations as the main basis of all their algorithms. However, it has long been known that traditional geometric ray tracing is an approximation that does not take diffraction into account, particularly when the aberrations are small and the lens is diffraction limited [3]. Imaging systems output sinusoidal intensity patterns, such as the distinctive Airy disc that is formed when an optical system images a point source [4, 5].

Figure 1.2 Representation showing an imaging system forming a sinusoidal image [6].

Calculation of the OTF requires consideration of the shape of the wavefront at the exit pupil, which must be done using wave theory analysis of the system. However, in the 19 th century, mathematical methods were such that although a number of optical systems had been worked out using wave theory and were found to be in accordance with observation, this involved lengthy integration and

was a slow process. Although Fourier published his breakthrough paper as early as 1807 [7], in which he modelled all functions using trigonometric series and Gauss demonstrated the first fast Fourier transform in 1805 [8], it was only in 1946 that Duffieux first demonstrated the use of the convolution theorem and the point spread function to calculate the OTF and his book on the subject was widely used as a practical source book in this field for many years [9].

Another important advance occurred in 1938 when Friesser suggested that instead of a bar chart, the object target should have a sinusoidal grey scale profile, instead of an abrupt black and white one, so that the form of the image would have the same form as the object and this is the concept that is still used in the OTF today [10].

In the 1950s the OTF was not yet in common use, but it was during this decade that instrumentation for measuring the OTF was greatly advanced. By the end of the decade 2 main methods for OTF measurement were considered the most useful to the industry. One method was to incoherently illuminate a sinusoidal object with a known variable spatial frequency and constant contrast. The object was brought into focus by the test lens and the contrast was measured in the image plane using a narrow slit and a photocell. The second method used a very similar setup, but the light was passed through the system in the reverse direction – the slit was illuminated and followed by a sinusoidal mask that chopped the image. Again, the result was detected by a photocell. Experimental complications arose using this method in the construction of truly sinusoidal object, including a test at 0 cycles/mm and measuring at low light levels, but also independent of room lighting. Due to these limitations, measuring the OTF was an inconsistent process and there were several attempts to standardize by taking a single test lens and having multiple laboratories measure the OTF. However, in e.g. 1968, the results were still too varied to conclude that it was sensible to use MTF as a standard measure of lens performance [11]. However, there was a move at this time to have a set of standard reference lenses with known OTFs so that the OTF could be used as a relative measure of performance if not an absolute one.

Figure 1.3 Spread of OTF data measured for a 50mm PCX lens by 13 separate laboratories [12].

Subsequent advances in computing and lens design software as well as further improvements in streamlining the data heavy Fourier transform techniques using new Fast Fourier Transform techniques meant that in the 70s and 80s the OTF was being used in the design and prototype stage

of lens design to specify optical performance [13]. The National Bureau of Standards also established an optical measurement service to provide standardised OTF measurements on lenses provided to them [14]. This allowed companies without the resources to do their own testing to get accurate test data. This also provided a resource for companies that did their own testing to check their measurements.

Subsequent advances in methodology were focused on increasing the speed at which OTF calculations could be performed. In 1976 Kintner and Sillitto published their method of calculating the OTF [15]. At the time there were two main methods employed in calculating the OTF, the first was the double- transform method, in which the pupil function undergoes an inverse Fourier transform to find the amplitude spread function, the modulus square of which is the intensity point spread function. The point spread function is then Fourier transformed into the OTF. The second method is the autocorrelation method, which allows the OTF to be calculated directly from the pupil function. These processes are outlined in Figure 1.4. Both methods required numerical integration in all but the simplest cases.

Figure 1.4 Schematic of the linear shift invariant system analysis of an optical system [16].

They developed a method to avoid the computationally heavy numeric integration, by expressing the pupil function as a series of Zernike polynomials. This allowed the authors to express the amplitude spread function in series form via a straightforward Fourier transform and then by cross-multiplying it by its complex conjugate, they obtain the PSF as a double series that can be converted into a single series via a summation and then Fourier transformed to obtain the OTF as a series. The cross multiplication step was further simplified in a later paper that outlined an analytic solution using a recurrence algorithm to calculate the coefficients of the cross-multiplied series [17]. A fully analytic solution to the Zernike expansion of the OTF in terms of the expansion coefficients of the pupil function was provided by Janssen [18] and an even faster method using an expansion of the OTF, where the coefficients are directly related to the wavefront errors and apodization parameters [19]. Being able to use optical parameters in a formula for calculating the OTF gives insight into the interplay between OTF, apodization and wavefront. It can allow optical designers to try and maximise or minimise particular coefficients in order to optimise OTF performance.

Concepts of the OTF

The OTF (and the MTF) both plot the spatial frequency in the object plane versus the contrast in the image plane. If you have a particular spatial frequency in the object plane, you can read off the contrast that would be achieved in the image plane. As discussed earlier, we know that any imaging system will output sinusoidal intensity patterns and that the OTF and the MTF both plot sinusoidal spatial frequencies.

Figure 1.5 Illustration of the difference between square and sinusoidal spatial frequencies [20].

Spatial frequency is defined as the number of cycles per mm, where a cycle is a line pair that consists of one white line and one black line in the square wave regime, or a wavelength in the sinusoidal regime.

Contrast is defined as:

And refers to the range of recognizeable shades of grey, from white to black, that is visible in the image. Due to the similarity between this definition and the definition of amplitude modulation in communication theory, this is often referred to as modulation contrast.

The PSF describes the system response to a point source of light, or the image of that point source. The image can never be as small or precise as the point itself since every physical optical system introduces inaccuracies or aberrations. These can result from diffraction, imperfections on optical surfaces, imperfections within the glass substrate and aberrations such as defocus, spherical, coma, astigmatism, etc. A well corrected system will produce an airy disc pattern from a point source of light and in a diffraction limited lens, (a lens where diffraction is the dominant aberration), the spread of the PSF is approximated by the diameter of the first null of the Airy disc.

D = 2.44 λ f/#

When the system is imaging a multiple frequency target, as it does in an OTF measurement, the ability of the lens to efficiently and accurately image the target decreases as the frequency of the target increases. Even in a “perfect” lens with no other aberrations, diffraction occurs and limits the accuracy. This results in a decrease in contrast as the spatial frequency increases. It is common practice to plot the diffraction limit on an MTF curve to see how close to ideal the lens is performing, (shown in black in Figure 1.6, in comparison to the expected performance of the lens). The OTF and the MTF allow you to read off the contrast at which your lens can image a particular frequency.

Figure 1.6 The MTF for a 12.5mm focal length micro video lens calculated in Zemax.

When the contrast, or modulus of the OTF, reaches zero, this is commonly known as the cut-off frequency, ξ c, and in a diffraction limited system, is defined as:

ξc = (λ f/#)⁻¹

At this frequency the lens can no longer resolve the dark and light parts of the target and the output is then a uniform grey in the image.

One interesting feature of the MTF is that you can track the phenomenon of spurious resolution, where resolution becomes inverted and dark lines appear light and vice versa. In Figure 1.7, it is possible to see this effect; there is a grey circle inside the star pattern, where the contrast has reached zero and then at higher resolution, the star pattern is visible again, but with the black and white lines inverted.

Figure 1.7 A defocused image of a star pattern with visible spurious resolution [21].

It is possible to understand how spurious resolution occurs using a simple diagram as seen in Figure 1.8. Essentially, as the system resolution decreases, the dark bars become wider in the image until they overlap. The overlapping parts are darker than the original bars, because two bars are contributing to that part of the image and the overlap occurs in the spaces between the bars, which should be white.

Figure 1.8 Diagram depicting how lower resolution results in spurious resolution [22].

In terms of the OTF, spurious resolution occurs when the OTF becomes negative. In the MTF we would see a discontinuity in the curve as it reaches zero, as seen in Figure 1.9.

Linear, Shift Invariant System Approach to the OTF

As shown in Figure 1.4, the OTF can be analysed from the perspective of a linear shift invariant, (LSI) system. In mathematics LSI systems are defined as follows; if L{·} is our LSI operator, it will act on a function, f(x) as follows:

L{f(x)} = g(x)

The operator must be both linear:

L{αf₁(x) + βf₂(x)} = αg₁(x) + βg₂(x)

And well as shift invariant:

L{f(x-x₀)} = g(x-x₀)

We can define an impulse response to the system, h(x), as:

L{δ(x)} = h(x)

Where δ(x) is the Dirac delta function. This impulse response function can be used to describe the system using Fourier transform and the convolution theorem. By definition:

f(x) * h(x) = g(x)

Where ‘*’ denotes convolution. If, F(ξ), G(ξ) and H(ξ) are the Fourier transforms of, f(x), g(x) and h(x) respectively, and f(x), g(x) and h(x) are related by convolution as above, then their Fourier transforms are related by simple multiplication:

F(ξ) H(ξ) = G(ξ)

This relationship is depicted in Figure 1.10.

Figure 1.10 Convolution theorem [23].

In a linear optical system, the effect of a lens on a propagating electric field can be described by a Fourier transform on that field. The PSF is the system response to a point source, which is analogous to the impulse response function, h(x), whilst the OTF is analogous to H(ξ). Once the PSF or OTF have been calculated, from the pupil function, as shown in Figure 1.4, the image, which in our analogy would be the output function, g(x), can then be calculated from the object field, f(x), by either convolution between the object field and the PSF or multiplying the Fourier transform of the object field by the OTF and then doing an inverse Fourier transform. Using this theory, the OTF is effectively a complete description of an optical system and can describe the optical response to any object.

Tolerancing Using the MTF

Tolerancing is a very important concept in optical design. An optical designer must take into account the limitations of lens manufacture and assembly in order to ensure that the performance of the finished design is still within the required metrics. Even small changes in the design can potentially result in large changes in the optical performance and so each and every design parameter, (i.e. lens diameter, radius of curvature, thickness, tilt, alignment etc.), needs to have a numerical +/- tolerance that will ensure the overall design remains within the specifications. It is of very little use to have a lens design where the tolerances are so tight that a lot of money is wasted in scrap material because the manufacturer is unable to meet the tolerances. The looser the tolerances can be the cheaper the lens will be for manufacture.

Traditionally, imaging systems would be toleranced using other metrics, such as geometric spot size, and they were also dominated by on-axis metrics, mostly because the calculation time is much faster, but on-axis spot size is not necessarily the best metric for an imaging system [24]. Often off-axis performance is very important for imaging applications and it is important to understand how an imaging system responds to a more complicated input and MTF is much better suited to that.

Optimizing and tolerancing with respect to the MTF is not always so straightforward. As noted above, the calculation time for MTF can be a lot slower than for other metrics despite the many advances in calculation time. If efficient algorithms are used it is feasible to use MTF as the optimization parameter, but it is not always wisest to do so. Especially at the beginning of a design, the MTF can oscillate wildly with the construction parameters and so it is often best used, when the lens is close to the final design stages and is already well corrected. However, this does mean that it is an ideal candidate for tolerancing concerns [25].

Using design software, like Zemax, allows you to tolerance directly using the MTF as the criteria, although it is generally the slowest of the options available and is only really recommended for high performance systems because the MTF may not be computable or meaningful, i.e. if it goes to zero below the chosen analysis frequency. There are several main modes for tolerancing in Zemax. Firstly, sensitivity mode, which allows you to specify the tolerances on any parameters and record the change in performance when those parameters are perturbed within the specified tolerances. Inverse sensitivity lets you specify how much a parameter is allowed to affect the merit function and then Zemax will compute the maximum and minimum allowable values for that parameter. Once the sensitivity analysis is complete and parameter tolerances have been specified, a Monte Carlo analysis is performed in order to simulate the effect of all the parameters varying at once. This gives you a good idea of the potential best, worst and most likely performance you would see for a given set of tolerances.

2. ASPHERIC LENSES

‘Aspheric surfaces, or “aspherics”, are optical surfaces that are neither spherical nor plane and are used in imaging and nonimaging systems. Mathematically an aspheric is generated by the rotation of an axisymmetrical plane curve about its axis. Spherical surfaces are a special case of aspherics in which the rotating curve is a circular arc.’ [26]

History of Aspheric Lenses

Spherical aberration occurs as a natural result of refraction, as described by Snell’s Law, of an EM wave as it passes through a lens with a spherical profile. If we imagine a spherical lens, such as the one pictured in Figure 2.1, the lens has variable thickness across its aperture meaning that some rays will have travelled through more or less glass resulting in different optical path lengths. The contribution from each surface can be calculated straightforwardly [27]. This optical path difference results in different focal lengths for rays passing through different parts of the lens aperture and gives rise to the distinctive caustic effect that is typical of spherical aberration.

Figure 2.3 Depiction of a lens displaying spherical aberration, on the left and a corrected asphere on the right [28].

Spherical aberration leads to spherical lenses having no common point of focus and instead of a perfect focal point, there is a blur circle. This has obvious implications in terms of limiting the accuracy of a lens and being detrimental to that lens’s performance in a given application. One of the common ways to try and correct these aberrations, is to deviate from a spherical surface, creating an aspheric lens.

Although Snell is often considered to have first fully described refraction in 1618, Roshdi Rashed credited Ibn Sahl with the discovery of the laws of refraction back in 964AD, in a manuscript for the court in Baghdad. He describes Sahl calculating the shapes of lenses that would have no geometric aberrations [29]. However early attempts at creating aspheric lenses were thwarted by the limited machinery and technical capabilities available at the time.

Despite understanding the need for aspheric surfaces, the first aspheric lenses were not made until the 17 th century. The earliest attempts are thought to have been made by René Descartes in around 1620, who used Cartesian ovals to form the shapes of his lenses, Figure 2.2. Descartes will have used a pin and string method to construct these ovals, but when Newton studied them beginning in 1664, he derived a more rigorous approach using differential equations.

If the oval has fixed points, P and Q and d(P,S) and d(Q,S) are the distances between P and Q to a variable point, S, then the oval is described as the locus of points that satisfy, d(P,S) + m d(Q,S) = a, where m and a are arbitrary real numbers. Descartes found that his ovals were intrinsically linked to the phenomena of spherical aberration, in that he could use the shapes to create an aplanatic lens, i.e. one with no spherical aberration, by choosing the ratio of distances from P and Q to match the

ratio of sines in Snell’s Law. Furthermore, when a spherical wavefront is incident on a spherical optic, the shape of the caustic that is formed is also a Cartesian oval.

Figure 2.4 Example of a Cartesian oval [30].

The first aspheric lens system, in the form of a telescope, was presented to the Royal Society in 1668 by Francis Smethwick [31]. It consisted of 4 lenses, 3 of which were aspheres and it was generally agreed by the society members present that the telescope outperformed current models, having both a larger angular field of view and a better accuracy. Huygens is also known to have experimented with aspheric lenses in the 1670s.

Around the same time, there were also several models of reflective telescope that were developed with conic mirrors. For instance the 1672 Cassegrain telescope, which has a parabolic primary mirror and a hyperbolic secondary mirror and the Gregorian telescope designed in 1663 and consisted of a parabolic primary mirror and an ellipsoidal secondary mirror [32, 33].

Aspheric surfaces were mainly used for correcting spherical aberration until the late 19 th century after Abbe discovered that they could also correct for astigmatism. People were able to develop methods to balance the correction of astigmatism and spherical aberrations, such as the ‘optical see-saw diagram’ [34].

More complicated aspheric surfaces could only be fabricated later, after the advent of computer assisted surface polishing in the 1970s [35]. Nowadays, aspheric surfaces are designed to correct many aberrations, such as spherical, coma, astigmatism, distortion and chromatic aberrations, with designers approaching the correction inclusively, correcting all aberrations, (including higher order aberrations), at the same time by considering all surfaces simultaneously. Although earlier methods, such as the Simultaneous Multiple Surfaces (or SMS) method, [36] were employed in order to find the best aspheric shape of a surface by solving differential equations, modern algorithms are able to make use of all these previous techniques to allow for optimum speed and better lens performance, especially since these methods do not always account for higher order aberrations.

When designing a compact imaging lens with the intention of using an aspheric surface within that lens there are a number of papers investigating which surface will typically result in optimum performance. Wasserman and Wolf found an aplanatic solution for two adjacent aspheric surfaces within a compound lens when the other surfaces are known, by created a stigmatic system and exactly solved for the sine condition to establish a one-to-one ratio between entrance and exit rays of a system [37]. This was expanded upon by Vaskas, who separated the two aspheric surfaces and had multiple standard surfaces between them [38]. Work by Frank Allen Lucy and Wolf and Preddy, used single aspheric correcting surface at either the front or the back of the system, eliminating axial spherical aberration [39, 40]. Miyamoto further generalised the problem by producing a method for deriving the aspheric form of a surface in an arbitrary position within a compound system [41].

Preferences for aspheric surface location in a compound system were investigated by Sloan and Hopkins, with specific reference to a double gauss lens [42]. They looked in particular at a design with f/2.1, F.L. 100mm and a total field of view of 42˚. They investigated 4 different lenses, firstly an optimized spherical design, secondly one with aspheres on the first and last surfaces, thirdly with aspheres on two surfaces either side of the stop and lastly one with an aspheric correcting plate at the stop.

Figure 2.5 Lens design #4 from Sloan and Hopkins 1967 paper.

They found that the best result was achieved with the correcting plate, but they did not rule out that this could have been due to that being the final lens that they designed and that it therefore profited from the experience they gained working with the previous three lenses. They thought that the third lens in particular had good axial correction and could probably have been improved upon.

In theory only two aspheric surfaces are required for complete correction [43] and conventional wisdom would suggest that one should be at the aperture stop or at a pupil, as this will allow correction of spherical aberration, which is rotationally symmetric and constant across all fields of view. Since the aperture stop location affects all fields of view simultaneously, it is the ideal location to correct that kind of aberration. Non-rotationally symmetric aberrations, such as coma and astigmatism would be best served by an aspheric surface at an intermediate image plane, where the field can be directly mapped onto that surface. However, there are problems with both of these ideal placements; the stop is not always fixed in a system and there is not always an intermediate image plane to take advantage of, (also if there is an intermediate image that can also be problematic, since any particles or defects on a surface at an image plane would be clearly visible in the final image). Instead, we can think of the surfaces close to the stop, close to the image plane and close to the entrance pupil as good candidate locations for an aspheric surface.

However, each lens presents its own distinct set of variables and the conventional wisdom may not always provide the best solutions. This can be seen in particular with the gauss lens in a paper by Scott Sparrold, where after evaluating multiple different design options, (Figure 2.4), the best solution was found utilising a double asphere on the final lens.

This is why modern optical design programmes such as Zemax include a ‘Find Best Asphere’ tool, which allows an optical designer to quickly test their optical system with a particular merit function. The compound lens system is scanned, placing the asphere on each surface and calculating the potential merit function that can be achieved. The user is then able to choose to keep the best aspherized design. However, even so, the designs found by the software may not be suitable for manufacture

and might also be improved upon significantly with further optimization, so it still requires a knowledgeable designer to make best use of these software tools.

Figure 2.6 Comparison of Double Gauss lens performance with variation in asphere placement [44].

Describing Aspheric surfaces

A spherical surface is defined by a single parameter, its radius of curvature. An asphere is loosely defined as a surface that departs from the standard spherical surface. The simplest of these (and the earliest to be manufactured were conic sections.

Figure 2.7 Table of conics, Cyclopaedia 1728 [45].

The equation for a conic section with its apex at the origin and tangent to the y axis is given by:

y² - 2Rx + (k + 1)x² = 0

Where R is the radius of curvature and k is the conic constant. In general the conic constant, which is the negative square of the eccentricity of the conic, can be used to determine the type of conic section; if k>0 the section is an oblate elliptical one, if k=0 you have a spherical surface, if 0

Because of the way they are manufactured, optical surface are more usually defined using the sagitta, (sag, Z(s)), which can be thought of as a measure of the glass removed from a reference flat surface. An even asphere is defined as follows:

Here r is the radial distance from the optical axis and A₂n are the higher order even aspheric coefficients.

There are several other ways to define an aspheric surface, such as an odd asphere, which uses the same basic formula, but uses the odd aspheric coefficients or a Forbes asphere, which can be described as follows:

Forbes aspheres use a conicoid as the base surface with r max as the maximum semi aperture, u is r/r max, Fˢᵗʳm are the Forbes strong coefficients and Qᶜᵒⁿm(u²) are a set of orthogonal polynomials. Forbes aspheres are specified with manufacturing in mind and are particularly useful for steep and challenging aspheres [46].

Manufacturing Aspheres

Knowing that aspheric surfaces can improve an optical system and being able to describe those surfaces is only part of what is necessary for using them in real life applications. Manufacturing aspheric surfaces was by far the biggest stumbling block when they were first being used in Cassegrain designs. Early grinding machines were unable to repeatedly achieve precision and so results tended to vary wildly. Descartes designed a system that allowed him to grind and polish a parabolic surface. Contact is made along the meridian line and the leverage system denotes a parabolic pathway. However, the system is limited by the accuracy of the bearings and machines like this one do not tend to produce good aspheres dependably.

Figure 2.8 Descartes’ aspheric grinding machine (1638) [47].

This technique is not dissimilar to modern day aspherical polishing, at least in terms of the overall idea. Modern polishing is clearly more advanced, but it also uses the same kind of single contact point polishing, where the contact point is moved on the desired aspheric pathway. Nowadays, the polishing tool is controlled by a computer and its location is automatically updated in real time, to polish away high areas that need to be removed.

Figure 2.9 Active fluid jet polishing by OptoTech [48].

This kind of technique does require each lens to be polished individually, but as opposed to moulding techniques, it only requires standard tooling, making it the best technique for low volume or prototype quantities. It can also achieve very high accuracies of up to λ/20.

Moulding techniques come in two basic types; moulding and hybrid moulding. Moulding can be done using a plastic or a glass substrate, (although glass moulding requires much higher temperatures), and both require a mould to be specially tooled from a durable material that will not change its shape during the moulding process. In the glass version, the glass is heated to a high temperature so that it becomes malleable and is then pressed into the mould. For plastic, specially treated molten plastic is fed into the mould and allowed to cool. The mould often needs to be made to a high standard of accuracy, taking into account the changes that the glass or plastic will undergo when cooled and can be expensive and difficult to make. However, once you have the mould the subsequent cost of making the aspheric lenses is more negligible, making this a great technique for high volume.

Hybrid moulding also requires a (usually) diamond turned mould, but is used with a spherical glass surface. A photopolymer is added to the mould and the spherical surfaced is pressed into it. After curing the once spherical surface has an aspheric overlay made from a polymer.

Figure 2.10 Hybrid moulding technique as utilised by Edmund Optics Inc.

Although manufacturing has come a long way, there are still certain surface types that are much easier to manufacture than others and certain guidelines should be taken into account by the designer when designing a system that includes aspheric surfaces so as to make sure that their design is feasible [49].

1. Convex or concave: many polishing machines capable of figuring aspheric surfaces have a minimum radius of curvature for concave surfaces since the polishing tool itself has a radius of curvature that must be less than the desired curvature of the workpiece. Convex surfaces do not have the same restrictions. As a rule of thumb, if the radius of curvature is less than 35mm, you should preferably aspherize the convex surface. 2. Conic section or higher order asphere: whether to choose a conic section is a question of three main variables. Firstly performance, which can be measured by a metric such as MTF or spot

diagrams, depending on the intended application. Secondly manufacturing cost, it does not necessarily follow that the asphere will be more expensive to manufacture because of the complexity of the sag equation. Instead it should be considered in terms of the aspheric sag departure from the spherical form; the greater the departure, the longer it will take to polish the spherical blank and thus the cost of manufacture will increase. Thirdly is the cost of testing the surface; a conic is much easier to test because it can be tested at the natural conic foci, but an asphere usually requires the manufacture of an aspheric null plate or a computer generated hologram. There have been improvements in interferometer testing, which allows for more precise and flexible characterisation of aspheric surfaces [50]. However one should always be aware of the availability and cost of such testing capabilities. 3. Aspheric slope: the more an asphere departs from a best fit sphere, the more challenging it is to manufacture. Surfaces with steep slope changes are also tough to characterise because the interferometer requires enough dynamic range to acquire continuous fringes. As a rough guideline, if aspheric departure is greater than 2µm per mm of the aperture, it will be difficult to manufacture as the polishing footprint becomes small and it is difficult to keep the surface smooth resulting in a lens that is very sensitive to alignment during testing. 4. Inflection point: an inflection point is a point where the surface moves from concave to convex, it is most obviously illustrated by gullwing lenses. Lenses with inflection points make form correction more difficult. Unless the lens being designed is a true gullwing lens and the inflection point represents a small departure, it should be possible to remove the inflection point without adversely affecting the performance too much. 5. Edge thickness: Since aspheric lenses spend a larger portion of time being polished than standard spherical lenses, it is advisable to have a 4-5mm polishing footprint outside of the clear aperture. 6. Asphere size: it is very important to know the lens diameter limitations of the polishing machine that you will be using. Exact limitations depend on the specific machinery being used and upper limitations can be anywhere from 90mm to 240mm. It is also important to note that testing equipment will also have limitations on diameter and even if you can make a larger asphere, if you cannot test it, you either have a worthless untested optic or you may incur additional costs having it tested by a third party. 7. Tolerancing: a designer can make a big impact on the cost of manufacturing by taking the surface figure accuracy into account during the design phase. If a 1µm, or looser, tolerance can be achieved then the surface can also be tested simply with a profilometer.

3. RESULTS

The purpose of this section is to take a compound lens that contains only spherical surfaces and improve its MTF performance by changing one or more of the standard spherical surfaces to an aspheric surface.

The Original Lens with all Spherical Surfaces

For the purpose of this project I will be using a compound fixed focal length lens from Edmund Optics, part number #59-870. It has a 16mm focal length, is designed for a 2/3” sensor format and has a maximum aperture of f/1.4. The design includes five singlet lenses and one cemented doublet, meaning there are 12 possible candidate surfaces for aspherization, (assuming we do not include the central surface of the cemented doublet), the general form of the design can be seen in Figure 3.1.

Figure 3.11 The original #59-870 with all spherical surfaces.

The setup in this case uses a working distance of 300mm and fields at 14.5° and 19.5°. For completeness, I am including a complete lens specification for the original design, Figure 3.2, but for the aspheric designs, please see Appendix 1 for lens design data.

Figure 3.12 Lens data for the #59-870.

As discussed in the first chapter of this report, OTF data (or more often MTF data) is an important metric for specifying lens performance and I will be using MTF plots, as calculated from Zemax to compare all the designs.

Figure 3.13 MTF for #59-870.

As can be seen in Figure 3.3 at 210 cycles per mm, (which I will be using as a benchmark throughout), this lens has an MTF of 0.285 on axis compared to a diffraction limit of 0.546, which means that there is substantial room for improvement with this lens. The Seidel coefficients suggest that distortion and field curvature are the most dominant aberrations, (Seidel coefficients of 0.030 and 0.011 respectively). This would suggest that an aspheric surface close to the image plane would be the best option. However there is also a significant amount of spherical aberrations, (0.0027), meaning that an aspheric surface close to the stop or close to the pupil should also significantly improve the performance.

Re-Design Using a single Aspheric Surface

First I will be adding a single aspheric surface to the design to see how each surface performs individually. I will work through each surface methodically, looking at the pros and cons of placing an asphere at that surface.

The first surface I used in Design #1 was the one at entrance pupil, which is denoted as surface 2 by Zemax. This is expected to be a good choice for this system as it is close to the entrance pupil.

Figure 3.14 Design #1 with the highlighted surface aspherized.

Figure 3.15 MTF data for Design #1.

As expected we see a much improved MTF with a value of 0.4508 at 210 cycles per mm.

Figure 3.16 Design #2 with the highlighted surface aspherized.

Figure 3.17 MTF data for Design #2.

Design #2 is aspherized on the concave side of the same lens as Design #1, or surface 3 in the Zemax design. In terms of manufacturability, this design is less favourable because the radius of curvature of the concave surface at 12.756mm is significantly less than the 35mm that acts as a guideline for manufacturability, meaning that it will be difficult to produce. The MTF is 0.3705 at 210 cycles per mm on axis and the off axis performance is also slightly worse than for Design #1.

Design #3 has the aspheric surface somewhere between the stop and the entrance aperture and is not expected to have a very good performance.

Figure 3.18 Design #3 with the highlighted surface aspherized.

Figure 3.19 MTF data for Design #3.

As expected, it has an MTF value of 0.4069 at 210 cycles per mm, which is somewhat worse than Design #1. So although this would be one of the easier surfaces to aspherize, we do not get the same performance improvement as for some other surfaces.

Figure 3.20 Design #4 with the highlighted surface aspherized.

Figure 3.21 MTF data for Design #4.

For Design #4 the on-axis MTF value of 0.4057 is achievable, which is again lower than for Design #1 that uses a surface close to the entrance pupil. However with both Design #3 and Design #4, the off axis performance is a little better.

Figure 3.22 Design #5 with the highlighted surface aspherized.

Figure 3.23 MTF data for Design #5.

Design #5 uses a surface approximately midway between the entrance pupil and the stop and is not expected to have a good performance, but surprisingly the MTF values of 0.4379 and 0.2548 on- and off-axis respectively is one of the more balanced results. It is also a good candidate surface in terms of manufacturability.

Figure 3.24 Design #6 with the highlighted surface aspherized.

Figure 3.25 MTF data for Design #6.

The MTF value of 0.4509 on axis, is one of the best, but off axis it performs particularly badly. Also designing with this surface is challenging and can yield some wild results with extremely large deviations, so extra restrictions need to be put in place. Also it is a concave surface with a radius of curvature of 38.5, which is only slightly over the preferred 35mm minimum.

Figure 3.26 Design #7 with the highlighted surface aspherized, the original design is on the left and the re-sized version on the right.

This design has an inflection in the surface, but since the clear aperture is much smaller than the size of the lens, I can simply reduce the lens size and cut out the part of the lens which would be expensive to manufacture. This step not only removes the inflection, but reduces the aspheric departure from 804.7µm to 0.5012µm, so it is a very significant cost saving exercise.

Figure 3.27 MTF data for Design #7.

The MTF value of 0.4464 on axis is pretty good, but it is out performed by other surfaces off-axis. It is a concave surface, with a radius of curvature of 38.07mm, so the machining this surface would certainly be possible, even if it is a little close to the 35mm limit. Exactly how difficult or easy this surface is to manufacture will depend on the limitations of the available machinery.

Figure 3.28 Design #8 with the highlighted surface aspherized.

Figure 3.29 MTF data for Design #8.

Design #8 has a relatively low MTF value of 0.3947 on axis, but does perform better off-axis. However the lens surface is concave with a radius of curvature of only 10.67mm and so is well below the recommended minimum. Also the aspheric deviation for this lens is higher than average at 14.33µm.

Figure 3.30 Design #9 with the highlighted surface aspherized.

Figure 3.31 MTF data for Design #9.

Design #9 uses the surface closest to the stop, so we would expect good correction using this surface, but the MTF value is only 0.4296, which is lower than what can be achieved working with some of the other surfaces. Also the aspheric deviation is 48.10µm, so it would be relatively expensive to machine this surface.

Figure 3.32 Design #10 with the highlighted surface aspherized.

Figure 3.33 MTF data for Design #10.

The MTF value of 0.4040 for Design #10 is not particularly high, but interestingly the off axis performance is very good at 0.2862.

Figure 3.34 Design #11 with the highlighted surface aspherized.

Figure 3.35 MTF data for Design #11.

Design #11 has a relatively high MTF value of 0.4443 and a very good off axis performance of 0.2851. The aspheric deviation is a little high at 18.85µm though.

Figure 3.36 Design #12 with the highlighted surface aspherized.

Figure 3.37 MTF data for Design #12.

Design #12 is expected to have a good performance since surface 12 is aspherized, which is the surface closest to the image plane. The MTF value of 0.4300 on axis is not bad, but it also has the best off axis performance of 0.2882 at full field.

Re-Design Using Two or More Aspheric Surfaces

After analysing the single surface results, there are 4 candidate surfaces of particular interest, giving good results and being manufacturable, namely surfaces 1, 5, 11 and 12, (or 2, 6, 15 and 16 in the Zemax design). So now I will try redesigning this compound lens with 2, 3 and 4 aspheric surfaces using a combination of these surfaces. It is not likely that there is much benefit in designing a lens with aspheric surfaces on both surface 11 and 12 since they are on the same element and would be expected to work against each other.

The best two surface design was found working with surface 1 and surface 11, (Design #16), but interestingly it did not out-perform Design #11, in which only surface 11 was aspherized. However, that design did have 18.85µm of aspheric deviation, so would be more expensive to manufacture. In general the 2-surface designs are well balanced with better performance on and off axis.

Most of the comparable designs using either surface 11 or surface 12 do better using surface 11. Although the single surface design for surface 11, (Design #11), had a large aspheric deviation, in all the multi surface designs that use surface 11, the aspheric deviation is a lot smaller. In fact there is only one design that has a large surface deviation, i.e. Design #20 has a 48.52µm deviation on surface 5. All the other multi-surface designs use viable surfaces which would be relatively cheap to manufacture.

The best 3 surface design is Design #18, but it is pretty comparable to Design #16, so there is not that much benefit in adding the 3 rd asphere.

I also attempted a 4-surface design, (Design #20), to see whether more surfaces could get a better result, but it does not out-perform the best 3-surface design, would be expensive because it requires 4 aspheres and also one of those aspheres has a large aspheric deviation.

Performance of each design can be compared in the table in Figure 3.28.

Figure 3.38 Table showing the MTF performance of all designs, listing the aspheric deviation in µm for all aspherized surfaces.

Which design is preferred will depend somewhat on the application and specific performance requirements. For example, if an on-axis performance of 0.4 and an off-axis of 0.25 is required, Design #5 would be a good solution, but if a better off-axis is required, Design #12 would be a good one to consider.

4. CONCLUSION

In general, the aspheric designs perform much better than the original spherical version. From looking at the Seidel coefficients, I expected that aspherizing a surface next to the image plane would provide the best correction for the system. Although surface 12, which is closest to the image plane, did give relatively good performance, surface 11 generally performed better, not just in the single surface design, but also in the multi-surface designs. I also expected surfaces close to the stop to provide good results, but surfaces 8 and 9 did not give a good lens performance. The other surface expected to give good results was surface 1, next to the entrance pupil and this surface did provide a good overall performance as well as being easily manufacturable, so I made that part of my multi-surface investigation. I did not expect to get such good results using surface 5, which is not in a good position for correcting the aberrrations.

There are several limitations to the work conducted for this report. Most problematic is that these designs were created with a view to improving the overall optical performance, but without a specific metric to aim for, which leaves specifics regarding tolerancing or manufacture of the design somewhat open ended. As discussed at the end of the Results, which design is the most suitable for a particular application would depend on the specific requirements of that application, but a performance comparison for the different surfaces would be the first step to deciding how to proceed with manufacturing that lens. The next step would be detailed tolerancing of the most promising designs, which would be dependent on the system requirements. This would be followed by an in depth costing analysis, which would depend on the chosen design, the available manufacturing techniques and the quantities to be made.

APPENDIX 1

1. Design #1

2. Design #2

3. Design #3

4. Design #4

5. Design #5

6. Design #6

7. Design #7

8. Design #8

9. Design #9

10. Design #10

11. Design #11

12. Design #12

13. Design #13

14. Design #14

15. Design #15

16. Design #16

17. Design #17

18. Design #18

19. Design #19

20. Design #20

REFERENCES

[1] Introduction to the Optical Transfer Function, Charles S. Williams, Orville A. Becklund, (2002)

[2] Multimodal instrument for high-sensitivity autofluorescence and spectral optical coherence tomography of the human eye fundus, K Komar, P Stremplewski, M Motoczyńska, M Szkulmowski, M Wojtkowski, Biomedical Optics Express, Vol. 4, Issue 11, pp. 2683-2695, (2013)

[3] On a Comparison between Wave Optics and Geometrical Optics by Using Fourier Analysis. I. General Theory, K Miyamoto, Journal of the Optical Society of America, Vol. 48, no. 1 (1958)

[4] On the Diffraction of an Object Glass with Circular Aperture, G. B. Airy, Trans. Cambridge Philos. Soc. 5, 283 (1835)

[5] Geometrical-optical Treatment of Frequency Response, H. H. Hopkins, Proceedings of the Physical Society. Section B, 12, Volume 70, Issue 12 (1957)

[6] Image hosted by http://www.edmundoptics.com/resources/application- notes/optics/introduction-to-modulation-transfer-function/

[7] Mémoire sur la propagation de la chaleur dans les corps solides, Joseph Fourier, pp215-221 (1807)

[8] Gauss and the history of the fast Fourier transform, Michael T. Heideman, Don H. Johnson and C. Sidney Burrus, Archive for History of Exact Sciences, Vol. 34, No. 3, pp. 265-277 (1985)

[9] L'Integrale de Fourier et ses Applications a l'Optique, P. M. Duffieux, Rennes: S.A. des Imprimeries Oberthur (1946)

[10] Photographic Resolution of Lenticular Films, H. Friesser, Zeitschrift für Wissenschaftliche Photographie 37, 261 (1938)

[11] The Calculation And Measurement Of The Optical Transfer Function (OTF), Robert E. Hopkins; David Dutton; Gary Noyes, SPIE 0013, Modulation Transfer Function (1968)

[12] Status of the OTF in 1970, L. R. Baker, Optica Acta, Vol. 18, no. 2, 81-92 (1971)

[13] Optical Transfer Function calculation by Winograd’s fast Fourier transform, D Heshmaty-Manesh, S. C. Tam, Applied Optics Vol. 21, Issue 18 (1982)

[14] The case for the pupil function, R. E. Swing, SPIE 0046 Image assessment and Specification, 104 (1974)

[15] A new ‘analytic’ method for computing the optical transfer function, E. C. Kintner, R. M. Sillitto, Optica Acta, Vol. 23, no. 8, 607-619 (1976)

[16] Parametric analysis of the effect of scattered light upon the modulation transfer function, J. E. Harvey, Optical Engineering 52, 7 (2013)

[17] An analytic recurrence procedure for computing the cross-multiplication coefficients in an analytic OTF method, E. C. Kintner, Optica Acta, 24:12, 1237-1246 (1977)

[18] New analytic results for the Zernike circle polynomials from a basic result in the Nijboer-Zernike diffraction theory, A. J. E. M. Janssen, Journal of the European Optical Society, 6, 11028 (2011)

[19] Relating wavefront error, apodization, and the optical transfer function: on-axis case, J. Schwiegerling, Journal of the Optical Society of America, Vol. 31, no. 11 (2014)

[20] The Mind’s Machine: Foundations of Brain and Behavior, N. V. Watson, S. M. Breedlove, Looseleaf (2012)

[21] Image hosted by http://www.dpreview.com/forums/post/57331531

[22] The Pinhole Camera, M Young, The Physics Teacher 648-655 (1989)

[23] Field Guide to Linear Systems in Optics, J.S. Tyo, A. Alenin, SPIE (2015)

[24] Assessment, Optimization, and Tolerancing Methods for Visual Instruments, A. Deslis, P. Mouroulis, Proceedings of SPIE (2000)

[25] MTF optimization in lens design, M. P. Rimmer, T. J. Bruegge, T. G. Kuper, Proceedings of SPIE (1990)

[26] Aspheric Surfaces, G. Schultz, E. Wolf, Progress in Optics (1988)

[27] Len Design Fundamentals, R. Kingslake, Elsevier (1978)

[28] Image hosted by http://www.edmundoptics.com/resources/application-notes/optics/all-about- aspheric-lenses/

[29] A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses, R. Rashed, Isis: A Journal of the History of Science 81:464-491 (1990)

[30] Image hosted by https://en.wikipedia.org/wiki/Cartesian_oval

[31] An Account of the Invention of Grinding Optick and Burning-Glasses, of a Figure not-Spherical, produced before the Royal Society, Philosophical Transactions of the Royal Society 3 (33–44): 631–2. 1668

[32] Cassegrain: un célèbre inconnu de l'astronomie instrumentale, A. Baranne, F. Launay, Journal of Optics 28. 4. 158-172 (1997)

[33] Optica promota, seu Abdita radiorum reflexorum & refractorum mysteria, geometrice enucleate, J. Gregory (1663)

[34] On Aspheric Anastigmatic Systems, C. R. Burch, The Proceedings of the Physical Society, 55, 6 (1943)

[35] Computer Assisted Optical Surfacing, R. Aspden, R. McDonough, F. R. Nitchie, Applied Optics, 11, 12, 2739-2747 (1972)

[36] Overview of the SMS design method applied to imaging optics, J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, A. Santamaría, Proceedings of SPIE 7429 (2009)

[37] G. D. Wassermann and E. Wolf, Proc. Phys. Soc. (London) B62, 2 (1949)

[38] Note on the Wasserman-Wolf method for designing aspheric surfaces, E. M. Vaskas, Journal of the Optical Society of America, 47, 7, 669-670 (1957)

[39] Exact and approximate computation of Schmidt cameras II. Some modified arrangements, F. A. Lucy, Journal of the Optical Society of America, 31, 5 (1941)

[40] On the determination of aspheric profiles, E. Wolf, W. S. Preddy, Proceedings of the Physical Society, 59, 4 (1947)

[41] On the design of optical systems with an aspheric surface, K. Miyamoto, Journal of the Optical Society of America, 51, 1 (1961)

[42] Design of Double Gauss systems using aspherics, T. R. Sloan, R. E. Hopkins, Applied Optics 1911, 6, 11 (1967)

[43] Aberrations of the symmetrical optical system, W. T. Welford, Academic Press (1974)

[44] Using aspheres to increase optical system performance, S. Sparrold, Photonics Tech Briefs (2010)

[45] Image hosted by https://en.wikipedia.org/wiki/Conic_section

[46] Lens design with Forbes aspheres, R. N. Youngworth, E. I. Betensky, Proceedings of SPIE, 7100 (2008)

[47] Aspheric Optics. How they are made and why they are needed, E. Heynacher, Phys. Technol. Vol. 10 (1979)

[48] Image hosted by http://www.optotech.de/en/active-fluid-jet-polishing-afjp

[49] Designing and specifying aspheres for manufacturability, J. Kumler, Proceedings of SPIE (2005)

[50] Interferometer for precise and flexible asphere testing, E. Garbusi, C. Pruss, W. Osten, Optics Letters, 33, 24 (2008)