Spherical Aberration Compensation Plates
Optical Components
Spherical aberration compensation plates
Scott Sparrold, Edmund Optics, Pennsburg, Pennsylvania, USA Anna Lansing, Edmund Optics GmbH, Karlsruhe, Germany
Imaging optics strive to obtain perfect or near diffraction limited performance. The main detriment to imaging systems is spherical aberration, and it costs money and effort to eliminate or minimize. This article will outline a new optical component for controlling this aberration.
Typical optical components are standard- ly, the article will explore compari- ized to focal length and diameters but sons between aspheres, spherical not so much to their level of aberration. lenses and spherical aberration Within the last decade aspheric lenses have plates in typical user applications. become a commodity, basically eliminating spherical aberration for a single wave- 1 Theory of spherical aberration lens produces “undercorrected” spherical length and specifi c imaging parameters. aberration, defi ned as the marginal rays The introduction of spherical aberration To obtain a mathematically perfect image, focusing closer to the lens than the paraxial plates provides another tool for construct- a mirror requires a parabolic surface while a rays. For a collimated beam and a spherical ing systems that can adequately control lens requires an elliptical surface (Lüneburg lens, equation 1 describes the deviation W and manipulate light. lens) [1]. For ease of manufacturing and from an ideal spherical wavefront [2]. This article will highlight typical scenarios testing, however, most optical elements W = W · ρ4 (Eq.1) involving spherical aberration and demon- have spherical surfaces. This introduces 040 strate how to control it. We’ll discuss the blur or a halo in the image due to spheri- where W040 is the wavefront aberration magnitude of the aberration via equations cal aberration, which results from differ- coeffi cient for spherical aberration (units and general rules of thumb. This will give ent focal lengths at different pupil zones are waves) and ρ is the normalized pupil the user a good understanding of how to (fi gure 1) such as the paraxial zone close radius. For a converging or diverging beam, handle spherical aberration when design- to the optical axis or the marginal zone at the spherical term is added to a quadratic ing with standard components. Additional- the edge of the lens. A typical spherical term (focus) to establish a wavefront devia- tion as shown in equation 2.
2 W = W020 · ρ (Eq.2)
where W020 is the wavefront aberration coeffi cient for focus (units are waves). Ray errors (deviation from a perfect point like image) at the image plane are a derivative of the wavefront deviation, W, with respect to the aperture, ρ. Assuming paraxial focus and neglecting diffraction effects, the total spot diameter, d, is simply the ray error of light from the edge of the focusing lens:
d = 16 · (F/#) · W040 · λ (Eq.3) with wavelength λ and the lens aperture (F/#) = focal length / lens diameter. This is not the RMS (root mean square) spot diameter that is commonly used to char- acterize imaging performance. There are Figure 1: Simple diagram explaining spherical aberration no simple equations to characterize the
32 Photonik international · 2011 Originally published in German in Photonik 2/2011
Optical Components
RMS spot size and it is usually left to ray tracing software to determine this value. d can be minimized by fi nding a medial focus between marginal (edge rays) and paraxial (center rays) focus, located close to “mid zone focus” in fi gure 1. A lens (fi gure 2) has two radii of curvature (front and back). For any given focal length there is an infi nite number of combina- tions, affecting aberration. A lens could be bent as a meniscus, an equiconvex or a plano convex lens. The shape factor K is
Figure 3: Singlet spherical aberration (Log) for an infi nite object versus lens bend- ing and index of refraction at 587 nm
Lord Rayleigh’s criteria states that a wave- 2 Spherical aberration plates Figure 2: Singlets can be “bent” from front should have a deviation < λ/4 to negative to positive K obtain a near diffraction limited image. Fig- Spherical aberration plates work well in ure 3 shows that a singlet can have several systems with little or no fi eld of view. defi ned by orders of magnitude of spherical aberration They are recommended for insertion in a and it shows that a singlet with the proper pure collimated beam. If they are to be K = R / (R – R ) (Eq.4) 2 2 1 bending is not suffi cient for good imaging. used in an imaging system with a fi eld With the index of refraction n and the Multiple spherical lenses can be used to fur- of view, it is recommended to put them lens diameter D, the amount of spherical ther reduce aberration. This modestly adds near an aperture stop or a pupil. The stop aberration in a singlet with an infi nite con- to cost but can greatly grow the size of the is the one aperture in the optical system jugate (object at infi nity) can be computed optical system. An asphere can be used to that limits the diameter of the beam. A by equation 5 [4] to completely eliminate spherical aberration in pupil is an image of the aperture stop.
+ the same size package as a spherical singlet. Placing a spherical aberration plate away = D 1 ⋅⋅ 1 ª 2 ()2 ⋅++⋅−⋅ n 2º W040 12 KnKn 128 ⋅ λ ()# 3 ()nF −1 2 ¬« n ¼» (Eq.5) An asphere is harder to manufacture and from a pupil will create induced coma test and it will therefore cost more than and astigmatism and it is not recom- Spherical aberration in a lens with a fi nite several spherical lenses. mended. conjugate is described in [3]. The discussion of spherical aberration is Spherical aberration plates have a sur- A positive lens (K>0) will introduce under- pertinent to zero fi eld-of-view systems face sag departure from a fl at plane that 4 corrected spherical aberration and W040 such as a centered point source at infi nity is characterized by pure ρ , where ρ is will always be greater than zero. There or laser systems. Imaging systems typically the semi-aperture. Spherical aberration are many things one can do to minimize image over a fi eld of view defi ned by the is sign dependent and can therefore be spherical aberration and consequently fi eld stop, usually the detector. As a fi eld convex or concave. A concave plate will shrink image size. To understand the vari- of view is added, more aberrations, such create negative or overcorrected spheri- ous trades, a plot of spherical aberration as coma and astigmatism become impor- cal aberration, while a convex plate will versus bending and index of refraction tant. These aberrations are beyond the create positive or undercorrected spheri- is shown in fi gure 3. Notice how there scope of this article. cal aberration. is always a minimum to the curve. Also notice that the higher the index of refrac- tion the lower the undercorrected spheri- Spherical aberration cal aberration. plate -25 λ There is a unique shape factor to mini- Convex-plano lens Asphere + convex-plano lens mize spherical aberration for each index of refraction. The fi rst derivative of equa- Relative cost 1 8 8 tion 5 with respect to shape factor, K, λ λ λ when set equal to zero will solve for the Spherical aberration 21.5 0 -3.5 optimum shape factor (equation 6). The Spot diameter 608 µm 1 µm 98 µm minimized spherical aberration is given by (paraxial focus) equation 7 and it is found by substituting Spot diameter the optimum shape back into equation 5. 400 µm 1 µm 36 µm (best focus) ()nn +12 K = (Eq.6) Optimum ()n + 22 Optical layout and D 1 4n2 n W040 minimized = 3 2 (Eq.7) Table 1: Spherical aberration content in a simple imaging example (F/3, 25 mm 512 ()F# ()n 1 ()n + 2 diameter)
Originally published in German in Photonik 2/2011 Photonik international · 2011 33
Optical Components
3 Application examples
3.1 Standard imaging components We will now contrast the use of spherical aberration plates to spherical and aspheric Figure 4: lenses. Since the spherical aberration plate Polarizing beam has no optical power and cannot focus sampler utilizes light, it has to be included with a spheri- spherical aber- cal lens. ration plates to The discussion will be constrained to all compensate for glass elements: Using common fabrica- overcorrected tion methods will yield better cost and spherical aberra- performance comparisons. Let us assume tion induced by a F/3 elements made from N-BK7 from cube beamsplitter Schott. Refer to table 1 for relative cost in a converging and spherical aberration content. The wavefront best solution in terms of optical perform- ance is a glass asphere. The lowest cost solution is a spherical element, but it has t 1 ()n2 −1 very poor performance. An intermediate W −= ⋅⋅ (Eq.8) tion. Using equation 7, the manufacturing 040 128 ⋅ λ ()4 n3 solution would be a spherical lens paired F # engineer calculates the exact amount of with a spherical aberration plate. It has overcorrected spherical aberration to be the same cost as the asphere but much 3.3 Beamsplitter -1.07 waves. He places a +1 wave spheri- improved performance over the spheri- When inserting a beamsplitter after a cal aberration plate in front of the plastic cal lens. focusing lens, the user either has to toler- asphere (fi gure 4). Evaluation in table 2 While this case doesn’t compel one to ate the added degradation or invest into shows that in this case the spherical aber- use spherical aberration plates versus a custom asphere to compensate for the ration plate provides the best cost versus a standard asphere, it highlights the beamsplitter’s induced spherical aberra- performance solution. benefi ts of a spherical aberration plate tion. Spherical aberration plates offer a added to an existing system plagued by new alternative. 3.4 Correcting for this aberration. Let us consider a plastic aspheric lens oper- non-standard conjugates ating at F/3.3 to focus green light onto Most off-the-shelf lenses are designed to 3.2 A window in a converging beam a detector. The amount of light provides work with specifi c conjugates. Given an Placing a window in a collimated wave- feedback for some fabrication process. object distance relative to the lens, there front will not induce any additional spheri- The manufacturing engineer realizes that is one unique image distance for paraxial cal aberration and is a preferable option. monitoring the different polarizing states rays. These two distances are said to be However, sometimes it is necessary to will provide increased fi delity to processing. conjugate to one another. A convex plano place a window after a focusing lens. Typi- This particular plastic lens has been used lens (K=1) is optimum for a collimated cal scenario could be a protective window in this production process for many years input, where the point object is located at over a detector or bandpass fi lter. Placing and management is hesitant to change the infi nity and the image is one focal length a plane parallel plate in a focusing or con- lens or use another one. The manufactur- away from the lens. A double convex lens verging beam will induce overcorrected ing engineer places a 25 mm broadband (K=0.5) is optimum for one-to-one imag- spherical aberration. polarizing cube beamsplitter after the lens ing, where the object-to-lens distance and The amount of spherical aberration induced and adds another detector. Unfortunately the lens-to-image distance are equal to in this case can be computed as follows the spot sizes have grown large and they double the focal length. Any other set-up [2,4], independent of where the window is overfi ll the detectors. The beam splitter has cannot achieve a minimum spherical aber- placed in the converging beam: induced overcorrected spherical aberra- ration with standard lenses. As an example let’s de-magnify a pinhole by a factor of 20x using a single F/1.7 lens Plastic asphere + with a pinhole-to-lens distance 20 times beam splitter and longer than the lens-to-image distance. Plastic asphere + Custom asphere +1 λ spherical Ray tracing with a standard convex plano beam splitter + beam splitter aberration plate lens shows there will be +290 waves of spherical aberration. Optimizing the Spherical aberration -1.07 λ 0 λ -0.07 λ shape factor via software shows +135 waves of aberration at optimum bend- Spot diameter 28 µm 0 µm 2 µm ing of K=0.86. This shape factor is not (paraxial focus) a standard lens and it would have to be Spot diameter (best 11 µm 0 µm 0.3 µm custom made. A standard convex plano focus) asphere will have +6.5 waves of spheri- Relative cost 1 9 1.5 cal aberration for this application. If this is too much spherical aberration, a costly Table 2: Optical performance for various solutions of a cube beam splitter placed custom asphere could be designed and after a focusing lens fabricated.
34 Photonik international · 2011 Originally published in German in Photonik 2/2011
Optical Components
Figure 5: Spherical aberration plates (here: -5, -1 and -0.5 wavelengths, in total -6.5 wavelengths, cf. text and table 3) used to correct for non-standard imaging conjugates
Instead, one can use a standard asphere [3] W.T. Welford, Aberrations of the Symmetrical Optical System, Academic Press, 1974 optimized for an infi nite conjugate with [4] W.J. Smith, Modern Optical Engineering, 3rd edi- spherical aberration plates to correct the tion, McGraw Hill, 2000 aberrations caused by a shift in conjugates (fi gure 5). Table 3 outlines the perform- Author contact: ance of the various optical confi gurations. Scott Sparrold 4 Conclusion Senior Optical Engineer Edmund Optics Inc. Spherical aberration is the zonal variation 601 Montgomery Avenue of focus and it costs money and effort to Pennsburg, PA 18073 control in an optical system. It is typically USA minimized through lens bending or using Tel. +1 (856) 547 3488 multiple spherical components or eliminat- Fax +1 (856) 573 6295 ed by aspheres for an added cost. If unac- eMail: [email protected] ceptable spherical aberration in an optical Internet: www.edmundoptics.com system cannot be eliminated by existing standard components, this either requires Anna Lansing a costly custom design or new solutions Sales and Applications involving spherical aberration plates. This Engineer provides an additional tool for the optical Edmund Optics GmbH system designer to control aberration by Zur Gießerei 19-27 off-the-shelf components. 76227 Karlsruhe Germany References: Tel. +49/721/62737-30 Fax +49/721/62737-33 [1] R.K. Lüneburg, Mathematical Theory of Optics, Cambridge University Press, 1964 eMail: [email protected] [2] J.E. Greivenkamp, Field Guide to Geometric Optics, Internet: www.edmundoptics.de SPIE Press, 2004
Custom Spherical ab- Standard bent erration plate convex- spherical Standard Custom -6.5 λ waves* plano lens lens asphere asphere and asphere Relative cost 1 12 8 120 30 Spherical 290 λ 135 λ 6.57 λ 0 λ 0.07 λ aberration Spot diameter 4250 µm 1900 µm 110 µm 0 µm 1.1 µm (paraxial focus) Spot diameter 2200 µm 1400 µm 53 µm <1 µm <1 µm (best focus)
Table 3: Spherical aberration content at non-standard conjugates (F/1.67, 25 mm diameter)
Originally published in German in Photonik 2/2011