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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.
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