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Lens Aberrations Aberration: a departure from the paraxial limit. Paraxial limit: (1) Rays are kept close to the optic axis. (2) Small angle approximation (1 st Order) 5 th Order 3 rd Order 1 Order 3 rd Order aberration theory: Keep the 3 rd order terms. Hope College, PHYS 352, Spring 2013

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Pages of Hecht, 4 th ed., treats refraction in the paraxial (1 st Order) limit. If the 3 rd order terms are kept in ℓ o and ℓ i (p. 154) we find that extended images don’t land on exactly the same focal surface (See Fig. 6.14, p, 254). ℓoℓo ℓiℓi sisi soso n2n2 n1n1 Path 1 Path 2 Paraxial Approximation: Path 1 and Path 2 have the same Optical Path Length Quantify the aberration: a(Q)=Path 2 – Path 1 a(Q)=(n 1 ℓ o +n 2 ℓ i )-(n 1 s o +n 2 s i ) a=0 means no aberration Q Hope College, PHYS 352, Spring 2013

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ℓoℓo ℓiℓi n2n2 n1n1 Q yiyi r Distortion Curvature of field Astigmatism Coma Spherical aberration The Five Monochromatic Seidel Aberrations yoyo y i and y o are paraxial image and object heights. (From Pedrotti, 3 rd Ed., Section 20.2) Hope College, PHYS 352, Spring 2013 Spread the image point

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1. Spherical aberration The only aberration that exists even for objects on the optic axis (no y i dependence). X Homework: Analyze this for a concave mirror. Positive SA: The marginal rays converge left of the paraxial image (positive lens) Negative SA: The marginal rays converge right of the paraxial image (negative lens) Hope College, PHYS 352, Spring 2013

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r2r2 r1r1 n The Coddington shape factor: Spherical aberration is minimized when (Shown in Jenkins & White, Sections ) Homework: For what value of n does a planar convex lens produce a minimum spherical aberration for an object located at infinity? Which side of the lens should the light enter? 1. Spherical aberration Minimizing spherical aberration requires using two different radii. Hope College, PHYS 352, Spring 2013

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2. Coma Depends on y i. e.g. It’s an “off-axis” aberration. Not symmetrical, which is the origin of the name. Negative coma: Marginal rays focus closer to the optic axis. Positive coma: …farther from.. off-axis object screen Figure 6.22a (Hecht, page 260) shows the formation of a comatic image from a series of comatic circles. The same lens designs minimize coma & spherical aberration. Hope College, PHYS 352, Spring 2013

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3. Astigmatism Astigmatism: rays from off-axis source do not strike the lens symmetrically (tangential rays versus saggital rays). Tangential rays Saggital rays See Fig (page 263): tangential and saggital rays will fan out and form line images of the point source at two different image surfaces. Hope College, PHYS 352, Spring 2013

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Circle of least confusion Hope College, PHYS 352, Spring 2013

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4. Curvature of Field Curvature of field: Tangential and saggital rays do not form images on the same surface. Very similar to astigmatism, but symmetric about the optic axis. Tangential rays Saggital rays S T (T left of S: Positive astigmatism) Hope College, PHYS 352, Spring 2013

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Elimination of astigmatism and curvature of field aberrations Engineer the lens curvatures or spacing so that tangential and saggital surfaces coincide. A focal surface that eliminates astigmatism to 3 rd order is called a Petzval surface. Flat P surface – eliminates both astigmatism and curvature of field aberrations. P T S P is always 3x farther from T than S Two lenses will have a flat P surface if n 1 f 1 =-n 2 f 2 Ultimately, the film must conform to P. Hope College, PHYS 352, Spring 2013

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5. Distortion Distortion aberration is caused by non-uniform lateral magnification and is often minimized using aperture stops. Hope College, PHYS 352, Spring 2013

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