Lens Design I Lecture 3: Properties of optical systems II 207-04-20 Herbert Gross Summer term 207 www.iap.uni-jena.de
2 Preliminary Schedule - Lens Design I 207 06.04. Basics 2 3.04. Properties of optical systems I 3 20.04. 4 27.04. Properties of optical systems II Properties of optical systems III Introduction, Zemax interface, menues, file handling, preferences, Editors, updates, windows, coordinates, System description, 3D geometry, aperture, field, wavelength Diameters, stop and pupil, vignetting, Layouts, Materials, Glass catalogs, Raytrace, Ray fans and sampling, Footprints Types of surfaces, cardinal elements, lens properties, Imaging, magnification, paraxial approximation and modelling, telecentricity, object distance and afocal image, local/global coordinates Component reversal, system insertion, scaling of systems, aspheres, gratings and diffractive surfaces, gradient media, solves 5 04.05. Advanced handling I Miscellaneous, fold mirror, universal plot, slider, multiconfiguration, lens catalogs 6.05. Aberrations I Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams, Aberration expansions, Primary aberrations 7 8.05. Aberrations II Wave aberrations, Zernike polynomials, measurement of quality 8 0.06. Aberrations III Point spread function, Optical transfer function 9 08.06. Optimization I 0 5.06. Optimization II 22.06. Advanced handling II Principles of nonlinear optimization, Optimization in optical design, general process, optimization in Zemax Initial systems, special issues, sensitivity of variables in optical systems, global optimization methods System merging, ray aiming, moving stop, double pass, IO of data, stock lens matching 2 29.06. Correction I 3 06.07. Correction II Symmetry principle, lens bending, correcting spherical aberration, coma, astigmatism, field curvature, chromatical correction Field lenses, stop position influence, retrofocus and telephoto setup, aspheres and higher orders, freeform systems, miscellaneous
3 Contents 3rd Lecture. Types of surfaces 2. Cardinal elements 3. Lens properties 4. Imaging 5. Special cases 6. Paraxial approximation
4 Important Surface Types Special surface types Data in Lens Data Editor or in Extra Data Editor Gradient media are descriped as 'special surfaces' Diffractive / micro structured surfaces described by simple ray tracing model in one order Standard Even asphere Paraxial Paraxial XY Coordinate break Diffraction grating Gradient Toroidal Zernike Fringe sag Extended polynomial Black Box Lens ABCD spherical and conic sections classical asphere ideal lens ideal toric lens change of coordinate system line grating gradient medium cylindrical lens surface as superposition of Zernike functions generalized asphere hidden system, from vendors paraxial segment
Cardinal elements of a lens Focal points:. incoming parallel ray intersects the axis in F 2. ray through F is leaves the lens parallel to the axis Principal plane P: location of apparent ray bending y principal plane P' f ' u' s BFL F' focal plane nodal planes s P' u N N' u' Nodal points: Ray through N goes through N and preserves the direction
Notations of a lens P principal point S vertex of the surface F focal point O n n n 2 s f intersection point of a ray with axis focal length PF y u F S P P' N N' S' u' F' r radius of surface curvature y' O' d thickness SS s f f' s' n refrative index f BFL s P s' P' f' BFL a d a'
Main properties of a lens Main notations and properties of a lens: - radii of curvature r, r 2 curvatures c sign: r > 0 : center of curvature is located on the right side - thickness d along the axis - diameter D - index of refraction of lens material n Focal length (paraxial) Optical power Back focal length intersection length, measured from the vertex point c r c 2 r yf ' f, f ' tan u F s n f f s n' f ' F ' ' H ' 2 y tan u'
Lens shape Different shapes of singlet lenses:. bi-, symmetric 2. plane convex / concave, one surface plane 3. Meniscus, both surface radii with the same sign Convex: bending outside Concave: hollow surface Principal planes P, P : outside for mesicus shaped lenses P P' P P' P P' P P' P P' P P' bi-convex lens plane-convex lens positive meniscus lens bi-concave lens plane-concave lens negative meniscus lens
Lens bending und shift of principal plane Ray path at a lens of constant focal length and different bending Quantitative parameter of description X: The ray angle inside the lens changes X R R 2 R 2 R The ray incidence angles at the surfaces changes strongly The principal planes move For invariant location of P, P the position of the lens moves P P' F' X = -4 X = -2 X = 0 X = +2 X = +4
Cardinal Points of a Lens Real lenses: The surface with the principal points as apparent ray bending points is a curved shell The ideal principal plane exists only in the paraxial approximation P' s' P'
Limitation of Principal Surface Definition The principal planes in paraxial optics are defined as the locations of the apparent ray bending of a lens of system In the case of a system with corrected sine conditions, these surfaces are spheres Sine condition and pupil spheres are also limited for off-axis points near to the optical axis For object points far from the axis, the apparent locations are complicated surfaces, which may consist of two branches
2 Limitation of Principal Surface Definition A r4eal microscopic lens has a nearly perfect spherical shape of the principal surface
Optical imaging Optical Image formation: All ray emerging from one object point meet in the perfect image point Region near axis: gaussian imaging ideal, paraxial Image field size: Chief ray field point O 2 chief ray pupil stop Aperture/size of light cone: marginal ray defined by pupil stop object axis marginal ray optical system O O' image O' 2
Single surface imaging equation Thin lens in air focal length Thin lens in air with one plane surface, focal length Thin symmetrical bi-lens Thick lens in air focal length ' ' ' ' f r n n s n s n 2 ' r r n f ' n r f 2 ' n r f 2 2 2 ' r r n d n r r n f Formulas for surface and lens imaging
Imaging equation s' Imaging by a lens in air: lens makers formula s' s f real object real image 4f' 2f' virtual image real image Magnification s m ' s - 4f' -2f' 2f' 4f' s Real imaging: s < 0, s' > 0 Intersection lengths s, s' measured with respective to the principal planes P, P' real object virtual image -2f' virtual object virtual image - 4f'
Magnification Lateral magnification for imaging Scaling of image size m y' y f tan u f ' tan u' principal planes object y focal point focal point F P P' F' z f f' z' image y' s s'
Angle Magnification Afocal systems with object/image in Definition with field angle w angular magnification tan w' tan w nh n' h' w' w Relation with -distance magnification m f f '
Object or field at Image in : - collimated exit ray bundle - realized in binoculars image image at Object in - input ray bundle collimated - realized in telescopes - aperture defined by diameter not by angle lens acts as aperture stop field lens stop eye lens object at collimated entrance bundle image in focal plane
9 Telecentricity Special stop positions:. stop in back focal plane: object sided telecentricity 2. stop in front focal plane: image sided telecentricity 3. stop in intermediate focal plane: both-sided telecentricity Telecentricity:. pupil in 2. chief ray parallel to the optical axis object object sides chief rays parallel to the optical axis telecentric stop image
20 Telecentricity Double telecentric system: stop in intermediate focus Realization in lithographic projection systems object lens f telecentric lens f 2 stop image f f f 2 f 2
2 Telecentricity, Infinity Object and Afocal Image.Telecentric object space Set in menue General / Aperture Means entrance pupil in Chief ray is forced to by parallel to axis Fixation of stop position is obsolete Object distance must be Field cannot be given as angle 2.Infinity distant object Aperture cannot be NA Object size cannot be height Cannot be combined with telecentricity 3.Afocal image location Set in menue General / Aperture Aberrations are considered in the angle domain Allows for a plane wave reference Spot automatically scaled in mrad
22 Infinity cases case object image entrance pupil exit pupil example sample layout Systematic of all cases Physically impossible:. object and entrance pupil in 2. image and exit pupil in 2 3 4 5 object telecentric image telecentric image telecentric relay metrology lens lithographic projection lens 4f-system afocal zoom telescopes beam expander metrology lens 6 camera lens focussing lens 7 eyepiece collimator 8 object telecentric microscopic lens 9 image telecentric metrology lens 0 impossible impossible 2 impossible 3 impossible 4 impossible 5 impossible 6 impossible
Paraxial Approximation Paraxiality is given for small angles relative to the optical axis for all rays Large numerical aperture angle u violates the paraxiality, spherical aberration occurs Large field angles w violates the paraxiality, coma, astigmatism, distortion, field curvature occurs
Paraxial approximation Paraxial approximation: ni n' i' Small angles of rays at every surface Small incidence angles allows for a linearization of the law of refraction All optical imaging conditions become linear (Gaussian optics), calculation with ABCD matrix calculus is possible No aberrations occur in optical systems There are no truncation effects due to transverse sized components Serves as a reference for ideal system conditions Is the fundament for many system properties (focal length, principal plane, magnification,...) The sag of optical surfaces (difference in z between vertex plane and real surface intersection point) can be neglected i x 2 All waves are plane of spherical (parabolic) R E( x) E0 e The phase factor of spherical waves is quadratic
Paraxial approximation Law of refraction nsin I n' sin I' Taylor expansion 3 5 x x sin x x... 3! 5! Linear formulation of the law of refraction ni n' i' Error of the paraxial approximation i'- I') / I' 0.05 0.04 0.03 0.02 n' =.9 n' =.7 n' =.5 ni i' I' n' I' nsin i arcsin n' 0.0 0 0 5 0 5 20 25 30 35 40 i
26 Paraxial approximation Taylor expansion of the sin-function Definition of allowed error 0-4 Deviation of the various approximations: - linear: 5 - cubic: 24-5th order: 542 sin(x) 0.8 0.6 0.4 0.2 exact sin(x) linear cubic 5th order 0 0 0 20 30 40 50 60 70 80 90 x = 5 x = 24 x = 52 deviation 0-4 x [ ]
27 Paraxial Approximation Contribution of various orders of the sin-expansion 3 i sini i 6 5 7 9 i i i... 20 5040 362880 0 0 0-2 st order i 3rd order i 3 /6 5th order i 5 /20 0-4 7th order i 7 /5040 0-6 9th order i 9 /362880 0-8 0-0 0-2 0 5 0 5 20 25 30 35 40 45 incidence angle i
28 Telecentricity, Infinity Object and Afocal Image.Telecentric object space Set in menue General / Aperture Means entrance pupil in Chief ray is forced to by parallel to axis Fixation of stop position is obsolete Object distance must be Field cannot be given as angle 2.Infinity distant object Aperture cannot be NA Object size cannot be height Cannot be combined with telecentricity 3.Afocal image location Set in menue General / Aperture Aberrations are considered in the angle domain Allows for a plane wave reference Spot automatically scaled in mrad
29 Miscellaneous. Coordinate reference Fixation of reference in menue: General / Misc Every surface vertex can be defined as global reference Helpful in constructing 3D-system geometries 2. Scale System In menue Tools / Miscellaneous / Scale Helpful in expoding/imploding all length scales Application: rescale patent systems Alternative option in menue Tools / Miscellaneous / Make focal, desired f realized 3. Add folding mirror Help command in menue Tools / Coordinates / Add fold mirror Automatically inserted coordinate break surface 4. Make double pass Help command in menue Tools / Miscellaneous / Make double pass Folding mirror and reversed system automatically generated
30 Cardinal Elements in Zemax Cardinal elements of a selected index range (lens or group)