Lens Design I. Lecture 11: Imaging Herbert Gross. Summer term

Transcription

1 Lens Design I Lecture 11: Imaging Herbert Gross Summer term

2 2 Preliminary Schedule Basics Properties of optical systrems I Properties of optical systrems II Properties of optical systrems III Advanced handling I Aberrations I Aberrations II Wave aberrations, Zernike polynomials 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, infinity distance and afocal, local/global coordinates Component reversal, system insertion, scaling of systems, aspheres, gratings and diffractive surfaces, gradient media, solves Add fold mirror, scale system, slider, multiconfiguration, universal plot, diameter types, lens catalogs Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams, Aberration expansions, Primary aberrations Aberrations III Point spread function, Optical transfer function Optimization I Principles of nonlinear optimization, Optimization in optical design, Global optimization methods, Solves and pickups, variables, Sensitivity of variables in optical systems Optimization II Systematic methods and optimization process, Starting points, Optimization in Zemax Imaging Fundamentals of Fourier optics, Physical optical formation, Imaging in Zemax Correction I Correction II Symmetry principle, Lens bending, Correcting spherical aberration, Coma, stop position, Astigmatism, Field flattening, Chromatical correction, Retrofocus and telephoto setup, Design method Field lenses, Stop position influence, Aspheres and higher orders, Principles of glass selection, Sensitivity of a system correction

3 3 Contents 1. Fourier imaging 2. Coherence 3. Phase imaging 4. Imaging in Zemax

4 Definitions of Fourier Optics Phase space with spatial coordinate x and 1. angle 2. spatial frequency in mm transverse wavenumber k x v x k 2v Fourier spectrum x k x k 0 A( v, v ) Fˆ E( x, y) x y structure k / g diffracted ray direction k T corresponds to a plane wave expansion i xkx yk y x, y, (,, ) A k k z E x y z e dx dy g = 1 / Diffraction at a grating with period g: deviation angle of first diffraction order varies linear with = 1/g sin 1 g v

5 Resolution of Fourier Components detail high spatial frequencies numerical aperture resolved frequencies point low spatial frequencies sum for low NA decomposition of Fourier components (sin waves) for high NA high spatial frequencies Ref: D.Aronstein / J. Bentley

6 6 Grating Diffraction and Resolution a) resolved incident light b) not resolved diffracted orders optical system Arbitrary expaneded into a spatial frequency spectrum by Fourier transform Every frequency component is considered separately To resolve a spatial detail, at least two orders must be supported by the system off-axis illumination g g sin m sin NA g Ref: M. Kempe 2 NA

7 Number of Supported Orders A structure of the is resolved, if the first diffraction order is propagated through the optical imaging system The fidelity of the increases with the number of propagated diffracted orders 0. / +1. / -1. order 0. / +1. / / -2. order 0. / / +2. / -2. / +3. / -3. order

8 Abbe Theorie of the Microscopic Resolution Diffraction of the illumination wave at the structure Occurence of the diffraction orders in the pupil Image generation by constructive interference of the supported orders Object details with high spatial frequency are blocked by the system aperture and can not be resolved source plane imaging lens pupil plane plane Ref: W. Singer

9 Fourier Filtering Imaging of a crossed grating Spatial frequency filtering by a slit: pupil complete open Case 1: - pupil open - Cross grating d Case 2: - truncation of the pupil by a split - only one direction of the grating is resolved pupil truncated by slit

10 Fourier Optics Point Spread Function Optical system with magnification m Pupil function P, Pupil coordinates x p,y p g psf ( x, y, x', y') N P x p, y p e ik x z p x' mx y y' my p dx p dy p PSF is Fourier transform of the pupil function (scaled coordinates) source point plane g ( x, y) N Fˆ P x, y psf p p plane point distribution

11 Fourier Theory of Incoherent Image Formation Transfer of an extended distribution I(x,y) In the case of shift invariance (isoplanasy): incoherent convolution Intensities are additive I I I inc inc ( x', y') ( x', y') g ( x', x, y', y) I( x, y) dxdy g ( x' x, y' y) I( x, y) dxdy psf psf ( x', y') I ( x, y)* I ( x, y) psf obj 2 2 plane plane intensity intensity single psf

12 Fourier Theory of Incoherent Image Formation intensity I(x,y) Fourier transform intensity spectrum I(v x,v y ) convolution product squared PSF, intensityresponse I psf (x',y') Fourier transform optical transfer function H OTF (v x,v y ) result result intensity I'(x',y') Fourier transform intensity spectrum I'(v x,v y )

13 Fourier Theory of Coherent Image Formation Transfer of an extended distribution I(x,y) In the case of shift invariance (isoplanasie): coherent convolution of fields Complex fields are additive plane E( x', y') g psf x, y, x', y' E( x, y) dx dy E( x', y') g psf x x', y y' E( x, y) dx dy x, y E( x, ) E( x', y') g psf y plane amplitude distribution amplitude distribution single point

14 Comparison Coherent Incoherent Image Formation incoherent coherent bars resolved bars not resolved bars resolved bars not resolved

15 Incoherent Image Formation in Frequency Space Incoherent illumination: No correlation between neighbouring points Superposition of intensity in the I ( x', y') In the case of shift invariance (isoplanasie): Incoherent imaging with convolution I I inc inc ( x', y') g ( x', x, y', y) I( x, y) dxdy g ( x' x, y' y) I( x, y) dxdy psf psf ( x', y') I ( x, y)* I ( x, y) psf obj 2 2 In frequency space: Product of spectra, linear transfer theory The spectrum of the psf works as low pass filter onto the spectrum Optical transfer function H ( v, v ) FT I ( x, y) I otf x y PSF ( vx, vy) Hotf ( vx, vy) Iobj( vx, vy)

16 Partial Coherent Imaging Complete description of an optical system: 1. Light source 2. Illumination system, amplitude response h ill 3. Transmission 4. Observation / imaging system with amplitude response h obs illumination field x s, y s plane x p, y p plane x i, y i source illumination system observation system pupil P h obs I s h ill I i I o sensor

17 Coherence Parameter Finite size of source : aperture cone with angle u ill Observation system: aperture angle u obs Definition of coherence parameter : Ratio of numerical apertures Limiting cases: coherent = 0 u ill << u obs sin u sin u ill obs incoherent = 1 u ill >> u obs source x o, y o lens x i, y i u ill u obs illumination observation

18 Imaging in Zemax Possible options in Zemax: Convolution of with Psf 1. geometrical 2. with diffraction Geometrical raytrace analysis 1. simple geometrical shapes (IMA-files) 2. bitmaps Diffraction imaging: 1. partial coherent 2. extended with variable PSF Structure of options in Zemax not clear Redundance Field definition and size scaling not good Numerical conditions and algorithms partially not stable 18

19 Imaging in Zemax Field size definitions Total field size in data (angle or length) Selected field index Relative size of structure in the total field Shown part of the field Not completly consistent in the different imaging tools y selected field (index) field heigth plotted size / pixel x Npix relative field size of structure x real maximum size of the field 19

20 General Image Simulation Field height: size of in the specific coordinates of the system - zero padding included (not: size = diameter) - size shown is product of pixel number x pixel size - can be full field or centre of local extracted part of the field PSF-X/Y points: number of field points to incorporate the changes of the PSF, interpolation between this coarse grid Object: bitmap PSF: geometrical or diffraction 20

21 Geometric Imaging I Geometrical imaging by raytrace Binary IMA-files with geometrical shapes Choice of: - field size - size, - wavelengths - number of rays Interpolation possible 21

22 Geometric Imaging II Geometrical imaging by raytrace of bitmaps Extension of 1st option: can be calculated at any surface If full field is used, this corresponds to a footprint with all rays Example: light distribution in pupil, at last surface, in 22

23 Partial Coherent Imaging Different types of partial coherent model algorithms possible Only IMA-Files can be used as s a describes the coherence factor (relative pupil filling) Control and algorithms not clear, not stable 23

24 Extended Diffraction Classical convolution of psf with pixels of IMA-File Coherent and incoherent model possible PSF may vary over field position 24

25 25 Geometric Raytrace 7x7 pixel IMA file raytraces From random position inside pixel To random position in entrance pupil Spot diagram Ref.: M. Eßlinger

26 Geometric Raytrace 1 ray per pixel 10 rays per pixel 100 rays per pixel 1000 rays per pixel Ref.: M. Eßlinger 26

27 27 Imaging Objects IMA (Image file) - Illumination brightness in each point of the - Zemax provides basic shapes like the letter F - ASCII format with 10 different grey values or binary with 256 grey values BIM (binary ) - like IMA, but 64bit (double precision) float values ZBF (Zemax beam file) - for sophisticated illumination optics - many features only available in Premium Version of Zemax BMP (bmp, jpg or png) - 3 x 8 bit RGB values ( raytrace with FdC: 656 nm, 587 nm and 486 nm) - for greyscale detector: raytrace with FdC, averaging on detector plane Ref.: M. Eßlinger

28 28 Imaging: Summary Advantages and Disadvantages of Geometric Raytracing + Easy to understand Object File type BMP IMA BIM ZBF conv. raytrace diffraction + Field dependent errors are considered automatically - Does not include Diffraction Limit - Requires large number of rays (slow) spatial variant aberr. - Coherent imaging is difficult (not possible with Zemax) pupil aberr. Coherence Image Simulation X X X Geometric X X X X X Geometric Bitmap X X X X X X Partially Coherent X X X Extended Diffraction X X X X coh. part. coh. incoh. Ref.: M. Eßlinger