Geometrical optics Geometrical optics is based o ray-tracig. The approach assumes that the dimesio of optical elemets is much larger tha light wavelegth, i.e.,. Vocabulary - Cojugate poits: object space ad image space - Covex (covergig, positive) les vs. cocave (divergig, egative) les - Focal poit ad focal legth / cojugate foci (frot FL, back FL) - Real image vs. virtual image - Aberratios (logitudial/lateral or traverse spherical, axial/logitudial chromatic, coma, astigmatism) - Diffractio-limited systems - Achromats (doublets, triplets) - Frauhofer-cemeted achromatic doublet (chromatic, spherical aberratios, coma) Geometrical optics: les Leses form images by refractig light. Leses are trasparet ad allow light through it ulike mirrors that are highly reflective. Leses usually have two faces. Each face ca be cocave, covex, or plao For leses, there are 2 focal poits i frot of ad behid the les. Real images are behid the les. (Images are opposite of the object) Virtual images are o the same side the object is located. http://mrscgriffi.wordpress.com/ap-physics/light-geometric-optics/ Geometrical optics Geometrical optics: les Les formula (Lesmaker s formula) R R ad f s i s o f Optical ceter: rays passig through ca be draw as straight lies. f i f o Magificatio M s s o i
Geometrical optics: les Sig covetio Positive (+) Negative ( ) R 1 Covex les Cocave les R 2 Cocave les Covex les s o Real object Virtual object s i Real image Virtual image f Covex les Cocave les M Erect image Iverted image F-umber: f f D typical camera leses have markigs of 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22 (fast to slow) Geometrical optics: mirror ad prism Mirror: formula for spherical mirrors s i s o R Prism: Right-agle prism ad dove prism Fiber: umerical aperture (NA) NA f Geometrical optics: image formatio (covex les) f f
Geometrical optics: image formatio (cocave les) Geometrical optics: image formatio Geometrical optics: cocave mirror
Geometrical optics: image formatio (cocave mirror) Geometrical optics: image formatio (cocave mirror) Geometrical optics: cocave mirror Geometrical optics: covex mirror
Geometrical optics: image formatio (covex mirror) Catalogues Catalogues Catalogues
Catalogues Catalogues Catalogues Aberratios: Chromatic aberratio: - is a fuctio of frequecy. - a les havig a differet refractive idex for differet wavelegths of light (the dispersio of the les)
Aberratios: Aberratios: Spherical aberratio (SA): - Spherical surfaces i geeral yield perfect imagery oly i the paraxial regio. - Image imperfectio that occurs due to the icreased refractio of light rays that occurs whe rays strike a les or mirror ear its edge, i compariso with those that strike earer the ceter. - Logitudial SA - Traverse/lateral SA - Wavefrot distortio Aberratios: Coma: - off-axis poit sources such as stars appearig distorted. - defied as a variatio i magificatio over the etrace pupil. Astigmatism: a optical system has differet foci for rays that propagate i two perpedicular plaes
Geometrical optics: paraxial approximatio Paraxial approximatio: - More geeral approach for ray-tracig through compoud leses - Agular deviatio of a ray from the cylidrical axis or the axis of propagatio (z) is small such that si or ta ca be approximated by. Based o this, the deflectio of a ray by a thi les ca be described as
Geometrical optics: paraxial approximatio i r i r o r o r r r o f r i From r i =r o ad, f r r o o f r r i i Geometrical optics: ABCD (or ray trasfer) matrix The ray propagatio through a series of optical compoets ca be described with matrix products. I equatio form, r r Ar Br Cr Dr or q r Ar Br r Cr Dr Aq Cq B D The matrix with the elemets of A, B, C, ad D is called ABCD matrix or ray-trasfer matrix. Note that the above discussio iheretly presumes plae wave. Geometrical optics: paraxial approximatio Similarly, the propagatio of a ray through a straight sectio of a homogeeous medium of legth d ca be traced with a matrix d. It ca be easily show that the matrix for the propagatio of a ray through a straight sectio followed by a thi les is give by, d d d f f f. Geometrical optics: A thick les = two spherical refractig surfaces + dielectric separatio - Les formula still holds with effective focal legth. f R R d R R Aalytical ray-tracig: use Sell s law. i
Catalogues d R R R f f Catalogues Geometrical optics: Gaussia beams For a Gaussia beam, startig from Maxwell s equatios with a few assumptios, it ca be obtaied that E w E i kz z r w z w x y z ik z R z where z w z w, z, ad z z z R z z z z w
Geometrical optics: Gaussia beams Also, ote that k = 2/. I this equatio, w(z) ad R respectively deote the beam spot size ad the radius of curvature of a Gaussia beam. Geometrical optics: Gaussia beams & ABCD The use of ABCD matrix is equally valid for Gaussia beams, except that q i the plae wave is replaced with z R z w z q i i the case of Gaussia beams. I other words, the propagatio of a Gaussia beam through a arbitrary umber of leslike elemets ca be traced by multiplyig ABCD matrices ad recoverig the beam radius R(z) ad spot size w(z). This is called the ABCD law. Geometrical optics: Gaussia beams The divergece agle G is give by G w which is a maifestatio of wave diffractio, idicatig trasverse spreadig i the far field for z>>z 0. Geometrical optics: eyes - Photoreceptors (rods; scotopic, ight, coes; photopic, day) - Accommodatio: myopia (egative leses), hyperopia/hypermetropia (positive leses), astigmatism (aamorphic leses), cataract
Geometrical optics: terrestrial telescope Geometrical optics: refractig telescope Geometrical optics: Galilea telescope
Geometrical optics: reflectig telescope Geometrical optics: Cassegrai reflector Geometrical optics: Newtoia reflector Geometrical optics: Schmidt-Cassegrai reflector
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