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Optics Spring 2016 ELEC E-5730 lectures: Toni Laurila, [email protected], office I431, Tel. 050-458 2460 excercises: Aigar Vaigu [email protected], office I427 Course web site on Aalto MyCourses • Exercise I question sheet and course schedule available • Course book chapters 1 and 2 available for downloading • course includes 6 compulsory laboratory works which will accomplished during two laboratory exercise sessions Last lecture – the Fundamentals • • Light as electromagnetic radiation Light can treated both as photons and waves Huygens’ principle Fermat’s principle (”principle of least time”) 1) In homogenous medium light travels in a linear fashion 2) Law of reflection: 1 = 2 3) Law of refraction or Snell’s law: n1 sin( 1) = n2 sin( 2) • Total internal reflection Today – Powerful set of tools for optical design • • • • • Derivation of the lens maker’s formula and thin lens equation Basics of ray tracing in optical systems Different types of lenses, magnification, numerical aperture, f-number Non-ideal lenses - aberrations Matrix formalism for ray tracing Why imaging optics is needed? optical elements Optics is required ”to collect” light scattered from the object to form a sharp, focused image Refractive power of a spherical surface optical path length According to the Fermat’s principle the variation of the path equals zero, that is, dL/d = 0. Refractive power of a spherical surface law of cosines applied to triangles SAC and ACP SAC: ACP: dL/d = 0 Here we do so called ’paraxial approximation’: Refractive power of a spherical surface All rays will refract through point P at si which is independent on the angle . Derivation of the lens maker’s formula and thin lens equation Lens maker’s formula 1 f 1 n 1 R1 1 R2 Thin lens equation 1 s 1 s' 1 f (s=so and s’=si) Assumptions used during the derivation: and cos 1. paraksial approximation (sin 2. lens is thin 1; 1st terms in the series expansions) Basics of lenses – positive lens Refraction of a ray is proportional to the distance from the optical axis. Basics of lenses – negative lens Refraction of a ray is proportional to the distance from the optical axis. Sign rules for lenses Ri is > 0 if the centre point of curvature is to the right of the surface Also the terms ”positive” ja ”negative” lens are used. Principal rays of a lens (3 rays) 1. Ray propagates through the centre of the lens without changing its direction. 2. Collimated ray travels through the focal point behind the lens (virtual focal point in front of a negative lens) 3. Ray going through the front focal point will be collimated after the lens (back virtual focal for a negative lens). Magnification M of a lens M yi/yo From equivivalent triangles S1S2O ja P1P2O it follows: M = yi/yo = si/so Magnification M M = yi/yo = si/so 1 so 1 si 1 f When the magnification is equal to 1? Or where the object should be placed to get an image having the original size? f-number and numerical aperture N.A. f-number (also known as f/#) = f/D For example, f-number = 2 is the same as f/2 numerical aperture characterises the light gathering power of a lens N.A. = n sin D n D/(2f) = n / (2 x f/#) f Ideal lens Validity of the paraxial approximation Further away from the optical axis the paraxial approximation is not good enough anymore. Aberrations or imperfections of lenses Ignoring the third and higher order terms in the series expansion of the sine and cosine functions causes aberrations. Typical aberrations for a monochromatic system are: • spherical aberration • astigmatism • coma • field curvature • distortion. For light containing several colours there are additional aberrations: • chromatic aberration • lateral colour Spherical aberration Ideal lens shape for imaging is not a spherical surface. However, a spherical surface is the easiest and most affordable to manufacture. In a bi-convex lens, for example, the rays of light propagating far away from the optical distance will refract closer to the lens than paraxial rays. With modern manufacturing systems aspherical lenses can already be manufactured at a reasonable cost by pressing lenses out of melt glass or plastic using a mold. Aspheric surface can compensate for the spherical aberration. Astigmatism Means lens imperfecton where a non-paraxial ray has different focal points in the horizontal and vertical directions. Field distortion Lens can distort the image in many ways even though the image is sharp or the focusing properties are good. Field curvature Non-paraksial rays do not focus on a place surface. Chromatic aberration Is caused by the fact that the refractive index n is actually not a constant but depends on the wavelength/frequency of light = material dispersion. n = n( ) = n( ) Chromatic aberration in a bi-convex lens Compensation of chromatic aberration achromatic doublet lens Lateral colour Matrix formalism for ray tracing Matrix formalism for ray tracing T1 R1 1 d1 0 1 1 1 r1 T2 0 1 nL T3 1 d2 0 1 d3 0 1 1 R2 1 1 r2 0 nL 1 Typical transformation matrices Principal planes of an optical system Principal planes of an optical system Principal planes of an optical system Principal planes of an optical system Principal planes of an optical system Summary We have discussed the fundamental concepts of geometrical optics: • • • • • • • lens maker’s formula for a thin lens thin lens equation ray tracing: • principal rays of lenses • sign convention concepts: magnification, numerical aperture, f-number lens aberrations matrix formalism for optical systems reduction of an optical system into a thin lens: principal planes