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Geometrical optics

Geometrical optics is based on ray-tracing. The approach assumes that the dimension of optical elements is much larger than light wavelength, i.e., l → 0.

Vocabulary- Conjugate points: object space and image space- Convex (converging, positive) lens vs. concave (diverging, negative) lens- Focal point and focal length / conjugate foci (front FL, back FL)- Real image vs. virtual image- Aberrations (longitudinal/lateral or traverse spherical, axial/longitudinal

chromatic, coma, astigmatism) - Diffraction-limited systems- Achromats (doublets, triplets)- Fraunhofer-cemented achromatic doublet (chromatic, spherical

aberrations, coma)

§Principles of image formation§Paraxial approximation§ABCD matrix§Geometrical optics for Gaussian beams§Aberrations§Eyes

Geometrical optics Geometrical optics: lens

Lenses form images by refracting light.

§ Lenses are transparent and allow light through it unlike mirrors that are highly reflective.

§ Lenses usually have two “faces”. Each face can be concave, convex, or plano

§ For lenses, there are 2 focal points – in front of and behind the lens.

§ Real images are behind the lens. (Images are opposite of the object)

§ Virtual images are on the same side the object is located.http://mrscgriffin.wordpress.com/ap-physics/light-geometric-optics/

Geometrical optics: lens

Lens formula (Lensmaker’s formula)

( ) ÷÷ø

öççè

æ--=

2

1

1

11

1

RRn

f fosis

111=+and

Optical center: rays passing through can be drawn as straight lines.

fo

fiso

si

Magnificationo

i

ss

M -=

Geometrical optics: lens

Sign convention

F-number:

typical camera lenses have markings of 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22 … (fast to slow)

Positive (+) Negative (–)R1 Convex lens Concave lensR2 Concave

lensConvex lens

so Real object Virtual objectsi Real image Virtual imagef Convex lens Concave lensM Erect image Inverted

image

Dff =/#

Geometrical optics: mirror and prism

Mirror: formula for spherical mirrors

Prism: Right-angle prism and dove prism

Rosis

211-=+

Fiber: numerical aperture (NA)

( ) ( )/#21sin max f

nNA =×= q

object between fand the lens (magnifying glass)

object at f

Geometrical optics: image formation (convex lens)

http://www.a-levelphysicstutor.com/index-optics.php

Geometrical optics: image formation (concave lens)

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Geometrical optics: image formation Geometrical optics: concave mirror

X) a ray parallel to the principal axis reflected through F (the principal focus)Y) a ray passing through C which is then reflected back along its original pathZ) a ray passing through F, which is then reflected parallel to the principal axis

Geometrical optics: image formation (concave mirror)

Reflection image from a spherical concave mirror

Geometrical optics: image formation (concave mirror)

The caustic is the name given to the region of a concave mirror where parallel rays of light come to different foci. This happens for rays away from the principal axis. The further away they are, the closer is the focus to the mirror.

A parabolic mirror produces one focus for all rays parallel to the principal axis, irrespective of their distance from it. Parabolic mirrors have uses in telescopes, solar furnaces, and car headlights/torches/floodlights.

Geometrical optics: concave mirror Geometrical optics: convex mirror

Geometrical optics: image formation (convex mirror) Catalogues Catalogues

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Catalogues Catalogues Catalogues

Catalogues Aberrations:

Chromatic aberration: - n is a function of frequency.- a lens having a different refractive index for different

wavelengths of light (the dispersion of the lens)

Aberrations:

Aberrations:Spherical aberration (SA):- Spherical surfaces in general yield perfect imagery only in the

paraxial region.- Image imperfection that occurs due to the increased refraction

of light rays that occurs when rays strike a lens or mirror near its edge, in comparison with those that strike nearer the center.

- Longitudinal SA- Traverse/lateral SA- Wavefront distortion

Aberrations:

Coma: - off-axis point sources such as stars appearing distorted. - defined as a variation in magnification over the entrance

pupil.

Astigmatism: an optical system has different foci for rays that propagate in two perpendicular planes

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Gravitational lensing Chambers, “Extrasolar planets: More giants in focus,“ Nature 405–406, 2010

A fresh analysis of data from gravitational microlensing surveys for planets orbiting stars other than the Sun finds that gas-giant planets similar to Jupiter are more common than previously thought.

a, When a faint star (red) passes in front of a more distant bright star (yellow), it focuses light from the distant object, causing it to brighten and fade over several weeks with a characteristic 'light curve'. b, The presence of a planet (brown) orbiting the nearer star causes additional brightness variations on a timescale of hours. Continual monitoring of a microlensing event can determine whether planets are present, and yields their mass and orbital separation from the parent star.

Geometrical optics: paraxial approximation

Paraxial approximation:- More general approach for ray-tracing throughcompound lenses

- Angular deviation of a ray from the cylindrical axis orthe axis of propagation (z) is small such that sinq ortanq can be approximated by q. Based on this, thedeflection of a ray by a thin lens can be described as

Geometrical optics: paraxial approximation

From ri = ro and ,frfrr ioi'' =+

', ii rr', oo rr

f

Focal plane

úû

ùêë

é×

úúû

ù

êêë

é-=ú

û

ùêë

é'' 11

01

i

i

o

o

rr

frr

Geometrical optics: paraxial approximation

Similarly, the propagation of a ray through a straightsection of a homogeneous medium of length d can betraced with a matrix

.

It can be easily shown that the matrix for the propagationof a ray through a straight section followed by a thinlens is given by,

.

úûù

êëé

101 d

( ) úûù

êëé

úû

ùêë

éúû

ùêë

é×-=-- 10

11101

111 d

ffd

f

d

Geometrical optics: ABCD (or ray transfer) matrix

The ray propagation through a series of opticalcomponents can be described with matrix products. Inequation form,

or

The matrix with the elements of A, B, C, and D is calledABCD matrix or ray-transfer matrix. Note that the abovediscussion inherently presumes plane wave.

''1

'1

nnn

nnn

DrCrrBrArr

+=+=

+

+

DCqBAq

DrCrBrAr

rr

qn

n

nn

nn

n

nn +

+=

+

+==

+

++ '

'

'1

11

Geometrical optics:

A thick lens = two spherical refracting surfaces +dielectric separation

- Lens formula still holds with effective focal length.

Analytical ray-tracing: use Snell’s law.

( ) ( )úû

ùêë

é -+--=

2121

11111RnRdn

RRn

f

Element Matrix Remarks

Propagation in free space or in amedium of constant refractive index d = distance

Refraction at a flat interfacen1 = initial refractive index

n2 = final refractive index.

Refraction at a curved interface

R = radius of curvature, R > 0 for convex (centre ofcurvature after interface)n1 = initial refractive indexn2 = final refractive index.

Reflection from a flat mirror Identity matrix

Reflection from a curved mirror R = radius of curvature, R > 0 for concave

Thin lens

f = focal length of lens where f > 0 for convex/positive (converging) lens.Only valid if the focal length is much greater thanthe thickness of the lens.

Single right angle prism

k = (cosψ/cosφ): the beam expansion factor, whereφ is the angle of incidenceψ is the angle of refraction, d = prism path length,n = refractive index of the prism material.This matrix applies for orthogonal beam exit.

Table of ray transfer matrices

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Catalogues Catalogues Geometrical optics: Gaussian beams

For a Gaussian beam, starting from Maxwell’s equationswith a few assumptions, it can be obtained that

where , , and

( ) ( ) ( )[ ] ( ) ( ) þýü

îíì

úû

ùêë

é+---=

zRik

zwrzkzi

zww

EzyxE2

1exp,, 220

0 h

( ) ÷÷

ø

ö

çç

è

æ+= 2

0

212

02

z

zwzw ÷

÷

ø

ö

çç

è

æ+= 2

201z

zzR ( ) ÷

÷ø

öççè

æ-=0

1tanz

zzh

l

p nwz 00 =

Geometrical optics: Gaussian beams

Also, note that k = 2pn/l. In this equation, w(z) and Rrespectively denote the beam spot size and the radiusof curvature of a Gaussian beam.

z axis

Phase fronts

Propagationlines

Geometrical optics: Gaussian beams

The divergence angle qG is given by

which is a manifestation of wave diffraction, indicatingtransverse spreading in the far field for z>>z0.

nwG0p

lq =

Geometrical optics: Gaussian beams & ABCD

The use of ABCD matrix is equally valid for Gaussianbeams, except that qn in the plane wave is replaced with

in the case of Gaussian beams. In other words, thepropagation of a Gaussian beam through an arbitrarynumber of lenslike elements can be traced by multiplyingABCD matrices and recovering the beam radius R(z) andspot size w(z). This is called the ABCD law.

( ) ( ) ( )znwi

zRzq nnn2

11p

l-=

Geometrical optics: eyes

- Photoreceptors (rods; scotopic, night, cones; photopic, day)- Accommodation: myopia (negative lenses),

hyperopia/hypermetropia (positive lenses), astigmatism (anamorphic lenses), cataract

θ: Angle subtended by an object at the eyeHi: Height of an image on the retina.We: the width of the eyeball between lens and retina

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Myopia (short-sightedness) Hypermyopia (long-sightedness) Floaters are deposits of various size, shape, consistency,refractive index, and motility within the eye's vitreous humour, whichis normally transparent. They may be of embryonic origin or acquireddue to degenerative changes of the vitreous humour or retina.http://www.wikipedia.org

Simulated image of floaters against a blue sky

Geometrical optics: refracting telescope

§ Angle subtended by the object: a = h/fo§ Angle subtended by the image: b = h/fe§ Angular magnification: Ma = b/a= fo/fe

Geometrical optics: terrestrial telescope

§ Additional convex lens to invert the intermediary image, while not affecting the image size.

§ The image is placed from the objective at 2f from the lens. § Another image is then produced at 2f on the other side of the lens. § This image is magnified by the eyepiece.

Geometrical optics: Galilean telescope

§ The Galilean telescope produces an erect image from a convex objective lens and a concave lens eyepiece.

§ The telescope is much shorter than both the astronomical and terrestrial telescopes.

§ Despite simple design, it is severely limited by having a very small 'field of view'.

Geometrical optics: reflecting telescope

§ Refracting telescopes are more compact than reflecting telescopes.§ Reflecting telescopes have mirrors that can be as large as a few

meters in diameter. Lenses are limited by their weight, which must be supported around the edges. Approximately one meter in diameter is the practical limit for a lens.

§ Mirrors can be modular.§ Mirrors do not produce chromatic aberration (no dispersion).§ Mirror glass need not be of as high a purity as a lens glass.§ Mirrors produce less spherical aberration than lenses.

Geometrical optics: Newtonian reflector

§ The primary image from the parabolic mirror is deflected at 45º from the principal axis by a planar mirror into an eyepiece located at the side of the telescope.

Geometrical optics: Cassegrain reflector

§ Light from the primary mirror is reflected backwards from a secondary, convex mirror and directed through the middle of the mirror. The image is further magnified by an eyepiece.

§ The Cassegrain reflector has an advantage over Newtonian by having a large f-number (ratio of focal length to primary mirror diameter). This allows much greater magnification to be attained.

§ Cassegrain reflector has a much shorter length of the telescope, typically less than half the length.

Geometrical optics: Schmidt-Cassegrain reflector

§ Along with the advantages of a Cassegrain reflector, a SCT reduces spherical aberration to a minimum using a 'corrector plate'. This is a specially designed lens, having properties of both convex and concave lenses.

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Hubble telescope Hubble telescope

Hubble telescope

Starry Bulges Yield Secrets to Galaxy Growth (Hubble and Ground-Based View)Astronomers have combined information from the NASA/ESA Hubble Space Telescope's visible- and infrared-light cameras to show the hearts of four spiral galaxies peppered with ancient populations of stars. The top row of pictures, taken by a ground-based telescope, represents complete views of each galaxy. The blue boxes outline the regions observed by the Hubble telescope.

http://www.spacetelescope.org/images/heic9902c/

CODE V, ZEMAX, TracePro, OSLO, BEAM3, …

telescope

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