optical systems - University of Arizonadial/ece425/notes8.pdf• Replace thin lens by effective...

$: optical systems - University of Arizonadial/ece425/notes8.pdf• Replace thin lens by effective refracting plane and trace rays ECE 425 CLASS NOTES – 2000 DR. ROBERT A. SCHOWENGERDT$
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s formed by

about a common

at any point to a for an

tics with no

θtan θ , θcos 1≅≅ ≅

DR. ROBERT A. SCHOWENGERDT [email protected]

GEOMETRICAL OPTICS

Used to find locations and sizes of imageoptical systems

• Assume:

• optical elements have rotational symmetry optical axis

• light paths are along rays, which are normalwavefront, a surface of constant time phaseelectromagnetic wave

• paraxial approximation:

• rays make small angles θ to optical axis

• plane or spherical wavefronts (“perfect” opaberrations)

θsin

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surface normal

α2 reflected ray

mirror


Reflection and refraction

• Snell’s Law of Reflection α1 = α2

α1incidentray

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Snell’s Law of efraction

where index of efraction

αsinβsin

-----------n2n1-----=

velocity of light in vacuumvelocity of light in medium-----------------------------------------------------------------=

cv--=


•R

r

• nair = 1

• 1.4 < nglass < 1.7

incident ray

refracted ray

αβ

index of refraction n1 index of refraction n2

refracting surface

surface normal

n2 > n1

n

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erent amounts

λ(nm)


Dispersion

• different wavlengths of light λ refract by diff

• typical dispersion from blue to green:

n

450 550

∼ 1%

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red (650nm)

blue (450nm)1

green (550nm)


• prism

n = 1 n = 1.5 n =

white light

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s

t distance of

n either side of


Thin Lens Image Formation

• thickness << radii of curvature of lens surface

• Gauss’ Formula (in air)

where

zi = image distance from lens (+ to right)

zo = object distance from lens (- to left)

focal length f = image distance for an objecinfinity; defines the focal point F

• thin lenses have two focal points, symmetric othe lens:

1zi---- 1

zo----- 1

f---+=

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iven:

image at ∞

image


• Ex: find focal length of a converging (+) lens, g

zi = fobject at ∞ image

F2

zo = -f

object

F1

zi = 3fzo = -6object

13 f------ 1–

6------

1f--- += f 4=⇒

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ft),

s

2

is distance

ion

etric to object)

image


• Newtonian Formula:

where s1 is distance of object from F1 (+ to leof image from F2 (+ to right)

• Ex: object 2f to left of + lens; find image locat

• Therefore, image is 2f to right of lens (symm

s1s2 f2

=

F2

s1 = fobject

F1

s2f

2

s1------ f= =

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image space and

10, ho = +5, zo = -

is inverted and 1/

zi

zo-----


Magnification

• lateral scale (normal to optical axis) between object space

• definition

where

hi = image height

ho = object height

• Ex: find location and size of image, given f = +50

• Gauss’ Formula —> zi = +12.5

• magnification m = 12.5/-50 = -0.25 (image 4 size of object)

mhi

ho-----= =

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2

+12.5-1.25


• image size hi = (-0.25)(+5) = -1.25

F-50

+5

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ject point to

t space,

al axis at lens

space, parallel to

and trace rays


Ray tracing

• Graphical way to find image location and size

• Any two of three rays needed from off-axis oblocate and size image:

• parallel ray - parallel to optical axis in objecintersects focal point F2 in image space

• chief ray - straight ray that intersects opticlocation (center of lens)

• focal ray - intersects focal point F1 in object optical axis in image space

• Replace thin lens by effective refracting plane

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ormula

?

fore doesn’t exist


• Use graph to check calculations from Gauss’ F

• What if the object is within the focal distance

• “virtual” image is left of the lens, and there

parallel ray

chief ray

focal ray

F1

F2object

image

parallel ray

chief ray

focal ray

F1

F2object

“virtual”image

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by another

e second lens,

tion), z1 = -4

s to right of lens

6 (2.86 units to


• virtual image can only be seen if re-imaged optical system, e.g. eye

Multiple, thin lenses

• Treat in serial fashion

• Image formed by first lens is the object for thand so forth

• Ex: 2 lenses, f1 = +3, f2 = +4, d = 2 (lens separa

• lens #1: Gauss’ Formula —> z2 = +12 (10 unit#2, “virtual” object)

• lens #2: z1 = +10; Gauss’ Formula —> z2 = 2.8right of lens #2)

Can you draw the ray trace for this system? (hint: find image of lens #1

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h

e


ignoring lens #2, and then re-image witlens #2)

• total magnification mtot = m1m2

Show that the total magnification of thabove system is mtot = -0.858

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rm spectral lass


Spherical Mirrors

• why use mirrors instead of lenses?

• folds optical path — makes smaller systems

• metallic reflecting coatings have more uniforeflectance over broad spectral range than g

• concave mirror

• converges parallel rays (f > 0)

• forms a real image

• convex mirror

• diverges parallel rays (f < 0)

• forms a virtual image

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vex) mirror)

measured in


• Spherical mirror equation

where R = radius of curvature

(R < 0 for + (concave) mirror, R > 0 for – (con

• Gauss’ formula for spherical mirrors

• real image has zi > 0 because image distancedirection of light propagation

fR–2

------=

1zi---- 1

zo----- 1

f---+ 1

zo----- 2

R---–= =

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equivalent“thin” mirrorreflectingplane (first-order)

zi = +2f/3

R


object@ zo = -2f

real image@ zi = +2f

Fcenter of curvature R

parallel ray

focal ray

+ mirror (f > 0)

– mirror (f < 0)object

@ zo = +2fvirtual image @

parallel ray

focal ray

F

chief ray

chief ray

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• Ex: concave (+) mirror, h = +2, zo = -10, R = -16

• From mirror equation, f = +8

• from Gauss’ formula, zi = +40

• therefore, image is real (left of mirror)

• magnification

• image height

m4010–

--------- 4–= =

hi mho 8–= =

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bject zo = -10

F

parallel ray

focal ray


• ray trace

o@

real image@ zi = +40

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ximation

, etc

icted by paraxial


Aberrations

• Expand sinθ in Maclaurin Series

• linear term in θ is paraxial, first-order appro

• cubic term in θ is third-order approximation

• Any deviations of real rays from the rays predapproximation are called aberrations

• Third-order abberations

• on-axis

• spherical aberration

θsin θ θ3

3!-----– θ5

5!----- …–+=

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ne ofnfusion”

paraxial focus

al focus


• focal length varies with ray height

pla“least co

margin

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ocus


• chromatic aberration (longitudinal)

• focal length varies with wavelength

blue light focus

red light f

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n2 > n1

ct n1 and n2 so that t focus at


• correctable at any 2 given wavelengths with an achromatic doublet

n1

n2

For example, selered and blue lighthe same point

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image of a pointsource

-like shape


• off-axis

• coma

• off-axis spherical aberration

paraxial focus

marginal focus

paraxial rays

marginal rays

comet

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ne orthogonal to off-axis angle) rays

S: sagittal focus

ngential focus

ptical axis


• astigmatism• different focal points for sagittal (in plaaxis angle) and tangential (in plane of off-

T: ta

off-axis point

o

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e, surface

Petzval) surface


• curvature-of-field• off-axis rays focus on a curved , not plan

focal (on-axis rays

off-axis rays

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l surface in correct

ortion

distortion


• distortion• off-axis rays do not intersect a flat focalocations

object

“barrel” dist

“pincushion”

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ays through a

ystem

nt) or within a multi-

ent of the image

y elements of the

ll elements before it


• Stops and pupils

• stops are physical apertures that limit the rsystem

• aperture stop

• controls the amount of light through a s

• usually located at the lens (single elemeelement system

• field stop

• limits the Field-Of-View (FOV), i.e. the ext

• usually located at the focal plane

• pupils are images of physical stops formed bsystem

• entrance pupil

• image of the aperture stop formed by a

• exit pupil

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ll elements after it

as an object and

perture stop at the

t at the image


• image of the aperture stop formed by a

• entrance and exit pupils are conjugate, justimage

• entrance and exit pupils coincide with the alens for a single, thin lens

• Defocus

• image is out-of-focus if receiving plane is nofocus

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th to aperture

in a design, for to N

to depth-of-focus

le d


• By geometry,

or

where N is the f-number (ratio of focal lengdiameter)

• For a given blur circle diameter (as specifiedexample), the depth-of-focus is proportional

• Depth-of-field is the object space equivalent

depth of focus ∆

blur circdiameter

aperturediameter D

f

Df---- d

∆ 2⁄----------=

∆ 2dfD

--------- 2dN= =

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ol the depth-of-y often use a low N f-focus

d (and et to the ’s image

be no more than and preferably less


• photographers use the f-number N to contrfield, e.g. when taking a portrait outside, the(say 2.8 or 4) to make the background out-o

• The maximum allowable blur circle diameterconsequently, the minimum N) is commonly smaximum acceptable spot size in the system

• For digital cameras, the maximum d shouldthe size of a single detector element (pixel),

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