matrix methods, aberrations optical systems

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Matrix methods, aberrations & optical systemsFriday September 27, 2002

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System matrix

o

o

f

f yDCBAy

oof

oof

DCy

BAyy

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System matrix: Special Cases(a) D = 0 (a) D = 0 ff = Cy = Cyo o (independent of (independent of oo))

yyoo

ff

Input plane is the first focal planeInput plane is the first focal plane

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System matrix: Special Cases(b) A = 0 (b) A = 0 y yff = B = Boo (independent of y (independent of yoo))

oo

yyff

Output plane is the second focal planeOutput plane is the second focal plane

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System matrix: Special Cases(c) B = 0 (c) B = 0 y yff = Ay = Ayoo

yyff

Input and output plane are conjugate – A = magnificationInput and output plane are conjugate – A = magnification

yyoo

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System matrix: Special Cases(d) C = 0 (d) C = 0 ff = D = Doo (independent of y (independent of yoo))

Telescopic system – parallel rays in : parallel rays outTelescopic system – parallel rays in : parallel rays out

oo ff

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Examples: Thin lens

L

LL

nPd

nn

nP

nnd

nPd

TL2

1

12

1''

1

Recall that for a thick lensRecall that for a thick lens

For a thin lens, d=0For a thin lens, d=0

''

01

nn

nPL

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Examples: Thin lens

LnPPdPPP 21

21

'''

2121 f

nfn

Rnn

RnnPPP LL

Recall that for a thick lensRecall that for a thick lens

For a thin lens, d=0For a thin lens, d=0

In air, n=n’=1In air, n=n’=1

2121

11111'

11RR

nRn

Rn

ffP L

LL

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Imaging with thin lens in air

oo’’

ss s’s’

yyoo y’y’

Input Input

planeplaneOutput Output planeplane

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01

fL

10

Imaging with thin lens in air

)()'( sTLsTS

101

10'1 s

DCBAs

S

DCsC

DsCssBAsCsADCBA '''

''''

For thin lens: For thin lens: A=1A=1 B=0B=0 D=1D=1 C=-1/f C=-1/f

y’ = A’yy’ = A’yoo + B’ + B’oo

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Imaging with thin lens in air

y’ = A’yy’ = A’yoo + B’ + B’oo

For imaging, y’ must be independent of For imaging, y’ must be independent of oo

B’ = 0B’ = 0

B’ = As + B + Css’ + Ds’ = 0B’ = As + B + Css’ + Ds’ = 0

s + 0 + (-1/f)ss’ + s’ = 0s + 0 + (-1/f)ss’ + s’ = 0

For thin lens: For thin lens: A=1A=1 B=0B=0 D=1D=1 C=-1/f C=-1/f

fss1

'11

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Examples: Thick Lens

nn nnff n’n’

yyoo y’y’

H’H’

h’h’x’x’

f’f’

’’

h’ = - ( f’ - x’ )h’ = - ( f’ - x’ )

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Cardinal points of a thick lens

0'

' oyDCBAy

''

'

0

0

fyCy

Ayy

o

LnPdPPP

nfC 21

21'1

'1

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Cardinal points of a thick lens

oo CyxAy

xy

''

''

CAx '

CAfxfh ')''('

CA

CA

Ch 11'

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Cardinal points of a thick lens

'

1 1

nPC

nPdAL

Pn

ndP

CAh

L

'1' 1

Recall that for a thick lensRecall that for a thick lens

PP

nndhL

1'' As we have found beforeAs we have found before

h can be recovered in a similar h can be recovered in a similar manner, along with other manner, along with other cardinal pointscardinal points

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Aberrations

A mathematical treatment can be developed by expanding the sine and tangent terms used in the paraxial approximation

ChromaticChromatic MonochromaticMonochromatic

Unclear Unclear imageimage

Deformation Deformation of imageof image

SphericalSpherical

ComaComa

astigmatismastigmatism

DistortionDistortion

CurvatureCurvature

n (n (λλ))

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Aberrations: Chromatic

Because the focal length of a lens depends on the refractive index (n), and this in turn depends on the wavelength, n = n(λ), light of different colors emanating from an object will come to a focus at different points.

A white object will therefore not give rise to a white image. It will be distorted and have rainbow edges

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Aberrations: Spherical This effect is related to rays which make large angles relative to the

optical axis of the system Mathematically, can be shown to arise from the fact that a lens has a

spherical surface and not a parabolic one Rays making significantly large angles with respect to the optic axis

are brought to different foci

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Aberrations: Coma An off-axis effect which appears when a bundle of incident rays all make the

same angle with respect to the optical axis (source at ∞) Rays are brought to a focus at different points on the focal plane Found in lenses with large spherical aberrations An off-axis object produces a comet-shaped image

ff

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Aberrations: Astigmatism and curvature of field

Yields elliptically distorted imagesYields elliptically distorted images

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Aberrations: Pincushion and Barrel Distortion

This effect results from the difference in lateral magnification of the lens.

If f differs for different parts of the lens,

o

i

o

iT y

yssM will differ alsowill differ also

objectobject Pincushion imagePincushion image Barrel imageBarrel image

ffii>0>0 ffii<0<0

M on axis less than off M on axis less than off axis (positive lens)axis (positive lens)

M on axis greater than M on axis greater than off axis (negative lens)off axis (negative lens)

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Stops in Optical SystemsIn any optical system, one is concerned with a number of things

including:

1. The brightness of the image

S S’

Two lenses of the same Two lenses of the same focal length (focal length (f)f), but , but diameter (D) differsdiameter (D) differs

More light collected More light collected from S by larger from S by larger lenslens

Bundle of Bundle of rays from S, rays from S, imaged at S’ imaged at S’ is larger for is larger for larger lenslarger lens

Image of S Image of S formed at formed at the same the same place by place by both lensesboth lenses

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Stops in Optical Systems Brightness of the image is determined primarily by

the size of the bundle of rays collected by the system (from each object point)

Stops can be used to reduce aberrations

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Stops in Optical SystemsHow much of the object we see is determined by:How much of the object we see is determined by:

(b) The field of View(b) The field of View

QQ

Q’Q’(not seen)(not seen)

Rays from Q do not pass through systemRays from Q do not pass through system

We can only see object points closer to the axis of the systemWe can only see object points closer to the axis of the system

Field of view is limited by the systemField of view is limited by the system

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Theory of Stops

We wish to develop an understanding of how and where the bundle of rays are limited by a given optical system

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Aperture Stop

A stop is an opening (despite its name) in a series of lenses, mirrors, diaphragms, etc.

The stop itself is the boundary of the lens or diaphragm

Aperture stop: that element of the optical system that limits the cone of light from any particular object point on the axis of the system

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Aperture Stop: Example

OO

AS