Ray Tracing and Aberrations - astro.cornell.eduRay_Tracing).pdf · A6525: Lecture - 02 1 Ray Tracing and Aberrations Astronomy 6525 Lecture 02 Ray Tracing 2 A6525 - Lecture 02 Outline

A6525: Lecture - 02

1

Ray Tracing and Aberrations

Astronomy 6525

Lecture 02

A6525 - Lecture 02Ray Tracing 2

Outline Ray Tracing

Laws of Geometrical Optics

Sketching Rules Thin Lenses, Multiple elements, Mirrors

Meridional Ray Trace Examples, Special Rays, Limits

Aspheric surfaces

Optical Aberrations On-axis and off-axis

A6525: Lecture - 02

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Ray Tracing

Ray Tracing - Allows study of the performance of an optical system

via geometrical optics.

Different types of rays: Paraxial - rays very close to the optical axis

Marginal - ray at the edge of the entrance pupil

Meridional - rays restricted to a plane containing the optical axis

Skew - rays traveling in any direction


Laws of Geometrical Optics

Law of Transmission Light travels in a straight line in a region of constant refractive

index

Law of Reflection The angle of incidence equals the angle of reflection.

Snell’s law ni sin θi = nt sin θt

opticalaxisθt

θi

θini < nt

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Total Internal Reflection

When ni > nt then we can have

1sinsin >= it

it n

n θθ

Which can’t happen Total Internal Reflection

opticalaxis

θi

θi

ni > nt


Lens Sketching Rules

1. Place object to the left of an optical system and trace rays from left to right

2. A light ray parallel to the optical axis will pass through the focus of the lens (red line).

3. A light ray through the focal point will be refracted parallel to the optical axis (blue line).

4. A light ray through the center of the lens is undeviated (green line).

f

f

Start on left

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Tracing Multiple Elements

f1 f1f2 f2

upright,virtual image inverted,

real image

ConcaveLens

ConvexLens

Combinations or multiple elements: An image of an element becomes the object for the next element.


Mirror Sketching Rules

1. Place object to the left of an optical system and trace rays from left to right

2. A ray parallel to the optical axis will be reflected through the focus of the lens (red line).

3. A ray through the focus will be reflected parallel to the optical axis (blue line).

4. A ray reflected at the vertex makes an equal angle w.r.t. the optical axis (surface normal) as the incident ray (green line).

5. A light ray through the center of the curvature is reflected back on itself (maroon line).

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Meridional Ray Trace

A

rNQ

P

C BrU

U

I

I

z

n

n'

OpticalAxis I = angle of incidence

I , U, Q = incident rayI', U', Q' = reflected/refracted ray

Meridional ray specified by: U = slope angleQ = ⊥ distance from vertex

Given theseWant these (equivalent sketch to above)

CN // BP: UrIrQ sinsin +=

Snell’s law: InIn ′′= sinsin

Eq. (1)

Eq. (2)

UrQI sinsin −=


Meridional Ray Trace (cont’d)

I = angle of incidence

Meridional ray specified by: U = slope angleQ = ⊥ distance from vertex

∠ PCA: UIUIPCA ′+′=+=

Primed versionof equation 1:

Eq. (3)

Eq. (4)

IUIU ′−+=′

[ ]UIrQ ′+′=′ sinsin

Using equations 1-4 we can determine U' & Q' from U & Qof the incident ray and the surface data r, n, and n'.

A

rNQ

P

C BrU

U

I

I

z

n

n'

OpticalAxis

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Planar SurfaceThe case of a planar surface must be considered separately.

UQ

UQY

′′

==coscos U

UQQcos

cos ′=′

Now

This plus Snell’s lawU

nnU sinsin′

=′

Define Q' and U' for a plane.

C = 1/r = surface curvature

If C = 0, use plane surface equations.

zOptical

Axis

UI = U

YQ


Transferring between surfaces

Thus we have Q and U for the next surface, etc.Also need1) Start-up: transfer from “near-field” or “far-field” object2) Finish: transfer to focal plane

Q2

U'1

U'1 zOptical

Axis

Q'1

d

Now :112 sinUdQQ ′+′= 12 UU ′=and

See Kinglake, pages 24-29.

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Examples: Out of Focus

Y Y'

Y'

Y

• Slope tells how far out• Quadrant tells direction

(inside or outside)

Usually the distance, L, from the last vertex to the intersection of the ray with x-axis is plotted vs. Y.

Thus the focus is automatically “taken out”.


Examples: Spherical Aberration

Y'

Y

Y Y'

• Get quadratic term forspherical aberration

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Special Rays

Paraxial Ray A ray that always stays near the optical axis

small U’s and I’s sin I → I, cos I → 1, etc. for U Called First Order or Gaussian Optics

Linear theory - the ray traces equations now have algebraic solutions.

Marginal Ray A ray passing through the edge of the first element

Spherical aberration can be evaluated with one paraxial and on marginal ray.

LA = spherical aberration = Lmarg - Lparaxial


Limit: U = 0, Paraxial ray

A

rQ'

P

C L'

nn'

OpticalAxis

I

II ′−=

InIn ′′= sinsin

(1)(2)

UrQI sinsin −=

(3)

(4) [ ]UIrQ ′+′=′ sinsin

rQI /=We have (sin I ~ I, etc):

& rQIIU 222 =′−==′

( ) ( ) rIIIrUIrQ =+−=′+′=′ 2

22

1r

IIr

UUIr

UQL =

′−′

+=′

′+′=

′′

=′

L' = distance of imagefrom vertex

I'

nn ′−=&

&

IUIU ′−+=′

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Limit: U = 0

A

rQ'

P

C

nn'

OpticalAxis

I

II ′−=

InIn ′′= sinsin

(1)(2)

UrQI sinsin −=

(3)

(4) [ ]UIrQ ′+′=′ sinsin

We have:

& IIU ′−==′ 22

( )UIrQ ′+′=′ sinsin

( )

−−=

−=

′′

=′21

1

2

11

2sin

sin1

sin rQr

IIr

UQL

L' = distance of imagefrom vertex

I'

nn ′−=&

&

rQI =sin

L'IUIU ′−+=′

Notice that L' depends on Q, the height above the optical axis


Aspheric Surfaces An aspheric surface can expressed as a departure from a

sphere of curvature, c (= 1/r)

( ) ⋅⋅⋅+++−+

= 66

442/122

2

11YaYa

YccYX

If the surface is a conic section then we have

( ) 021 222 =+−− YrXXe

Where c is the vertex curvature and e is the eccentricity, Κ = - e2 is the conic constant

( )[ ] 2/1222

2

111 eYccYX

−−+=or

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Conic SurfacesSurface Eccentricity Conic Constant

e Κ = - e2

Hyperbola > 1 < -1

Parabola 1 -1

Prolate spheroid(small end of ellipse)

< 1 -1 < Κ < 0

Sphere 0 0

Oblate spheroid(side of ellipse)

-- > 1

abae

22 +=Ellipse: abae

22 −= Hyperbola:a = semi-major axis, b = semi-minor axis

For some plots and info on optical application of conics seehttp://www.telescope-optics.net/conics_and_aberrations.htm

A6525 - Lecture 03Optical Aberrations 20

Aberrations

On-axis: Chromatic

Spherical

Off-axis Coma

Astigmatism

Field curvature

Distortion

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Chromatic Aberration Spherical and chromatic aberrations are the only “on-axis” aberrations.

Chromatic aberration occurs for lenses only.

Blue Red

( )

−−=

21

111

1

rrn

f

( )λnn =

f

0.4 0.6 0.8λ (μm)


Optical Dispersion Note: dispersion and absorption are not just inconvenient,

but necessary consequences of refractive index and also necessarily larger for larger refractive index (c.f. Drude-Lorentz model and Kramers-Kronig relations)

A “perfect” glass with n=constant and no absorption does not satisfy Maxwell’s equations


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Longitudinal Spherical Aberration

Spherical Aberration: Light rays striking the entrance aperture different heights but parallel to the optical axis focus at different places.

Longitudinal Aberration (LA): LA = L - Lparaxial

Y

LA


Transverse Spherical Aberration

TA = distance of ray from the axis

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ComaThe effective focal lengths and transverse magnifications differ for rays transversing off-axis regions of the lens

Coma ∝ y2θ


Coma Diagram

For an oblique bundle of rays, coma occurs when the intersection of the rays is not symmetrical, that is, shifted w.r.t. the axis of the bundle.

Focal length and magnification are different for each “ring” on lens.

Want paraxial and marginal magnifications to be the same.

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Abbe Sine Condition For an optical system to be free of coma, it must obey the Abbe

sine condition which for a very distant object is:

CU

h =′sin

h = height of ray before it enters the system

U′ = angle between ray and optical axis as it travels towards focus

C = constant

h U′


Astigmatism

The foci in the tangential (meridional) and sagittal planes are at different location.

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Field Curvature

With a finite aperture the image plane is a curve.

Also called Petzval field curvature


Distortion

Happens if the transverse magni-fication is a function of the off-axis image distance.

Each point may be sharply focused but the image is distorted.

Introduction of a stop can cause distortion.

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Aberrations Expanding beyond the linear approximation give the third-

order Seidel aberration terms. The angular aberrations are:

322

2

2

3

3

θθθθdfcacs a

Rya

Rya

Rya

RyaAA ++++=

AA = angular aberration (e.g. arcsec or radians)ai = constantsR = radius of curvaturey = height of rayθ = angle of incidence of rays from object at infinity

distortion

field curvatureastigmatism

coma

spherical aberration


Aberrations (cont’d) Taking R = fny then:

322

23max θθθθd

nfc

na

nc

n

s af

af

af

afaAA ++++=

AAmax = AA for y = ymax = Dfn = f-numberD = aperture diameter

distortion

field curvatureastigmatism

coma

spherical aberration

Note: the “faster” theoptical system, thegreater the aberrations!

Documents

Ray Tracing and Aberrations - astro.cornell.eduRay_Tracing).pdf · A6525: Lecture - 02 1 Ray Tracing and Aberrations Astronomy 6525 Lecture 02 Ray Tracing 2 A6525 - Lecture 02 Outline