aberration theory optical system designecee.colorado.edu/~ecen5616/webmaterial/10 aberration theory...

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Aberration Theory&

Optical System Design

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Contents• Introduction

• Apertures and Stops

• Paraxial Quantities and Terminology

• Rays as Wave Normals

• Transverse Ray Aberrations

• Wave Aberration

• Relationship between Wave aberration and Transverse Ray Aberration

• Longitudinal focal shift

• Transverse focal shift

• Equation of a wavefront

• Normalized form of T.R.A. formulae

• Wave Aberration function of axial object

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Aberration

Failure of an image forming system to produce a true image; i.e. a point object as a point image.

The reason of the failure is that real optical design work is based on Snell’s law

which for any reasonable aperture and field is a non linear function.

Five monochromatic aberrations were recognized by the German mathematician Ludwig Von Seidel and are referred to as third-order or Seidel aberrations, compared to first-order optics that is based on the Gaussianapproximation with which you can calculate object and image distances, EFL, magnification, the Optical Invariant, Cardinal Points, etc.

Two additional aberrations, chromatic aberrations, result from the wavelength dependence of the refractive index of glass materials. Purely reflective systems do not suffer from chromatic aberrations.

Introduction

'sin'sin θθ nn =

''θθ nn =

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IntroductionThe total of third-order aberrations are:

1. Spherical aberration

2. Coma

3. Astigmatism

4. Field curvature

5. Distortion

6. Longitudinal color (chromatic variation of focus)

7. Transverse color (chromatic variation of magnification)

The monochromatic aberrations can be further divided into two groups depending on whether or not they cause the image to deteriorate and become blurred. Image blur occurs in spherical aberration coma and astigmatism.

Distortion causes a displacement of the point image (resulting in deformation of an extended image) but no blur. Field curvature also results in a perfect point image provided the image surface is not restricted to lie on a plane.

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Apertures and StopsThe APERTURE STOP is defined to be that stop or lens rim which physically limits the solid angle of rays passing through the system.

Thus it controls the brightness of the image, as well as the resolving power of the optical system.

The Aperture Stop is not always the first element of the system.

The Image of the aperture stop in the object space is the Entrance pupil

The image of the aperture stop in the image space is the Exit pupil

The Field STOP is the aperture that controls the field of view. A field STOP is often located in the image plane (Photographic film, detectors) or it may be located in the object plane (as in the case of a slide projector where the opaque mount of the slide limits the field)

The Image of the field stop in the object space is the entrance window

The Image of the field stop in the image space is the exit window

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Apertures and Stops

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Apertures and Stops

Entrance PupilExit Pupil

ApertureSTOP

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Paraxial Quantities and Terminology

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Paraxial Quantities and TerminologyA marginal ray is that ray from the axial object point which passes through the very edge of the entrance pupil and therefore through the edge of the aperture stop.

The chief ray (or pupil ray, or reference ray or principal ray) is that ray from the edge of the field which passes through the center of the entrance pupil and hence close to the center of the stop (not necessarily exactly through its center because of pupil aberrations). The chief ray is very important because it is used as an origin of position and direction for aberration calculations in the image space.

In a pencil of rays from an object point at any ray other than the chief ray may be termed as an aperture ray.

The meridian section is that plane containing the object point and the optical axis.Rays laying in this section are meridian rays, other rays are skew rays.

Both chief and marginal rays are meridional rays. To distinguish them, heights and angles of the chief ray are barred, heights and angles of the marginal rays are unbarred.

Whenever a marginal ray crosses the optical axis an image is located.

Whenever a chief ray crosses the optical axis a pupil is located.

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RAYS AS WAVE NORMALS

In Figure 3, a spherical wave ABC is diverging from a point source O. The wavefronts are spheres with center at O, and OAP, OBQ and OCR, are ray normals to the wavefront at A, B, C . These rays meet an interface between media of refractive index n and n’ at the points P, Q, R.

The transmitted wavefronts will not generally be spherical, so that the wave-normals (rays) PA’, QB’, and RC’ will not pass through a single point.

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RAYS AS WAVE NORMALSWe may regard each ray as the path followed by the corresponding element of the wavefront.

The wave element at A travels along AP, then along PA’ to A’, and similarly for all other elements of the wavefront at ABC.

Now in wave optics, the refractive index, for a given wavelength, is the ratio of the velocity in vacuum, c, to the velocity, V in a given medium. Where

V=c/n

is the velocity of the wave in a medium of refractive index n. The time of travel from A to A’ is therefore given by

We define the product n(AP)=[AP] to be the optical path length of the segment (AP); and always using square brackets to denote the optical path length, we have for the time of flight from A to A’.

{ })'(')(1')'()(

' PAnAPncV

PAVAPtAA +=+=

]'...[1')'()(

' AAcV

PAVAPtAA =+=

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RAYS AS WAVE NORMALS

The wave disturbances at A, B and C are all in phase, since ABC is a wavefront. For A’B’C’ to be a wavefront, having disturbances of the same phase at A’, B’, C, the times of travel along APA’, BQB’ and CRC’ must all be equal. That is, the optical path lengths along all rays between the two wavefronts must be equal; for example,

[A…A’]=[B…B’]=[C…C’]

It is this method which is usually employed to find the form of a wavefront after transmission through any optical system. Rays are traced from each object point, and optical path lengths are calculated along them.

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Transverse Ray Aberration

Aperture ray

O

η

'η

O’

Chief Ray

'δη

'δξ

The intersection of the chief ray with the image plane is taken as the origin for measuring the transverse ray aberration (TRA). If the chief ray intersection is (O’, ) and that of an aperture ray is then the two rectangular components of the TRA of that aperture ray are . The point (O’, ) is sometimes called the geometrical image point.

'η )'','( δηηδξ +

)','( δηδξ 'η

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Wave Aberration

The shape of an imaging-forming wavefront for an aberration free system is a perfect sphere.

The departure of an actual wavefront from this ideal shape is a measure of the aberration of the system. The wave aberration associated with a ray is the optical path length between the wavefront and the Ideal or Reference Sphere measured along the ray. For this purpose the Reference sphere and wavefront are coincident at the exit pupil and the sphere is centered on the paraxial image point.

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Wave Aberration

In Figure 5 EBo is a sphere centered on O. The sphere E’Bo’ is centered on O’ (the paraxial image point) and is the reference sphere. The surface E’B’ is the aberrated wavefront; it is chosen to coincide with the reference sphere at E’. The wave aberration is the optical length [Bo’B’] and is taken to be positive when the wavefront leads the reference sphere.

Clearly

[E…E’]=[Bo…B’]

[Bo…B’]=[Bo…Bo’]+W

Thus W= [Bo…B’]-[Bo…Bo’]

This is the general expression for W which formed the basis of the calculations of W by subtraction of optical path lengths. In fact W sometimes is noted as OPD (Optical Path Difference)

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Relation between wave and TRA

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Normalized form of TRA formulae

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Longitudinal Focal shift

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Longitudinal Focal Shift

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Longitudinal Focal Shift

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Transverse Focal Shift – Tilt of Reference Sphere

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Transverse Focal Shift

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Transverse Focal Shift

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Wave Aberration Polynomial for a rotationally symmetric system

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From (3)

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Wave Aberration Polynomial

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0 Tilt of reference sphere

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and

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Petzval SumLet us assume that we have a spherical interface and that the object for the time being is lying on a spherical surface rather than a plane, and is ρas seen in the figure below.

Then by using the simple formula for a single surface

And substituting ρ=s-r and ρ’=s’-rr

nnsn

sn )'('' −

=−

n n’

r

ρ

ρ’object

Image

For example r=10, s=20

n=1 and n’=1.5

Then s’=-30

rnnnn

nn ''1

''1 −

=−ρρ

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Petzval Sum

For k surfaces it can proved that the Petzval sum is given by

And therefore for a thin lens with two surfaces and n=1 for air

Rearranging terms and taking ρ1=infinity we get that for a thin lens the

Petzval Sum=

∑ −=−

rnnnn

nn kk ''1

''1

1ρρ

]11['1'

''

''

''11

212112 rrnn

rnnnn

rnnnn

rnnnn

nn−

−=

−+

−=

−=− ∑ρρ

∑Φ=

'1

nρ

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Reading and HW

• New Homework is out on website

• Read Chapter 6 (aberrations) in Smith

aberration theory optical system designecee.colorado.edu/~ecen5616/webmaterial/10 aberration theory...

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