2009 Class Summary Essay - Vineeth Abraham

OPTI 517 Lens Design Course Summary

Vineeth Abraham College of Optical Sciences University of Arizona 09 December 2009

OPTI 517 Lens Design: Course Summary

Contents Chapter

Page

1

-

What is Imaging

3

2

-

First order Optics

4

3

6

Overview of Aberrations

4

10

Reading Aberrations

5

16

Chromatic Aberrations

6

20

Diffractive Optical Elements (DOE)

7

22

Image Quality

8

-

Classical Lenses

26

9

-

Tolerancing

29

College of Optical Sciences, University of Arizona

2

OPTI 517 Lens Design: Course Summary

References

31

Chapter 1 What is Imaging? Maxwellian Definition: Every ray from a point of the object must converge to or diverge from a single point after passing through the system Image of a plane object normal to the axis will also be perpendicular to the axis Image in that plane will be similar to the object Collinear Transformation Definition: Defines imaging as a one-to-one mapping between object and image spaces The mappings are X '=

a1 X + b1Y + c1Z + d1 a0 X + b0Y + c0 Z + d 0

Y'=

a2 X + b2Y + c2 Z + d 2 a0 X + b0Y + c0 Z + d 0

Z'=

a3 X + b3Y + c3 Z + d 3 a0 X + b0Y + c0 Z + d 0

These are simplified for 1-D and 2-D cases (projection of a point to a point and a line to a line) X '=

a1;X a0 X + b0

X '=

a1 X a0 X + b0Y + c0

Y'=

a2Y a0 X + b0Y + c0

Symmetry: The collinear transformation equations are greatly simplified by assuming symmetries in the system. Plane symmetry (symmetric about the YZ plane) and Double plane symmetries (symmetric about both YZ and XZ planes) simplifies the system. The Axial symmetry case is the most useful. In axially symmetric systems, and z' are functions of and z. Assuming origins are ρ Z 1 ρ' 1 Z' Z' f' f = 1− m= = 1− m m = 1− + =1 f m 1− Z \ f f' f' Z' Z at the principal planes, we obtain the Gaussian equations. College of Optical Sciences, University of Arizona

3

OPTI 517 Lens Design: Course Summary

;

;

;

; Assuming origins at the focal points gives the Newtonian equations. Z 1 =f m

;

ZZ ' = ff '

;

Z' =-m f'

Scheimpflug Condition: When the object plane is not perpendicular to the axis, the image plane is also tilted in an axially symmetric system. The tilts of the object and image planes are related by the Scheimpflug Condition, tan(θ ') - Z'\Z tan(θ ) = 0.

Chapter 2 First order Optics In all current design works, both the paraxial and real rays are traced by computer programs. But paraxial raytracing is still used in many design tasks since it gives us substantial information about the system by tracing just two paraxial rays, the chief ray and the marginal ray. Hence it is important to learn first order optics. Paraxial Image Formation In the paraxial region, Snell’s law is given by: n i = n’ i’ Using this relation, we can write the power of a surface as: ϕ = (n’ – n)C = (n'-n)r

The unit of power is Diopters. This allows us to write the equations of a refracted paraxial ray in the form: n’u’ = nu – yϕ Refraction equation y’ = y + u’t’ Transfer equation Using these equations for an arbitrary point located at z in object space and its image z’ in the image space, we obtain the relations, n'z' + nz = ∅ And the relations for the front and back focal distances are, College of Optical Sciences, University of Arizona

4

OPTI 517 Lens Design: Course Summary

f’ = n'∅ and f = - n∅ Also, Transverse magnification, m=nn' z'z Invariants Optical Invariants are quantities that are non-changing on refraction and transfer. These are given by, n’u’y – n’u’y = nuy - nuy Invariant on Refraction n’u’yτ – n’u’yτ = nuy - nuy

Invariant on Transfer

In the above equations, the two rays are arbitrary. But when these rays are the Marginal and Chief rays, the invariant is called the Legrange Invariant. It is denoted as, Ж = nuy – nuy The Aperture Stop A stop in the system that limits the amount of light passing through the system is called the aperture stop. The image of this aperture stop in object space is called the Entrance pupil and the image in image space is called the Exit pupil. The aperture stop introduces order into the system by controlling stray light and unwanted fields. Also, in any system, the exit pupil is an important point of consideration since it is the ideal point to define wavefront deformation and diffraction effects are minimized at this point.

College of Optical Sciences, University of Arizona

5

OPTI 517 Lens Design: Course Summary

Cardinal Points The main cardinal points used in optical systems are: Focal points: Conjugates of the infinite points. Conjugate of infinity in object space is the rear focal point (F’) and conjugate of infinity in image space is the front focal point (F). Nodal points: They are conjugate points that are the centers of perspective. A ray passing through one nodal point will pass through the other nodal point too (N and N’) Principal points: They are conjugate points too of unit magnification (P and P’)

College of Optical Sciences, University of Arizona

6

OPTI 517 Lens Design: Course Summary

Chapter 3 Overview of Aberrations Aberrations are interesting phenomenon which can help in design of optical systems. Aberrations can be looked into using ray and wave approach. The wave approach helps to understand superposition of errors more easily. Aberrations are caused due to the deformation of the ideal wavefronts. Ideally wavefronts are spherical and homocentric. Hence, wavefront deformation is measured as the distance between the reference wavefront and the aberrated wavefront along the actual ray path at the exit pupil.

Now if we look at on-axis images, we realize that our system is axially symmetric. So we can have only quadratic and 4th order terms in the aberration. These two terms give rise to Focus and Spherical aberrations, which are only on-axis aberrations possible. But when we go off-axis, we reduce our symmetry to plane symmetry and brings up the other basic aberrations. Wave Aberration Function The wave aberration function is a function of the scalar product of the field and aperture vectors expressed in the form, W(H,ρ,cosθ) = k,l,mWk,l,m Hk ρl cosmθ College of Optical Sciences, University of Arizona

7

OPTI 517 Lens Design: Course Summary

The various aberration functions are also called Seidel sums since they were defined by Seidel. They are: Spherical Aberration Coma -----------------Astigmatism --------Field Curvature ----Distortion -----------Axial Chromatic Aberration ∂λ W020 = 12 CL C L = Ay ∆(∂nn) and ∆(∂nn) = n-1ν n Longitudinal C A ∂λ W111 = CT CT = Ay ∆(∂nn) and ∆(∂nn) = n-1ν n The variables used in the equation are, A = nu +nyc = ni A = nu + nyc = n i and c = 1r P = c ∆(1n) ν = (nd- 1)(nf- nc) Aspheric Surfaces Aspheric surfaces can be considered as caps on the spherical surface. So the aberrations contributed by aspheric surfaces can be modeled separately. The nett aberration will be calculated as the sum of the aberrations by the spherical surface and the aspheric cap.

The aberration contributions by the aspheric cap is given by, College of Optical Sciences, University of Arizona

8

OPTI 517 Lens Design: Course Summary

Stop shifting Stop shifting refers to changing the position of the stop from the element to another point along the optical axis, but at the same time, changing the size of the stop to maintain the f-number of the system.

Changing the stop position affects the off-axis aberrations. The variation of aberrations with stop shifting is given by,

Here, δy is calculated at the original stop position for convenience since here, the initial y will be zero and hence δy = y after stop shifting. College of Optical Sciences, University of Arizona

9

OPTI 517 Lens Design: Course Summary

Structural coefficients This is when the Seidel aberration coefficients are written in terms of y, the marginal ray height at the principal plane, ∅, the power and ℵ, the Legrange invariant. σ are the structural coefficients.

For a thin lens, these can be written as,

Bending of a lens

College of Optical Sciences, University of Arizona

10

OPTI 517 Lens Design: Course Summary

Bending is the process by which the power of the lens is maintained but the shape factor, X as defined above, is varied. This is useful since the aberrations depend on the shape factor. For example, spherical aberration is a quadratic function of the shape factor and coma is a linear function.

Bending of a lens

College of Optical Sciences, University of Arizona

11

OPTI 517 Lens Design: Course Summary

Chapter 4 Reading Aberrations Since the lens design programs display the performance of the system using OPD curves, it is very important that the user should know what the aberrations are from the OPD curves. The good thing about wave fans is that knowing the aberrations in terms of the waves helps to identify the diffraction limits very easily. The shape and nature of the various fourthorder aberrations are as given below: W020 Defocus Defocus is a second order aberration and is caused due to the rays coming to focus at different points along the optical axis from different points in the aperture. Thus it is a variation of focal length with aperture. It does not have any field dependence. W020 ∝ ρ2 The wave fan shape for 1 wave of defocus is shown below.

All pictures courtesy of Sample report by Matthew Lang

W111 Tilt This term represents the tilt in the image plane and has a linear dependence on Field and Aperture. W111 ∝ H ρ cosθ

College of Optical Sciences, University of Arizona

12

OPTI 517 Lens Design: Course Summary

W040 Spherical Aberration Spherical aberration (SA) is a fourth order aberration caused by the variation of focal length with aperture and it has a quadratic dependence on the aperture and varies as the third power of optical power. Spherical aberration and Defocus are the only aberrations that are possible on-axis. W040 ∝ ρ4

Correction of Spherical Aberration ➢ Focus can be used to compensate spherical aberration. But this makes the system very sensitive since the depth of focus becomes very shallow. ➢ Splitting the lens – Since SA varies as the third power of optical power, it can be reduced by splitting the lens. By splitting a lens into two equal halves, we reduce the power by half and the SA to a quarter. ➢ Bending the lens – Since SA is a quadratic function of the shape factor of the lens, contro; of SA is also done by varying the shape factor by keeping the power constant, which is known as lens bending. ➢ Refractive index – A higher refractive index means the lens can afford to have lesser curvature for a constant power. Thus a smaller curvature translates to lesser SA. ➢ Aplanatic surface – An aplanatic surface is a surface which satisfies the condition, ∆(un) = 0. As a result SA will be zero as this term is present in the seidel sum expression of SA. ➢ Merte Surface – A merte surface is a system with a strong index break and a strong radius of curvature. But the system has nearly no optical power. ➢ Using an aspheric surface - An aspheric surface with a conic constant equal to the index of refraction of the medium can completely remove SA. ➢ Meniscus lens – A meniscus lens with nearly zero power can bend the rays at the edge of the pupil outwards which helps to counter the effect of spherical aberration.

College of Optical Sciences, University of Arizona

13

OPTI 517 Lens Design: Course Summary

W131 Coma Coma is an off-axis aberration and is caused by the variation of magnification with pupil position. The field and aperture dependence is given by, W111 ∝ H ρ3 cosθ

From the seidel sum expression, we can deduce the ways to eliminate coma.

Correction of Coma

➢ Stop location – Placing the stop at the natural stop position eliminates coma. It uses the fact that y=0 at the stop position. At the natural stop position, y=0 for both on-axis and off-axis beams. ➢ A or A can be made zero by having the surface concentric to wavefront. Under both of these conditions, coma will be zero. ➢ Coma will be eliminated also if ∆un= 0, which means the surface is aplanatic. These four cases are shown in figure below, where, B = A ➢ Bending is another method for reduction of coma.

College of Optical Sciences, University of Arizona

14

OPTI 517 Lens Design: Course Summary

Abbe Sine Condition – Abbe proposed that the condition for zero coma is

that the paraxial magnification be equal to the real ray marginal magnification since coma can be considered as variation of magnification with pupil position. This can be formed into an expression as shown below. uu' = sinUsinU' where u and u’ are the paraxial ray angles and U and U’ are the real ray angles W222 Astigmatism Astigmatism is a non-axially symmetric aberration since it has different aberration characteristics along the two axes (tangential and sagittal). Astigmatism results in the image coming to focus at two different points for the tangential and sagittal fields. As a result we get two field curves. When the tangential and Sagittal field curves are separated, we can say that there is Astigmatism in the system. It is very difficult to correct and normally, the tangential field curve is taken as the imaging plane for calculation purposes.

College of Optical Sciences, University of Arizona

15

OPTI 517 Lens Design: Course Summary

Correction of Astigmatism As seen from the seidel sum expression for Astigmatism, there is three ways to eliminate astigmatism. ➢ Y = 0 by making the image plane at the surface ➢ Make A zero by making the surface concentric with the incoming wavefront. ➢ Make the surface aplanatic which will satisfy ∆un= 0 The figure below explains these three cases.

➢ Astigmatism is also eliminated by placing the stop at the natural stop position. Thus as in previous case, y =0 and hence at the natural stop position, we have zero coma and astigmatism. ➢ Being an odd aberration, astigmatism can also be corrected by introducing negative astigmatism to cancel out the positive astigmatism. This a done in the cooke triplet as shown in figure below

W220 Petzval Field Curvature

College of Optical Sciences, University of Arizona

16

OPTI 517 Lens Design: Course Summary

Field curvature occurs due to the thickness variation of the element. As a result, the rays are focused onto a spherical surface instead of a plane surface. This spherical surface which has the optimum image is known as the petzval field. Thus we can say that if we compress all the elements of the system and nett glass thickness is a parallel plate, then we will not have any field curvature. This principle is used in most of the correction methods.

Correction of field Curvature

There are four classical ways for the correction of field curvature. ➢ A thick meniscus lens can be used since it contributes optical power even when both surfaces have the same radii of curvature. Thus we have a curved parallel plate with power. ➢ Separated thin lenses – Having a separated system of thin lenses that introduces bulges and constrictions into the beam path helps to reduce field curvature. Basically, when all the components of the system are compressed together, we will get a parallel plate of glass. ➢ Using a field flattener lens near the image plane helps to introduce field curvature which helps to cancel the field curvature of the system. But this lens do not contribute to spherical aberration, coma or astigmatism which is very useful. ➢ New achromat – The design of the achromat can be modified to get system with nearly similar radii of curvature for both surfaces. This design is based on the expression of sag of the petzval surface with a higher refractive index for the crown and a lower refractive index for the flint. 1ρ'k = ∅n

College of Optical Sciences, University of Arizona

17

OPTI 517 Lens Design: Course Summary

➢ Another method of field correction is by using a curved image plane. But mounting a ffilm on a curved surface is always very hard and expensive since special kinds of films and mounts might be needed. W311 Distortion Distortion is caused due to the change of magnification with field. It has a cubic dependence on the field. It can be thought of as the spherical aberration of the chief ray. Hence most of the correction methods applied for spherical aberration can be used here too. The two classifications of distortion are Barrel Distortion and Pincushion distortion which are shown below.

As can be seen, due to the change in magnification, the squares at the edges are either magnified or demagnified.

Correction of Distortion College of Optical Sciences, University of Arizona

18

OPTI 517 Lens Design: Course Summary

➢ Symmetry – Since distortion is an odd aberration, we can cancel it by using symmetry about the stop ➢ An aspheric can be used near the image plane that compensates for distortion ➢ A field flattener near the image plane can be bend to reduce distortion since distortion is affected by lens bending

College of Optical Sciences, University of Arizona

19

OPTI 517 Lens Design: Course Summary

Chapter 5 Chromatic Aberrations Chromatic aberration in general refers to the change in lens aberrations as a function of wavelength. In general we can define two first order aberrations that change with wavelength; the focus, W020 and magnification W111. The former is called axial chromatic aberration and the latter, lateral chromatic aberration. Since refractive index is the most important factor when talking about chromatic aberration, we will now look at the properties of various glasses. Properties of Glasses The important properties of glass are: Refractivity: nd – 1 Mean Dispersion: nF – nC Partial Dispersion: nd – n C Abbe number ν = Partial Dispersion Ratio P = The standard for specifying glass type is using six digits: abcdef where, nd = 1.abc and ν = de.f Glasses are also divided into many categories based on these properties. • Crown Glass – has a ν > 50 • Flint Glass – has a ν < 50 • Normal glass – it is made up of soda-lime, silica and lead. The glass catalog describes the two important properties of glass, the nd/ν plot. Shown below is typical glass catalog from the manufacturer [1]

College of Optical Sciences, University of Arizona

20

OPTI 517 Lens Design: Course Summary

Also, it is impossible to find out the refractive index of a glass for all wavelengths. So interpolation formulas like Hartman, Conrady, KettlerDrude, Sellmeier, Herzberger and Schott’s own formula are used which calculates ‘n’ from known data points. Third order theory We can now define the actual variation of first order coefficients with change in wavelength. •

δλ W020 = 12 Ay* Δ(δnn)

•

δλ W111 = By* Δ(δnn)

where, δnn = (n-1)ν n

Also, on stop shifting, the change in axial and lateral chromatic aberrations will be, •

Δ δλ W020 = 0 unchanged)

(This is because, on stop shift, the on-axis beam is

•

Δ δλ W020 = 2 (δyy) * δλ W020

And, for a thin lens, these can be written as, •

δλ W020 = 12 y2f ν

•

δλ W020 = y*yf ν College of Optical Sciences, University of Arizona

21

OPTI 517 Lens Design: Course Summary

Achromatization It is the process of eliminating chromatic aberration which is normally done for two wavelengths, the F and C wavelengths. Achromatic Wedge: For a wedge, the desired condition would be deviation without any dispersion. Two wedges are used normally, one being crown glass and the Deviation other a flint glass. Dispersion Secondary dispersion

Hence,

and,

Achromatic Doublet (treated as two wedges): An achromatic doublet lens can be designed by assuming that the 2 lenses are wedges. But this assumption breaks down for very fast lenses. The design equations for the two lenses are:

Single glass achromats There are a various forms of achromatic devices that achieve chromatic correction using only one type of glass. Some of them are:

College of Optical Sciences, University of Arizona

22

OPTI 517 Lens Design: Course Summary

The Huyghenian eyepiece – It has two plano convex lenses arranged as shown

The image is formed in between the lenses and the marginal ray height changes from positive at the first lens to negative at the second lens. Thus by designing the values of y, f, and ν, appropriately, we can cancel lateral chromatic aberration.

The Maksutov meniscus – This design makes use of the fact that a meniscus lens can be thought of as a plano convex lens, a parallel plate and a plano concave lens. Since the focal lengths are opposing, we can make the axial chromatic aberrations cancel.

Shupman dialytes – A schupman dialyte has a positive and a negative lens combination as shown below

Here also the focal lengths are opposite while the marginal ray height will be positive since it is the square. So the objective achieves axial chromatic aberration correction. But the problem with this design is that the image is virtual and in between the lenses. This is not very useful for a telescope

College of Optical Sciences, University of Arizona

23

OPTI 517 Lens Design: Course Summary

objective. Hence the design was modified to develop the Schupman medial telescope. The Schupman medial telescope works on the same principle as a Schupman dialyte. But the difference is that, to deal with the diverging beam from the negative element, the exit surface if the element is mirrored to focus the light back. This negative lens-mirror combination is called a Mangin mirror. The setup is shown below which corrects for axial chromatic aberration.

But this design can be improved by adding a field lens at the intermediate image plane. In such a configuration, y = 0 at the Objective and the Mangin mirror and y = 0 at the field lens. As a result, the system gets corrected for lateral chromatic aberration also. Chromatic correction techniques

Achromatize all elements - This is the easiest design solution but it makes the system redundant since we are varying a lot of parameters when we need to have only two degrees of freedom to correct for the two chromatic aberrations.

Buried Surface – In the buried surface technique, an achromat with nearly the same nd but different abbe numbers is used.

College of Optical Sciences, University of Arizona

24

OPTI 517 Lens Design: Course Summary

Phantom stop – In the phantom stop technique, we use some of the properties of axial and lateral color to control the two aberrations. Firstly, since lateral color is an odd aberration, we place the stop at such a position so that the symmetry of the lens system will cancel the lateral chromatic aberration. So, CL = 0. Now at this position, we correct for axial chromatic aberration. Once we obtain minimum axial chromatic aberration, we shift the stop to its original position. Since stop shifting does not change Lateral color and as the value of CL at the previous position was zero, both lateral and axial colors remain the same.

College of Optical Sciences, University of Arizona

25

OPTI 517 Lens Design: Course Summary

Chapter 6 Diffractive Optical Elements (DOE) Commonly used terminology: Kinoform – This is a phased Fresnel lens. The individual zones of the diffractive element is nicely arranged and the phase modulation is achieved from the topography of the element. Binary optics – These are diffractive elements made by electronic fabrication techniques, which mean that the ideal staircase is replaced by staircases. Hybrid lens – It is a lens that uses both refractive and diffractive properties to deviate the light. Fresnel lens A Fresnel lens is different from a diffractive optical element. Some of the differences are: • Scale is larger – the size of the zones is in the order of mm • Wavefronts are discontinuous – There appears to be breaks in the wavefront after diffraction • Due to the broken wavefronts, the pattern at the image plane will be a random pattern instead of an airy disc Fresnel Zone Plate: A Fresnel zone plate is used amplitude modulate the incoming wave and focus it to a point. Rings which contribute with destructive interference are made opaque. The radius of the rings in a Fresnel zone plate for a focal length of f is given by: rn ≈ nλf Amplitude and Phase: Amplitude modulated DOE will modulate the light by placing opacities in the path of light. But this results in loss of light. A phase modulated design is more efficient as it uses the phase modulation produced by the glass of a particular thickness. Blaze is defined as the geometry that is used in phase modulated elements. It has a saw-tooth structure in most elements. It determines the amount of energy that goes into each order. Zone boundary determines the shape of the wavefront emerging Chromatic properties For a refractive lens, ∅refractive = n-1R College of Optical Sciences, University of Arizona

26

OPTI 517 Lens Design: Course Summary ∂∅=∅d nf- ncnd - 1 ϑrefractive = ∅∂∅

For a diffractive lens, ∅diffractive = ∅refractive λreconstructionλconstruction ∂∅diffractive = ∅refractive ∂λreconstructionλconstruction ϑdiffractive = ∅diffractive∂∅diffractive

Modeling DOE Two point construction model This was done by having two point sources and letting the waves interfere. The resulting pattern will emulate a DOE. This was done in the earlier lens design programs where the user could specify the two points, the wavelength and the order at which work is done. The order ‘m’ and the wavelengths at which work is done, λconstruction and λreconstruction can be specified

Point B(x,y,z)

Point A(x,y,z)

Phase Model In this method, the deformation introduced by the DOE is modeled by a polynomial of the form, where ρ is the radial distance from the axis. Sweatt’s Model Sweatt’s model was first introduced to overcome the problem of modeling DOE in lens design programs. He found that by increasing the refractive College of Optical Sciences, University of Arizona

27

OPTI 517 Lens Design: Course Summary

index to a very high value, a DOE behaves similar to a refractive element. Thus, by assuming a very high ‘n’ for a diffractive element, it could be modeled in a conventional program used to model refractive elements. Aberrations in DOE The aberrations introduced by DOE can be found by substituting the Sweatt’s model in the structural coefficients. Thus by assuming refractive index to be infinity in the equations for the structural coefficients of a thin lens, we can obtain the expressions for diffractive aberrations. Efficiency The ideal curve of a DOE is nearly impossible to fabricate. Hence it is constructed in multi-level microstructures. The efficiency of the DOE is dependent on the number of levels used for fabrication of the curve. This dependence is given by the formula, η1N = [sinπN(πN)]2 where N is the number of levels and 1 is the diffraction order

College of Optical Sciences, University of Arizona

28

OPTI 517 Lens Design: Course Summary

Chapter 7 Image Quality Image quality is important since it decides the resolution and hence the readable information content in an image. Image Quality (IQ) Specifications It can be based on Geometry, Diffraction or other parameters like uniformity, F-theta linearity etc. But there should always be a spec when designing. Always work towards end-product specs based on performance, like MTF, ensquared energy etc. but it is very hard to optimize based on these. Hence design can be done using IQ metrics specs like ray aberration plots and field plots. Choice of the correct IQ metric depends on the application. For example, in astronomical telescopes the object is a point source and ensquared energy or RMS wavefront might be appropriate. Wavefront error For error budgeting, it is always better to utilize RMS error rather than peakto-valley error.

(Figure from lecture by Richard Juergens)

Surface error College of Optical Sciences, University of Arizona

29

OPTI 517 Lens Design: Course Summary

The size of a bump is defined in wave terms where, σ = Δt/λ , where Δt is the physical size of the bump and λ is th wavelength of light. Hence the wavefront error produced by a bump will be, WFE = σ Δn where Δn is the difference in refractive index between the two media and for a mirror, WFE = 2σ. Rayleigh Criterion Image quality is acceptable if wavefront error does not exceed λ/4 peak-tovalley. This corresponds to 0.07 waves of RMS error (Φ) and a Strehl ratio of 0.8 Strehl ratio = e-(2πΦ)2 Ray Aberration curves Maps the change in image position of the rays in a fan from the chief ray image position v/s the pupil position of the rays. Plots are done in the tangential direction (y-axis) and the sagittal direction (x-axis). (Figure from lecture by Richard Juergens)

It can be Transverse or Wavefront ray aberration curves. The former is a derivative of the latter. It is the same data presented in a different form. For example, below is the wavefront ray aberration curve and transverse ray aberration curve of a lens having spherical aberration.

College of Optical Sciences, University of Arizona

30

OPTI 517 Lens Design: Course Summary

Spot Diagrams The problem with ray aberration curves is that it does not give any information about the skew rays. Hence we use spot diagrams. But the problem with spot diagrams is that it does not give information about the intensity, if more than one ray is falling at the same point. Diffraction The Point Spread Function is used when diffraction also comes into the picture. PSF gives the image of a point object. Diffraction makes the PSF of a point object into an airy disc. But the effect of aberrations is to spread the energy from the central order to the outer rings. This is shown in the figure below. Aberrations also change the shape of the PSF. But at 0.8 Strehl ratio, the PSF due to the different aberrations are comparable. (Figure from lecture by Richard Juergens)

Modulation transfer Function (MTF) College of Optical Sciences, University of Arizona

31

OPTI 517 Lens Design: Course Summary

The Modulation Transfer Function (MTF) is a measure of the image quality of a system based o the size of the object which is expressed in terms of the spatial frequency. For computing the MTF, a sine wave target is used and the amount of modulation in the sine wave target is plotted for different spatial frequencies. This is clear from the image shown below.

11:12:33

Hence for an aberration free system with a round pupil, the MTF will be of the form, DIFFRACTION MTF DIFFRACTION LIMIT

MTF ( f ) =

WAVELENGTH 500.0 NM

WEIGHT 1

2 λf and 0.00000 f is the spatial frequency in [ϕ − cosϕ1.0sinϕ ] where ϕ13-Oct-02 = cos−1 ( f / f co ) = cos−1 ( )DEFOCUSING π 2 NA lpmm AXIS

Thus, we get an MTF of the form given below. 0.9 0.8 0.7

M O 0.6 D U L 0.5 A T I O 0.4 N

Cutoff frequency fco = 1 / (λ f/#)

0.3 0.2 0.1

50

150

250

350

450

550

650

750

850

950

SPATIAL FREQUENCY (CYCLES/MM)

An example MTF curve for a normal system would be as shown below. As shown, there are separate curves for sagittal and tangential fields. This is due to the fact that coma is present only in the tangential direction. As a result, the tangential MTF deteriorates more than the sagittal MTF.

College of Optical Sciences, University of Arizona

32

OPTI 517 Lens Design: Course Summary

Mathematically, MTF is defined as the autocorrelation of the pupil error function. So, as the spatial frequent is increased there will be a point at which the MTF reaches zero. This frequency is called the cut-off frequency fc.

College of Optical Sciences, University of Arizona

33

OPTI 517 Lens Design: Course Summary

Chapter 8 Classical Lenses Landscape lens The landscape lens, used in the Brownie camera is a simple one element plano convex lens. The stop is places at the natural stop location such that the first surface is aplanatic and the second surface is concentric with the stop. This helps to reduce aberrations like spherical aberration, coma and astigmatism. But the lens has a large amount of field curvature and since it is very hard to image on a curved surface, the landscape lens is not an optimal solution. So the next step was to correct for the field curvature. But the number of degrees of freedom at our disposal is very few. So Wollaston came up with the idea that by moving the stop from the natural stop position to introduce negative astigmatism to flatten the tangential field. Another solution for the Landscape lens is to put the stop behind the lens. Even though this solution is not as good as the stop-in-front design, the lens at the front helps to protect the stop, shutter and other camera parts from the outside elements. The lens-in-front solution has a spot radii of around 40-160 µm for an HFOV of 30º for an f/15 design. Even though this is not a perfect solution, it works pretty well for landscape cameras due to the large depth of field and fairly large FOV. The image quality is also enough to produce spot sizes on the photograph which are smaller than the human visual resolution limit of 1 arc minute. Chevalier lens Chevalier had designed a double glass solution which corrected for chromatic aberration in telescopes, which was naturally brought over to the photographic realm to improve the chromatic performance of the landscape lens. The Chevalier lens was an achromatic solution having the crown and flint glasses put together and it even reduced spherical aberration and coma. But field curvature, distortion and astigmatism were issues of this lens and the image quality was poor. Also size of the lens was a problem since for

College of Optical Sciences, University of Arizona

34

OPTI 517 Lens Design: Course Summary

a 200mm focal length a lens of 60mm diameter was needed. But the lens helped in the taking the design of the achromatic doublet one step further. Petzval portrait lens The problem of the portrait lens was that it was slow. At f/15, it was very hard to take photographs of people due to the movement of the subject. Hence, Petzval designed an f/3.3 lens system which was fast enough to take portrait photographs. It consisted of two elements which were achromatised individually with the stop in between. Splitting the lens helps to reduce spherical aberration and the symmetry about the stop helps to control coma. Distortion was in control since the lens had a relatively small field of view of 15º.

Periscopic lens Made by Wollaston, it consisted of two meniscus lenses with stop symmetrically in between the two. This has the ability of correcting the odd aberrations like distortion very nicely but unfortunately, the image quality of the lens was comparatively same as that of the normal landscape lens. Rapid Rectilinear lens The periscopic lens helped to bring the idea of doubling the lens into lens design. In this method, the rear lens is designed with a flattened tangential field and then a symmetrical lens is placed in front with respect to the stop. Using the same principle, Dallmeyer and Steinheil came up with a faster lens than the landscape lens with a smaller field of view. The lens also had the problem of field curvature and the tangential field was artificially flattened. Schroeder Lens In all the earlier lenses we have seen, there was the problem of astigmatism being used to correct for field curvature. To get rid of both of these, schroeder took the new achromat lens which were then doubled. The speciality was that since the new achromat decreases the field curavature, College of Optical Sciences, University of Arizona

35

OPTI 517 Lens Design: Course Summary

we need only a small amount of astigmatism to flatten the tangential field. The symmetry takes care of the odd aberrations. Protar The Protar lens was made by Paul Rudolf of Zeiss Optics and the lens used a low power old achromat meniscus in front of the stop along with the new achromat. This meniscus helps to reduce the spherical aberration caused by the new achromat. As a result the lens has a well corrected field, and very good spherical aberration and astigmatism control. Since the lens is symmetrical, it cancels out distortion too to an extent. Cooke Triplet Cooke triplet is a classical lens that used the principle of symmetry to perfection. Two opposite power lenses were used which meant the petzval sum was zero. Power was introduced by controlling the spacing between the elements. But the system still does not have enough degrees of freedom to correct for all the fourth order aberrations. SO the two positive elements were split into two. This meant there was 3 powers, 2 air gaps and 3 shape factors which was just enough to solve for the 5 fourth order aberrations, axial and chromatic color and focal length. Planar / Double Gauss The design of the lens was done in steps by Alvan Clark and Paul Rudolph. It started with the normal landscape lens which has negative coma and spherical aberration. So a thick meniscus was added which had positive coma and corrected spherical aberration. Clark doubled this lens to arrive at a patented design. Rudolph carried the design further by adding a buried surface which corrected for axial chromatic aberration without adding any spherical aberration. So now the lens had a flat field due to the meniscus shape, no astigmatism due to the symmetry of the stop position, correction of spherical aberration due to the shape factor of the meniscus and odd aberration cancelled due to symmetry. The lens was very fast and performed well over an acceptable FOV and the spot size was nearing the diffraction limit.

College of Optical Sciences, University of Arizona

36

OPTI 517 Lens Design: Course Summary

Chapter 9 Tolerancing From what I learnt in the lens design class until now, designing of lenses is an art. It is a skill acquired over a long period of time with practice since there is a lot of parameters that needs to be controlled and there a lot of interdependence. But an equally challenging task awaits a designer even after he designs his lenses. That is the process of tolerancing. Tolenrancing of lenses is needed since a lens cannot be perfectly manufactured. Errors during manufacture in radius, index, thickness, spacing etc., lead to a decrease in the merit function and image quality. Hence a designer should always mention tolerances within the constraints of which an acceptable image quality will be obtained. Simple tolerancing techniques include having compensators in the system, which are parameters of the system that can be varied to compensate to any error accumulating due to manufacturing flaws. Some of the tolerancing limits as explained by R. Shannon in his book is as given below.

College of Optical Sciences, University of Arizona

37

OPTI 517 Lens Design: Course Summary

Using Statistics It is known that when a system is manufactured with all the allowable tolerance values, the performance distribution of these systems take a bellshaped curve.

Also, each system parameter has its own probability distribution function (PDF) for performance, and making use of the central limit theorem, we can link these individual PDFs to the overall system PDF. Finally by integrating the PDF we can estimate how many systems will meet a given performance. Sensitivity analysis In this method, we will be given the design value of system parameters like radius, thicknesses etc., and each will have a tolerance range. So we change each of the parameters within the tolerance values for the surfaces and analyze the change in the system merit function. Thus we are finding the College of Optical Sciences, University of Arizona

38

OPTI 517 Lens Design: Course Summary

sensitivity of the system for these changes and this is called the sensitivity analysis. Inverse sensitivity analysis The inverse sensitivity analysis works backwards from the sensitivity analysis. Here, we change the system merit function within the allowed limits and monitor the change in the surface parameter values. In this method the tolerance values of the surface obtained will be the worst case scenario for that particular surface and if all the surfaces are manufactured at these limits, naturally the system merit function drops significantly below the allowed tolerance limit. Monte Carlo As can be expected, it is possible to know the performance distribution by changing all the possible variables randomly. This is what is dome in a Monte Carlo simulation. It is in a way a brute force method in which the system goes through iterations changing the variables at random and calculating the performance. As expected, when the number of iteration increases, it gives a better idea of the performance distribution. But for complex systems, this method can be extremely demanding on the processors and timeconsuming.

College of Optical Sciences, University of Arizona

39

OPTI 517 Lens Design: Course Summary

References ➢ 517 Introduction to Lens Design – Class notes and Lectures – Jose Sasian ➢ Image Quality Lecture – Richard Juergens ➢ OPTI 517 Class Summary by Matthew Lang ➢ http://images.pennnet.com/articles/lfw/thm/th_lfw30238-20.gif

Acknowledgements This class summary was prepared based on the knowledge obtained in lens design from the OPTI 517 course and I would like to thank the TA, Lirong Wang at this time who was always willing to clear any doubt I had, however trivial it was. This was especially important for me since I had taken the course without taking the Aberrations course, OPTI 509 which was considered to be a prerequisite. I would also like to thank Prof. Jose Sasian for sharing his immense knowledge in the interesting field of lens design. This course helped me to gain an insight into the challenges and subtleties of this field which also one of my favorite among the optics research areas.

College of Optical Sciences, University of Arizona

40