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© FEMTO Project, 2014

Imaging

Document that explains how to design experimental set-ups to illustrate various optical

aberrations

This document explains how to build experimental set-ups which illustrate different optical aberrations. It details the steps required to build, operate and understand

the different experimental set-ups used to highlight various optical aberrations. Some parts of this document are from A.-S. Poulin-Girard’s Masters’ thesis1. 1 Poulin-Girard, A.-S. (2011). Banc de caractérisation pour lentilles panoramiques, mémoire de maîtrise, Université

Laval, 106 pages.

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Optical aberrations and imaging

Imaging systems such as the photographic camera have come a long way since the first prototype proposed by Daguerre at the French Academy of Sciences in

1839, called the Daguerreotype. In over a century, photography has matured and adapted to customer needs. Nowadays, it is possible to buy an easy-to-use digital camera with great resolution for a very decent price. However, the design of an

optical system able to collect and image light from an object with the least amount of deformation is a considerable challenge. To design an optical system to

satisfactory requirements, an optical engineer has to consider multiple parameters such as the field of view, the focal length, the luminosity and many other factors. Either way, it has never been as easy as it is now to capture the most memorable

moments in a picture.

Imaging systems

There are many different kinds of optical systems. Zooms for photo and video cameras come easily to mind. However, imaging systems are not limited to

cameras. They range from a simple lens such as a pair of eyeglasses to a more complex lens such as a microscope objective and huge ones such as telescopes.

Figure 1 shows the optical layout of different optical systems.

Figure 1 : Different optical system layouts: (a) 10X microscope objective, (b) mechanical zoom, (c) Cassegrain reflector 1 and (d) the 1.6 m Ritchey-Chrétien telescope from Mont Mégantic Observatory.

These optical systems are made of lenses and mirrors, each of them fulfilling a

specific task. Together they allow us to produce devices able to image objects from just across the room to very far away.

1 Schematics from LAIKIN, M. (2006). Lens Design, Fourth Edition. Boca Raton : CRC Press, 473

pages.

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Characteristics of an optical imaging system

The simplest imaging system is a single lens. Just like more complex optical

systems, the behaviour of a single lens can be defined by a few parameters such as its thickness e, a refraction index n and two radii of curvature R1 and R2. A lens is

also characterized by three different focal lengths: the front focal length (FFL), corresponding to the distance between the front focal point and the lens’ front vertex; the back focal length (BFL), corresponding to the distance between the back

focal point and the back vertex; and an effective focal length (EFL) as illustrated in figure 2.

Figure 2 : The figure above shows the parameters that characterize a simple optical system. A single lens of thickness e, radii of curvature R1 and R2 and a material of refraction index n is characterized by different focal lengths. The front focal length (FFL) refers to the distance between the front focal point and the front vertex while the back focal length (BFL) refers to the distance between the back focal point and the rear vertex. One can also characterize a lens by its effective focal length (EFL).

The purpose of an optical system is to collect light from an object in order to redirect it onto an image sensor and obtain a sharp image. The front and back focal

lengths, defined earlier, are not the only distances that define an optical system. Other variables such as the entrance pupil, the exit pupil and the stop are defined by tracing the chief ray (to learn more, see appendix A).

Thin lens

Simple imaging systems, such as those mentioned in this protocol, can be

considered as thin lenses and are defined only by their effective focal length. The relationship between the object distance s, the effective focal length f and the image

distance s’ can be expressed in equation (1).

1 1 1

s s f

(1)

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Effects of optical aberrations

All optical instruments are subject to optical aberrations. These aberrations

describe the difference between the real image and the image predicted by the paraxial theory, also known as small angle approximation. Paraxial optics is based

on the assumption that the angle of incidence is small. Using this approximation, it is possible to rewrite Snell-Descartes’ law, which links the refraction index ni to the

propagation angles i, by replacing sini by the first term of its Taylor series.

1 1 2 2n n (2)

This approximation is commonly assumed to be valid up to an angle of incidence

of 20°. For a greater angle of incidence, it is necessary to consider more terms in

the Taylor series for sini.

Monochromatic aberrations

The first person to consider more terms in the Taylor series approach to ray tracing was Ludwig von Seidel. Using the two first terms in the Taylor series, he was

able to estimate the final position (x’,y’) of a monochromatic ray incoming at an

initial height h from the optical axis. Using the polar coordinates () at the entrance pupil, he was able to consider five monochromatic aberrations, also known

as Seidel aberrations. One can express the final position (x’,y’) as a function of the paraxial solution (x,y) corrected by optical aberrations. Even though equations (3)

and (4) give a good estimate of the real trajectory of a monochromatic ray, it remains an approximation since only the first and third order aberrations are considered.

3 2 2

1 1 2 3 4' sin sin sin(2 ) ( ) sinx A B B h B B h (3)

3 2 2 3

1 2 1 2 3 4 5' cos cos 2 cos(2 ) (3 ) cosy A A h B B h B B h B h (4)

In the above equations, the paraxial positions x and y are given respectively by

the terms and . The following terms are corrections caused by

optical aberrations of the third order. In the above equations, the coefficients B1 to B5 relate to: spherical aberration (B1), coma (B2), astigmatism (B3), Petzval curvature (B4) and distortion (B5). We will split these five aberrations into two

distinct categories. While spherical aberration, coma and astigmatism degrade the image quality, Petzval curvature and distortion only warp the image without

degrading it.

1 sinA 1 2cosA A h

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Spherical aberrations

Spherical aberration corresponds to a variation of the focal length as a function of the aperture, which is the height at which a ray enters the optical system. The closer the rays are to the optical axis, the closer to the paraxial focus their image

will be. Spherical aberration can affect an imaging system in two different ways. It can either be a transverse aberration (TA) or a lateral aberration (LA), as shown in

figure 3.

Figure 3: Spherical aberration affects an optical system by shifting the focal point closer to the lens as a function of the ray height h. It can either be transverse (TA) or lateral (LA).

Coma

Another common monochromatic aberration is the coma. It refers to the variation in the magnification of an image as a function of the incoming ray height.

This causes parallel rays entering an optical system to be focused at a different position than a ray crossing the center of the lens.

Figure 4 : Coma’s effect on an optical system. It corresponds to an off-axis aberration that manifests itself through a variation in the magnification of an image as a function of the incoming ray angle.

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Astigmatism

Astigmatism is the last of the three third order aberrations that degrade image quality. A system affected by astigmatism has two different focal lengths, one for the sagittal plane and the other for the tangential plane. Thus, such a system has

two foci: the sagittal focus and the tangential focus.

Figure 5 : Astigmatism’s effect on an optical system. A system subject to astigmatism will focus both sagittal and tangential planes at different points. The figure above shows two focal lines rather than a single point as a focus. 2

Petzval curvature (field curvature)

In a system free of coma, spherical aberration and astigmatism, the Petzval

curvature is the curved image plane formed by focusing a flat object normal to the optical axis. The greater the angle of incidence, the farther it will be focused from

the paraxial focal point.

Figure 6 : Petzval curvature’s effect on an optical system. Rays from a perpendicular plane to the optical axis are focused on a curved surface called Petzval surface in a system free of astigmatism, coma and spherical aberration.

2 Figure from : Smith, W. J. (2008). Modern Optical Engineering. New York: McGraw-Hill, 754 pages.

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Distortion

Distortion is a monochromatic aberration that deforms an image without affecting its quality. It is caused by a variation of the focal length and magnification as a function of the field of view. This effect displaces the image point from the

position estimated by the paraxial approximation. The two types of distortion are shown below: the pincushion distortion (positive) and barrel distortion (negative).

Figure 7 : (a) A distortion free grid can either be affected by (b) pincushion distortion or (c) barrel distortion.

Chromatic aberrations

The refraction index of a glass decreases as a function of wavelength. This

causes blue and red rays to be focused at different focal points and is referred to as chromatic aberration. We will split chromatic aberrations into two distinct categories: axial aberrations and transverse aberrations.

Axial chromatic aberration

Shorter wavelengths are affected by a greater deviation. This causes blue to be focused closer to the lens’ back vertex than red. Axial aberration thus refers to the

difference in the focal lengths for blue and red. In contrast, transverse aberration corresponds to the distance between blue and red on the image plane, as shown in

figure 8.

Figure 8 : Axial chromatic aberration for different wavelengths. Chromatic aberrations are the result of the different focus for two different wavelengths.

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Transverse chromatic aberration

Transverse chromatic aberration is defined by the difference in the final position of two chief rays having the same initial trajectory, but different wavelengths. The resulting effect is a different magnification for both wavelengths as shown in figure

9. This effect is also called lateral color.

Figure 9 : Transverse chromatic aberration. Each wavelength of a polychromatic chief ray is refracted differently. The difference in optical path, measured at the image plane, is often called lateral color.

Image quality indexes

There are many different ways to characterize the quality of an imaging system. These different indexes allow an optical designer to determine the performance and quality of an imaging system. The choice of an index is determined by the

requirements of a project and is chosen by the optical designer to help customize and optimize a design in order to meet the required specifications. In this

document, we will focus on the point spread function (PSF). Other methods may be used to estimate the imaging quality of an optical system such as the Modulation Transfer Function (MTF). See appendix B for more information on MTF.

Point Spread Function (PSF)

In an optical system, a distant point source should be imaged on a sensor as a point. However, real systems are subject to limitations. The resolution of an optical

system is first limited by diffraction, which produces an Airy disk instead of a point. Other imperfections such as small defects in lenses or misalignments may cause limitations in the resolution of an imaging system. As a result, a distant point source

will not be imaged as a point on the sensor and is what we call the point spread f (PSF). PSF is the response function of an imaging system to a distant point source

and characterizes its deformation. Figure 10 shows the effect of spherical aberration, coma and astigmatism on a distant point source limited by diffraction (Airy disk). Although we look at these aberrations separately, it is assumed that a

real system may be subject to all of these aberrations at once.

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Figure 10 : (a) The ideal distant point source will always be limited by diffraction and as the shape of an Airy disk. The distant point can be modified by various aberrations: (b) spherical aberration, (c) coma and (d) astigmatism.

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Required hardware

Here we list all of the different parts required to build the experimental set-ups presented in the following sections.

Qty. Piece number and description Furnisher Price

1 66 mm Construction Rail, L=1000 mm (XT66-

1000) Thorlabs 122.40$

1 HeNe Laser, 632.8 nm, 0.8 mW (HNL008R) Thorlabs 1076.34$

1 XY Adjustable HeNe Mount for 60 mm Cage System (HCM2)

Thorlabs 125$

1 Quartz Tungsten-Halogen Lamp (QTH10) Thorlabs 156$

1 Post-Mountable Viewing Screen (EDU-VS1) Thorlabs 18$

1 Lens Mount for ø1’’ Optics, One Retaining Ring Included (LMR1)

Thorlabs 15.23$

1 SM1 Spanner Wrench, Length=1’’ (SPW606) Thorlabs 26$

1 Filter Holder for 2" Square Filters (FH2) Thorlabs 18.90$

1 Post-Mounted Iris Diaphragm, ø20.0 mm Max Aperture (ID20)

Thorlabs 45.90$

6 200 mm Long Double Dovetail Clamp (XTC66C1) Thorlabs 16.83$

1 40 mm Long Double Dovetail Clamp (XT66C2) Thorlabs 18.67$

6 ¼’’ -20 Tapped Dovetail for 66 mm Rail, 20 mm

Long (XT66DE1) Thorlabs 9.69$

1 Mounting Platform, Three ¼’’ (M6) Counterbore, 50 mm Long (XT66D2-50)

Thorlabs 16.32$

7 Post Holder with Spring Loaded Hex-Locking Thumbscrew, L=2.00’’ (PH2)

Thorlabs 7.70$

7 ø½’’ X 3’’ Stainless Steel Optical Post (TR3) Thorlabs 5.42$

1 N-BK7 Bi-Convex Lens, ø1’’, f=150.0 mm (LB1437)

Thorlabs 21.20$

1 N_BK7 Plano-Convex Lens, ø1’’, f=75 mm, Uncoated (LA1608)

Thorlabs 20.60$

2 Rotation Mount for Ø1" Optics, One SM1RR Retaining Ring Included (RSP1)

Thorlabs 82.72$

2 f=100 mm, ø1’’, N-BK& Mounted Plano-Convex Round Cylindrical Lens (LJ1567RM)

Thorlabs 96.08$

1 Double-Convex Lens, 5mm Dia. x 5mm FL (63-

530)

Edmunds Optics

34.00$

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The objects such as the cross, the ruler and the grid are cellulose acetate prints and are custom made. They are well identified on the different experimental set-up schematics.

Lambda Scientific offers a nice educative bundle to build the various experimental set-ups presented in the following section (LEOK-5). This bundle offers

all the required parts except the cylindrical lenses and a few cellulose acetate prints. It is a very versatile kit with a good price value. It is strongly recommended to anyone that wishes to build the following experiments.

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Experimental set-ups

Here we will detail the experimental set-ups that we will use in order to shed light on the various optical aberrations. Most of the optical aberrations will be

explained.

It is highly suggested to fix the laser and the lamp at the two extremities of the optical bench, as this makes it more versatile. The schemes will assume such a

placement for both the laser and the lamp.

Monochromatic aberrations

A red laser will be used to illustrate spherical aberration, coma, astigmatism and distortion. Since these effects are not a function of the wavelength, we will use a monochromatic light source.

Spherical aberration

Figure 11 shows the experimental set-up to demonstrate spherical aberration. In order to properly observe this effect, place the screen just behind the paraxial focus

(see figure 3), which corresponds to the position where the focus is best.

Figure 11 : Experimental set-up to highlight spherical aberration

Place the iris right behind the lens (use parts XT66C2 and XT55D2-50). It is now possible to observe the difference in the image quality by varying the size of the aperture. The image on the screen is blurry when the iris is wide open. However, as

we close the iris, the image quality improves because the iris blocks the rays that are focused before the paraxial focus (see figure 3).

While the geometry of a lens greatly influences the extent of spherical aberration, its orientation is also of a great importance. To easily highlight this, place the screen at the paraxial focus and remove the iris. A plano-convex lens will

exhibit much more spherical aberration if the rays hit its flat surface first. Changing the orientation of the lens will highlight this effect.

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Figure 12 : Spherical aberration of a plano-convex lens as a function of its orientation

Coma

The following set-up makes use of a laser, a screen, a 75 mm plano-convex lens and a 4.5 mm lens to spread the laser beam and to show the effect of coma.

Figure 13 : Experimental set-up used to highlight coma

The first lens must be as close as possible to the laser output and is used as an object in this experiment. The screen must be located behind the focus in order to

be illuminated by a large circular area. It is then possible to observe a comet shape image on the screen by tilting the plano-convex lens around a vertical axis. This

comet shape is the main feature of the coma.

Astigmatism

Astigmatism can be shown by using a cross shaped object and two cylindrical lenses as shown in figure 14. A cylindrical lens focuses light in only one axis.

Figure 14 : Experimental set-up used to highlight astigmatism

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In the above experiment, each of the two lenses images a part of the cross. By moving the screen, it is possible to identify two distinct foci. One position will correspond to a sharp horizontal line and blurry vertical line, while the other focus

position will correspond to a sharp vertical line and a blurry horizontal line. Such an optical system is strongly affected by astigmatism.

Distortion

It is easier to highlight pincushion distortion than barrel distortion in a low budget set-up. In order to observe this distortion, a grid will be used as an object.

Figure 15 : Experimental set-up highlighting pincushion distortion

In order to show the effect of pincushion distortion, place the 4.5 mm lens close

to the laser output and fully illuminate the grid with the spread laser light. Place the grid near the 50 mm focal length lens. Placing the screen at the paraxial focus

(sharpest image) will show the effect of pincushion distortion.

Chromatic aberration

In order to illustrate chromatic aberration, a white lamp will be used as a source.

Chromatic aberration

Chromatic aberration manifests itself by a changing focus as a function of the wavelength. For example, this difference in focus is usually a few millimeters

between red and blue. The following set-up shows how to observe this aberration.

Figure 16 : Experimental set-up used to highlight axial chromatic aberration

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In this set-up, the object is a closed iris imaged through a 150 mm focal lens. There is a position where the spot imaged on the screen is the smallest: we will call this position the optimal focus. However, when one displaces the screen towards the

lamp, it is possible to observe a position where the luminous spot is contoured by a fine red line. This fine red line is caused by chromatic aberration and is an

illustration of the smaller focus for red than for blue. On the other hand, if one would place the screen behind the optimal focus, a thin blue line would contour the luminous spot.

Other aberrations

As mentioned earlier, some aberrations are harder to observe and isolate than others. While they are not covered here, it is possible to build set-ups able to highlight these effects.

Field curvature is a monochromatic aberration. To show this effect, one would require a lens with a large field of view in order to cause an important field

curvature. It will then be impossible to have both the center and the side of the image sharply defined by refocusing the strongly curved field. The difference in focus between the center and the side of the image is caused by the field curvature.

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MATLAB demonstration

A MATLAB program has been built in order to visualize the effects of optical aberrations on an image. It is perfectly analogous to real photography, but in a

much more exaggerated fashion. This program is available for you to download at www.femto.ca, and requires the installation of the conv2fft function, available for download at MATLAB central.

In order to use this program, place the files in the current folder and execute the Demonstration_aberrations.m script. The following interface should show up:

Figure 17 : Graphical user interface for the MATLAB program to simulate optical aberrations on real photographs.

Now, just choose an image and play with the four aberration sliders. The Calculer button shows the simulated effect on the photography as well as its PSF.

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Appendix A : Complex optical systems

A complex optical system is made up of a multitude of lenses. While such a system is characterized by its effective focal length, other parameters are used to

fully characterize it such as the entrance pupil, the exit pupil and the stop. The stop is the aperture, real or not, where the ray cone angle is the smallest. It is bordered by marginal rays, and the chief ray crosses the optical axis in the middle of the

stop. By placing a variable iris at the stop, it is possible to affect the amount of luminosity in a system without affecting the field of view.

Figure 18 : The stop, the entrance and the exit pupils in a simple telescope. The stop can be a physical aperture, in this case an iris, limiting the amount of light in this optical system. The entrance and exit pupils are the object and image planes respectively. Marginal rays are identified by the full black lines.

The entrance and exit pupils are the images of the stop as seen by either the object plane or the image plane. The chief ray crosses the optical axis at these three

positions, and their diameter is directly related to the stop diameter.

The F/# (f-number) measures the responsiveness of a lens. A small F/#

indicates a fast system while a large F/# will be a slow system. This number

corresponds to the ratio between the effective focal length f and the entrance pupil

diameter D.

#f

FD

(5)

The depth of field increases with the F/#. This means that a bigger F/# benefits systems that cannot adjust focus. However, the greater the F/# is, the less the

amount of light that will enter the system.

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Appendix B : Modulation transfer function

Modulation Transfer Function (MTF)

As an object is imaged by an optical system, the contrast and sharpness will

degrade. This is because light collected by the multiple lenses is subject to monochromatic and polychromatic aberrations.

No matter how good the design of an optical system may be, some aberrations

will always remain. Manufacturers also have to set a tolerance for the production of lenses. This means that there will be some variation in the production of a lens of a

given focal length and thickness. A tolerance in assembly also needs to be considered, as distances between each lenses will not be exactly identical to that of the design. These variables are not under the control of the optical designer and

cause other unwanted aberrations.

In order to evaluate the performances of an optical system, one can use a series

of parallel black lines of equal spacing and thickness. The spatial spacing frequency f corresponds to the inverse of the period T. It is then possible to calculate the frequency modulation of the imaged object by taking the ratio between maximum

and minimum intensities. One period is defined as two consecutive black and white lines, and is measured in units of cycles/mm.

max min

max min

I IModulation

I I

(6)

Figure 19 shows how such a pattern is affected when imaged by an optical system. One can see the effect of the imaging system on the contrast for two

different spatial frequencies.

Figure 19 : Contrast degradation of a periodic pattern affected by an imaging system for two different spatial frequencies. A periodic pattern of spatial frequency f=1/T is collected by an imaging system. The contrast between black and white in the imaging system is limited. The contrast will generally be better for (a) a low spatial frequency and (b) a high spatial frequency.

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It is generally desirable to analyse the degradation of the image for many different spatial frequencies. The relation between the modulation function and spatial frequency is called MTF. Figure 20 shows the MTF for tangential (T) and

sagittal (S) planes for 3 different objects sizes of a microscope objective similar to the one shown in figure 1(a).

Figure 20 : Polychromatic MTF for a 10X microscope objective as shown in figure 1(a) as a function of object height. This figure has been generated by the ZEMAX software.

In an imaging system, the MTF values are multiplicative. This means that the MTF of a system made of a camera and a commercial lens will be the product of the MTF for each individual components.

système lentilles senseurMTF MTF MTF (7)

MTF and PSF are also related through the Optical Transfer Function:

MTF OTF (8)

OTF PSF (9)