VII–2 Basic Optical Elements and Instruments

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VII–2 Basic Optical Elements and Instruments

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Main Topics

• Refraction, Dispersion and Refraction Optics.

• Thin Lenses. Types and Properties.

• Combination of Lenses.

• Basic Optical Instruments• Human Eye

• Magnifying Glass

• Telescope

• Microscope

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Refraction I

• Another important basic optical effect is refraction appearing when rays pass from one material to another. Transparent materials may differ in their optical density.

• The more dense material the lower is the speed of light in it. Optical density is characterized by the absolute refraction index: n = c/v • c is the speed of light in vacuum

• v speed in the particular material.

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Refraction II

• We can again use the Fermat’s principle to find the law of refraction.

• To find which ray makes it first from S to P is a similar problem as if we want to safe a drowning person in the shortest time, taking into account that we run much faster than swim.

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Refraction III

• We use the more general definition that the correct ray is the stationary one. In other words, if we take some neighboring ray its time of flight will be (roughly) the same.

• Let the point:• S be in a space where the light travels with the

speed v1 = c/n1 and

• P in the space where the speed is v2 = c/n2.

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Refraction IV

• Now, let the SCP be the correct ray for and the SXP some neighboring ray. Should the time of flight be the same: EC/v1 = XF/v2

• We use : EC = XCsin1 and XF = XCsin2 substitute for v1 and v2 and get the

• Snell’s law:

2211 sinsin nn

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Refraction V

• We see that the higher is the optical density or the slower is the speed of light the smaller is the refraction angle.

• If the angle of incidence from the less dense material is 90° the refracted angle is given:

• sinc = n1/n2 the maximum refracted angle or the critical angle.

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Refraction VI

• If the beam would try to pass from the optically dense material under an incident angle higher than the critical angle it would not get through the boundary but rather be totally reflected.

• The effect of total (internal) reflection is used for instance in fiber optics.

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Dispersion I

• Transparent materials have an important property that the speed of light and thereby their refraction index depends on the wavelength of the applied light.

• The higher energy (lower ) the stronger interaction and thereby higher optical density and higher deflection from the original direction.

• This means that light of every wavelength or color is refracted under a (little) different angle.

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Dispersion II

• The effect of dispersion complicates design of optical systems and has to be compensated by using more lenses of different materials.

• On the other hand it gives us the possibility to decompose the visible light and near IR and UV regions into different wavelengths.

• That is important for instance for studies of properties of matter by spectroscopic methods. The matter can be very far away in the universe!

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Refraction Optics I

• The effect of refraction is used to build optical components and systems.

• If we have a point S in the medium n1 and the point P in the medium n2 > n1 we may use the Fermat’s principle to find the shape of the boundary between the media so the points are conjugated or the optical system is stigmatic for them.

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Refraction Optics II

• If we compare a time of flight of some refracted ray with the one directly connecting both points we find a relation:

l1n1 + l2n2 = s1n1 + s2n2 • We readily understand from here, why the

optically denser media must be convex.• The corresponding surface is of the fourth

order, so called, Cartesian ovoid.

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Refraction Optics III

• If we move one of the points S or P into infinity the surface becomes second order, either elliptical or hyperbolical.

• This can be in principal used to construct lenses - optical components from some material, which allow that the object as well as the image are in the same media.

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Refraction Optics IV

• Ideal lenses are for instance double hyperbolic or planar-hyperbolic.

• Although, recently they can be, in principle, machined, for the same reasons, as in the case of mirrors aspherical surfaces are approximated by cheaper spherical ones.

• But they can be successfully used only in the paraxial region.

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Refraction Optics V

• Spherical surface can be shown to be stigmatic for points on the optical axis in the paraxial approximation.

• Let the the ray come from the point O in matter n1 under an angle and hit the spherical surface in the point P, which is seen from the curvature center C under an angle and deflects to the point I in material n2, where it arrives under an angle .

1 and 2 will be the incident and refracted angles.

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Refraction Optics VI

• From triangle PIC : = + 2; OPC : 1 = +

• In paraxial approximation the angles are small, so we can write: n11 = n22

= h/d0; = h/R; = h/di; where h is the height of the point P from the optical axis.

• We can show that the angle dependence vanishes:

R

nn

d

n

d

n

nnnn

i

)(

)(

122

0

1

1221

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Refraction Optics VII

• It is important to obey the following convention:

• If C in on the same side as the light comes from, it is negative.

• If O in on the same side as the light comes from, it is positive.

• If I in on the same side as the light comes from, it is negative.

• We can see that the reciprocity principle is valid!

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Thin Lenses I

• Very important lenses are those which can be considered as thin.• All their properties can be characterized by a

single parameter the focal length f.

• It is the distance from the optical center to the focal points F.

• There is one focal point in front and one behind the lens, both equally distant from the center.

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Thin Lenses II

• The, so called, lensmaker’s equation can be derived which relates the focal distance of a thin lens with the radii of its spherical surfaces

• Sign conventions must be obeyed.• Note that the focal length is the same on

both sides even if the radii are different.

)11

)(1(1

21 RRn

f

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Thin Lenses III

• It is possible to make converging lenses with positive focal length when the positive radius of curvature is smaller or diverging lenses with negative focal length when the negative radius of curvature is smaller.

• Optometrist and ophthalmologist use the power P = 1/f to specify lenses. Its unit is diopter (D), 1D = 1m-1.

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Thin Lenses IV

• To find an image of some point, we can again use two of three special rays.

• A ray passing in any direction through optical center is not deflected.

• A ray arriving in parallel with the optical axis will pass through the image focus if f is positive or appear to leave it, if f is negative.

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Thin Lenses V

• A ray passing through the object focus if f is positive or heading towards it, if f is negative, will continue in parallel with the optical axis on the other side of the lens.

• If the imaging is stigmatic (sharp) all other rays leaving the object point must appear in the image point as well. But they can’t be used to find it.

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Thin Lenses VI

• The lens equation which relates the distances of the object and image with the focal distance can be easily derived:

1/do + 1/di = 1/f

• and lateral magnification is defined as the ratio of the image height to the object height

m = ho/hi = - di/do

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Thin Lenses VII

• To comprehend functioning of almost any optical instrument it is necessary to fully understand the importance of the focal planes of the lenses.

• For converging lens a bunch of parallel rays coming under some angle with the optical axis will pass through a point in the image focal plane, which is on the other side of the lens.

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Thin Lenses VII

• We can locate this point using the ray passing the optical center and the one passing through the object focus.

• Using the lens equation we can verify that an object producing an image in the focal plane (di = f) must be in infinity.

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Thin Lenses VIII

• For diverging lens all beams heading towards a point in the object focal plane, which is now behind the lens, will run as a bunch of parallel rays after the lens.

• We can find their direction using the ray passing the optical center and the one coming in parallel with the optical axis.

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Thin Lenses IX

• We can again verify this using the lens equation. If the object is in the object focal plane (do = f , both negative now) the image must be in infinity.

• We can produce parallel bunches of rays by both types of thin lenses if the object is in the object focal plane. For the diverging lens the object distance is, however, negative!

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Combination of Lenses

• We start from the lens closest to the object. • We display the object by this lens only.• The image of produced by the first lens will

be the object for the second lens.• Then we display the new object by the

second lens only. And so on.• The sign convention must be strictly obeyed

since now object distance may be negative!

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The Human Eye I

• Most of the focusing (refraction) is done by the cornea (n = 1.376). The lens does just the ‘fine tuning’.

• The quality of focusing and the depth of focus depends on the iris. The smaller the aperture the better.

• Normal eye has the near point at 25 cm and the far point in infinity.

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The Human Eye II

• In the case of nearsightedness (myopia) the far point is not infinity. This has to be corrected by a diverging lens.

• In the case of farsightedness (hyperopia or presbyopia – developed by age) the eye can’t focus on near objects. This has to be corrected by a converging lens.

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The Human Eye III

• The eye is relaxed if it watches the far point so eyepieces usually produce parallel rays.

• Some other optical instruments produce a virtual image in the conventional length equal to the standard near point at 25 cm.

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Magnifying Glass

• Magnifying glass is used:• either the object is in the focal plane and we

watch it by relaxed eye.• or the lens is close to the eye (Sherlock

Holmes) and a virtual image is produced in the conventional distance.

• Magnification is the angle magnification – we see objects as big as is the angle of their image on the retina.

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Telescopes I

• Astronomical refractive telescopes have two lenses an objective (with longer f) and an eyepiece, which share the same focal plane.

• The eyepiece can be either a converging lens or a diverging one, then the shared focal plane is behind the eyepiece.

• The angle magnification in both cases is minus the ratio of the focal lengths.

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Telescopes II

• Important are reflecting telescopes:• Large mirrors are easier to produce and support

• Mirrors don’t suffer from color aberration.

• But we have to realize that although reflection is not influenced by dispersion, it is still a complicated process and reflectivity of any material isn’t ideal .

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Compound Microscope

• The principle of a microscope can be shown also using two lenses. The objective (now with very short f) produces a real image. It is watched by the eyepiece, which usually produces the imaginary image in the conventional distance.

• Good microscopes are complicated since it is important to compensate aberrations.

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Homework

• The last homework is due tomorrow!

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Things to read and learn

• This lecture covers

Chapters 33 – 5, 6, 7, 8; 34

• Advance reading

• Chapters 35, 36

• Please, read and try to understand even the parts which were not dealt with in detail in the lecture. You should have far enough background knowledge to understand everything!

Lens Equation

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