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Slide 1 |
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In this tutorial, a few concepts regarding the way light rays interact with thick lenses, that is, real lenses we use, are discussed. Next, the different kinds of magnification of images produced by lenses and the different kinds of telescopes are discussed. Finally, there will be an overview of the optics instruments that are used in daily practice.
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An important principle is that as an object moves in relation to a lens, so will its image. Both object and image go in the same direction. Also, as a lens moves in relation to an object, the image will also move in the same direction. For example, if an object is 2 m from a +3.00 D lens, where will its image be located? The following equation can be used: U + P = V. Here, U = -1/2 meters = -0.5 D (minus because the light rays are diverging at the lens), P = +3.00, so V = -0.50 + 3.00 = +2.50 D. The image will be found 1/V, 1/+2.5, away from the lens, or 40 cm, to the right of the lens since the light rays are converging (have plus vergence) as they exit the lens (Slide 1).
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When the object is moved to 1 m from the lens, since U + P = V now is -1.00 + 3.00 = +2.00, the image will be 1/2 m, or 50 cm, to the right of the lens (Slide 2). When the object is placed 33 cm from the lens, the image will be located at infinity (-3.00 + 3.00 = 0), and this is the position of the primary focal point of the lens. If the object is moved closer to the lens, for example 25 cm, the image will be found on the left of the lens, since the vergence of the image will be negative (-4.00 + 3.00 = -1.00) and the light will be diverging (Slide 3). When the object is moved relative to a stationary lens, the image will always move in the same direction, though not the same distance that the object moved. The same would be true of images formed by a stationary object if the lens is moved. You can also do the ray tracings and math to confirm that the result is the same for minus lenses.
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What happens if the object is at infinity and the lens is moved? By definition, an object at infinity will have zero vergence and will be focused at the secondary focal point of a lens. If the +3.00 D lens is used with an object at infinity, the image will be located 33 cm to the right of the lens (Slide 4). If the lens is then moved 25 cm to the left, the image of the object at infinity will still be focused 33 cm to the right of our lens, but it will have moved 25 cm to the left of its previous position (Slide 5). Notice that, if the object remains at infinity, its image moves in the same direction and exactly as much as the lens moves. This explains why there is such importance placed on the vertex distance of a spectacle lens when prescribing glasses of greater than 5 D, plus or minus. It is essential, when writing a prescription for a high refractive error, that you specify the distance from the cornea to the lens (vertex distance) which existed during your refraction, because if this distance is changed, objects at infinity (all our distance vision targets) will no longer be focused on the retina and will be seen blurred. If the vertex distance of the spectacles is smaller than during your refraction, distant objects will be focused behind the retina, producing a false hyperopia, whereas if the glasses are positioned farther out than your refraction's vertex distance the focus will be in front of the retina and the patient will be falsely myopic.
This is a great lead into the subject of effective lens power. Most of the previous discussions have been about imaginary thin lenses, but our spectacles are not true thin lenses. Therefore, the front power of a lens must be measured from the primary focal point to the primary principal plane. Likewise, the back surface power of a lens is measured from the secondary focal point to the secondary focal plane. These two measurements together determine the effective power of a lens in air. It is fortunate that we have the luxury of the lensometer, which measures this for us.
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There are different types of magnification commonly encountered in optics. I will discuss linear magnification (also called transverse or lateral), angular magnification, and axial magnification.
Linear Magnification
Linear magnification is
probably the easiest to understand, but perhaps the least useful. Linear
magnification is simply the ratio of image size to object size:
If a ray is drawn from the peak of the object through the nodal point of the lens, it will not be deviated and it will pass through the peak of the image (Slide 6). A pair of similar right triangles are formed with their apices at the nodal point. Following the rules of geometry of similar triangles:
Linear magnification does not take into account what the magnification appears to be to an observer. It has no relevance when the object is at infinity, because the object vergence is zero. A more useful way of describing magnification is the next type — angular magnification.
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Angular Magnification
Angular magnification occurs
as an object is brought closer to the observer. We know that even a large
object in the far distance (a mountain, for example) appears small, but as we
get closer to it, it seems larger. This is because as an object moves closer to
the eye it subtends a larger angle on the retina (Slide
7). Without going through all the calculations and assuming we are dealing
with relatively small angles, the angular magnification of a lens is:
When considering angular magnification in an eye, take into account the added lens power effect of accommodation by adding the accommodative effort at the position of the object to the power of the lens and divide by 4.
Axial Magnification
Axial magnification attempts to
take into account the stress placed on accommodation when looking at an object.
It describes the apparent height of the object when magnified by a lens. The
axial magnification is the square of the linear magnification. Therefore, if an
object's linear magnification is 4, then its axial magnification is 16.
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Lens systems are discussed in general in the Basic Geometric Optics tutorial . Following is an overview of two specific lens systems — the Galilean and astronomical telescopes.
Galilean Telescope
The Galilean telescope uses a
weak plus lens and a high minus lens. The minus lens is placed in back of the
plus lens such that the primary focal point of the minus lens is at the
secondary focal point of the plus lens. Therefore, the minus lens "sees" the
image of the plus lens as its object. Light rays from this object (which is the
image the plus lens formed) will be refracted by the minus lens and exit the
telescope as parallel rays. The resulting image, focused at infinity, may be
viewed by the eye. "Great," you may say, "the telescope can now be pointed at a
distant target at optical infinity and see it as if it is at infinity." But
what is the advantage? It will become apparent if we look at two rays passing
through the system, those that pass through the nodal points of the lenses (Slide 8). The angle from the horizontal of the ray
exiting from the minus lens (θ1) is
greater than that of the ray entering the plus lens (θ). Therefore, the
image of this lens system subtends a greater angle on the retina of an
observer's eye than would the object seen without the telescope. The lens
system has produced angular magnification of the object. The amount of
magnification is:
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Therefore, the magnification of the Galilean telescope lens system is the ratio of the negative of the power of the eyepiece (P2) over the power of the objective lens (P1).
Astronomical Telescope
The second type of telescope
is the astronomical, or Keplarian, telescope (Slide
9). In this lens system, a weak plus lens is used as the objective and a
high plus lens is the eyepiece. Again the eyepiece lens is placed so that its
primary focal plane is coincident with the secondary focal plane of the
objective lens. Angular magnification is produced with this arrangement, as it
was with the Galilean telescope, and the ray tracings are similar. The
magnification of the astronomical telescope is described by the same formula as
the Galilean, but both lenses are plus power, so the result is a negative
magnification signifying that the image is inverted. It is important to
remember that negative magnification does not imply minification, but rather
that the image is inverted from the orientation of the object; minification is
indicated by a number less than 1.
The direct ophthalmoscope is easy to use. This tool uses the patient's eye as a magnifier and the ophthalmoscope provides a light source and a viewing port. If the power of the cornea is estimated to be about +40 D and the power of the human lens about +20 D, then the total power of the eye is approximately +60 D. The angular magnification achieved by viewing through these lenses is therefore 60/4 = 15 X. The inside of the eye appears 15 times larger than it would if viewed with your eye alone after cutting away the front of the eyeball. The dial on the direct ophthalmoscope is used to correct any refractive error in either the observer's eye or the patient's eye, or both.
The lensometer is a device that is simple in design. If an object is placed behind a lens and at its image is viewed with a telescope focused at infinity, when the object was perfectly in focus we would know the refracted light rays from the lens were parallel. Therefore, the object would be at the secondary focal point of this lens. If we then measure this distance and take its reciprocal, we would have the dioptric power of this lens. This is the concept behind the lensometer. The only problem with this approach is the length the instrument would have to be to measure weak lenses (e.g., 4 M to measure a 0.25 D lens). Furthermore, the relationship between focal length and lens power is not linear, and the measuring scale would have to be long for low power lenses and condensed for high powers.
If one places a +5.00 D lens in front of the object (in this case the object is the reticule lines seen in the lensometer) and places that +5.00 D with its focal point on the back surface of the unknown lens, when the telescopic eyepiece "sees" the object clearly, its image must be at the focal plane of the lens being evaluated. That image is much closer than it would be if the +5.00 D lens was not present, and the instrument can therefore be a manageable length. By arranging the lensometer with this +5.00 D "field" lens, however, the distance between target (reticule) and the +5.00 D lens turns out to be directly proportional to the power of the lens you are measuring (Badal Principle). The measurement scale on the device can be simplified, since every millimeter will have the same dioptric value.
Finally, let's take a look at how a keratometer works. If we were able to take an object with known dimensions and measure the dimensions of its image created by a mirror, we would be able to determine the distance that object is from the mirror. Following is the linear magnification equation:
If that object is at optical infinity for the mirror, then its image will be at the mirror's focal plane. The moist corneal surface is our mirror, and optical infinity for such a high power mirror is not very far away, so the object will be focused at the cornea's focal plane which is half its radius of curvature (r). The keratometer is arranged so that the distance from the object to its reflected image is fixed, and since the radius of the cornea is so small, that distance is almost the same as the distance of the object from the corneal surface. Based on the following equations, the corneal radius of curvature is directly proportional to the image size.
The keratometer allows us to measure the image size, and converts the result into a report of the corneal power (curvature) in diopters, based on the assumption of a corneal index of refraction of 1.336. We know that the power of a curved surface is n2 - n1/r. We know the index of refraction of air is 1.00. Therefore, 1.336 - 1.000/r = 0.336/r. Now just plug in the radius of curvature and you obtain the power of the surface of the cornea.
