An optometer is a device for measuring refractive error. The simplest possible optometer (if you're a myope) is just a detailed picture tacked to a wall. Once the picture is on the wall, walk away from it, so long as it still appears sharp. When you've reached the furthest you can go, while the picture is still sharp, the picture is at your far point. Then your ocular refraction (or refractive error) is just one over the distance between your eye and the picture.
This procedure works fine if you have no astigmatism. If you have astigmatism, you'll need a picture which is designed to show it. One such is in Figure 1(a) below. This shows a set of radial lines. Again, you tack this picture to the wall, and walk away. This time, however, you're looking for a distance where some of the lines at one angle appear sharp and the rest look blurred ( Figure 1(b)). This gives you the far point of your eye at one meridian. Then continue walking away until the sharp and blurred lines have swapped. This is your far point along the other meridian. The angle of the sharp lines gives you the angle of the meridians.
The problem with these simple optometers is that they are difficult to use, and don't work at all for hyperopes. Nor, in fact, do they even diagnose emmetropia, because an emmetrope has to walk very far away indeed, at which point the picture is not so much blurred as too tiny to see.
An optometer which works for myopes, hyperopes, and emmetropes is the Badal optometer (invented by Jules Badal, in 1876). A Badal Optometer consists of a target object and a positive lens. By moving the target, we can create a range of vergences leaving the lens. A Badal optometer is shown in Figure 2, using a \( +30\text{D} \) lens. (Badal optometers typically use lower powered lenses, but \( +30\text{D} \) is needed here simply to fit it all in a small figure.) The patient's eye is one focal length away from the positive lens. A target is initially placed at the opposite focal point of the lens, and the patientmoves the target away from them until it blurs, or towards them until it looks sharp.
The vergence of the light hitting the patient's eye can be worked out using the thin lens equation and the stepalong procedure. This vergence is the patient's ocular refraction, or refractive error.
If the myopic eye is used in Figure 2 by clicking the myope button, you will have to move the target about \( 0.011\text{m} \) to the right of the focal point of the \( +30\text{D} \) lens. You can work out the vergence of the light hitting the patient's eye using the thin lens equation and the stepalong procedure, as follows:
The distance from the target to the lens is the focal length of the lens plus the distance the target moves. If the target moves \( 0.011\text{m} \), then the distance from lens to target is \( -1/F+0.011=-1/30+0.011= \) \( -0.02233\text{m} \).
Thus \( V_{in}=1/(-0.02233)=-44.78\text{D} \)
From the thin lens equation, \( V_{in}+F=-44.78+30=-14.78\text{D} = V_{out} \)
The distance from the lens to the image is \( 1/V_{out}=1/(-14.78)= -0.0677\text{m} \)
To figure out the vergence when it hits the eye, we use the stepalong procedure, and subtract the distance from the lens to the eye ( \( 1/30\text{m} \)) from the distance in step (3) above to get \( -0.0677-1/30=-0.101\text{m} \).
The vergence entering the eye is then \( 1/(-0.101)=-9.9\text{D} \)
Thus, the ocular refraction, or refractive error, of the myopic patient is \( -9.9\text{D} \)
Instead of working this out every time we use the Badal optometer, we can derive a simple relationship between the distance the patient moves the target and the vergence of the light hitting the patient's eye. Suppose the lens has power \( F \), and the patient's eye is placed \( 1/F \) to the right of the lens. The target is moved a distance \( t \) from the left-hand focal point.
| Step | Reason | Statement |
|---|---|---|
| 1 | The distance from target to lens | \( -1/F+t \) |
| 2 | \( V_{in} \) for the lens | \( V_{in}=\dfrac{1}{-1/F+t} \) |
| 3 | Multiply top and bottom of the fraction by \( F \) | \( V_{in}=\dfrac{F}{-1+tF} \) |
| 4 | \( V_{out} \) from the lens is \( V_{in}+F \) | \( V_{out}=\dfrac{F}{-1+tF}+F \) |
| 5 | Put both parts over common denominator | \( V_{out}=\dfrac{F}{-1+tF}+\dfrac{F(-1+tF)}{-1+tF} \) |
| 6 | Add parts together | \( V_{out}=\dfrac{tF^2}{-1+tF}=\dfrac{tF^2}{tF-1} \) |
| 7 | Stepalong (1): the distance \( d \) to the image is \( 1/V_{out} \) | \( d=\dfrac{1}{V_{out}}=\dfrac{tF-1}{tF^2} \) |
| 8 | Stepalong (2): subtract distance from lens to eye | \( \dfrac{tF-1}{tF^2}-\dfrac{1}{F} = \left(\dfrac{tF-1}{tF^2}-\dfrac{tF}{tF^2}\right) \) |
| 9 | Add the two parts together | \( \dfrac{tF-1}{tF^2}-\dfrac{tF}{tF^2} = \dfrac{-1}{tF^2} \) |
| 10 | The vergence hitting the eye (\( K \)) is one over step (9) | \( K=-tF^2 \) |
Let's try this. In Figure 2, with the myope, we have \( F=30\text{D} \) and we moved the target \( t=0.012\text{m} \), so the refractive error is \( K=-0.011\times30^2=-9.9\text{D} \). If you click the hyperope button, you have to move the target about \( -0.014\text{m} \) to put a sharp image on the retina, so their refractive error is \( -(-0.014)\times 30^2=+12.6\text{D} \).
Because we're measuring \( t \) in metres, things get even simpler if the lens in the Badal optometer has a power of \( F=+10\text{D} \). Then \( K=-tF^2=-100t \); but if we're measuring \( t \) in metres, then \( 100t \) is the number of centimetres that the target has moved. So, with a lens of \( +10\text{D} \), we can simply read off the patient's refractive error by measuring how many centimetres the patient moves the target, which is really easy. For example, if the patient moves the target \( +2\text{cm} \) to the right, the refractive error is \( -2\text{D} \). if they move it \( -3\text{cm} \) to the left, the refractive error is \( +3\text{D} \) ; and so on.
The main advantage of the Badal optometer is that calculations about ocular refraction are very simple: using a \( +10\text{D} \) lens, \( 1 \) cm of movement of the target corresponds to \( 1\text{D} \) of ocular refraction. The other advantage is that the target does not change its size as it is moved towards or away from the lens. This reduces the effects of accommodation, which might otherwise be triggered by the object changing size (and so appearing to come closer or further away).
This is shown in Figure 3. All rays that are parallel to the optic axis of the \( +10\text{D} \) lens will be refracted to pass through the focal point of the lens. But because the lens is one focal length from the patient's eye, the focal point is at the centre of the patient's cornea. So all rays parallel to the optic axis hit the centre of the patient's cornea. In Figure 3, a target has been placed at two different distances (1) and (2). In both cases, a single ray of light from the top of the target, parallel to the optic axis, hits the centre of the patient's cornea at the same angle. This means that, regardless of the distance, both rays will be refracted to the same point on the patient's retina, and so the target will appear to be the same size.
Accommodation is only an issue when trying to work out the patient's far point. The Badal optometer can also be used to work out the near point, by simply moving the target closer and closer until it again can no longer be seen clearly.
One of the drawbacks of the Badal Optometer is that the patient needs to make a judgement about the degree of blur. While this is not too hard, it would be easier if they had to make a simpler judgement. The Scheiner Disc Optometer is a way of achieving this.
If, however, the eye is myopic or hyperopic, the top and bottom ray bundles do not converge to the same point, and so two points are seen.
The Scheinder disc is just an opaque barrier with two tiny holes in it, as shown in the middle of Figure 4. When parallel light is sent towards the disc, the holes only let through a very small number of rays. These rays, when using the emmetropic eye, converge to the same point and so the person sees just a single point of light.
However, if the eye is myopic (obtained by clicking the myope button), something different occurs. Now the rays cross before they hit the retina, and continue to spread out. This time the person sees two spots of light. A similar thing occurs when the eye is hyperopic. So an emmetrope sees a single spot when looking through a Scheiner disc, but a myope or hyperope sees two. We can also work out which kind of ametropia is present by blocking one of the holes. If the upper hole is blocked, a myope sees the upper spot disappear, whereas a hyperope sees the bottom spot disappear.
A Scheiner disc can be combined with a Badal Optometer. This is called a Coincidence Optometer (because if the patient sees two points, they move the target to make them coincide on the retina, and form one point). A Coincidence optometer is shown in Figure 5. The patient moves the target forwards or backwards until they see only a single spot. The patient's refractive error is worked out the same way as for the ordinary Badal Optometer; the only thing that changes is the judgement the patient makes.
The coincidence optometer makes it fairly easy to diagnose and measure astigmatism as well. With astigmatism, the patient will not see points of light, but elongated ovals. If the two holes of the Scheiner disc lie in one of the astigmatic meridians, then the patient can move the target so the two ovals formed overlap and form a single line image. The Badal equation then gives the power along that meridian.
If the two holes do not lie on a meridan, however, the two ovals will never overlap. So the patient also has to be able to rotate the Scheiner disc to get a single line image. The angle of rotation gives the meridian of astigmatism. By finding the refractive error along the two meridians, the coincidence optometer gives you a power cross for the eye's refractive error.
The two optometers discussed above are still subjective refraction devices. That is, they rely on the patient making a judgement about what they see, although in the case of the Scheiner disc, that judgement is easy (i.e. one spot or two). We could remove that element of subjectivity if we could somehow look at what is on the patient's retina.
We can look at the patient's retina using an ophthalmoscope. If we projected an image onto the retina using a Badal optometer, we could then use an ophthalmoscope to see it, and adjust the optometer until we see a sharp image, or (if using a Scheiner disc) a single point image on the retina. This would be an "objective" optometer, since we would no longer rely on the patient's judgement to decide the position of the target in the Badal optometer. To combine a Badal optometer and an ophthalmoscope, you'd have to have some way of ensuring the two devices don't get in each other's way.
The design of real autorefractors is quite complex, to maximize their performance and minimize their cost, but we can get a good idea about how autorefractors work by looking at a simplified design. The autorefractor we will consider here is based on a Badal optometer.
Figure 6 shows a simplified Badal autorefractor. A semi-transparent mirror and an image screen have been introduced into the optometer, and move in synchrony with the target. The light from the target passes through the mirror and goes into the patient's eye, just as the ordinary Badal optometer in Figure 2.
Some of the light hitting the patient's eye is reflected (click the reflected light button to see it). Some of this reflected light leaves the patient's eye and retraces the path backwards through the optometer. When it hits the semi-transparent mirror, some gets reflected down towards the image screen. We can adjust the postion of the target to form a sharp image on the screen. When we have done so, we know that there is also a sharp image on the patient's retina.
To understand how this optometer works,
To make this an autorefractor, we need some device which can tell whether the image on the screen is sharp or not, and a motor to move the target+mirror+screen back and forwards to find a sharp image.
In order to get enough light to reflect off the patient's retina and make it all the way back to the image screen, the Badal light source has to be quite bright. The problem with a bright light is it will reduce the pupil size (thus reducing the amount of light leaving the eye) and will be very uncomfortable to view. For that reason, autorefractors use an infrared light source, which we can't see. Although this solves the problem of an uncomfortably bright light, it creates a few new problems of its own: