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Module 19 |
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Module 19: |
Clinical Optics Part 3 |
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Measurements, Calculations, Adjustments, and Formulae |
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Frame PD and Lens Decentration The Axis Cross and the Power Cross Focal Length and Diopter Power Focal Distance for Non-parallel Light Converting K Readings to Cyl Correction |
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The distance between the pupillary centers, or the interpupillary distance (PD), is an important measurement to know when choosing glasses frames, aligning optical centers, evaluating induced prism, and adjusting binocular optical instruments. |
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The pupillometer has been proven to be the most accurate method for measuring the PD. Use one if it is handy. If not, the light reflex method is simple, sufficiently accurate, and inexpensive (mm ruler + penlight). | |||
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Procedure 1) Face the patient, arms length away. 2) Hold your mm ruler across the bridge of the patient’s nose with the "0" mark approximately lined up with the patient’s right pupil. 3) Close your right eye and fix the patient’s right eye with your left eye. Instruct the patient to look at your left eye. 4) Hold the penlight directly under your left eye and shine it at the patient’s right eye. 5) Align the "0" mark of the mm ruler directly under the light reflex on the cornea of the patient’s right eye. The reflex should appear to be in the patient’s pupil. |
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6) Without moving the ruler, shift the penlight to a position directly under your right eye, and shine it at the patient’s left eye. Instruct the patient to look into your right eye. 7) Open your right eye and close your left eye. Observe the position of the light reflex in the patient’s left pupil. Read the mm mark directly below the light reflex. This is the PD reading. |
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The procedure will become smooth and quick with a little practice. If the patient’s face is asymmetrical it may be necessary to measure monocular PDs. The light reflex method is still used, but the measurement for each eye is taken from the center of the nasal bridge to the light reflex.
Frame PD and Lens Decentration
When a patient is fit with glasses, if the patient's pupil does not line up very near the center of the lens, then the glasses lens must be decentered to line up the optical center of the lens with the patient's pupil. This results in an unwanted increase in the lens thickness. (See Module 18, Section 1 for more information)
There are two formulas used to figure lens decentration:
Frame PD = A-box measurement + Distance Between Lenses (DBL)
Lens Decentration = Frame PD - Patient PD
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The spherical equivalent is an important optical concept for the technician to understand. It is used in the refractometric process, in figuring corrections for visual field exams, and in contact lens power calculations. Mathematically, the spherical equivalent is computed by algebraically adding half the cylinder power to the sphere power of a spherocylinder. It represents the average of the two powers that make up the spherocylinder. Minus cylinder example: +1.00-3.00x90 The spherical equivalent is— +1.00+(-1.50) = -0.50 or +1.00-1.50 = -0.50 Plus cylinder example: +1.00+3.00x90 The spherical equivalent is— +1.00+1.50 = +2.50 Optically, the spherical equivalent represents the Circle of Least Confusion, or the Circle of Most Confusion as I like to call it, because it is not an easy concept to grasp. The Circle of Least Confusion is part of the Conoid of Sturm.
You can read about the Conoid of Sturm, if you have trouble going to sleep, in any book on optics. For our purposes, a simplified explanation will do. |
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An image that falls on the retina can be thought of as being made up of many dots, just like a photo in the newspaper. If an ametropic eye (needs glasses to see well) is not optically corrected, then the image will consist of many blur circles instead of sharp dots. The more out of focus the image is, the larger the circles are. If an astigmatic eye is not optically corrected, the blur circles will be distorted into ellipses. For example, if an emmetropic person is looking at a cross, it may be represented as an image with sharp dots, like this— |
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A minus 2D myope, without correction, may see the image like this— |
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If a person with an Rx of -2.00-2.00x90, without his glasses, may see the same image like this— |
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Sometimes we do not supply the patient with his full astigmatic correction: such as patients with 1.00D or less cylinder correction when performing a visual field exam, or perhaps the soft contact lens patient with 1.00D or less astigmatism in one eye. In these situations we use the spherical equivalent. These patients will not see the sharply focused dot image. The spherical equivalent, representing the "Circle" of Least Confusion, provides a blur circle instead of a blur ellipse. The basic idea is that if the image is going to be a little blurry, it is better that it not be distorted also. |
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Transposing a glasses prescription is simply converting the prescription from minus cylinder notation to plus cylinder notation. The optical properties of the prescription remain the same.
Procedure:
Example 1— Transpose the following prescription:
Example 2 —
Why are there plus and minus cylinders? Plus cylinder lenses exist only in phoropters and trial lenses. Glasses lenses are made in minus cylinder. Plus cylinder phoropters are popular because it is easier to teach retinoscopy in plus cylinder. Ophthalmologists typically learn retinoscopy and refraction in plus cylinder. Optometrists typically use minus cylinder.
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The Axis Cross and The Power Cross
The power cross is a graphical representation of the power orientation of any glasses prescription. It is useful when calculating induced prism. The axis cross is a graphical representation of the axis orientation of any glasses prescription. It is useful when performing retinoscopy with loose lenses.
The power and axis crosses for -3.00-1.50x90 are pictured below: |
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The key to constructing a cross from a given prescription is to first transpose the prescription:
-3.00-1.50x90 transposed is -4.50+1.50x180
The diopter powers on the cross are the sphere powers from each form of the transposition (-3.00 and -4.50). For the axis cross, the minus cylinder axis is 90, corresponding to the most minus number. The plus cylinder axis is 180, corresponding to the most plus (or least minus) number.
Remember that the lens power (thickness) is 90 degrees from the axis. The power cross will have the diopter powers reversed from the axis cross.
How the power cross is used:
When calculating induced prism from a PD that does not match an OCD, we are interested in the power of the lens in the horizontal meridian. The power cross will give us this number. In the example above, -4.50 D is the lens power in the 180 meridian.
When calculating the need for a slab-off, we are interested in the lens power in the 90 degree meridian. In our example the lens power at 90 degrees is -3.00 D.
How the axis cross is used:
An infant or your child will not sit behind a phoroptor. In order to perform retinoscopy in such a situation, lose lenses are used. Retinoscopy is performed with the usual techniques, but by holding only sphere powers up in front of the child's eye. In our example, with motion neutralized while steaking the 180 degree meridian (plus axis 90) with a -4.50 sphere. There was still with motion when streaking the 90 degree meridian (plus axis 180), and this neutralized with a -3.00 sphere. Draw a cross on paper and simply match the neutralizing sphere power with the orientation of the streak. The axis are labeled by estimation. The prescription is derived from starting at either diopter power and traveling on the number line to the other power. For example, if -3.00 is our sphere power, then we travel -1.50 to get to -4.50 and the axis of the -4.50 power is 90, -3.00-1.50x90.
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The Lens Power at 180 or 90
When calculating induced prism from an OCD that is different than the PD, we use the lens power in the 180 degree meridian. When figuring a possible slab-off, we are interested in the lens power in the 90 degree meridian. The power cross is useful for this calculation, but what if the axis of the prescription is not 180 or 90? We would then need to calculate a percentage of the total lens power, the percentage that has effect at either the 180 degree meridian, or the 90 degree meridian.
Let's stay with our prescription example, but change the minus cyl axis to 135. Below is pictured the power cross for -3.00-1.50x135. |
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Graphically, we can see that the power in the 180 degree meridian, and the 90 degree meridian, is going to be half way between -3.00 and -4.50. The power at 90 and 180 is -3.75. Let's take another example, one that is not so obvious. Suppose the minus cylinder axis is 75? Below is pictured the power cross for -3.00-1.50x75. |
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From the graph we can see that the power at 90 is going to be a little more than -3.00, perhaps -3.25. We can also see that the power at 180 is going to be a little less than -4.50, perhaps -4.25. For practical purposes there is nothing wrong with estimating from the graph. There is, of course, a way of calculating the exact power at 90 and 180 for any glasses correction.
Let's stay with our lens power of -3.00-1.50x75
Step one: Transpose so you have both cylinder forms.
-3.00-1.50x75 and -4.50+1.50x165
Step two: Subtract the lower axis number from 90.
90 - 75 = 15
Step three: Divide the step 2 answer by 90.
15 / 90 = .16
Step four: Multiply the step 3 answer by the cylinder power.
.16 x 1.50 = .24
Step five: Subtract the answer in step four from the sphere power of the minus cylinder Rx, and add it the sphere power of the plus cylinder Rx.
-3.00 - .24 = -3.24 and -4.50 + .24 = -4.26
Axis 75 is closest to 90, so we match up -3.24 with 90. Axis 165 is closest to 180, so we match up -4.26 with 180. If in doubt, chart it out!
Answer: For the prescription -3.00-1.50x75, the power in the 90 degree meridian is exactly -3.24 D, and the power in the 180 degree meridian is exactly -4.26 D. |
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The distance from the back surface of the glasses lens to the front surface of the eye (the vertex distance) can affect the effective power of the lens, especially in higher powered prescriptions. |
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The vertex distance changes the effect of plus lenses and minus lenses in opposite directions, as shown below:
1) Increasing the vertex distance of plus lens will increase the effective power of the lens. 2) Decreasing the vertex distance of a plus lens will decrease the effective power of the lens. 3) Increasing the vertex distance of a minus lens will decrease the effective power of the lens. 4)Decreasing the vertex distance of a minus lens will increase the effective power of the lens.
In order for the glasses prescription to have exactly the same effective power as the refraction, the vertex distance of the phoroptor or trial frame must match the vertex distance of the lenses in the frames that the patient will wear. A difference in the two vertex distances only becomes significant if the diopter power of the prescription exceeds 6 diopters.
Vertex Compensation Formula
The formula for the needed compensation per millimeter of displacement, per diopter of lens power, is as follows:
diopters squared, divided by 1000
The answer is multiplied by the millimeters of displacement. The result is added or subtracted from the diopter power according to the following set of conditions:
1) Plus lens moving closer - add to increase the diopter power 2) Plus lens moving farther away - subract to reduce the diopter power 3) Minus lens moving closer - subtract to reduce the diopter power 4) Minus lens moving farther away - add to increase the diopter power
Spherical lens example:
Consider a -12.00 Sph Rx that was refracted at 13mm. The lens in the patient's new glasses will sit 10mm away from the patient's eye.
12 squared = 144, 144/1000 = .14
The movement is 3mm closer to the patient's eye, with a minus lens.
3 x .14 = .42, so .5 D is subtracted from -12.00 to reduce the lens power to -11.50 D.
Cylindrical lens example (same vertex change as above):
If the Rx has a significant cylinder power (at least 1 D), we must perform the calculation for the primary meridians of power (remember the power cross?).
Consider a +12.00+3.00x180 lens that was refracted at 14mm. The lens in the patient's new glasses will sit 10mm away.
Transpose to +15.00-3.00x90
+12.00 and +15.00 are the powers we must adjust.
For +1200:
12 squared is 144, 144/1000 = .14
The movement is 4mm closer to the patient's eye, with a plus lens.
4 x .14 = .48, so .5 D is added to +12.00 to increase the lens power to +12.50.
For +15.00:
15 squared is 225, 225/1000 = .225
4 x .225 = .9, so 1 D is added to +15.00 to increase the lens power to +16.00.
Our adjusted Rx is:
+12.50+3.50x180 or +16.00-3.50x90
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Focal Length and Diopter Power
The focal length of a lens is the distance at which the lens brings light to a focus. For a plus lens, the focal point is real, meaning light is brought to a focus in front of the lens. |
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For a minus lens, the focal point is virtual, meaning, since the light is diverged by the lens, the focal point is behind the lens. |
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A lens diopter is defined as the lens power that will bring light to focus at a focal distance of 1 meter.
If the lens power is know, the focal distance can be found by the following formula:
F(m) = 1 / D
Note that the focal distance is in meters.
If the focal distance of a lens is know, the diopter power can be found by this formula:
D = 1 / F(m)
Example: What is the diopter power of a lens with a focal distance of 20 centimeters?
We first convert 20 centimeters to meters by moving the decimal point two places to the left.
D = 1 / .2 D = 5 diopters
Example: What is the focal length in millimeters of an 8 diopter lens?
F(m) = 1 / D F(m) = 1 / 8 F(m) = .125 meters
We convert .125 meters to 125 millimeters by moving the decimal point 3 places to the right.
These formulas are very useful when calculating the add power needed for a particular reading or intermediate viewing distance. The results of the most used calculations are marked in diopters, centimeters, and inches on the reading rod supplied with the phoroptor.
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Focal distance for non-parallel light
For parallel light we can use the f(m) = 1 / D to figure the focal length of a lens, but not all light that enters a lens is parallel. If light is being bounce off of an object 1 meter away, the light will be diverging as it enters the lens. There is a formula that lets us compute the focal point for objects closer than infinity.
U + P = V
U = vergence of object rays in diopters P = lens power V = vergence of the rays from the lens in diopters
Example 1: An object is 2 meters from a +4 D lens. How far from the lens does the image focus?
U = 1 / f(m), U = 1 / 2, U = -.5 D, U is negative because light reflected from an object closer than infinity is divergent.
U + P = V, -.5 + 4 = 3.5, V = 3.5
Since V is in diopters, we will need to convert to focal distance.
f(m) = 1 / D, f(m) = 1 / 3.5, f(m) = .28 meters or 28 centimeters
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One prism diopter is defined as the prism power needed to deviate a ray of light 1 centimeter at a distance of one meter. |
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Formula: If deviation and distance are know, the power of the prism can be calculated by this formula:
PD = deviation (cm) / distance (m)
Example: What is the power of a prism that will deviate light 2 cm at a distance of 50 cm.
We convert 50 cm to .5 meters by moving the decimal point two places to the left.
PD = 2 / .50 = 4 prism diopters
Variations of the formula:
deviation (cm) = PD x distance (m) distance (m) = deviation (cm) / PD
Example: How far will a 6 D prism deviate light at a distance of one meter?
deviation (cm) = PD x distance (m) deviation = 6 x 1 = 6 cm
Example: At what distance will a 10 D prism deviate light 4 cm?
distance (m) = deviation (cm) / PD distance = 4 / 10 = .4 meter
Induced prism is created when the patient is not looking through the optical center of a glasses lens. The most common glasses problems caused by induced prism are:
1) Horizontally induced prism is caused when the optical center distance (OCD) in a pair of glasses does not match the patient's PD. For a discussion of this situation, see Module 17, Section 3.
2) Troublesome vertically induced prism is caused when the patient looks through the bifocal of a pair of glasses having lenses that are of opposite powers in the 90 degree meridian. For a discussion of this situation, see Module 18, Section 2.
Induced prism is calculated using the Prentice Rule: Induce prism (in diopters) = Lens power (in diopters) x Displacement (in centimeters).
The tricky part about this equation is that displacement is in centimeters in the equation but is always read in millimeters. So the millimeter reading must be converted to centimeters (move the decimal point one place to the left) for use in the formula. An alternative would be to use the following form of the formula for millimeters:
Induced prism (in diopters) = Lens power (in diopters) x Displacement (in millimeters) / 10.
Example:
The patient is looking 3mm away from the optical center of a 4 diopter lens. Calculate the amount of induced prism.
4 x .3 = 1.2 diopters of induced prism
As you can see from the result, it takes a fairly strong lens and good deal of displacement to create a significant amount of induced prism. Of course most induced prism problems are more complicated than this simple calculation. You usually have to think about two lenses, the total power in the affected meridian, and the direction of the prism effect.
Convert K readings to cylinder correction
Corneal cylinder power equals the difference between the diopter power of the steepest corneal meridian and the diopter power of the flattest corneal meridian.
Example: Ks OD 44.50 x 90 / 43.50 x 180 OS 45.00 x 105 / 42.00 x 15
The astigmatism for the right eye would be 44.50 - 43.50 = 1 D. For the left eye it would be 45.00 - 42.00 = 3 D. Translating to a refractive notation is as follows:
For minus cylinder: Take as the axis, the axis of the more minus number. For plus cylinder: Take as the axis, the axis of the more plus number.
In the above example:
OD is either -1.00 x 180 or +1.00 x 90 OS is either -3.00 x 15 or +3.00 x 105
Radius of curvature and diopters
The diopter power of a lens can be expressed in terms of radius of curvature if the index of refraction is known. For the cornea, the formula is:
R = 337.5 / D or D = 337.5 / R
Example: What is the diopter power of a cornea measured to have a radius of curvature of 8.0?
D = 337.5 / 8.0 = 42.18 D
The base curve, ocular curve, and lens power are related as follows:
Base Curve (BC) = Lens Power (P) - Ocular Curve (OC)
Ocular curve (OC) = Lens Power (P) - Base Curve (BC)
Example: If the lens power is -6.00 and the base curve is +2, what is the power of the ocular curve?
OC = P - BC
OC = -6.00 - 2 = -8
Example: A patient is wearing a +3.00 Sphere lens with a base curve of 6 D. If the prescription changes to +1.00, what will the new base curve have to be to keep the ocular curve the same.
First we calculate the ocular curve of the current glasses:
OC = P - BC
OC = 3 -6 = -3
Now we plug this into the BC version of the formula:
BC = 1 - (-3) = 4
We will need to specify a base curve of +4 in the new glasses to keep the ocular curve the same with a lens power of +1.00.
These have been the formulas and the calculations for some of the key concepts in clinical optics. When these formulas are not used often, it is easy to forget how they are applied. That's why we developed the Clinical Optics Calculator. Copy the MSExcel file and put it on a computer at work. Be sure to read the help PDF file for tips on using the calculator.
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