Chapter 38 The Cross Cylinder STUART E. WUNSH Table Of Contents |

DETERMINATION OF CYLINDER AXIS THEORY OF AXIS DETERMINATION DETERMINATION OF CYLINDER POWER PRESBYOPIA ACCOMMODATION CROSS CYLINDERS AND OTHER REFRACTIVE TECHNIQUES ASTIGMATISM REFERENCES |

The invention of the cross cylinder can be traced back to the early 19th
century. In 1837, the astronomer Sir George Biddle Airy^{1} made a cylindric lens and used it to correct astigmatism. Astigmatism
was not readily recognized in those days, and when instances were found
they were reported in the literature. The trial lens sets of that era
contained no cylindric lenses; when such lenses were prescribed, they
could be obtained only on special order from a few laboratories.In response to the growing recognition of astigmatism and the need to correct
it, a device called the Stokes lens was introduced in 1849. In 1885, W.S. Dennett improved the lens mounting and tried to popularize
the instrument. Although Jackson did describe his findings, it remained for W.H. Crisp, a
more prolific writer, to bring this device to worldwide attention. + 0.12 D cyl × 90°-0.12 D cyl × 180° Written in this form, it is clear that we are dealing with cylinders (cyl). The
spherical notation (sph) may be used ( The question of the possible spherical component of the cross cylinder
and its effect on the final correction has not been settled. This question
has nothing to do with the notation of the lens or whether it is
ground as two cylinders or as one sphere and one cylinder, but it is concerned
with the total effect of the lens on the object viewed. When
it is introduced before the tentative correction, does the cross cylinder
change the spherical power, or does it merely expand the conoid of
Sturm? Jackson I believe that the cross cylinder, even though it can be expressed in spherical form, is not a -0.50 sphere with a plus cylinder axis 90° away, but rather an infinite series of powers at an infinite number of meridians, resulting in an enlargement of the conoid of Sturm. Some day, someone will design a three-dimensional model that can be viewed on a screen; this might better clarify the issue. In point of fact, the question of spherical power in the cross cylinder
is moot. Refraction is a dynamic process. |

DETERMINATION OF CYLINDER AXIS |

The following definition of terms will aid in understanding the mechanical
technique of determining cylinder axis with the cross cylinder. The
term twirl means a “flipping” of the lens before the eye in such a way
that the side of the cross cylinder facing the patient at the beginning
of the maneuver comes to face the examiner at the conclusion of the
maneuver. The term rotate refers to a clockwise or counterclockwise motion of the cross cylinder
in front of the patient, in the plane of the spectacle lens, about an
axis parallel to the line of sight. The term correcting cylinder refers to the cylindric lens in the trial frame. This may represent either
the tentative or the final correction, depending on the stage of
the refraction. The term resultant denotes the total dioptric power obtained when different lenses are superimposed.The first step in determining cylinder axis is to choose a tentative sphere
and cylinder. The sphere is then adjusted so that the patient can
read 20/40 (6/12)* or better. At this point, there should be a state of Metric equivalent given in parentheses after Snellen notation. The cross cylinder is now placed in front of the correcting cylinder, with
its handle parallel to the axis of the correcting cylinder. In this
position, the axes of the cross cylinder straddle the axis of the correcting
cylinder at 45° each. It is explained to the patient that
both images to be presented may be slightly blurred, but one may be clearer
than the other. The patient is to state which position of the cross
cylinder presents the clearer image (position 1 or position 2) or
whether they appear alike. The The cross cylinder is then twirled in front of the correcting cylinder. This maneuver reverses the positions of the plus and minus axes of the cross cylinder. If both positions appear equally clear (or equally blurred), then the axis of the correcting cylinder is at the proper meridian and the test for axis is complete. If one position is clearer, the correcting cylinder is rotated toward the axis on the cross cylinder having the same sign as that of the correcting cylinder. Assume that a minus cylinder is being used as our correcting cylinder, and the patient selects the clearer position of the cross cylinder. We locate the minus axis of the cross cylinder in that position and rotate the correcting cylinder toward that axis 5° or 10°. (If plus cylinders were used for correction, we would rotate the correcting cylinder toward the plus axis of the cross cylinder in the preferred position.) The reason for this maneuver is explained later in the chapter. The cross cylinder is then relocated so that its axes once more straddle the axis of the correcting cylinder in its new position, and the test is repeated. When twirling the cross cylinder produces equal clarity (or blurring), the end point of the procedure has been reached and the axis of astigmatism has been identified. |

THEORY OF AXIS DETERMINATION | |||||||||||

The cross cylinder is particularly sensitive in assisting the patient in
the selection of the proper axis of his or her astigmatism. When the
cross cylinder straddles the presumed cylinder axis in the trial frame
in one position, it creates a new cylinder whose axis is approximately 40° away
from the original axis and increases the power of the
cylinder; however, in the other position, the axis will jump 40° in
the opposite direction but reduce the power. The patient easily perceives
this change, seeing the lower powered cylinder as clearer, and the
appropriate steps may be taken by the examiner. Many texts at this
point refer to the complexity of understanding the mathematics that show
why the foregoing occurs and pass onto another topic. With regard to the cross cylinder and axis determination, comprehending
the first paragraph is all you will need to be a successful refractionist. The
full power of the cylindric lens lies 90° from its axis. For
example, a + 1.00 D cyl × 180° has for its power at
the 180° meridian 0 D and at the 90° meridian + 1.00 D. What
may not be generally appreciated is that between these two meridians (180° and 90°) the lens has dioptric power. In the example
just mentioned, this begins with 0 D at 180° and increases at each
meridian to the maximum power, + 1.00 D at 90° (Fig. 1). Of course, this applies to any cylinder regardless of power, sign (plus
or minus), or axis. A cylinder will have a power of 0 D at its axis
and increasing plus or minus power at each meridian, reaching a maximum (
The power at each meridian of the lens is a function of the sine of the angle between the given meridian and the axis of the cylindric lens. To review, the sine of an angle in a right triangle is determined by the ratio between the length of the side opposite the angle and the length of the hypotenuse of the triangle. For a given angle, , it may be written as follows (Fig. 2).
The sine of 0° is 0, the sine of 90° is 1, and the sine of 180° is once again 0. Between 0° and 90°, the values of sine increase from 0 to 1; they show a corresponding decrease between 90° and 180°. The sines of angles from 0° to 180° may be plotted on a graph, as shown in Figure 3 (the round points), and give rise to the sine curve. Many wavelike or cyclic phenomena may be described by a sine curve. In our current example, the changing powers of a cylindric lens seem to follow this curve.
Two additional factors slightly complicate this discussion. First, the
relationship between the angle and the cylinder power is more accurately
described by the square of the sine. That is, Dm (the dioptric power
at any meridian of a cylindric lens) is equal to D (the maximum power
of the cylinder) multiplied by the sine squared of the given angle. If two cylinders of equal strength but opposite sign are placed one before the other with axes superimposed, they will neutralize each other. For example, if a -1.00 D cyl × 180° is placed before a + 1.00 D cyl × 180°, the resultant power is zero. The -1.00 D, which is the power in the vertical meridian of one lens, is neutralized by the + 1.00 D in the same position in the other lens. In addition, at each meridian between 0° and 180°, the powers present are equal but of opposite sign, thus neutralizing each other as well. If, however, the axes were not perfectly aligned, we would have to add, algebraically, the powers present at each meridian and arrive at a resultant power. For example, beginning with a cylinder + 1.00 D × 180°, as pictured in Figure 1, we will superimpose a -1.00 D cyl but place its axis at 170° (10° “off axis”) (Fig. 4). Note that at 90° in Figure 1, we have + 1.00 D; at 90° in Figure 4, we have -0.970 D. The resultant in this meridian is + 0.030 D; this calculation is repeated at each 10° meridian.
The resultant is pictured in Figure 5. We have created a spherocylinder, the formula of which may be written + 0.174 D
sph-0.348 D cyl × 130°. Note the following: our -1.00 D cylinder
no longer neutralizes our + 1.00 D cylinder. A new cylinder has
been formed, in which the axis (using minus cylinder notation) is approximately 40° removed
from the axes with which we began (
Suppose the correcting cylinder were displaced by an even greater amount. Again, begin
with a + 1.00 D cyl × 180° and superimpose
a -1.00 D cyl, but this time displace the axis 30° (
The power of the resultant spherocylinder has increased from the previous example (see Fig. 5), but the shift in the resultant cylinder axis is not much greater than that produced by shifting the correcting cylinder 10° “off axis.” By moving the correcting cylinder 20° farther off axis (from 170° to 150°), we have moved the resultant cylinder only 10° (from 130° to 120°). It can be shown that further deviation of the correcting cylinder axis produces proportionately smaller shifts in the resultant cylinder axis. The conclusion to be drawn is that a small error in the placement of the correcting cylinder produces almost maximal shifts of the resultant cylinder axis. (This large shift in axis can be perceived by the patient and may be used in determining the correct cylinder axis in refraction. This will be considered in more detail later in the chapter.) Apply this knowledge to an actual refraction, using an eye that requires
a -1.00 D cyl × 180° for correction of ametropia. We have
shown previously that a -1.00 D cyl × 180° neutralizes a + 1.00 D
cyl × 180° in all meridians. Therefore, we may assume
the eye thus corrected contains a + 1.00 D cyl × 180°. That
is, it is 1.00 D myopic in the 90° meridian, with progressive
decrease in myopia to plano in the 180° meridian, and is corrected
by the -1.00 D cylinder. The refracting power of the eye is “too
strong” in the 90° meridian, and the image is in the
vitreous. This + 1.00 D cyl × 180° we can call the We have now set the stage for understanding the use of the cross cylinder. In
considering the example given of a patient's eye with its
cylinder of + 1.00 D × 180°, if we assume that this eye
cylinder axis is actually at 170° and place our -1.00 D correcting
cylinder before the eye at this axis, a secondary astigmatism is created. (This
has been previously considered in detail in this chapter.) It
is this secondary astigmatism that is actually examined with the cross
cylinder. The cross cylinder is placed in front of the correcting
cylinder so that its axes straddle the axis of the correcting cylinder
at 45° each. (I have chosen a + 0.25 D cyl-0.25 D cyl for our example.) The cross cylinder is twirled, presenting
each of the two sides to the patient and thus reversing the positions
of the plus and minus axes. Figure 5 diagrams the secondary astigmatism. Figure 8 represents a 0.25-D cross cylinder, axes straddling 170°; the minus
axis is marked as a double line. If we superimpose Figures 5 and 8 and add the powers at each meridian, we arrive at Figure 9, which represents the approximate
Figure 10 shows the cross cylinder in the second position with axes reversed; the minus cylinder axis is represented by a double line. Superimposing Figure 5 (the secondary astigmatism) and Figure 10 yields Figure 11, the tertiary astigmatism produced with the cross cylinder in the following position: + 0.083 D sph-0.166 D cyl × 30°. Notice that in the second position of the cross cylinder, the powers seem to neutralize each other at each meridian, whereas in the first position they augment each other. The patient will readily choose the position of the cross cylinder that produces the weaker tertiary astigmatism as being less distorting or clearer. This is the position of the cross cylinder depicted in Figure 10. Rotating the axis of the correcting cylinder toward the minus axis of the cross cylinder will bring the correcting cylinder axis closer to 180°. This is the reason for the following rule: “The correcting cylinder is rotated toward the axis of the cross cylinder of similar sign in the clearer position.” If a plus cylinder were used for correction, the cylinder would be rotated toward the plus axis of the cross cylinder; similar results would be obtained.
When the correcting cylinder arrives at 180° the secondary astigmatism disappears. The cross cylinder itself now forms a secondary astigmatism as it straddles the axis of the correcting cylinder (which is now at the correct axis). When the blurring produced in the two positions of the cross cylinder is equal, the end point is established. Notice that the cross cylinder produces little shift in the axis of the secondary astigmatism. It instead augments the power of the tertiary astigmatism produced in one position and diminishes it in another. In summary, we have done two things: - By misplacing the axis of the correcting cylinder, we have created a secondary
astigmatism.
- By using the cross cylinder, we have augmented or diminished the power
of this new astigmatism in order to ascertain which direction the correcting
cylinder must be rotated to approach the true axis of astigmatism.
You may wonder why we become involved with “tertiary astigmatism” when
we might use the blurring induced by the secondary astigmatism
to determine the correct axis. The following method may be used - Make a temporary overcorrection with the trial cylinder.
- Rotate the trial cylinder in the frame a few degrees to either side of
the assumed axis.
- Ask the patient to choose the clearer position; he or she will readily
recognize the sudden shift in the resultant cylinder axis as the correct
axis is passed, forming a secondary astigmatism.
There are, however, pitfalls to this procedure. Rotation of the correcting cylinder before the eye to determine the correct axis of astigmatism is highly accurate when the correct or slightly stronger cylinder power is selected as the trial lens and the Reagan-Lancaster dial is used as the test target. No similar reservations need be made regarding the use of the cross cylinder in determining the astigmatic axis. It is a highly accurate and reproducible technique. |

DETERMINATION OF CYLINDER POWER |

When the correct cylinder axis has been found, attention is turned to determining
the proper cylinder power. The cross cylinder is placed in
front of the correcting cylinder with either axis (plus or minus) parallel
to and superimposed on the axis of the correcting cylinder. Snellen
letters, preferably 20/40 or better, are used. The cross cylinder is
then twirled. The plus and minus axes of the instrument successively
come to overlie the axis of the correcting cylinder. Cylinder power is
changed according to the patient's selection. For example, if the
plus axis overlying the correcting cylinder axis produces the clearer
image, more plus cylinder or less minus cylinder is placed in the trial
frame. This is continued until equal clarity (or equal blurredness) is
noted on twirling the cross cylinder. The following simple example demonstrates exactly how this occurs. In the previous section, our model needed a -1.00 D cyl × 180° for total correction. From our retinoscopy, we have decided that the correcting cylinder is -0.75 D cyl × 180°. Once we have determined the correct axis, we perform the maneuvers outlined earlier. With the plus axis of the cross cylinder at 180°, we have the following: -0.75 D cyl × 180° With the minus axis of the cross cylinder at 180°, we have the following: -0.75 D cyl × 180° With the plus cylinder axis of the cross cylinder at 180°, the resultant
differs from the required 1.00 D cyl × 180° by -0.50 D
cyl × 180°-0.25 D cyl × 90°; however, with the minus axis of the cross
cylinder at 180°, this difference is only 0.25 D cyl × 90°. The
patient will choose the second position, which offers more minus
cylinder at 180°. We now add -0.25 D cyl × 180° and the
correction is exact ( It has been determined Jackson, however, used the cross cylinder apparently satisfactorily with
the eye fogged. |

PRESBYOPIA | |

The use of the cross cylinder in determining presbyopia and in prescribing
the necessary corrective lenses was described by Jacques, an optometrist.^{2} The target he used was a cross printed on a card (Fig. 12); each limb of the cross was made up of three heavy black lines. The test
may be performed monocularly or binocularly. The Jacques target and
the appropriate cross cylinder are incorporated in the AO Phoropter (Reichert
Optical Instrument Division, Leica Inc., Buffalo, New York).
The full distance correction is determined, and the cross target is placed 40 cm
in front of the patient. A cross cylinder (+ 0.50 D × 180°-0.50 × 90°) is introduced before each eye. (The minus cylinder
axis should be at 90°.) If the target is conjugate with the retina ( The test is based on the assumption that the patient will suspend his accommodation between the bars of the cross when the cross cylinder is introduced. The test is not valid if the patient actively accommodates on one set of lines. One investigator Theoretically, this should be a valid test for the detection and correction
of presbyopia. In practice, however, patient responses for some authors
were variable. |

ACCOMMODATION |

The cross cylinder may be used to measure the amplitude of accommodation.^{2} It is a variation of the test for presbyopia. The patient wears his full-distance
correction, and a 0.25-D or 0.50-D cross cylinder with the
minus axis at 90° is placed before the patient. A Jacques blur-point
cross is then placed 33 cm from the patient. If the patient is not
presbyopic, the horizontal and vertical lines will appear equally clear. The
card is then slowly moved toward the patient's eye, and he
is asked to report when the horizontal lines appear blacker than the
vertical lines. At this point, the amount of accommodation is read from
a Prince rule, or it is determined from the distance between the patient's
eye and the target. If the patient is presbyopic, a + 3.00-D
lens is introduced, and the test is performed again; the 3 D
is subtracted from the final result. As the card is brought closer to
the patient's eye, he or she is forced to accommodate to keep the
lines equally clear. When the accommodative reserve is exhausted, this
dual clarity cannot be maintained. The lines parallel to the plus axis
on the cross cylinder come to lie on or near the retina. Those parallel
to the minus axis of the cross cylinder are focused behind the retina
and are less clear. |

CROSS CYLINDERS AND OTHER REFRACTIVE TECHNIQUES |

Cross cylinder technique may be combined with other refractive techniques
for the determination of axis and strength of the correcting cylinder. The Reagan-Lancaster dial may be used with the cross cylinder instead of Snellen letters. After a tentative spherocylinder is selected, the patient is presented with the dial figure. The cross cylinder is placed with its axes straddling the correcting cylinder axis at 45° each and is twirled. (The techniques and principles set forth in the sections on determination of cylinder axis and power apply to the dial and cross cylinder. The difference is only in the choice of target.) The correcting cylinder is rotated toward the axis of similar sign in the clearer position of the cross cylinder. The end point is equal clarity or blurring of the lines of the dial in both positions of the cross cylinder. The cross cylinder is then rotated so that its axes successively come to overlie the correcting cylinder axis. Cylinder power is determined using the dial figure. The end point is equal clarity of all the lines of the dial. The rotating cross may be substituted for the dial. Cylinder power is determined by placing one limb of the cross parallel to the axis of the correcting cylinder. Cylinder power is changed appropriately until equal clarity of the bars of the cross is obtained. The foregoing methods are not intended to be the primary technique used for the detection and correction of astigmatism. They appear to be most useful in evaluating small astigmatic errors. Linksz An objective method for determining and correcting astigmatism has also
been described. This, of course, is a variation of the technique of determining cylinder axis with the cross cylinder. We know that in one position of the cross cylinder we augment the tertiary astigmatism and in the other we diminish it. The position that diminishes the astigmatism is the one that the patient says is clearer, and this is the position discussed earlier. At the correct axis, the tertiary astigmatism is neutralized, leaving only the power of the cross cylinder. This is “plus” power at the minus axis and thus gives us a “with” motion on retinoscopy. |

ASTIGMATISM |

The cross cylinder may be used to determine astigmatism,^{2}^{,}^{3} particularly when its presence is in doubt. The cross cylinder is introduced
over the correcting sphere with the plus cylinder first at 90° and
then at 180°. If neither position provides any increased clarity
for the patient, the oblique meridians are explored in a similar
fashion. If some enhancement of clarity is found in any position, a
correcting cylinder may be introduced at the appropriate axis and refining
procedures can then be carried out. |

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