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**Theme 6: Optics of Refractive Error**

Concept of Refractive Error. Types of Refractive Error: Axial and Cylindrical. Clear vision zone of an ametropic eye. Retinal image of an ametropic eye. In this theme we begin to study refractive error from an optical point of view. We define the ocular condition of ametropia, study the different types that exist and classify them according to different criteria. Finally we analyze what is the zone of clear vision for an ametropic eye and we study the main characteristics of the image that forms on the retina of this kind of eye.

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**Concept of Refractive Error**

Emmetropic Eye: Vision of objects at infinity is clear without accommodation. Ametropic Eye: Vision of objects at infinity is blurry without accommodation An eye is emmetropic when it is at rest, or not using accommodation (A = 0), it is focused on infinity, the image coincides with the retinal plane. As a result this eye can see far objects clearly without using accommodation. On the contrary, an ametropic eye does not meet this condition and as a result, without accommodation, sees distant objects as blurry. In the figure on the left is an example of a distant landscape as seen by an emmetropic eye (clear) and on the right as seen by an ametropic eye (blurry). EMMETROPIC AMETROPIC

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**Concept of Refractive Error**

Classification SPHERICAL REFRACTIVE ERRORS: -All of the surfaces of the eye are spherical - Equal power for all meridians Refractive errors can be classified mainly as spherical and astigmatisms. A refractive error is spherical when all the surfaces of the eye are spherical and as a result, the power of all the meridians is the same.

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**Concept of Refractive Error**

Classification REFRACTIVE ERROR ASTIGMATISMS: - Part of the surface of the eye is not spherical - Different power in the different meridians A refractive error is astigmatic when a surface of the eye (normally the first surface of the cornea) is not spherical, and as a result, the power of the meridians of the eye is different. There are other causes that can produce astigmatism, but the first surface of the cornea is the most usual. In this theme we will only deal with spherical refracive errors, while the astigmatisms and the different causes that can produce them will be studied in theme 8.

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**Concept of Refractive Error**

Classification SPHERICAL ERRORS: MYOPIA: F’ in front of the retina HYPEROPIA: F’ behind the retina There are two types of spherical errors: myopia (near sightedness) and hyperopia (far sightedness). To define them, we can go to the position of the focal image point with respect to the retina or the value of refraction. We will begin with the position of the image focal point. With myopia the focal image point (F’) is in front of the eye. The image of an objected situated at infinity, when the eye does not accommodate, is formed in front of the retina and as a result this object is seen as blurry given that the image on the retina will by blurry. In the figure on the left we can see the diagram of ocular condition of myopia with the focal image point in front of the retina. In hyperopia the image focal pin it behind the retina, which means that if an image of an object is at infinity, when the eye does not accommodate, it will form the image behind the retina and as a result, the object will be seen as blurry given that the image on the retina is blurry. In the figure on the right we can see the diagram of the ocular condition of hyperopia with the focal image point behind the retina. MYOPIA HYPEROPIA

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**Concept of Refractive Error**

The far point is the conjugate of the retina for nul accommodation X’ = X + P As seen in theme 5 the far point is defined as the conjugate point of the retina when accommodation is nul. The vergence of the far point is called refraction (R) and the numeric value will characterize the refractive error. Remember that when a distance is measured in meters, the vergence is expressed in diopters. Keeping in mind the definition of far point, if we apply the relation of conjugation that indicates that the vergence image (X’) is equal to the sum of the vergence object (X) and the power of the system (P), we have the vergence image of the far point (the image is on the retina) equal to the vergence of the far point (Refraction, R) plus the power of the eye (P0a) and finally we can obtain an expression of the Refraction (R) that is equal to the vergence of the retina (n’/x’) minus the power of the eye (P0a).

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**Concept of Refractive Error**

For the far point: MYOPIA: far point in front of the eye (R<0) Using the value of refraction R, or the position of the far point, we can define the spherical refractive errors in the following way: Myopia: corresponds to the refractive error where the far point is in front of the ey. As the distance to the far point in this case is negative, the vergence (Refraction) also is (R<0). So, a myope corresponds to an eye that has a negative refraction. Hyperopia: is the refractive error when the far point is behind the eye. As the distance to the far point in this case is positive, the vergence (Refraction) will also be (R>0). So, a hyperope is an eye that has a positive refraction. In this case the far point (conjugate of the retina) is virtual. HYPEROPIA: far point behind the eye (R>0)

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**Concept of Refractive Error**

MYOPIA As we have already explained, the myopic eye, without accommodation, forms the image of a far object behind the retina, as can be seen in the figure on the left of the slide. Because the image forms behind the eye, if the eye accomodates the image it moves even farther from the eye, as can be observered in the figure on the right. As a consequence, a myopic eey can neat see far objects clearly, even though it accommodates.

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**Concept of Refractive Error**

MYOPIA In these photographs we simulate the vision of a myopic eye. The myopic eye sees far objects as blurry, but can see near objects clearly as it has the remote point in front of the eye. As a consequence, the objects that are placed o its far point are seen clearly wihtout the need to accommodate. As far as closer objects, they can also be seen clearly up to a determined distance, depending on the capacity to acccommodate. The photograph on the left of the transparency shows how near objects (people) are seen clearly, while the background is blurry. In the photograph on the right we see how the myope sees the butterfly clearly, as it is near to the obeserver, whiile the sailboat and sea are blurry as this image corresponds to far vision.

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**Concept of Refractive Error**

HYPEROPIA As previously indicated, the hyperopic ee, without accommodation, forms the image of a far object behind the retina, as can be seen in the slide. As the image forms behind the retina, if the eye accommodates the image can be formed on the retina. As a result the hyperope can see far objects clearly with accommodation. For this to happen the hyperopia should not be very great and the eye should have a sufficient capacity to accommodate.

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**Concept of Refractive Error**

HYPEROPIA In these photographs far sightedness (Hyperopia) is simulated. In general we can state that a far sighted eye can see far objects clearly (to do so it should use some degree of accommodation) and sees near ojbects as blurry (it does not have sufficient capacity to accommodate, given that part of it is already being used to see far). In the photograph on the left we can see how the landscape is seen clearly, while the near objects (people) are blurry. In the photograph on the right the butterfly is blurry while the sailboat and sunset are clear, as they correspond to far vision. Near Far

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**Types of refractive error: Axial and Spherical**

Optic Formula for refractive error For an unaccommodated eye, the far point and the retina are conjugate points independent of the refraction value. As such we can write the relation of conjugation as (n’/x’)=R+Poa where n’ is the refractive index of the vitreous humor, x’ the distance from the main image plane to the retina, R the refraction and Poa the power of an eye with refractive eror (see figure). For an emetropic eye, as the refraction is zero, whereby the far point is at infinity, the same relation is expressed as (n’/x’)=Po, with Po as the power of the emetropic eye. For the emmetropic eye R=0

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**Types of refractive error: Axial and Spherical**

Optic Formula for refractive error Refractive component The optic formula for refractive error is obtained by reducing both expressions and reordering the terms. As can be seen in the last expression in the slide, the refraction is the sum of two components: axial and refractive. The axial component describes the difference between the longitudinal axis of the emmetropic eye, while the refractive component describes the differences between the power of the ametropic eye and the emmetropic eye. Axial component

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**Types of refractive error: Axial and Spherical**

Optic formula for refractive error Axial Component Refractive P L R M. Ax. Poa=Po x’>xo R<0 M. Ref. Poa>Po x’=xo H. Ax. x’<xo R>0 H. Ref. Poa<Po With the equation obtained in the previous slide and that we repeat here, we can deduce two possible origins for spherical ametropias. When the power of the ametropic eye coincides with the power of the emmetropic eye, or Poa = Po. The refractive component is cancelled and the resulting error is purely axial, which is to say, the ametropia has an origin which is only due to the difference in longitude between the ametropic and emmetropic eye. When the longitude of the emmetropic eye and the ametropic eye are equal, and as such x0=x0’, in this case the axial component is cancelled and the resulting error is purely refractive, and in this case, the origin of the ametropia is the difference of power of the ametropic eye with respect to the emmetropic eye. In the table the we can see the characteristics of the power and the longitudinal axis for each case of ametropia (Myopia R<0, Hyperopia R>0) in the cases of axial or refractive ametropia. An axial myope (first row of the table) has a power equal to the emmetropic eye (Poa=Po) and an axial longitude greater than that of the emmetropic eye (so that x’>x’0). A refractive myope (second row of the table) has a longitude equal to the emmetropic eye (so that x’=x’0) and a greater power than the emmetropic eye (Poa > Po ). An axial hyperope (third row in the table) has a power equal to the emmetropic eye (Poa=Po) and a greater axial longitude (so that x’<x’0). A refractive hyperope (fourth row of the table) has a longitude equal to the emmetropic eye (so that x’=x’0) and a greater power (Poa < Po ). In the majority of situations the ametropia not due only to one factor, but to a combination of the axial and refractive component.

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**Types of refractive error: Axial and Spherical**

Relation between axial longitude and refraction The eye’s longitude is one of the most important parameters for determining the ocular refraction and also presents the advantage of being able to be measured with biometric instruments. As such, it is interesting to understand how the variation of the longitude influences ocular refraction. Suppose we have two eyes, one emmetropic and the other ametropic, equal except for the axial longitude. We define the ocular lengthening as the difference of longitude in both eyes x’ = x’ – x’0 (remember that distances x’ and x’0 are taken in respect to the principal image plane, but can be considered as equivalents to the longitude of the eye, especially in the case of working with a reduced eye model. We apply the relation of conjugation (see slide 12) for the two eyes (emmetropic and ametropic). Emmetropic Eye: Ametropic Eye:

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**Types of refractive error: Axial and Spherical**

Relation between axial longitude and refraction Reducing the expressions of x’ and x0 we obtain the value of ocular lengthening as x’=-n’R/P(R+P). Now, keeping in mind that the refraction value R is much smaller than the power of the eye, we can ignore R in the denominator of the equation and we obtain a much easier equation for ocular lengthening, that is given as x’=-n’R/P2. So that, for example, for an eye with a power of 60 D (a typical value for eye power), we can consider an ametropia of 1 D supposes a variation of 0.36mm in the longitude of the eye and so a variation of 1mm of longitude of the eye corresponds to an ametropia of about 3D. P=60 D n’=1.336 R=1D x’=0.36mm R=3D x’ 1 mm

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**Clear vison zone of an ametropic eye**

Amplitude and accommodation range of the ametropic eye The amplitude of the ametropic eye is practically the same as the emmetropic eye The amplitude of the ametropic eye can be considered to be equal to that of an emmetropic eye. The amplitude is the difference between the refraction and vergence of the near point. For an ametrope the vergence of the near point will be given by the difference between the refractions and amplitude (P=R-Am), while an emmetropic eye has a refraction of zero, the vergence of the near point is made by changing the sign of the amplitude. Ametrope Emetrope

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**Clear vison zone of an ametropic eye**

Amplitude and range of accommodation of an ametropic eye Range of accommodation Emetrope Although the amplitude of accommodation is independent of the value and type of ametropia, the range of accommodation can sharply vary with the refraction. For an emmetropic eye, eh far point is always at infinity and the near point is calculated as the inverse of the amplitude of accommodation, changing the sign. For example, an emmetrope with an amplitude of accommodation of 5 diopters will have a range of accommodation between infinity and 20cm in front of the eye (remember that the distances in front of the eye have a negative sign). If the amplitude of accommodation is 3D, the range of accommodation is between infinity and 33cm in front of the eye. R Am ZVN 5 -, -20cm 3 -, -33cm

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**Clear vison zone of an ametropic eye**

Range of Accommodation Myope R Am ZVN -2 5 -50, -14cm 3 -50, -20cm -4 -25,-11cm -25,-14cm For a myopic eye the range of accommodation will always be finite and real, given that the far point and near point are found in front of the eye. The slide indicates the expressions to calculate the far point and the near point. For example, a myope of 2 diopters has their far point situated at 50cm in front of the eye and with an amplitude of accommodation of 5 diopters, the near point is placed 14cm in front of the eye, while with an amplitude of accommodation of 3 diopters the near point will be situated at 20cm in front of the eye. In the table there is also an example for a myope of 4 diopters.

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**Clear vison zone of an ametropic eye**

Range of Accommodation Hyperope For the hyperopic eye, the range of accommodation varies in function of if the value of refraction is greater or lesser than that of the amplitude of accommodation. The slide shows the expressions for calculating the far point and the near point. In the table of the slide there are two examples in which the refraction has a value greater than the amplitude of accommodation and as such, the range of accommodation of the hyperope is virtual. So, if the refraction is + 2 D and the amplitude of accommodation of 1 D the hyperope has a virtual range of accommodation between 50 cm and 100 cm behind the eye. If the refraction is +4 D and the amplitude of accommodation is 3 D, the virtual range is between 25cm and 100 cm behind the eye. R Am ZVN +2 1 +50, +100cm +4 3 +25,+100cm

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**Clear vison zone of an ametropic eye**

Range of Accommodation Hyperope When the amplitude of accommodation is greater than the value of refraction, there is a part of the range of accommodation that is real. For example, if the refraction is +2 D and the amplitude of accommodation of 5 D the hyperope will have a real range of accommodation between infinity and 33 cm in front of the eye (between 50cm and infinity behind the eye will be virutal). This means that the person accommodating two diopters can see objects at infinity clearly. In the table there are results for the case of an eye with 4 diopters of hyperopia and 5 diopters of amplitude of accommodation. R Am ZVN +2 5 +50, -33cm +4 +25,-100cm

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**Retinal image of an ametropic eye**

Point Object Point image if the object is situated in the clear vision zone As the final part of this subject we will study the size of the retinal image of an ametropic eye. The retinal image of a point object will be a point if the object is withing the clear vision zone of the person. If the object is outside of the clear vision zone, the image on the retina will be a blur circle and as a result the subject will see it as blurry. Blur Circle if the object is outside the clear vision zone

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**Retinal image of an ametropic eye**

Point Object Blur Circle The diameter of the blur circle that corresponds to the image of a point object when this is situated outside of the clear vision zone of an ametropic eye, is given in the equation in the slide. When the reraction value is greater, the greater the diamter of the blur circle and as a result, the person sees the object as blurrier. PE diameter pupil entrance X object vergence P Eye Power R Refraction Si R

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**Retinal image of an ametropic eye**

Extended Object CLEAR IMAGE Lagrange-Helmholtz Invariant (applied to H and H’) Just like for a point object, in the case of an extended object there are also two situations: that the image will be clear or blurry. If the image is clear, which means that the object is in the clear vision zone, applying the Lagrange - Helmholtz invariant to the principal planes of the eye we obtain u’=nu/n’. To do this it is important to keep in mind the property of the principle planes that the lateral increase is the unit and so yH=y’H’

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**Retinal image of an ametropic eye**

Extended Object CLEAR IMAGE As can be seen in the diagram u’=y’/x’ (by triangulation) and as n’/x’=R+P and the expression of u’ from the previous slide, we get the equation for the size of the retinal image: y’=un/R+P

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**Retinal image of an ametropic eye**

Extended Object BLURRY IMAGE If the image is blurry, which means it is outside of the clear vision zone, the size of the image, as can be seen in the figures, is made of two parts: the pseudo image (b) and the diameter of the blur circle. The size of the pseudo-image coincides with that of the clear image, while the blur circle was determined earlier in slide 23. b: pseudo-image size : blur circle size

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**Retinal image of an ametropic eye**

Relation of emmetropic eye and ametropic eye Comparing the size equations of the clear image of an emmetropic eye and an ametropic eye, we can get the relation between both images: y’am/y’em=Po/R+Poa. In the case of a refractive ametropia R + Poa = Po and so y’am/y’em=1, which is to say, the size of the retinal image is the same for the emmetrope and for the refractive ametrope, independent of the value of the ametropia. Refractive Ametropia (R + Poa = Po)

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**Retinal image of an ametropic eye**

Relation emmetropic eye – ametropic eye Axial Ametropia (Poa = Po) R yam/yem R<0 R>0 1 1.017 0.984 5 1.091 0.923 10 1.200 0.857 In the case of an axial ametropia, the power of the emmetropic and ametropic eye coincides: Poa = Po. For an axial myope (R<0), the size of the retinal image is greater than that of the emmetrope, while for an axial hyperope (R>0) the size of the retinal image is less than that of the emmetrope. In the table are the values of the retinal image quotients for axial myopes and hyperopes of 1, 5, and 10 diopters.

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