VISUAL OPTICS LABORATORY POWER MEASUREMENTS Prof.Dr.A.Necmeddin YAZICI GAZİANTEP UNIVERSITY OPTİCAL and ACOUSTICAL ENGINEERING DEPARTMENT http://opac.gantep.edu.tr/index.php/tr/ 1
SURFACE GEOMETRY 2 The height (or depth) of the curvature of a surface, at a given diameter, is referred to as the sagitta, or simply sag, of that curve. Simply, the sag is considered to be the difference between a flat surface and the curved surface. This value will vary with both the radius of curvature of the surface and the diameter, as illustrated in below figure. The radius of the surface may be determined indirectly by measuring the sag. The sagitta is directly related to the curvature, and for a given set of indices the curvature is directly related to the dioptric power. FIGURE: The sagittae for both convex and concave curves are shown. Using the figure, a right triangle can be formed using the radius r as the hypotenuse, ½ Ø as one leg, and the quantity (r - s) as the other leg. This is the basis of the sag formula.
SURFACE GEOMETRY 3 With the help of this diagram, we can show that the sagitta s of a lens surface can be computed for a given diameter Ø (chord length) which is equal to (Ø=2h), and radius of curvature r, using the Pythagorean theorem. This is called the sag formula: This equation is an exact relationship between the sagittal length s, the half chord length h, and the radius of curvature of the spherical interface or wavefront. From calculus, the square root can be approximated by a Taylor series expansion: FIGURE: The sagittae for both convex and concave curves are shown. Using the figure, a right triangle can be formed using the radius r as the hypotenuse, ½ Ø as one leg, and the quantity (r - s) as the other leg. This is the basis of the sag formula.
SURFACE GEOMETRY 4 When h 2 is much smaller than r 2, only the first two terms of the expression are significant, or The square root is equivalent to an exponential of 1/2, which is the source of the 2 in the denominator of the second term. When we substitute the above equation into below equation, we obtain Ten, if s and Ø will be relatively small compared to r, an approximation of the sag s of a lens in millimeters can be found with Furthermore, since the curvature R of a sphere is equal to the reciprocal of the radius r, it follows from the sagittal approximation that
Surface Geometry 5 The relationship between the radius and the sag may be expressed as where: r = the radius of the surface s = the sag of the surface to the chord y = one half of the chord length (see below figure) Figure: Radius-sag relationship. When working with contact lenses (small diameters, short radii), above equation must be used to determine the radius-sag relationship. However, for conventional lenses, the sag is usually much smaller than the radius, and therefore the second term in above equation does not change the value of the radius significantly. Thus if s «r, then above equation becomes
Surface Geometry 6 Example: A lens has one curved surface with a radius of 50 cm and one flat surface, as shown in below figure. The blank diameter of the lens is 60 mm. What is the minimum center thickness of the lens? In this figure, the sag (s) is equal to the center thickness (ct) of the lens. This is true when the lens has a knife edge or zero edge thickness. The diameter (d) of the blank and half the chord length (y) are also labeled. Because the sag for the entire lens is required, the chord is half the diameter of the lens. Using the folowing equation and rearranging terms, you can solve for the sag: Therefore the center thickness is 0.90 mm.
Surface Geometry 7 Example (Cont.): What if the lens had an edge thickness? This would mean that the edge thickness must be added to the minimum center thickness to determine the total center thickness. This is shown in below figure. Notice from the diagram that the relationship between the sag (s), center thickness (ct), and edge thickness (et) can be written as a simple equation: Let's assume an edge thickness of 2 mm for this example. The sag calculated above would represent the minimum center thickness that must be added to the edge thickness to obtain the actual center thickness: This procedure may be used for surfaces that are curved inward (concave surfaces) and for lenses with two curves. However, each problem must be diagrammed and labeled so that the relationship between sag, center thickness and edge thickness can be determined.
Surface Geometry 8 Example: (a) A spherical convex glass surface has a radius of curvature of 10.00 cm. What is the sagittal length for a chord length of 1.80 cm? (b) Use the sagittal approximation to calculate s for the surface. (a) The half chord length h is 0.9 cm. Then (b) From below equation which is the same value obtained with the exact equation.
9 For the lens measure to provide a direct reading of diopters of refracting power of a surface, it is necessary to state the relationship between refracting power (F or P), index of refraction (n), sagitta (s), and chord length (h being equal to one-half the chord length). From the following equations: Surface Geometry In this formula, the distance s is measured from the arc to the chord, and is positive if measured toward the right and negative if measured toward the left. From the definition of surface power, we can show that the radius r, in millimeters, is equal to The result is the approximate sag formula:
Surface Geometry The sign (±) of the value is not important, since we are only concerned with the magnitude of the sag, and not the direction. This last formula shows a specific relationship between the sagittal depth and the surface power of a lens. If the refractive index is known, either value may be utilized for surfacing calculations. If we are given the sagitta of each of the surfaces of a lens, together with the chord length and the index of refraction of the material, we will be able to determine the refracting power of each of the lens surfaces and, by algebraically adding the two surface powers, to determine the approximate power of the lens. If the sagittae for both surfaces of a lens are known, it is also possible to determine the final thickness of the lens at a given diameter. The sag is directly proportional to the surface power and will increase as the power of the surface increases. FIGURE: The sag s S represents the sag at a smaller diameter, while s L represents the sag at a larger diameter. 10
Surface Geometry 11 The sag will also increase as the diameter of the lens increases, as shown in below figure for a minus lens. When considering the curvature of one of the surfaces of an ophthalmic lens, the chord length (equal to 2h) is considered as the aperture, or width, of the lens.
Surface Geometry 12 Example: A +8.50 D convex curve is ground on a lens material with a 1.500 index of refraction. What is the approximate sag value of the curve at a 70-mm diameter? s = 10.41 Sagitta is 10.41 mm. EXAMPLE: Calculate the sagitta of a glass surface having a refracting power of 15.00 D, 42 mm in diameter, and having an index of refraction of 1.523, first by using the exact formula and then by using the approximate formula. By using the exact formula Since By using the approximate formula, s = h 2 /2r
Surface Geometry 13 EXAMPLE: Given a round lens (see below figure) having the sagittae s 1 = 4.8 mm and s 2 = 2.8 mm, a diameter (chord length) of 40 mm, and an index of refraction of 1.50, find the approximate power of the lens. For the front surface of the lens, and for the back surface, We find the approximate power by adding the two surface powers, FIGURE: Determining the approximate power of a lens on the basis of the sagittae and chord length.
Measuring Surface Curvature 14 If the separation of the outer pins of the lens is 20 mm (½Ø = 10 mm) and then assign an arbitrary refractive index of 1.530, our formula simplifies to: Example A lens surface is measured with a lens measure and causes the center pin to rise a full 1.0 mm. What is its 1.530-based surface power? 1.530-based surface power is 10.64 D. Since surface power is directly proportional to the sagitta measured by the device, this lens measure will indicate 10.64 D of power for every 1.0 mm the pin is moved. If the pin is only moved a fourth of that distance or 0.25 mm the surface power indicated will be one-fourth of 10.64 D: ¼ 10.64 = 2.66 D.
MEASURING SURFACE CURVATURE 15 These curves (sag) on the lens surface can easily be measured with an instrument called a spherometer or lens measure or lens clock, or lens gauge. A lens clock actually measures the sagitta of spherical surface. It is geared internally so that the sag measurement is converted directly to a power. A lens clock should be firstly calibrated directly in terms of the dioptric power of an assumed index of refraction. The tools for making spectable lenses are usually calibrated for a 1.53 index of refraction. A lens clock, calibrated for index 1.53, reads the true dioptric power on an airglass SSRI provided 1.53 is the index of the glass or plastic is spectable lens material. When the lens clock is used on an SSRI with an index different from the calibrated index, then it gives an incorrect reading. The true power may be calculated directly from the lens clock reading and the differences in indices. Actually, in the case of changing media while the curvature is unchanged, the radius can be algebraically eliminated and an equation derived that relates the old dioptric power to the new dioptric power.
MEASURING SURFACE CURVATURE 16 Hence, the old dioptric power (P old ) is used as the lens clock reading, and the new dioptric power (P new ) as the true dioptric power of the surface, i.e., The true dioptric power is larger in magnitude than the lens clock reading when the true index difference is larger than the assumed index difference, and vice versa.
Measuring Surface Curvature 17 EXAMPLE: A lens clock, calibrated for a 1.53 index, reads + 14.00 D on a hard resin plastic (n = 1.49) spherical surface. To the nearest quarter diopter, what is the true dioptric power of the interface? Before calculating, do you expect the true dioptric power to be larger or smaller? The difference in indices is smaller than the difference assumed for the lens clock, so the true dioptric power is also smaller.
MEASURING SURFACE CURVATURE 18 The typical lens clock has three legs: two stationary outer legs one movable middle leg. A lens measure has three points of contact which are placed on the lens surface to measure its curve. The outer two points are stationary while the inner point moves in or out to measure the sagittal depth of the lens. A scale on the instrument indicates the position of the middle leg. The outer pins serve as the measuring aperture of the device, allowing the center pin to measure the sagitta of a lens surface at a given diameter as shown in figure. FIGURE: The device measures the sagitta s of a lens surface from the plane of its fixed diameter Ø, or measuring aperture. The middle leg moves in or out of the base of the instrument and the difference between the middle leg's position on a flat surface and its position on a curved surface is the sag.
Measuring Surface Curvature The spherometer is first zeroed by placing it on a flat surface and adjusting the middle leg until all four legs touch the surface (see below figure-a). The scale reading of the middle leg is noted. Figure: (a) The spherometer in the zero position on a flat surface. Note that the scale position of the middle leg is approximately 10 units, (b) The spherometer on a convex surface with the middle leg adjusted (13 units) so that all four legs touch the surface. The difference in the scale readings (13-10 = 3 units) for the two positions is the sag. 19
Measuring Surface Curvature The spherometer is then placed on the curved surface, and the middle leg is moved again so that all four legs contact the surface (see below figure-b). The scale position of the middle leg is again noted. The difference between the two readings is the sag. Figure: (a) The spherometer in the zero position on a flat surface. Note that the scale position of the middle leg is approximately 10 units, (b) The spherometer on a convex surface with the middle leg adjusted (13 units) so that all four legs touch the surface. The difference in the scale readings (13-10 = 3 units) for the two positions is the sag. 20
21 Measuring Surface Curvature From the sagittal depth (by measuring the surface curvature) the instrument indicator displays the curve in diopters, with plus (+) curves shown in one direction and minus (-) curves in the other. FİGURE: The lens clock on a flat surface, a convex surface, and a concave surface. The sag is directly converted to power (in diopters) by an internal gearing system. The shape of the surface must be known to read the proper scale (i.e., a convex surface is positive; a concave surface is negative).
Measuring Surface Curvature This direct power measurement is based on these assumptions: The curvature of the surface is found by measuring its sag at a given diameter. Then, by assuming an arbitrary index of refraction, the surface power can be determined. The instrument is generally calibrated for lenses made of crown glass (refractive index 1.523) and a correction factor must be applied in the case of lenses made of materials of different refractive indices. The surrounding media is air with an index of 1.00. The chord length (i.e., the distance between the two outer legs). For most lens clocks; the chord length is a fixed distance of approximately 2.0 cm. FIGURE: A typical lens measure. 22
Measuring Surface Curvature 23 When using these devices: 1. The correct scale should be read when measuring concave versus convex surfaces. For instance, certain lens measures employ black letters on an outer scale for reading convex surfaces and red letters on an inner scale for reading concave surfaces. 2. Lens measures should be held perpendicularly to the lens surface when taking measurements. Tipping the pins with respect to the surface will produce inaccurate readings. 3. Lens measures and sag gauges should be periodically checked for accuracy using either a perfectly flat surface or reference standard of known curvature.
Measuring Surface Curvature 24 Example: Show the mathematics involved in the lens clock by developing a formula that can be used to solve directly for the power given the sag. Substitute the following equation-1 into equation-2 for the radius and substitute the known values:
25 Measuring Surface Curvature Example: A lens clock measures a power of - 8.50 D on a plastic surface (n = 1.49). What is the radius of the surface? What is the actual power of the surface? If a lens clock is used to measure a surface consisting of a material other than crown glass, the power readings will be in error. This is easily corrected if the material has a known index. Given the power reading, the radius of the measured surface may be determined using the following equation. Using the calculated radius, substitute back into the above equation with the appropriate index of refraction, and calculate the power. From the lens clock reading, using the assumed values for the index, solve the above equation for the radius: Using the calculated radius and the index of the surface, the actual power is calculated using the following equation:
Measuring Surface Curvature 26 EXAMPLE: A lens clock calibrated for an index of 1.53 reads -4.50 D on a spherical interface with a 1.71 index. What is the true dioptric power to the nearest quarter diopter? As expected, the true dioptric power is larger in magnitude that the lens clock reading since the 71 was multiplied in and the 53 was divided out. Note that with a little practice, the line containing 71/53 can be written down immediately.
Measuring Surface Curvature 27 EXAMPLE: Suppose we have a +2.00 D single spherical refracting interface (SSRI) between water (n = 1.33) and glass (n = 1.53), and the water is drained off. What is the dioptric power of the resulting air-glass interface? Let us assume that the light is initially incident in the water, hence n 1 = 1.33 and n 2 = 1.53. For the air-glass interface, The dioptric power of the air-glass interface is +5.30 D, which is considerably higher than the +2.00 D of the water-glass interface. Since the curvature is unchanged, the increase is due to the change in the index difference.
Measuring Surface Curvature 28 EXAMPLE: A +2.00 D water (n = 1.33)-glass (n = 1.53) interface had the water drained off. Since the index difference increases, what is the new dioptric power? The old indices are: n 2 = 1.53 and n 1 = 1.33. The new indices are: n 2 = 1.53 and n 1 = 1.00. Then
Measuring Surface Curvature 29 The lens measure can also be used to determine whether a lens surface is spherical or toric by placing the lens measure on the optical center of a lens and rotating the instrument about the center. If the indicator does not move while rotating, the surface is spherical. If the indicator changes when the lens measure is rotated, the lens surface is toric, with the minimum and maximum readings corresponding to the meridians of power.
Measuring Surface Curvature Because of the geometry of the lens measure and the relationship between surface tilt and prism, the amount of slab-off prism can be read directly from the lens measure. The amount of slab-off prism in a lens can be determined by taking a measurement of the curvature in the distance portion of the lens surface containing the slab-off, and comparing it to a measurement vertically straddled (ata biner gibi oturmak) across the slaboff line. (The center pin of the lens measure should be positioned on the slab line.) The difference in curvature measurements between the two readings indicates the amount of slab-off prism present, in prism diopters. This is demonstrated in below figure. For instance, consider a plastic lens with a slaboff on the rear surface. If the lens measure reads -4.00 D in the distance portion and -1.00 D across the slab line, the amount of slab-off prism present is roughly 3 Δ (base up), or 4.00-1.00 = 3 Δ. FIGURE: To determine the amount of slab-off with a lens measure, first take a reading in the distance portion of the lens surface containing the slab-off. Then take a reading across the slab-off line of the surface ensuring that the center pin of the lens measure is positioned on the slab line. The difference between the two lens measure readings indicates the amount of slab-off prism present, in prism diopters. 30
The Focimeter 31 A focimeter is used to measure the vertex power of a lens, the axes and major powers of an astigmatic lens and the power of a prism. It consists of two main parts, a focusing system and an observation system (see below figure). The focusing system comprises an illuminated target and a collimating lens. In the focimeters, green light is used to eliminate chromatic aberration. The position of the collimating lens is fixed but the target may be moved relative to it. The eyepiece contains a graticule (enlem - boylam şebekesi) and a protractor (açıölçer) scale for measuring the axes of cylindrical lenses and prismatic power. Figure: The focimeter.
The Focimeter 32 Before use, the instrument should be set to zero and the eyepiece adjusted until the dots and the graticule are sharply focused. The lens being tested is placed in a special rack (raf) which lies at the second principal focus of the collimating lens. The focimeter measures the vertex power of the lens surface in contact with the lens rest. Movement of the target allows the vergence of light emerging from the collimating lens to be varied. The target is moved until the light entering the observation telescope is parallel in which case a focused image of the target is seen by the observer. The distance through which the target is moved is directly related to the dioptric power of the lens under test.
LENS THICKNESS 33 Thickness usually becomes an important factor for lens prescriptions in the neighborhood of about ±4.00 D or more. An approximate formula relating the power of an ophthalmic lens to the difference between center thickness and edge thickness may be derived with the help of below figure. FIGURE: Diagram for derivation of a formula relating the power of an opthalmic lens to the difference between the center thickness and the edge thickness.
Lens Thickness 34 To determine the final, maximum thickness of the lens t MAX, use the formula: which, after substituting for s, gives us t MAX = s + t MIN t c = s + t p s = t c - t p where t MIN is the minimum thickness required for the lens. Below table shows the sag values for a range of surface power and lens diameter combinations. These values are based upon an arbitrary refractive index of 1.530. Tables like this can be used to determine the exact thickness of a lens. TABLE: Sag values at various diameters.
Lens Thickness 35 In such a case, the sagittal values of s 1 and s 2 in the formula must be calculated by the exact formula, EXAMPLE Given a round lens, 50 mm in diameter, having a power of +1.00 D and an index of refraction of 1.523. What is the difference between the center thickness and edge thickness (t c - t p ) of this lens?
Lens Thickness 36 EXAMPLE A lens has the following parameters: F 1 =+12.00 D, F 2 =-6.00 D, n=1.523, edge thhickness (t P ) = 1.0 mm, and lens diameter = 60 mm. What will be the center thickness, t C, (1) using the approximate sagitta formula and (2) using the exact sagitta formula? 1. Using the approximate sagitta formula,
37 EXAMPLE (Cont): 2. Using the exact sagitta formula, Lens Thickness By comparing the approximate value (6.14 mm) and the exact value (7.63 mm), we can deter mine the percentage error resulting from the use of the approximate formula:
LENS THICKNESS 38 The maximum thickness of the lens is the center thickness t CNTR of plus lenses and the edge thickness t EDGE of minus lenses. FIGURE: For a meniscus plus lens, the center thickness t CNTR = s 1 - s 2 + t EDGE. To determine the final center thickness t CNTR of a plus lens, use the formula: t CNTR = s 1 s 2 + t EDGE To determine the final edge thickness t EDGE of a minus lens, use the formula: t EDGE = s 2 s 1 + t CNTR Example A meniscus lens (convex front, concave back) has a front sag of 6.0 mm, a back sag of 2.0 mm, and an edge thickness of 1.0 mm. What is the center thickness and a plate height of the lens? t CNTR = 6.0 2.0 + 1.0 t CNTR = 5.0 mm p = 6.0 + 1.0 p = 7 0 mm Center thickness is 5.0 mm and the plate height is 7.0 mm.
PLATE HEIGHT 39 The plate height is the height of a lens as measured from a flat plane, upon which the lens rests, to a plane tangent to the apex of the front surface. The plate height p of a lens can be found by adding the sag of the back curve to the center thickness of the lens, or by adding the sag of the front curve to the edge thickness: p = s 2 + t CNTR or p = s 1 + t EDGE TABLE-1: +4.00 D lens FIGURE: The plate height can be found by simply adding the center thickness to the sag of the back curve, or by adding the edge thickness to the sag of the front curve. where s 1 is the sag of the front curve, s 2 is the sag of the back curve, t CNTR is the center thickness, and t EDGE is the edge thickness of the lens. TABLE-2: -4.00 D lens All six of these lenses in table 1 and 2 have been computed for a 1.500 index of refraction (CR-39) on a 70-mm blank size.
40 Lens Thickness Thickness Calculations for Cylindrical and Sphero-Cylindrical Lenses For a plano-cylindrical lens, thickness has a constant value along the axis meridian whereas along the power meridian thickness changes in the same manner that it does for a spherical lens. If th edge thickness is known the center thickness may be found by applying the following equation to the power meridian of the lens. To determine the thickness at a point in the periphery of a sphero-cylindrical lens, it is necessary to consider both the spherical and cylindrical components. Using F s to designate the spherical power and F c to designate the cylinder power, the thickness due to both the spherical and cylindrical components, at a peripheral point P located at a distance h from the center of the lens along a meridian oriented at a degrees from the cylinder axis, may be found by the use of the formula FIGURE: For a pianocylindrical lens, thickness has a constant value along the axis meridian, but changes in the same manner as for a spherical lens in the power meridian.
Lens Thickness 41 EXAMPLE Given a 46-mm round lens, centered in the eye wire, having a power of +5.00 DC axis 90, an index of refraction of 1.523, and an edge thickness of 1.0 mm at the thinnest point along the edge. Find the thickness at a point, P, located at the edge of the lens and such that a line drawn from this point to the center of the lens subtends an angle (a) of 30 from the cylinder axis (see below figure). We must first find the center thickness, t C :. Note that for a plus cylinder the thinnest point along the edge of the lens will be the point where the cylinder power meridian meets the edge of the lens, specified as point A. Therefore, for the power meridian (the 180 meridian), FIGURE: Diagram used to find the thickness at point P
42 EXAMPLE (Cont): The distance CP, or h, is found as follows: Lens Thickness Since all points along the axis have the same thickness, that is, we can find the thickness at point t P as follows: Second way: The power along OP is found as follows: Since the distance OP is 23 mm, we can find the thickness at P by
Abbe Refractometer 43 An Abbe refractometer is a bench-top device (tezgah üstü cihaz) for the highprecision measurement of an index of refraction. Abbé refractometer can be used to measure both refractive index of liquids and solids. Liquid samples must be non corrosive, to not damage surface of the prisms. Abbé refractometers come in many variants, that differ in details of their construction. Abbé refractometer working principle is based on critical angle.
Abbe Refractometer In the Abbe' refractometer the liquid sample is sandwiched into a thin layer between an illuminating prism and a refracting prism. The refracting prism is made of a glass with a high refractive index (e.g., 1.75) and the refractometer is designed to be used with samples having a refractive index smaller than that of the refracting prism. FIGURE: Abbé type crtitical angle refractometer. 44
Abbe Refractometer In both cases refractive index of the substance must be lower than the refractive index of the glass used to made measuring prism. A light source is projected through the illuminating prism, the bottom surface of which is ground (i.e., roughened like a ground-glass joint), so each point on this surface can be thought of as generating light rays traveling in all directions. FIGURE: Abbé type crtitical angle refractometer. 45
Abbe Refractometer 46 Light enters sample from the illuminating prism, gets refracted at critical angle at the bottom surface of measuring prism. Then the telescope is used to measure position of the border between bright and light areas. Telescope reverts the image, so the dark area is at the bottom, even if we expect it to be in the upper part of the field of view. In original design whole telescope was rotated around stationary sample and scale. In modern designs telescope position is fixed, what moves is an additional mirror between sample and telescope.
Abbe Refractometer 47 Knowing the angle and refractive index of the measuring prism it is not difficult to calculate refractive index of the sample. Surface of the illuminating prism is matted, so that the light enters the sample at all possible angles, including those almost parallel to the surface. A detector placed on the back side of the refracting prism would show a light and a dark region.
Abbe Refractometer While the image above already explains the basic principle, it is not yet a complete desing of the Abbé refractometer. Refractive index of a substance is a function of a wavelength. If the light source is not monochromatic (and in simple devices it rarely is) light gets dispersed and shadow boundary is not well defined, instead of seeing sharp edge between white and black, you will see a blurred blue or red border. FIGURE: Abbé refractometer. 48
Abbe Refractometer 49 In most cases that means measurements are either very inaccurate or even impossible. To prevent dispersion Abbé added two compensating Amici prisms into his design. Not only telescope position can be changed to measure the angle, also position of Amici prisms can be adjusted, to correct the dispersion. In effect edge of the shadow is well defined and easy to locate. In the late 1990s, Abbe refractometers became available with the capability of measurements at wavelengths other than the standard 589 nanometers. These instruments use special filters to reach the desired wavelength, and can extend measurements well into the near infrared. Multi-wavelength Abbe refractometers can be used to easily determine a sample's Abbe number.
Abbe Refractometer 50
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TEŞEKKÜRLER Prof.Dr.A.Necmeddin YAZICI University Of Gaziantep, Optic and Acoustic Engineering mail:yazici@gantep.edu.tr 52