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Chapter 7 Lenses Chapter 7 Lenses © C. Robert Bagnell, Jr., Ph.D., 2012 Lenses are the microscope’s jewels. Understanding their properties is critical in understanding the microscope. Objective, condenser, and eyepiece lenses have information about their properties inscribed on their housings. Table 7.1 is an example from objective lenses. This chapter will unfold the meaning and significance of these symbols. The three main categories of information are Numerical Aperture, Magnification / Tube Length, and Aberration Corrections. Zeiss Leitz Plan-Apochromat 170/0.17 63X/1.40 Oil Pl 40 / 0.65 ∞/0.17 Nikon Olympus Fluor 20 Ph3 ULWD CDPlan 40PL 0.75 0.50 160/0.17 160/0-2 Table 7.1 – Common Objective Lens Inscriptions Numerical Aperture & Resolution Figure 7.1 Inscribed on every objective lens and most 3 X condenser lenses is a number that indicates the lenses N.A. 0.12 resolving power – its numerical aperture or NA. For the Zeiss lens in table 7.1 it is 1.40. The larger the NA the better the resolving power. Ernst Abbe invented the concept of numerical aperture in 1873. However, prior to Abbe’s quantitative formulation of resolving power other people, such as Charles Spencer, intuitively understood the underlying principles and had used them to produce 1 4˚ superior lenses. Spencer and Angular Aperture 95 X So, here is a bit about Charles Spencer. In the mid N.A. 1.25 1850's microscopists debated the relationship of angular aperture to resolution. Angular aperture is illustrated in 110˚ Pathology 464 Light Microscopy 1 Chapter 7 Lenses figure 7.1. The 3 X lens has an angular aperture of 14˚ compared to 110˚ for the 95 X lens. Some people thought that magnification was the sole contributor to resolution. Charles Spencer (figure 7.2) - a lens Figure 7.2 maker in New York - defended the idea that angular aperture was Charles Spencer equally important. The fact that larger angular aperture improved resolution was empirical. The finer details of diatoms were visible when using lenses of larger aperture but comparable magnification. Abbe and Numerical Aperture In 1873, Ernst Abbe, working with Carl Zeiss in Germany, devised a method for determining the resolving power of lenses. Abbe took into account the refractive index of the medium between the objective and the cover glass (n), the natural sine of one half the angular aperture (α) and the wavelength of light (λ). The first two he multiplied together to produce a value he called numerical aperture: NA = n sine α. Abbe's formula is: Resolving Power (D) = λ/2ΝΑ. This applies when the NA of the condenser lens is equal to or greater than the objective’s NA and when the illumination consists of nearly parallel rays formed into a cone of light whose angle matches the objective lenses angular aperture. All of these criteria are fulfilled in Köhler illumination. Today, the accepted formula for resolution is D = 1.22λ/ΝΑcondenser + NAobjective. The constant 1.22 comes from the work of Lord Rayleigh in 1898. For a more complete explanation of how Abbe made this discovery and how Rayleigh extended it, see the footnote at the end of this chapter. Typical numerical apertures for dry objectives range from 0.04 for a 1 X objective to 0.95 for a 60 X objective. (The largest possible NA for a dry objective is around 0.95. Here is why based on Abbe’s formula. The refractive index of air is 1. The largest possible angular aperture is 180 degrees. This would put the lens on the specimen so a slightly smaller angular aperture is assumed. Half the largest angular aperture is 90 degrees. The sine of 90 degrees is 1 and NA = n sine α so 1 X almost 1 = almost 1 (i.e., ~ 0.95).) Now, here is where refractive index (n) comes in. The largest possible NA for liquid immersion objectives depends on the refractive index of the immersion medium. From Abbe’s formula you can see that as n gets bigger so does NA. Table 7.2 lists the refractive index for several common optical materials. Numerical apertures may be as high as 1.40 or 1.50 for a 60 X or a 100 X oil immersion lens. Pathology 464 Light Microscopy 2 Chapter 7 Lenses Optical Material RI - Sodium D Line Air 1.00029 Fused Quartz 1.45845 Crown Glass 1.52300 Dense Flint Glass 1.67050 Cover Glass 1.52500 Immersion Oil 1.51500 Water 1.33300 Table 7.2 - Refractive Index of Common Optical Materials Table 7.3 shows resolution versus NA based on Abbe’s formula assuming a wavelength of 0.5 µm (blue-green) and assuming the condenser fully illuminates the objective’s aperture. Numerical Aperture Resolution In µm Typical X - Y Magnification 0.04 7.62 1 X 0.08 3.81 2 X 0.20 1.52 4 X 0.45 0.67 10 X 0.75 0.40 20 X 0.95 0.32 40 X 1.30 0.23 40 X 1.40 0.21 60 X 1.50 0.20 100 X Table 7.3 - Numerical Aperture and Resolution Index of Refraction Numerical aperture also takes into account Refractive Figure 7.3 I = angle Index (RI). RI is extremely important in light microscopy so of incidence, R = angle let’s take a closer look at it. of refraction. Blue lines are normal to the glass Light bends from its normally straight path when it surface. passes from a medium of one density into a different density medium. This is called refraction. A common example is the apparent bend in a pencil inserted into a glass of water. The Glass pencil looks bent because light travels a different path in the n=1.51 water than in the air. Lenses work by refraction. As illustrated 5 R in figure 7.3, when going from a less dense to a more dense medium light bends toward a line drawn normal to the surface I Air of the medium at the point of incidence. When going from a n=1. more dense to a less dense medium, light bends away from the 0 normal. In 960 AD, the Muslim scholar Al Hazan attempted to derive a mathematical relationship between a light ray’s angle of incidence and its angle of refraction. He was unsuccessful, as was everyone else for the next 600 or so years. The correct law of Pathology 464 Light Microscopy 3 Chapter 7 Lenses refraction was derived in 1621 by Willibrord Snell of Holland Figure 7.4 and is known as Snell's law of sins. Snell's law of sins states: when a ray of light passes from one transparent medium into another, its direction will change n = 1.52 by an amount that depends on the refractive index of the new n=1.00 n=1.52 medium. The refractive index of a substance is equal to the sine Air Oil of the angle of incidence divided by the sine of the angle of Cover n=1.52 Glass refraction. n=1.00 Air Air The refractive index of a substance will vary for different wavelengths of light. Shorter wavelengths are refracted more strongly than longer wavelengths. This is what causes chromatic aberrations in lenses. Unless otherwise stated, a refractive index is based on the yellow sodium D line of 589 nm. A practical example of the effect of index of refraction Figure 7.5 can be seen in the use of oil versus air immersion lenses. Figure Robert Tolles 7.4 is a classic illustration of this principle. An American working in New York, Robert B. Tolles (student of Charles Spencer) described the homogeneous immersion principal and made immersion lenses in 1874 (figure 7.5). This method connects together the condenser, the slide with its specimen, and the objective lens using a medium of refractive index close to that of glass. Tolles' immersion medium was Canada Balsam (RI 1.535). In 1878 Ernst Abbe created the type of homogeneous immersion objective used today. His immersion medium was cedar wood oil (RI 1.515). Critical Angle There is a case in which refraction cannot occur. Figure 7.6 This is shown in figure 7.6. When light is traveling Air n=1.00 Air from a denser to a rarer medium and strikes the 41.3˚ Glass interface at an angle greater than the critical angle, the n=1.515 61.7˚ Glass light is totally reflected back into the denser medium. This is called total internal reflection. The critical angle n=1.33 Water Water is the angle of incidence for which the angle of refraction is 90˚. The critical angle for glass to air is Glass 67.5˚ 61.7˚ Glass 41.3˚, and for glass to water it is 61.7˚. There is no critical angle for glass to oil. The angle 67.5˚ in figure 7.6 is equivalent to a numerical aperture of 1.40. Dark Oil n=1.515 field condensers use total internal reflection, as do some Glass polarizing prisms such as the Nicol prism. Total internal reflection produces an evanescent wave along the reflective interface that penetrates about Pathology 464 Light Microscopy 4 Chapter 7 Lenses 100 nm into the lower refractive index medium. It can be used to reveal features attached to the surface of the interface such as adhesion plaques of motile cells. The technique known as TIRF (total internal reflection fluorescence) can be utilized to visualize structures far below the resolution limit of the microscope.
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