   Medicine

# Light and optics

Optical methods investigation medical-biological of system.

Light and optics

The Law of Reflection

1.               Light as an Electromagnetic Wave Light is an electromagnetic wave; a transverse wave that travels through empty space at the speed of or 186.000 miles/s. The relation between the wavelength , frequency , and speed c of the light is given by the fundamental equation of wave propagation as The wavelength of visible light varies from about for violet light to the longer wavelength of red light at . Since the wavelength of light is so small, we commarly use the nanometer (nm), where  So visible light varies from about 380.0 nm to about 720.0 nm. A nonstandard unit is the angstran abbreviated . So visible light is from 3800 to 7200 .

A great deal of research went into optics, the study of light, before its electromagnetic character was known. A light wave is represented in figure The electric and magnetic vectors are not shown and the magnetic portion of the wave is completely missing.

If a monochromatic point source of light (one of a single wavelength) is turned on at a particular instant, then a spherical wave emanates from the source. A two dimensional view of the wave is shown in figure.

Far away from the source of light the circular fronts look more like plane fronts. The waves then called plane waves. These plane waves can then be shown in figure.

A line drawn perpendicular to the wave front is called a ray of light and represent the direction of propagation of the light wave. Note that the ray of light travels in a straight line. The wavelength of light is so small that in a relatively short distance away from the point source of light, the waves appear plane. Even if the source of light is not a point, then we are sufficiently far away from the source, the waves are effectively plane. In all the subsequent discussions we will assume that all the light waves are plane waves.

This analysis of light into waves and rays allows us two different descriptions of light.

When only the light rays are dealt with in the analysis of an optical system, the description is called geometrical optics. When the analysis of an optical system is done in terms of waves, the description is called wave optics or physical optics. Still another description of light is possible by treading light as little bundles of electromagnetic energy, called photons. Such a description is called quantum optics.

Huygens’ principle states that each point on a wave front may be considered as a source of secondary spherical wavelets. These secondary wavelets propagate in the forward direction at the same speed as the initial wave. The new position of the wave front at a later time is found by drawing the tangent to all of these secondary wavelets at the later time. 2.    The Law of Reflection

Consider a plane wave advancing toward a smooth surface such as a glass or a mirror, as in figure The First Law of reflection says that the angle of incidence is equal to the angle o reflection r.

The Second Law of reflection says that the incident ray, the normal, and the reflected ray all lie in the same plane.

3.               The Plane Mirror

An object O is placed a distance p (called the object distance) in front of a plane mirror RT, as shown in Figure As you know, if you are the object O, you will see your image in the mirror. How is this image formed and how far behind the mirror is the image located?

From figure you can see that q=p. This is, image is as far behind the mirror as the object is in front of it.

In general, to describe an optical image three words are necessary: its nature (real or virtual), its orientation (erect, inverted, perverted), and its size (enlarged, true, or reduced). Recal that when you look into a mirror your image is reversed. That is, if you hold up your right hand in front of the mirror, your image appears as though the left hand was held  up. This inversion of left-right symmetry is called perversion. Thus, a plane mirror produces a virtual, perverted, true image.

4.               The Concave Spherical Mirror

A spherical mirror is a reflection surface, whose radius of curvature is the radius of the sphere from which the mirror is formed. Eve C is the center of curvature of the mirror and R is its radius of curvature. The line going through the center of the mirror, the vertex, is called the principal axis, or optical axis, of the mirror. Light rays that are parallel and close to the principal axis of the concave mirror converge to a point called the principal focus F of the mirror.

Focal Length of a Concave Spherical Mirror

The distance VF from the vertex of the mirror to the principal focus F is called the focal length f of the mirror.

It was proved that  The Mirror Equation  It shows the relation between the focal length f of the mirror, the object distance p, and the image distance q.
Magnification

Linear magnification M of the mirror is the radio of the size of the image to the size of the object .

That is, .

Thus, the magnification tells how much larger the image is than the object. We can rewrite this as ###### Diffraction Diffraction manifests itself in the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Diffraction Grating

When there is a need to separate light of different wavelengths with high resolution, then a diffraction grating is most often the tool of choice. This "super prism" aspect of the diffraction grating leads to application for measuring atomic spectra in both laboratory instruments and telescopes. A large number of parallel, closely spaced slits constitutes a diffraction grating. The condition for maximum intensity is the same as that for the double slit or multiple slits, but with a large number of slits the intensity maximum is very sharp and narrow, providing the high resolution for spectroscopic applications. The
peak intensities are also much higher for the grating than for the double slit. Polarization by Reflection

Polarization by Scattering The scattering of light off air molecules produces linearly polarized light in the plane perpendicular to the incident light. The scatterers can be visualized as tiny antennae which radiate perpendicular to their line of oscillation. If the charges in a molecule are oscillating along the y-axis, it will not radiate along the y-axis. Therefore, at 90° away from the beam direction, the scattered light is linearly polarized. This causes the light which undergoes Rayleigh scattering from the blue sky to be partially polarized.

# Diffraction Grating A diffraction grating is the tool of choice for separating the colors in incident light. The condition for maximum intensity is the same as that for a double slit. However, angular separation of the maxima is generally much greater because the slit spacing is so small for a diffraction grating.

Interference The wave properties of light lead to interference, but certain conditions of coherence must be met for these interference effects to be readily visible. Thin Films

The optical properties of thin films arise from interference and reflection. The basic conditions for interference depend upon whether the reflections involve 180 degree phase changes.    Reflection Light incident upon a surface will in general be partially reflected and partially transmitted as a refracted ray. The angle relationships for both reflection and refraction can be derived from Fermat's principle. The fact that the angle of incidence is equal to the angle of reflection is sometimes called the "law of reflection".

Polarization by Reflection

Since the reflection coefficient for light which has electric field parallel to the plane of incidence goes to zero at some angle between 0° and 90°, the reflected light at that angle is linearly polarized with its electric field vectors perpendicular to the plane of incidence. The angle at which this occurs is called the polarizing angle or the Brewster angle. At other angles the reflected light is partially polarized.

From Fresnel's equations it can be determined that the parallel reflection coefficient is zero when the incident and transmitted angles sum to 90°. The use of Snell's law gives an expression for the Brewster angle.   when  by Snell’ law "Blackbody radiation" or "cavity radiation" refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only, not dependent upon the type of radiation which is incident upon it. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating.  The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. The best of classical physics suggested that all modes had an equal chance of being produced, and that the number of modes went up proportional to the square of the frequency.

But the predicted continual increase in radiated energy with frequency (dubbed the "ultraviolet catastrophe") did not happen. Nature knew better.
Cavity Modes

A mode for an electromagnetic wave in a cavity must satisfy the condition of zero electric field at the wall. If the mode is of shorter wavelength, there are more ways you can fit it into the cavity to meet that condition. Careful analysis by Rayleigh and Jeans showed that the number of modes was proportional to the frequency squared. From the assumption that the electromagnetic modes in a cavity were quantized in energy with the quantum energy equal to Planck's constant times the frequency, Planck derived a radiation formula. The average energy per "mode" or "quantum" is the energy of the quantum times the probability that it will be occupied (the Einstein-Bose distribution function): This average energy times the density of such states, expressed in terms of either frequency or wavelength   gives the energy density , the Planck radiation formula.

Energy per unit

volume per unit frequency

Energy per unit

volume per unit wavelenght

The Planck radiation formula is an example of the distribution of energy according to Bose-Einstein statistics. The above expressions are obtained by multiplying the density of states in terms of frequency or wavelength times the photon energy times the Bose-Einstein distribution function with normalization constant A=1.

To find the radiated power per unit area from a surface at this temperature, multiply the energy density by c/4. The density above is for thermal equilibrium, so setting inward=outward gives a factor of 1/2 for the radiated power outward. Then one must average over all angles, which gives another factor of 1/2 for the angular dependence which is the square of the cosine.

Rayleigh-Jeans vs Planck

Comparison of the classical Rayleigh-Jeans Law and the quantum Planck radiation formula. Experiment confirms the Planck relationship.

Blackbody Intensity as a Function of Frequency ### The Rayleigh-Jeans curve agrees with the Planck radiation formula for long wavelengths, low frequencies.  ## Microwave Interactions

The quantum energy of microwave photons is in the range 0.00001 to 0.001 eV which is in the range of energies separating the quantum states of molecular rotation and torsion. The interaction of microwaves with matter other than metallic conductors will be to rotate molecules and produce heat as result of that molecular motion. Conductors will strongly absorb microwaves and any lower frequencies because they will cause electric currents which will heat the material. Most matter, including the human body, is largely transparent to microwaves. High intensity microwaves, as in a microwave oven where they pass back and forth through the food millions of times, will heat the material by producing molecular rotations and torsions. Since the quantum energies are a million times lower than those of x-rays, they cannot produce ionization and the characteristic types of radiation damage associated with ionizing radiation. Infrared Interactions

The quantum energy of infrared photons is in the range 0.001 to 1.7 eV which is in the range of energies separating the quantum states of molecular vibrations. Infrared is absorbed more strongly than microwaves, but less strongly than visible light. The result of infrared absorption is heating of the tissue since it increases molecular vibrational activity. Infrared radiation does penetrate the skin further than visible light and can thus be used for photographic imaging of subcutaneous blood vessels. ## Visible Light Interactions

The primary mechanism for the absorption of visible light photons is the elevation of electrons to higher energy levels. There are many available states, so visible light is absorbed strongly. With a strong light source, red light can be transmitted through the hand or a fold of skin, showing that the red end of the spectrum is not absorbed as strongly as the violet end. While exposure to visible light causes heating, it does not cause ionization with its risks. You may be heated by the sun through a car windshield, but you will not be sunburned - that is an effect of the higher frequency uv part of sunlight which is blocked by the glass of the windshield.

Ultraviolet Interactions

The near ultraviolet is absorbed very strongly in the surface layer of the skin by electron transitions. As you go to higher energies, the ionization energies for many molecules are reached and the more dangerous photoionization processes take place. Sunburn is primarily an effect of uv, and ionization produces the risk of skin cancer. The ozone layer in the upper atmosphere is important for human health because it absorbs most of the harmful ultraviolet radiation from the sun before it reaches the surface. The higher frequencies in the ultraviolet are ionizing radiation and can produce harmful physiological effects ranging from sunburn to skin cancer.

# Star Temperatures

Stars approximate blackbody radiators and their visible color depends upon the temperature of the radiator. The curves show blue, white, and red stars. The white star is adjusted to 5270K so that the peak of its blackbody curve is at the peak wavelength of the sun, 550 nm. From the wavelength at the peak, the temperature can be deduced from the Wien displacement law.

Wien's Displacement Law When the temperature of a blackbody radiator increases, the overall radiated energy increases and the peak of the radiation curve moves to shorter wavelengths. When the maximum is evaluated from the Planck radiation formula, the product of the peak wavelength and the temperature is found to be a constant. This relationship is called Wien's displacement law and is useful for the determining the temperatures of hot radiant objects such as stars, and indeed for a determination of the temperature of any radiant object whose temperature is far above that of its surroundings.

Basic Concepts and Formulas in Microscopy

In order to realize the full potential of the optical microscope, one must have a firm grasp of the fundamental physical principles surrounding its operation. Important topics for understanding the microscope, such as resolution, numerical aperture, depth of field, image brightness, objective working distance, field of view, conjugate planes, and the useful magnification range, are discussed in the review articles linked below.

Conjugate Planes in Optical Microscopy

In a properly focused and aligned optical microscope, a review of the geometrical properties of the optical train demonstrates that there are two sets of principal conjugate focal planes that occur along the optical pathway through the microscope. One set consists of four field planes and is referred to as the field or image-forming conjugate set, while the other consists of four aperture planes and is referred to as the illumination conjugate set. Each plane within a set is said to be conjugate with the others in that set because they are simultaneously in focus and can be viewed superimposed upon one another when observing specimens through the microscope. Presented in Figure 1 is a cutaway diagram of a modern microscope (a Nikon Eclipse E600), which illustrates the strategic location of optical components comprising the two sets of conjugate planes in the optical pathways for both transmitted and incident (reflected or epi) illumination modes. Components that reside in the field set of conjugate planes are described in black text, while those comprising the aperture set are described in red text. Note that conjugate planes are illustrated for both observation and digital imaging (or photomicrography) modes. Table 1 lists the elements that make up each set of conjugate planes, including alternate nomenclature (listed in parentheses) that has often been employed and may be encountered in the literature. A minor difference exists in the relative location of the field and condenser apertures between the incident and transmitted modes of illumination, which will be explained later.

Conjugate Focal Planes

 Aperture or Illuminating Conjugate Plane Set Field or Image-Forming Conjugate Plane Set Microscope Exit Pupil (Eye Iris Diaphragm) (Ramsden Disc) (Eyepoint) Retina of the Eye Camera Image Plane Objective Rear Focal Plane (Objective Rear Aperture) Intermediate Image Plane (Eyepiece Fixed Diaphragm) Condenser Aperture Diaphragm (Condenser Front Focal Plane) Specimen Plane (Object Plane) Lamp Filament Field Diaphragm (Field Stop) (Köhler Diaphragm)   Table 1 In normal observation mode (using the eyepieces), the conjugate set of object or field planes can all be simultaneously viewed when the specimen is in focus. This observation mode is referred to as the orthoscopic mode, and the image is known as the orthoscopic image. Observing the other conjugate set of aperture or diffraction planes requires the ability to focus on the rear aperture of the objective, which may be accomplished by using an eyepiece telescope in place of an ocular, or a built-in Bertrand lens on microscopes that are so equipped. This observation mode is termed the conoscopic, aperture, or diffraction mode and the image observed at the objective rear aperture is known as the conoscopic image. Although the terms orthoscopic and conoscopic are scattered widely throughout the literature, many microscopists favor using normal mode and aperture mode because the latter nomenclature more clearly relates to the operation of the microscope. Planes belonging to the pair of conjugate sets alternate in succession through the optical train from the light source filament to the final microscope image produced on the retina or the image plane of an electronic sensor. A thorough understanding of the relationships between these conjugate plane sets, and their location within the microscope, is essential in understanding image formation and carrying out correct adjustment of illumination. In addition, the location of principal conjugate planes is often a key factor in the proper placement of optical components such as phase plates, differential interference contrast (DIC) Wollaston prisms, polarizers, modulators, filters, or graticules.

The imaging and illumination ray paths through a microscope adjusted for Köhler illumination are presented in Figure 2, with the focal conjugates of each plane set indicated by crossover points of the ray traces. Illustrated diagrammatically in the figure is the reciprocal nature of the two sets of conjugate planes that occur in the microscope. The optical relationship between the conjugate plane sets is based upon the fact that, in the illuminating ray path (shown in red), the spherical wave fronts converge and are brought into focus onto the aperture planes, while in the imaging ray path (shown in yellow), the spherical waves converge into focused rays in the field planes. Light rays that are focused in one set of conjugate planes are nearly parallel when passing through the other set of conjugate planes. The reciprocal relationship between the two sets of conjugate planes determines how the two ray paths fundamentally interact in forming an image in the microscope, and it also has practical consequences for operation of the microscope.

Illumination is perhaps the most critical factor in determining the overall performance of the optical microscope. The full aperture and field of the instrument is usually best achieved by adjusting the illumination system following the principles first introduced by August Köhler in the late nineteenth century. It is under the conditions of Köhler illumination that the requirements are met for having two separate sets of conjugate focal planes, field planes and aperture planes, in precise physical locations in the microscope. The details of adjusting a given microscope to satisfy the Köhler illumination conditions depend to some extent upon how the individual manufacturer meets the requirements, and are not discussed here.

The basic requirements of Köhler illumination are very simple. A collector lens on the lamp housing is required to focus light emitted from the various points on the lamp filament at the front aperture of the condenser while completely filling the aperture. Simultaneously, the condenser must be focused to bring the two sets of conjugate focal planes (when the specimen is also focused) into specific locations along the optical axis of the microscope. Meeting these conditions will result in a bright, evenly illuminated specimen plane, even with an inherently uneven light source such as a tungsten-halogen lamp filament (the filament will not be in focus in the specimen plane). With the specimen and condenser in focus, the focal conjugates will be in the correct position so that resolution and contrast can be optimized by adjusting the field and condenser aperture diaphragms. The concept that specific planes in the optical path of the microscope are conjugate indicates that they are equal. That is, whatever appears in focus in one plane of a conjugate set will appear in focus in all the other planes belonging to the same set. On the other hand, the reciprocal nature of the two sets of microscope conjugate planes requires that an object appearing in focus in one set of planes, will not be focused in the other set. The existence of two interrelated optical paths and two sets of image planes characterize Köhler illumination and is the foundation that allows the various adjustable diaphragms and aperture stops in the microscope to be used to control both the cone angle of illumination, and the size, brightness and uniformity of illumination of the field of view. The planes belonging to the field set are sometimes referred to as the field-limiting planes because a diaphragm placed in any one of these planes will limit the diameter of the image field. Figure 3 illustrates each of these planes, which will be focused and coincident with the specimen image. The aperture planes may be considered aperture-limiting because the numerical aperture of the optical system can be controlled by fixed or adjustable (iris) diaphragms inserted at any of these positions. Components containing the set of aperture planes, which are not in focus with the specimen image, are presented (removed from their locations in the microscope) in Figure 4. A microscopist may not be aware that in the normal observation mode, the specimen image is actually a combined view of four in-focus conjugate image planes (including the specimen plane) whose optical characteristics are determined or modulated by another set of four out-of-focus aperture planes. In a properly adjusted microscope, this fact can be easily ignored. However, the mechanism of the image formation becomes much more apparent, and of immediate practical interest, if there is something seriously wrong with the image. Knowledge of the location and image-forming function of each of the conjugate image and aperture planes is critical in troubleshooting problems that arise in the microscope image. It is also necessary for proper placement and use of filters, stops, phase rings, DIC prisms, and other optical components so that they do not introduce problems. Modern microscope design takes into consideration the location of the conjugate planes and the need for the microscopist to have access to them.

One of the more common ways in which the conjugate nature of the various field planes (including the specimen) and aperture planes (including the illumination source) is exploited, is in the placement of filters and other illumination-modifying optical components. These optical elements are likely to be contaminated with dust or fingerprints, due to frequent handling, and are often not optically designed to be included with the objective, oculars, and other elements of the optical path that are intended to form the focused final image. Therefore, auxiliary components should never be placed in any plane conjugate with the specimen, because any defects, dust, or debris may appear in the final image of the specimen formed at the retina or at a camera imaging plane.

In contrast, measurement graticules, scales, pointers, and other devices that are intended to be in focus and registered on the specimen must be placed at one of the field conjugate planes in order to be imaged with the specimen. For practical reasons, most of the commercially available measuring graticules are designed to be placed at the eyepiece field stop, coincident with the real intermediate image plane. Figure 5 illustrates a microscope viewfield containing three simultaneously focused conjugate planes that will appear, in focus, on the retina of the eye or the imaging plane of a camera. Phase plates, and their corresponding annuli, are intended to control the light path (not to appear in the specimen image), and are placed at conjugate planes in the aperture series. The phase plate is usually placed at the rear aperture (at or very near the rear focal plane) of the objective, so placing the correct annulus at the previous conjugate plane of the aperture series (the front aperture of the substage condenser) will ensure that the images of the two components are superimposed in the illumination optical path. Likewise, Wollaston prisms, which shear and recombine light beams in differential interference contrast microscopy, are placed in the condenser front focal plane and objective rear focal plane for the same reason. Polarizers and analyzers, utilized in polarized light and DIC microscopy, are generally placed far away from either conjugate plane set to avoid introducing contaminating dust or fingerprints into the specimen image or degrading the illumination conditions. If problems arise that affect the image quality obtained in the microscope, two avenues should be explored when attempting to correct them. In the normal microscope view through the eyepieces, the conjugate field planes are observed in focus. If dust or other imperfections appear distinct and in sharp focus with the image, then the source of the problem is probably one of the glass components located in or very near the field planes of the microscope optical path. The most likely components to contain dust and debris (that can be observed with the specimen) are the specimen slide itself, the eye lenses of the eyepieces, the objective front lens, the condenser top lens, graticules, and any lenses near the field diaphragm. Often, it is a simple matter to rotate the eyepieces, objective, and condenser and determine if the offending debris rotates in the viewfield, thereby confirming its location. If this technique does not identify the problem, then the best way to continue troubleshooting is by observing the rear focal plane of the objective to evaluate possible obstructions or misalignments in the illumination path. This aperture, or conoscopic, view is obtained by replacing a normal eyepiece with a specialized eyepiece telescope or using a Bertrand lens (built into the microscope observation head). Although, it is possible to simply remove an eyepiece and peer down the observation tube at the rear aperture of the objective, the image is too small to easily evaluate without a telescope. Observing the aperture planes in this manner will reveal any obstructions such as poorly-centered lenses, dirt or contaminants in the lenses, illumination irregularities such as a poorly centered lamp filament, air bubbles that might be present in immersion oil, and improper adjustment of the condenser diaphragm. The condenser aperture diaphragm determines the effective numerical aperture of the objective-condenser combination, and should normally be adjusted to fill about 70 to 80 percent of the objective rear aperture with light, as an optimum compromise between maximum resolution, depth of field, and adequate contrast.

As stated previously, use of the incident (reflected light) illumination mode requires a minor change in the arrangement of the field and aperture diaphragms in the microscope. This adjustment is necessary because, with incident illumination, the objective plays a dual role and also functions as the condenser, which requires an aperture diaphragm that does not lie in the imaging ray path between the objective lens and the eye or camera. To meet this requirement, lenses in the illumination system are employed to create a conjugate plane between the lamp and the field diaphragm for placement of an aperture diaphragm that is conjugate with the lamp filament. The aperture diaphragm is then projected by relay lenses and mirrors into the rear focal plane (exit pupil) of the objective where these two conjugate planes become coincident. Figure 6 illustrates the illuminating ray path (shown in red) and the imaging ray path (shown in yellow) in an incident light system.

The best image quality can be achieved in the microscope only with a thorough understanding of the central role played by the two sets of reciprocal conjugate focal planes. For optimum results, the microscopist must be familiar with their interrelationships and where they are located in the microscope when the conditions required for Köhler illumination are met. With this understanding, the primary optical components of the microscope, as well as any accessories that might be added, can be utilized in a manner that exploits the reciprocal nature of the conjugate plane sets to maximum benefit.

Microscope Alignment for Köhler Illumination - Perhaps one of the most misunderstood and often neglected concepts in optical microscopy is proper configuration of the microscope with regards to illumination, which is a critical parameter that must be fulfilled in order to achieve optimum performance. The intensity and wavelength spectrum of light emitted by the illumination source is of significant importance, but even more essential is that light emitted from various locations on the lamp filament be collected and focused at the plane of the condenser aperture diaphragm. This interactive tutorial reviews both the filament and condenser alignment procedures necessary to achieve Köhler illumination.

Depth of Field and Depth of Focus - The depth of field is the thickness of the specimen that is acceptably sharp at a given focus level. In contrast, depth of focus refers to the range over which the image plane can be moved while an acceptable amount of sharpness is maintained. The two concepts are often incorrectly used interchangeably when referring to the depth of field of a microscope objective.

Field of View - The diameter of the field in an optical microscope is expressed by the field-of-view number, or simply the field number, which is the diameter of the view field in millimeters measured in the intermediate image plane. In most cases, the eyepiece field diaphragm opening diameter determines the view field size.

Refractive Index (Index of Refraction) - Refractive index is a value calculated from the ratio of the speed of light in a vacuum to that in a second medium of greater density. The refractive index variable is most commonly symbolized by the letter n or n' in descriptive text and mathematical equations.

Numerical Aperture - The numerical aperture of a microscope objective is a measure of its ability to gather light and resolve fine specimen detail at a fixed object distance. All modern microscope objectives have the numerical aperture value inscribed on the lens barrel, which allows determination of the smallest specimen detail resolvable by the objective and an approximate indication of the depth of field.

Resolution - The resolving power of a microscope is the most important feature of the optical system and influences the ability to distinguish between fine details of a particular specimen. As discussed in this section, the primary factor in determining resolution is the objective numerical aperture, but resolution is also dependent upon the type of specimen, coherence of illumination, degree of aberration correction, and other factors such as contrast enhancing methodology either in the optical system of the microscope or in the specimen itself.

Useful Magnification Range - The range of useful magnification for an objective/eyepiece combination is defined by the numerical aperture of the system. There is a minimum magnification necessary for the detail present in an image to be resolved, and this value is usually rather arbitrarily set to a value between 500 and 1000 times the numerical aperture (500 or 1000 x NA) of the objective.

Working Distance and Parfocal Length - Microscope objectives are generally designed with a short free working distance, which is defined as the distance from the front lens element of the objective to the closest surface of the coverslip when the specimen is in sharp focus. The parfocal length represents the distance between the specimen plane and the shoulder of the flange by which the objective is supported on the revolving nosepiece.

Image Brightness - Regardless of the imaging mode utilized in optical microscopy, image brightness is governed by the light-gathering power of the objective, which is a function of numerical aperture. Just as the brightness of illumination is determined by the square of the condenser working numerical aperture, the image brightness is proportional to the square of the objective numerical aperture.

Coverslip Correction - Non-immersion high-dry microscope objectives having a numerical aperture exceeding 0.75 are prone to introduction of aberration when imaging through coverslips that deviate from standard thickness and refractive index. To prevent artifacts, many objectives are equipped with correction collars that help compensate for coverslip thickness variations.

Adjustment of Objective Correction Collars - Most microscope objectives are designed to be used with a cover glass that has a standard thickness of 0.17 millimeters and a refractive index of 1.515, which is satisfactory when the objective numerical aperture is 0.4 or less. However, when using high numerical aperture dry objectives (numerical aperture of 0.8 or greater), cover glass thickness variations of only a few micrometers result in dramatic image degradation due to aberration, which grows worse with increasing cover glass thickness. To compensate for this error, the more highly corrected objectives are equipped with a correction collar to allow adjustment of the central lens group position to coincide with fluctuations in cover glass thickness. This interactive tutorial explores how a correction collar is adjusted to achieve maximum image quality.

Focusing and Alignment of Arc Lamps - Mercury and xenon arc lamps are now widely utilized as illumination sources for a large number of investigations in widefield fluorescence microscopy. Visitors can gain practice aligning and focusing the arc lamp in a Mercury or Xenon Burner with this interactive tutorial, which simulates how the lamp is adjusted in a fluorescence microscope.

Linear Measurements (Micrometry) - Performing measurements at high magnifications in compound optical microscopy is generally conducted by the application of eyepiece reticles in combination with stage micrometers. A majority of measurements made with compound microscopes fall into the size range of 0.2 micrometers to 25 millimeters (the average field diameter of widefield eyepieces). Horizontal distances below 0.2 micrometers are beneath the resolving power of the microscope, and lengths larger than the field of view of a widefield eyepiece are usually (and far more conveniently) measured with a stereomicroscope.

### 1.1 Interference due to Differing Distances

Before the 1800's, the view that light had a particle nature rather than the wave nature described by Huygens, held sway over many opinions. The difficulties with Huygens model are rather obvious. The light being emitted into directions other than the wavefront were deemed "to weak to see", the difficulty of believing that waves could propagate across the space between the earth and the sun without an intervening medium, and so on. The first experimental "proof" that light behaves as a wave was provided by Thomas Young in 1801. That experiment shows that light undergoes interference effects. Interference is a classic hallmark of the behavior of waves and is a phenomenon not seen to be intrinsic to matter until the advent of quantum mechanics in the 20th century. Before considering Young's experiment, let's note a relatively simple example of interference with mechanical waves. Since mechanical waves experience interference (consider that we introduced the notions of complete constructive and destructive interference with mechanical waves), it is necessary that electromagnetic waves should as well. For the simple example, we consider two "clappers" that hit the water in a shallow tank at periodic intervals. Since the frequency of the two "clappers" is the same and they have a fixed phase relationship, we say that they serve as coherent wave sources. Coherence simply implies same frequency/fixed phase relationship. To see this phenomenon with water waves, see figure 14.1.
Figure 14.1: Two strikers hit the water a certain distance apart and create transverse water waves in phase, i.e. at the same time. The distinctive interference pattern of the two waves is shown. The cause of this pattern is easy to determine if we remember how constructive/destructive interference can come about. To review what happens when waves overlap, look again at figure  5. In the case of the clappers, we note that they are different distances away from most positions in the tank of water. We see for one dark area and one light area how the distances from one clapper (r1 and r1
) differ from the corresponding distances of the other clapper (r2 and r2). The different distances traveled mean that even if the waves started out in phase from the two clappers, they will be out of phase by the time they reach the same destination.  Figure 2: The two waves from the clappers are in phase, but become out of phase as they travel different distances to get to the same destination. The waves superpose at every position in the tank, but those phase differences due to distance differences to a point which differ by an integral wavelength lead to a pileup of water at that position (and hence a dark spot) while those points which lead to distance differences which are a non-even integer of half-wavelengths lead to a low spot of water (and hence a brighter spot).

We can quantitatively describe this effect by stating that constructive interference occurs when the path difference, r2 r1, satisfies the following relationship:

 r2  r1 = m   (m = 0, 1, 2, 3, ...)

(14.1.1.1)

i.e. an integer number of wavelengths for the path difference has the two waves looking "back" in phase and completely constructive interference occurs. If the path difference is an odd integer of half-integral wavelengths, then completely destructive interference occurs since the phase difference is an odd integer multiple of /2 and so troughs for one wave correspond to crests of the other and vice versa. The formula for this is

 r2  r1 =    m + 1 2       (m = 0, 1, 2, 3, ...).

(14.1.1.2)

### 1.2  Young's Experiment

We can now understand Young's experiment in that we can see that any two coherent sources can generate phase differences over a space due to distance differences. Young produced his coherent sources starting with an incoherent light source by letting monochromatic (i.e. one frequency) light fall on a slit. This is necessary since light, even monochromatic light, from a source is generally not synchronized in phase from different parts of the light source. Allowing the light to fall on a slit means that only the light from one small region of the light source gets through. This light is now allowed to fall onto two other slits. Light from these slits forms two coherent sources and interference from these two can be seen by allowing the light from both slits to fall onto a screen. We can see the result in figure 14.3 and in this Java applet. Figure 14.3: Young's experiment showing constructive and destructive interference for light. Constructive interference corresponds to the regions where the bright colored bars appear. The dark areas in between are the regions of destructive interference. The graph at the far right end of the picture shows the intensity of the light. The quantitative description of interference for Young's experiment is characterized by the distance d between the two slits, the wavelength of the light, and the angle
between the horizontal and the line from the top slit to the point of interest on the screen. We find experimentally that

 dsin
 =
 m   (m = 0, 1, 2, 3, ...)  constructive interference

 dsin
 =
    m + 1 2       (m = 0, 1, 2, 3, ...)  destructive interference

(1.3)

The light and dark bands are referred to as interference fringes. The center of the pattern, corresponding to m = 0, is equidistant from the two slits. All other points on the screen are further away from one of the slits than the other, hence the constructive and destructive interference effect works the same way here as it did for the clappers in the water wave tank. The "extra" distance traveled by light from one of the slits to point P on the screen is easily shown to be about dsin, as seen in figure 14.3, if we also assume that the distance L from the slits to the screen is much larger than d. This requirement is necessary so that we can assume that  is approximately the same angle that the line from the bottom slit to the point P makes with respect to the horizontal. Then the vertical position of each bright interference fringe is given by

 ym = Ltanm

(2.4)

where m identifies which bright fringe from the center we are talking about. Notice that ym can be either positive or negative depending on the value of m. If L >> d, then, for the first few values of m, we have m as a small angle. In that case,

 tanm  sinm ym = Lsinm = L m d .

(2.5)

#### Phase Differences and Path Differences

As a general rule, any path length difference between two coherent sources can be expressed as a phase difference with the following association:

  2 = r2  r1   = 2  (r2  r1) = k(r2  r1)

(2.6)

where k is the wave number. We can go further: any path difference which results in a different number of wavelengths to go from the source to a point in space in comparison to the number of wavelengths to go from another coherent source to the same point generates a phase difference. Other than path length differences what could result in such a wavelength difference? It turns out that electromagnetic waves can change their wavelength (compared to the wavelength in vacuum) inside a material with an index of refraction other than 1.0. The frequency of electromagnetic waves is unaffected by passage through most materials, so if the speed of light in those materials is smaller than in a vacuum, this must be the result of a reduction in wavelength! Hence even if the distance traveled by the waves from two coherent sources is the same, they may still experience a phase difference as shown in figure  14.4. Figure 14.4: Wavelength changes in a material can cause phase differences even for coherent waves which travel the same physical distance.

This is incorporated into our general scheme for calculating phase changes by letting the wavelength, , be given by

  = 0 n  k = nk0

(2.7)

where 0 is the wavelength in vacuum and n is the refraction index of any material an EM wave goes through. Finally, if the two coherent sources are separated by a distance d, then the combination of their waves at a point (call it P) which is distant from either of the two sources gives a path difference which is

 r2  r1 = dsin

(2.8)

with  defined as the angle of the line from one of the slits to the point P compared to a line from the center of the slits to point P. In this case we can define the general phase difference as

  = k(r2  r1) = kdsin = 2d  sin

(2.9)

with  and k defined as in equation 7.

Intensity for Phase Difference Superposition

For mechanical waves, we showed that we can use trigonometric identities to find the result of superposing two waves with different phases. Suppose the electric fields for two waves are described as follows:

 E1(t)
 =
 Ecos(t)

 E2(t)
 =
 Ecos(t + )

(2.10)

Then the sum of these two waves yields a resultant wave which is described by

 EP =    2Ecos 1 2     cos    t + 1 2    

(2.11)

Since, for electromagnetic waves, we have seen that the intensity at any point and time is proportional to the square of the amplitude, we have

 I
 =
 I0cos2  2

 I0
 =
 Sav, max. = EP, max.2 20c = 1 2 0cEP, max2 = 1 2 0c(4E2) = 20cE2

(2.12)

where I0 is the maximum intensity and occurs at positions where the phase difference between the two original waves is zero (i.e. = 0). Hence we can use our expression for phase differences from equation 9 to predict that the intensity far from two sources is given by

 I = I0cos2  2 = I0cos2    1 2 kdsin    = I0cos2    d  sin   

(2.13)

and the directions of maximum intensity occur for

 d  sin = m   (m = 0,1, 2, ...)

(2.14)

## 2  Diffraction

### 2.1  Why Diffraction Occurs

Diffraction is another instance in which waves do something that is unique to their nature. The phenomenon of diffraction is easily explained qualitatively in terms of Huygen's theory. As a wavefront propagates along, it produces point sources which emit spherical waves. As long as all these point sources start at the same time and are allowed to emit waves without interruption, the wavefront along the direction of propagation is a straight line. If an obstruction or a barrier with a slit intrudes, however, then some of the spherical waves are blocked and cannot contribute to the wavefront. The response is that the wavefront becomes curved. To see this, consider the static images below that show a plane wave moving into a barrier with a slit or a just a barrier. As long as the obstruction or opening is much larger than the wavelength as shown in figure 1, the behavior of the waves is what we expect, namely the part of the wavefront that is allowed to continue does so with the wavefront remaining straight. Figure 1: Plane wavefronts approach a barrier with an opening or an obstruction. Both the opening and the obstruction are large compared to the wavelength.

If, however, the size of the opening becomes comparable to the wavelength, the waves proceed to "bend through" or around the opening or obstruction as shown in figure 2. Figure 2: Now the plane wavefronts impinge on a barrier with an opening or an obstruction which is not much larger than the wavelenth. The wavefront is not allowed to propagate freely through the opening or past the obstruction but experiences some retardation of some parts of the wavefront. The result is that the wavefront experiences significant curvature upon emerging from the opening or the obstruction.

Finally, as the obstruction or opening equals the wavelength, the diffraction becomes quite pronounced as shown in figure .3. We refer to this phenomenon as diffraction. Figure 15.3: As the barrier or opening size gets smaller, the wavefront experiences more and more curvature.

### 2.1  Types of Diffraction

Given the mathematical complexity, only a part of what constitutes the theory of diffraction can be discussed in detail in the text. Just to introduce the nomenclature though, note that cases in which the source of radiation or the screen are close to the obstruction causing the diffraction are termed Fresnel diffraction. Cases in which the source and screen are far from the obstruction are termed Fraunhofer diffraction. The text describes only Fraunhofer diffraction in quantitative detail as Fresnel diffraction is beyond the mathematical scope of the text. To see the effect of Fraunhofer diffraction in action for visible light.

As described in the text, the minima of such diffraction is given by

 a sin = m

(2.1) where a is the size of the obstruction and  is the angle relative to the horizontal as shown in figure 15.4.
Figure 4: Defining the angle
for calculation of diffraction of light through an aperture of width a.

The condition for interference minima is given by considering, just as for Young's experiment, the "extra" distance traveled by waves from one part of the aperture versus waves from another part. If, for example, we divide up the aperture into two parts and look at a ray from the top half vs. a ray from the corresponding location in the bottom half in getting to point P in figure 15.5, then we see that the condition for destructive interference is

 a 2 sin =   2 .

(2.2) Figure 15.5: If the distance from the aperture to the screen, x, is much greater than the size of the aperture, a, then the distance from the top half of the aperture to point P on the screen is less than the distance from the bottom half of the aperture to point P by approximately (a/2)sin
.

So, we expect dark fringes at values of   which satisfy

 sin = m a (m = 1, 2, 3, ...)

(2.3)

since we could divide the slit into quarters, eighths, 16th's, etc. and repeat the argument of having interference minima for each adjacent pair of intervals.

This is the formula for single slit diffraction minima. Note that there is no central minimum. The center of a diffraction pattern is always a maximum just as it is for interference in Young's experiment. The first minimum for diffraction therefore corresponds to m = 1 rather than zero as in Young's experiment. The vertical position of these minima is given approximately as

 ym = xtan  x m a

(2.4)

for y << x. The maxima or bright fringes for diffraction are approximately halfway in between the minima.

#### Intensity for Diffraction

We just state that the intensity for the many subdivions of the aperture that we can make can be expressed in terms of the following formula:

 I = I0    sin(/2) (/2)    2

(2.5)

with

  = 2a  sin

(2.6)

and I0 is the intensity at = = 0.

#### Circular Apertures

It should be remarked at this point that the Fraunhofer diffraction described up to this point assumes a rectangular obstruction or opening in a barrier. If the aperture is circular, then the pattern of maxima and minima on the screen is disk-shaped with a central bright spot surrounded by a series of bright and dark rings. The first minimum occurs at an angle 1 which satisfies the equation

 sin = 1.22  D

(2.7)

with D being the diameter of the obstruction or aperture.

Oddsei - What are the odds of anything.