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 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.
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.
Soap film example Anti-reflection coating
example
Interference maxima and
minima
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
"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.
Planck Radiation Formula
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
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The Rayleigh-Jeans curve agrees with the Planck radiation
formula for long wavelengths, low frequencies.
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You
may click on any of the types of radiation for more detail about its particular
type of interaction with matter. The different parts of the electromagnetic spectrum have very different effects upon interaction with matter. Starting with
low frequency radio waves, the human body is quite transparent. (You can listen
to your portable radio inside your home since the waves pass freely through the
walls of your house and even through the person beside you!) As you move upward
through microwaves and infrared to visible light, you absorb more and more strongly. In the lower ultraviolet range, all the uv from the sun is absorbed in a thin outer layer of
your skin. As you move further up into the x-ray region of the spectrum, you become transparent again, because most of
the mechanisms for absorption are gone. You then absorb only a small fraction of the radiation, but
that absorption involves the more violent ionization events. Each portion of
the electromagnetic spectrum has quantum energies appropriate for the excitation of certain types of physical processes.
The energy levels for all physical processes at the atomic and molecular levels
are quantized, and if there are no available quantized energy levels with
spacings which match the quantum energy of the incident radiation, then the
material will be transparent to that radiation, and it will pass through.
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.
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.
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
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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.
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:
|
(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
|
(14.1.1.2) |
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
|
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
|
(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,
|
(2.5) |
As a general rule, any path length difference
between two coherent sources can be expressed as a phase difference with the
following association:
|
(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
|
(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
|
(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
|
(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:
|
Then the sum
of these two waves yields a resultant wave which is described by
|
(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
|
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
|
(2.13) |
and the
directions of maximum intensity occur for
|
(2.14) |
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.
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
|
(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
|
(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
|
(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
|
(2.4) |
for y
<< x. The maxima or bright fringes for diffraction are approximately halfway
in between the minima.
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:
|
(2.5) |
with
|
(2.6) |
and I0 is
the intensity at =
=
0.
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
|
(2.7) |
with
D being the diameter of the obstruction or aperture.