Holographic Images - Part 1
When waveforms are distorted
Objects have
the effect of distorting or altering the waveforms that engage such objects. The manner in
which a wave form is altered serves as an index or signal of the character of the object
encountered. In a sense, the nature of the alteration of the waveform is sort of like a
lingering trace of the character of the object engaged by the waveform.
For example,
if a given object has the property of absorbing a certain range of wavelengths, then, when
it meets a complex waveform, the object will 'extract' those energies which it is capable
of absorbing from the waveform complex. Those wavelengths in the waveform complex falling
outside the object's absorption range will be reflected. By extracting certain
wavelengths, the object has altered the character of the waveform, and the nature of the
alteration provides an index for one of the properties of the object involved. We
usually refer to this property as color.
Objects with
a penchant for absorbing all manner of wavelengths will appear dark or black because
little of the original waveform is reflected back or permitted to be further transmitted
due to the absorption property. On the other hand, objects possessing little capacity for
absorbing any of a range of wavelengths in an encountered waveform complex will appear to
be whatever color happens to predominate in the wavelengths of the waveform complex being
engaged.
Thus, if the
entire spectrum of wavelengths is present, the object will appear to be white. However, if
the wavelengths in the waveform complex are dominated by those corresponding to the blue
region of the spectrum, then, the object will appear bluish, and so on.
Beside the
property of color, objects also will alter the character of encountered waveform complexes
as a function of a variety of other features. These other features include general shape,
texture, surface contours, and so on.
The energy
associated with a given form of electromagnetic radiation is directly proportional to
wavelength. The shorter the wavelength of the radiation, the greater will be
the energy of that radiation. Furthermore, the shorter the wavelength of a given form of
radiation, the greater will be the frequency or cycles per unit of time of such radiation.
Finally, as
the wavelength of a given form of electromagnetic radiation becomes smaller, the amplitude
of the waveform increases - that is, the peaks of this radiation's waveform
become higher, and the valleys or troughs become deeper. The intensity of a waveform is
directly proportional to the height of its amplitude. One should keep in mind, however,
that although frequency and amplitude are functionally linked in the various forms of
electromagnetic radiation, these two characteristics are independent in other kinds of
waveforms such as in the case of sound waves and water waves.
The central role of amplitude and phase
No matter
what kind of waveform one is dealing with, one can define that waveform completely by
considering only its amplitude and phase. In mathematical terms, amplitude and
phase constitute the essential variables in the function describing a given waveform,
whether simple or complex.
Phase refers
to the portion of a cycle that a wave has passed through at a given moment. The term
'cycle' is used because waveforms can be mapped onto points along the circumference of a
circle. This provides one with the opportunity to describe the waveform in
mathematical terms.
More
specifically, the circumference of a circle covers an angle of 360 degrees. Since
frequency is the rate at which a waveform repeats itself per unit of time, the 360 degrees
circumscribed by the circumference of a circle can be used as a unit measure for the
number of cycles completed per unit of time by a given waveform of a certain frequency.
A function's
value depends on what happens to some other value. When the circumstances surrounding this
other, independent value change, then, the value of the function will also change in an
appropriately dependent fashion.
Sines are
numerical values ranging from 0 to 1 and from 1 to 0 as the value of an acute angle
varies, respectively, from 0 degrees to 90 degrees and from 90 degrees to 0 degrees.
Cosines are also numerical values. However, as acute angles vary from 0 to 90 degrees and
from 90 to 0 degrees, cosines range, respectively, from 1 to 0 and from 0 to 1.
When the
sine value is maximum, the cosine is 0, and when the cosine is at its maximum value, the
sine value is 0. Sine and cosine are opposite in value, both with respect to magnitude as
well as sign, so if one of the two is positive, the other will be negative in value.
If one draws
a unit circle, in which the radius of the circle remains constant at 1, then, any right
triangle one inscribes in the circle with angle A's vertex at the center, will have a
constant hypotenuse of 1. On the other hand, one moves the right triangle around the unit
circle, the values of 'A', 'x' and 'y' all will change.
As 'A'
changes from quadrant to quadrant (one should envision the unit circle with diameters
running from top to bottom and from side to side, forming a perpendicular axis), one gets
two non-zero values for 'x' and 'y' of the right-triangle. That is, one gets two non-zero
values for the sine and cosine of the right triangle.
If one
constructs a graph, plotting values of sine and cosine (fluctuating between +1 and -1 and
forming the y-axis) against the corresponding degree readings of the unit circle (ranging
from 0 degrees to 360 degrees and which will form the x-axis), one gets a wave form. In
the case of the sine wave, one starts off at 0 (for the sine value) versus 0 degrees.
As the sine
value approaches a maximum of +1, the degree value approaches a maximum of 90 degrees. At
180 degrees, the sine value becomes 0 again. As the sine value approaches a value of -1,
the degree value works toward 270 degrees. Finally, when the degree value is 360 degrees,
the sine value once again returns to 0, and the wave cycle has been brought to its
original starting point of a 0 sine value and a 0 degree value.
In the case
of the cosine wave, one starts off with a cosine value of +1 and a degree value of 0. As
the cosine value approaches 0 for the first time, the degree value comes closer to 90
degrees. When the cosine value reaches a value of -1, the degree value is at 180 degrees.
When the cosine value reaches 0 for the second time, this corresponds to 270 degrees.
Finally, as the unit circle completes its cycle at 360 degrees, the cosine value once
again approaches its initial value of +1.
The formula
for the circumference of a circle is 2-pi r. In the case of a unit circle, however, where
r = 1, then, the formula for the circumference becomes merely 2-pi. If one translates
degree values into ‘pi’ values, 90 degrees, which corresponds to 1/4 of the
circumference, converts into 1/4 x 2-pi = 1/2-pi. 180 degrees becomes 1/2 x 2-pi = pi, and
270 degrees translates into 3/4 x 2-pi = 1 1/2-pi. With each new cycle, one merely adds 2
(which represents one complete circumference or cycle) to all the ‘pi’ values
for the corresponding degree values. Thus, 450 degrees (that is, 90 degrees into the
second cycle) becomes 2 1/2-pi, and so on.
The value of
+1 represents the highest point of amplitude for either a sine or cosine wave. However,
the value of +1, in and of itself, does not inform one whether one is dealing with a sine
or cosine wave (or some form of wave in between a sine and cosine wave), nor does it tell
one exactly where one is in the cycle.
The aspect
of phase enters in at this point, for in giving the phase spectrum with the amplitude
value, one is providing a means of locating where a given amplitude value occurs in a
cycle, relative to some identifiable point of reference such as 0 degrees, or the starting
point of a cycle.
A sine wave
reaches a maximum of +1 at 1/2-pi, 2 1/2-pi, 4 1/2-pi, etc.. A cosine wave, on the other
hand, reaches a maximum amplitude of +1 at 0-pi, 2-pi, 4-pi, and so on.
If one has a
wave, for example, of amplitude +1, with a phase spectrum of 1/2-pi, 2 1/2-pi, or 4
1/2-pi, one knows that one is dealing with a sine wave. As long as one has both an
amplitude and a phase spectrum, one has the basic components for defining a regular wave.
In short:
(a)
amplitude and phase define sine and cosine waves;
(b) sine and
cosine waves define regular waves;
(c) a series
of sine and cosine waves can define a compound wave;
(d)
amplitude and phase define compound waves.
In a sense,
the cycle of a waveform marks the transitions in amplitude that the waveform undergoes
over time, ranging from zero, to maximum, and back to zero again. If one wishes to inquire
about the character of the amplitude at any given point in the cycle, then, one will have
to engage the cycle at an appropriate point in time during which the aspect of the cycle
in which one is interested is being expressed. The precise stage of transition of the
wave's amplitude at that point in time constitutes the wave's phase.
Amplitude
gives expression to a quantitative measure of the energy of a wave. For example, a wave
has maximum energy at the crest point and minimum energy at the trough point. Phase, on
the other hand, locates or places a particular manifestation of a given waveform relative
to the structure of the entire cycle of transitions which such a waveform goes through
over time.
The
character of transition in amplitude referred to earlier does not refer to the absolute
magnitude of the amplitude at a given point. It refers to whether the amplitude is
increasing or decreasing as well as whether the amplitude is approaching or leaving: (a) a
zero point in amplitude; (b) a maximum point in amplitude, or (c) a minimum point in
amplitude. These themes of whether the amplitude is increasing or decreasing
- together with the nature of the relationship of this increasing/decreasing activity with
the maximum/minimum points of the cycle - describes how the current expression of
amplitude (as a pure magnitude) stands in relation to the structural character of the
waveform as a whole.
Thus, phase
constitutes the facet of the waveform's structural character being engaged at a given
point in time. Phase is the waveform's amplitude orientation to the world at a given
instant of engagement or manifestation. As such, phase is not something which can be
weighed with scales or measured, calibrated and scanned with instrumentation.
Phase is
essentially relational in character. Therefore, it requires a reference point against
which it plays off in order to establish its orientation within the structure of which
phase is an expression. For the most part, the relational character of phase is expressed
as a function of time and/or angles.
As long as
one knows where to place 0-pi, which serves as a point of reference, one has a means of
determining both amplitude and phase. However, if one has no means of
identifying the point of reference through which one starts the ‘pi’ scale, one
really has no means of establishing whether a regular wave is a sine wave or a cosine wave
or some other form of regular wave.
As a general
principle, one might argue that any methodology involves, as part and parcel of its being
a methodology, a means or technique for locating or establishing a point of origin or a
reliable point of reference. This sort of point of reference is one that is rooted in, or
is purported to be rooted in, the structural character of reality or that which reflects
an aspect of such structural character. Through this point of reference, one can locate or
orient oneself in relation to a wave's or latticework's (considered as a complex or
compound waveform structure) current expression of its phase spectrum.
As long as
one's methodology is unsuccessful in establishing this referential point of engagement,
one will have no means of locating, identifying, determining or establishing what the
phase spectrum of a latticework is or where one is in that phase spectrum when one
experientially engages that latticework. Moreover, if one selects an incorrect, distortive
or problematic point of reference as a basis through which to engage a given latticework,
the difficulties surrounding that initial selection will be transmitted throughout the
whole subsequent engagement and orientation process.
Interference and relative phase in relation to holography
Even if one
is not able to establish an absolute point of reference for locating where the n-scale
begins, relative phase can still be given a determinate characterization under certain
circumstances. For example, this can be done when one has two waves which are out of phase
with one another by a specifiable amount of ‘pi’.
In other
words, when one looks at the phase difference between two waves, one has a means of
engaging the waves in a relative manner that permits one to orient oneself with respect to
them to a certain extent. The phase difference between two waves is usually calculated as
an angle.
Interference
involves two or more waves that are interacting through their phase differences.
For instance, if one considers two waves of different amplitudes but which are in phase,
when the two waves interact with one another, they will tend to produce a wave with higher
crests and lower troughs than either of the original waves considered individually. This
is a case of constructive interference in which there is a relative phase difference of 0.
However, if
two waves of the same amplitude, but opposite phase, interact with one another, the result
will be a wave in which troughs and crests coincide and, therefore, cancel out to have
zero amplitude. This is a case of destructive interference in which the relative phase
difference of the two waves is a non-zero value.
As the
relative phase difference approaches a maximum value when the two waves are precisely
opposite in phase character, the crests of the daughter waves will become increasingly
less than either of the parent waves and the troughs of the daughter waves will become
increasingly less than either of the parent waves. In short, both the crest and trough of
the wave will approach the horizontal axis of the graph as a limit.
A definite
phase relationship must be established between two or more sets of waves in order for an
interference pattern to be created. A phase relationship which is well-defined is referred
to as being "in step".
On the other
hand, when the phase relationship is not well-defined, then, the waves are said to be
"out of step". Out of step waves cannot produce interference patterns.
Therefore, even in the case of destructive interference, there must be some degree of
well-definedness to the phase relationship of the waves involved.
The
situation becomes more complicated if one keeps the amplitudes of the interacting waves
equal but allows the phase difference to have values less than n or 180 degrees. Under
these circumstances, the waves sometimes will manifest constructive interference and, at
other times, will give expression to destructive interference, depending on the value of
the relative phase difference. Nonetheless, for each specific relative phase difference,
there will be a unique daughter wave whose shape is a reflection of that specific relative
phase difference.
If we permit
one more complicating factor to be introduced (namely, variable amplitudes for the
interacting waves), in addition to a relative phase difference of less than ‘pi’
, one will generate a daughter wave which has a unique size (as a function of the
interacting amplitudes) and unique shape (as a function of the interacting phase
differences). In other words, the magnitude of the daughter waves will be a
function of the amplitudes of the parent waves, while phase differences will determine
where and when constructive and destructive interference will occur.
In general,
the magnitude and shape of the daughter wave produced by the interaction of n-waves will
be completely determined by the amplitudes and relative phase differences of the
interacting waves. Therefore, any compound wave can be represented as a summation series
of amplitudes and phases of a set of interfering waves.
In an
optical hologram, information is stored in the form of alternating zones and bands of
light and dark. These alternating bands are the telltale signs of the
presence of interference. The density of these interference regions depends on the
intensity of the light being used to make the hologram.
As indicated
earlier, the intensity of the light wave is an index of the wave's amplitude. Therefore,
density of the interference patterns provides one with a means of deriving information
about amplitude.
This is one
of the two factors necessary to be able to give a complete description of a given
waveform. The other factor enabling one to describe a waveform is the relative phase.
Such
information is reflected in the rate at which transitions occur in relation to the shifts
in constructive and destructive manifestations of interference as one moves from one point
or zone of the hologram to another contiguous point or zone in the hologram. This rate of
transition carries the phase code.
In
simplified terms, objects alter the structural character of those light waves interacting
with it. This alteration affects both the amplitude and phase character of the waveform.
These altered characteristics will be transmitted to, and given expression in, the pattern
of interference that develops when the light which has encountered an object meets up with
light waves which have not encountered such an object.
Photographs
of a conventional sort record data about amplitude but not about phase. Holograms also
record and keep track of data on phase relations as well.
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