I
N T E R D I S C I P L I N A R Y
C O N T E N T S
1. Introduction
2. Further Introduction:
Two Important Design
Features
3. The Problem
Requirements
4. The Annular Data
Area and Track Pitch
5. Diffraction Limits for
Track Pitch and
Linear Data Density
6. CAV Versus CLV Disc
Drives
7. Linear Data Density
8. Digital Video
Compression
Appendix A:
Glossary of Terms
Appendix B:
Notes for the Teacher
Appendix C:
Sample Solution and
Analysis
L
I V E L Y
A
P P L I C A T I O N S
P
R O J E C T
Red&
Blue
Laser
CDs:
How Much Data
Can They Hold?
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Interdisciplinary Lively Applications Project
INTERDISCIPLINARY LIVELY APPLICATION PROJECT
TITLE:
RED AND BLUE LASER CDS:HOW MUCH DATA CAN THEY HOLD?
AUTHORS: BRUCE ACKERSON, PHYSICS, OKLAHOMA STATE UNIVERSITY
DENNIS BERTHOLF, MATHEMATICS, OKLAHOMA STATE UNIVERSITY
JAMES CHOIKE, MATHEMATICS, OKLAHOMA STATE UNIVERSITY
EMILY STANLEY, BIOLOGY, OKLAHOMA STATE UNIVERSITY
JOHN WOLFE, MATHEMATICS, OKLAHOMA STATE UNIVERSITY
EDITORS: JOSEPH MYERS
DAVID ARNEY
MATHEMATICAL REQUIREMENTS: BASIC GEOMETRIC PROBLEM SOLVING,
OPTIMIZING A FUNCTION OF ONE VARIABLE, ARC LENGTH IN POLAR
COORDINATES, APPROXIMATION.
DISCIPLINARY CLASSIFICATION: LASER DISC TECHNOLOGY, PHYSICS.
PREREQUISITE SKILLS: INTEGRAL CALCULUS, THROUGH ARC LENGTH.
PHYSICAL CONCEPTS EXAMINED: DIFFRACTION, LASER DISC DESIGN.
INTERDISCIPLINARY LIVELY APPLICATIONS PROJECT IS FUNDED
BY THE NATIONAL SCIENCE FOUNDATION, DIRECTORATE OF
EDUCATION AND HUMAN RESOURCES DIVISION OF
UNDERGRADUATE EDUCATION, NSF GRANT #9455980
© COPYRIGHT 1998 THE CONSORTIUM FOR MATHEMATICS AND ITS APPLICATIONS
(COMAP)
NSF INITIATIVE:
MATHEMATICS SCIENCES AND THEIR APPLICATIONS THROUGHOUT THE CURRICULUM
(CCD-MATH)
Red & Blue Laser CDs: How Much Data Can They Hold?
1. INTRODUCTION
Philips
and Sony pioneered the CD
audio disc in 1982. To date
over 120 million CD players
and 3 billion CDs have been
sold in the United States
alone. The introduction of the CD laser
technology offered unheard of levels of information storage — about 74
minutes of music or about 680 million bytes (megabytes) of computer
programs and data.
However, for storing video the CD is just not big enough — about 20
minutes of video is the maximum capacity for a CD (5 to 7 CDs are
needed for a movie). A national phone directory for the United States
with 112 million listings requires 6 CDs.
Since 1982 significant advances in enabling technologies (laser diodes,
disc manufacturing, digital coding and compression algorithms, integrated circuits, drive mechanisms) have opened the way for the current generation of laserdiscs—the digital video disc or DVD.
The DVD, physically the same size as the audio CD, can contain 7 times
the data of an audio CD on just one layer of the 4 available layers. Over 2
hours of video (including multiple sound and subtitle tracks) will fit on
one layer of one side of a DVD.
For this project you are to present an analysis of the data capacity for different types of laser disc technologies. Two design factors will be important: laser wavelength (infrared, red or blue) and disc drive mechanics
(CLV or CAV). A brief introduction to the two design factors will be given
in section 2 so that the problem requirements can be stated in section 3.
Perhaps surprisingly, the laser color or wavelength is a determining factor
in laser disc data capacity. The wave phenomena called diffraction is the
key to this mystery. Diffraction, technical data and background information about disc drive mechanics are discussed in later sections.
Appendix A is a glossary; included terms are identified by bold italic font
when they first occur in the text.
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2.FURTHER INTRODUCTION:TWO IMPORTANT DESIGN FEATURES
Just
enough information about laser color or wavelength and disc drive mechanics is given in this
section so that problem requirements can be stated
in the next section. See later sections for a more
complete discussion of these topics including essential technical data.
The first design feature is laser wavelength. On a laser disc, digital data is
encoded in a spiral track consisting of pits of varying lengths. Surface
features for a CD and a DVD are depicted in the next figure.
1.6 µm spacing
0.74 µm spacing
0.83 µm
minimum
Figure 1:
SURFACE FEATURES OF
CD VERSUS DVD
0.4 µm
minimum
Disc technology is based on a focused laser beam which must distinguish
data tracks and recognize data pits on the reflective disc surface. How
finely a laser beam can be focused is a limiting factor for data storage
capacity. Due to diffraction, this focusing limit is related to the wavelength or color of the laser used. The common CD audio discs are based
on an infrared laser diode. The recent DVD is based on a red laser diode.
In 1996, lasermakers reached a long-sought milestone with the development of blue-light lasers from semiconductor chips made from galliumnitride. The shorter wavelength blue lasers may one day support a new
generation of compact discs storing more data than the current DVD.
Physical disc properties for laser technology based on infrared (CD), red
(DVD) and blue (hypothetical) lasers are presented below.
The second design feature is disc drive mechanics. Laser discs can be
designed for either constant angular velocity (CAV) or constant linear
velocity (CLV) disc drives. In order to maintain a constant speed of the
read head over the data track, CLV drives must rotate faster when reading
the inner (shorter) tracks. In general, CLV discs have greater storage
capacity. However, CAV discs offer one performance advantage—faster
access time. For CLV discs there is a wait time for the disc to speed up or
slow down as the read head moves between inner and outer tracks. For
applications involving random access to data (e.g., a disc based phone
directory or dictionary) a CAV drive offers a performance advantage. CLV
discs with their larger data capacity offer better performance for such
applications as video and/or music.
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Red & Blue Laser CDs: How Much Data Can They Hold?
3.THE PROBLEM REQUIREMENTS
Broadly
speaking, your problem is to
present an analysis of data
capacity for laser discs based
on two independent design
features: laser color/wavelength
(infrared, red or blue) and disc
drive mechanisms (CAV or CLV). Your report must address each of the
following requirements.
The first requirement concerns a purely mathematical problem—how
long is a spiral?
Comments on this requirement:
• The needed physical dimensions for the track based on the different
wavelengths is given below in section 4.
• What kind of function appropriately models a spiral track? How do you
find its length?
• Can you think of other ways to model the length of a spiral, in particular,
ways for which the length is easy to compute? What about a series of
concentric circles? What about a direct relationship between the cross
sectional area of a roll of toilet paper and the total length of the roll?
REQUIREMENT 1A:
Using an appropriate functional model for a spiral, calculate the exact length of
an ideal mathematical spiral based on the three laser wavelengths.
REQUIREMENT 1B:
Present at least one (but possibly more) alternate ways to model the spiral
track which give approximate values for the track length. Calculate the length
based on the three laser wavelengths for these approximate methods.
REQUIREMENT 1C:
Compare the values found in 1a and 1b above. Are they very different or surprisingly close? Evaluate and discuss this comparison.
The second requirement focuses on a difference between discs designed for
CLV versus CAV disc drives.
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Audio CDs and all other discs designed for CLV drives use the same area of
the disc for data storage. The dimensions of this area are shown in the next
section. Surprisingly, for discs designed for CAV drives, using less of the disc
surface for data increases the data capacity. You will address the “less is more”
issue of how much of the surface area should be used for maximizing data on
a disc for a CAV drive in requirement 2.
COMMENTS:
• At first the significance of the difference between CLV and CAV is illusive. Insights and background information in section 6 might help clear
up ideas presented in brief outline form here.
• For a constant angular velocity (CAV) disc, the data capacity is determined by the length of the inner cycle of the spiral track with all other
cycles containing the same amount of data. Thus the “effective” length of
the data track is the length of the innermost cycle of the track times the
number of cycles in the spiral.
• If the full area is used, there are more cycles; however, the capacity of
each cycle is smaller due to the shorter innermost cycle of the track.
• On the other hand, if, say, only the outer third of the disc area is used,
there are fewer cycles but more information per cycle.
REQUIREMENT 2:
Determine how much of the data area to use for a disc designed for a CAV
drive in order to have the largest “effective” data track length (and thus largest
data capacity).
REQUIREMENT 3:
For each of the three laser wavelengths you must evaluate data capacity for
both CAV and CLV discs. Present your findings regarding data storage capacity
in terms of two measures. First report capacity in terms of the minutes of high
resolution video which can be stored (this measure is meaningful for video
applications). Secondly, report capacity in terms of the number of millions of
data records which can be stored (this measure is meaningful for database
applications).
In summary, you are to present an analysis of different mixes of CAV/CLV and
laser wavelength technology in terms of their feasibility to provide the storage
capacity needed for high resolution video and database storage applications.
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Red & Blue Laser CDs: How Much Data Can They Hold?
4 . T H E A N N U L A R D ATA A R E A A N D T R A C K P I T C H
Figure 2 below indicates the dimensions of a standard compact disc and,
for a CLV disc, the annular region used for data storage.
Figure 2:
DIMENSIONS OF DATA AREA ON CLV LASER DISCS
A compact disc has a diameter of 120 mm. The data surface is an annular
region 35.5 mm wide with an inner radius of 22.5 mm and an outer
radius of 58 mm. A 2 mm band on the outside of the disc is not used for
data.
With a metric ruler and any audio CD you can verify these dimensions
which are established by international convention.
Figure 3:
SPIRAL SHOWING TRACK PITCH
A linear spiral track is pictured in Figure 3. The track pitch, indicated
by the letter p, is the fixed distance between successive cycles of the spiral.
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5 . DIFFRACTION LIMITS FOR TRACK PITCH AND LINEAR DATA DENSITY
In
this section the value for the track pitch for each of the three
laser types is given. Background information on diffraction and
focusing/resolution problems and their relationship to track
pitch are presented. In addition, linear data density is defined
and values given for the three laser types.
For conventional audio CDs, the track pitch is 1.6 microns (millionths of
a meter). The pit size is small enough that on average each millimeter of
track can encode about 121 bytes of information. A laser beam is focused
on the moving data track on the reflective CD surface. The intensity of
the reflected light is modulated by the presence (or absence) of data pits
which enable the processor to pick up digital data from the disk. The circular image on the CD surface of the laser beam is approximately 2
microns in diameter. The track pitch and laser beam image are depicted
in Figure 4.
Figure 4:
TRACK PITCH AND LASER
IMAGE FOR LASER DISC
These dimensions are tiny—the smallest entities which are routinely
manufactured by current technology. A typical speck of dust (40 microns
wide) would cover 20 tracks on a CD. Track pitch for a DVD is even
smaller (see below for the value).
To read these tiny pits, a light beam must be sharply focused—which
confronts laser makers with the phenomenon called diffraction. To better
describe the focusing or resolution problem, Figure 5 shows a common
configuration of optical elements. Light is incident on a lens after passing
through some restrictive opening (diaphragm) and then is focused to a
“point” on some substrate or surface.
Diaphragm
Lens
ll
D
Figure 5:
LASER CONFIGURATION
SHOWING BEAM WAIST
Beam
Waist
l
f
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Red & Blue Laser CDs: How Much Data Can They Hold?
This configuration might portray a simple camera where the substrate is
film, or the human eye where the diaphragm is the pupil and the substrate is the retina. This setup is part of the photolithography equipment
for making computer microcircuitry; the diaphragm is the mask containing the printed circuit diagram, and the substrate is the microchip surface
on which the circuit will be etched. In the case of CDs the diaphragm is
simply the width of the laser diode beam and the substrate is the CD disk
surface.
Imagining the rays of light as a stream of particles (photons), as Isaac
Newton did, suggests that a beam of light can be focused to an infinitely
small spot, a point. However, in this situation where dimensions are
microscopic, diffraction becomes a limiting factor. Conceptualizing light
as a wave provides a model consistent with physical observation and
experiment. In fact, focusing to a point, even with perfect equipment, is
not possible. For a given focusing configuration, the size of the smallest
possible spot of light, called the beam waist l, is determined by the
relationship l= 1.22 ×
λf
D
where f is the focal length of the lens, D
is the opening of the diaphragm and λ is the wavelength of the light in
the beam. The technological implications of the information in this formula are quite surprising. A small value of the beam waist is critical for a
sharp photographic image, a high density printed circuit or, in our case,
spotlighting the tiny data pits on a laser disc. The beam waist could be
reduced by enlarging the diaphragm. This forces the lens to be larger
which poses technical challenges. Also, the focal length will necessarily
get longer as the lens gets larger and fatter, because of precision constraints on grinding curvature. Adjusting the wavelength (say, from red
toward blue or even beyond the visible spectrum) would also reduce the
beam waist. For this reason, the development of blue-light lasers from
gallium-nitride semiconductor chips was an important technology milestone in 1996. The final design of the laser configuration for a laser disc
(with the resulting beam waist diameter) is an optimization with respect
to many variables and technical factors.
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The next table contains values for the beam waist of a focused laser beam
for infrared (CD), red (DVD) and blue (hypothetical) laser technologies.
Laser Color
Table 1:
PHYSICAL PROPERTIES FOR THE
THREE LASER WAVELENGTH
TECHNOLOGIES
Wavelength
Beam Waist
Linear Data Density
infrared
780 nanometers
2 microns
121 bytes/mm
red
640 nanometers
0.925 microns
387 bytes/mm
blue
410 nanometers
0.4 microns
800 bytes/mm
The beam waist of the focused laser beam controls both the track pitch
and the density and size of the small pits which encode the data and
make up the track.
For the purposes of this project you may assume that the track pitch is
0.8 times the beam waist.
The linear data density (data per unit length of data track) depends on
the size of the pits and how closely they are packed together. These
dimensions are also controlled by the resolution of the focused laser
beam, i.e., the beam waist. Values for linear data density are listed in the
last column of Table 1 for infrared, red and blue laser technologies.
Linear data density is discussed further in section 7 below.
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Red & Blue Laser CDs: How Much Data Can They Hold?
6 . C AV V E R S U S C L V D I S C D R I V E S
Data
storage discs can be designed for either constant angular velocity (CAV) or constant linear velocity (CLV) disc drives. The choice
between these two design types will effect
the data capacity of discs. In addition, the size of the annular area on the
disc used for data storage is different for the two types.
CAV (constant angular velocity) devices rotate at a fixed number of revolutions per minute. Floppy discs and the old 33 1/3 rpm records rotate at a
constant rate of speed and are CAV devices. During each revolution one
cycle of the spiral track will pass under the read head. However, outer
cycles are much longer than inner cycles so the length of track scanned
during each revolution will depend on the inner or outer location of the
read head. CAV discs normally pack the same amount of data in each
track cycle since this provides a constant data rate through the read head
to the microprocessor. Data pits are packed tightest on the innermost
track; pits and spaces are elongated on the outer tracks. The constant
amount of data per cycle depends on the linear data density (constant for
a given laser wavelength and physical configuration) and on the length of
the shortest track. Furthermore, the total data capacity will depend on the
data capacity of the innermost track and the total number of tracks.
By contrast, CLV (constant linear velocity) devices are designed so that the
track passes under the read head at a uniform rate. The rotation rate will
vary, slower when reading the outer (longer) tracks and faster when reading the inner (shorter) tracks.
Perhaps imagining a cassette tape drive (a kind of CLV device) will clarify
this situation. The pinch roller moves the cassette tape over the read head
at a constant rate. However, you can hear how the take-up reel and feed
reel vary their rotation speed with the emptier reel revolving quickly and
the fuller reel rotating at a more leisurely pace.
The standard audio CD drive is a CLV device which varies its rate of rotation to maintain a uniform flow of track under the laser read head. The
DVD drive characteristics are uncertain as of the writing of this manuscript. All DVD drives will handle CLV discs such as movie videos; however, some DVD drives (especially those designed for computer applications)
are expected to be capable of operating in both CLV and CAV mode.
Data capacity is different for the two kinds of laser discs. CLV discs can
hold more data since data density is the same on all parts of the track,
outer (and longer) cycles of track hold more information than inner
(shorter) track cycles. On the other hand, for CAV discs the maximum linear data density supported by the technology is used only on the innermost cycle of the spiral track. The outer tracks on CAV discs have the
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same amount of data as the innermost cycle and, thus, the information is
spread out more thinly on the outer, longer track cycles. CLV discs have a
larger data capacity.
If data capacity were the only issue, all disc drives would be CLV drives.
However, computer manufacturers had compelling reasons to design floppy disc drives as CAV devices rotating at a fixed speed just like old record
players. The performance advantage for CAV devices is improved random
access time, the time required for the read head to move from one portion
of the disc surface to another. A serious drawback for CLV discs is that
there is a wait time for the discs to speed up or slow down as the read
head moves between inner and outer cycles of the track. In contrast,
although CAV discs hold less data there is no speed-up or slow-down
time needed when jumping around for data on the disc.
There is a trade-off between data capacity and access time for the two
technologies. For applications where large capacity is vital and data is normally accessed in a fixed sequence (audio and video applications) CLV
discs are preferred. Database applications (phone number or dictionary
look-up, for example) and interactive computer applications perform better when using CAV discs and drives.
One final design feature of CAV and CLV discs affects data capacity, namely the size of the annular area on the surface of the disc which is used for
data storage. This area is visible on both CDs and floppy discs.
Since CLV disc data capacity depends on total track length, the larger the
area the better. The dimensions shown in section 4 above have been
adopted by international convention between manufacturers of CLV discs.
The choice of data area for a CAV disc is a critical part of this project.
Leaving a 2 millimeter unused ring on the outer edge of the disc is necessary for the physical protection of the data pits. The best radius for the
inner boundary for the data area needs to be determined. If the entire disc
surface is used, there will be an increased number of track cycles but
there will not be much data per cycle—the innermost track will be very
short and the pits and spaces on the outer ring will be elongated and inefficient. On the other hand, using less of the surface gives more data per
ring but fewer rings. A mathematical analysis of the size of the annular
area giving the optimal storage capacity is needed. The reasonableness of a
mathematical analysis of this problem can be judged by checking that the
actual measured data area on floppy discs is consistent with your conclusions—surely disc manufacturers for these CAV discs thought through
this issue when designing floppy discs.
In summary, data capacity for a CAV disc depends on the length of the
innermost track cycle and the number of cycles since all cycles contain
the same amount of data. Data capacity for a CLV disc depends on the
total length of the spiral track. The size of the annular area on the disc
surface used for data storage is set by convention for CLV discs. For CAV
discs the data area is chosen to maximize data capacity.
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7 . L I N E A R D ATA D E N S I T Y
The
linear data density, expressed in bytes per millimeter,
reflects the amount of information which can be
stored per fixed length of the spiral data track on a
CLV disc or per fixed length of the inner ring on a
CAV disc. More data can be packed per millimeter of track on a DVD than
on a CD as can be seen from the depiction of the two disc surfaces in
Figure 1. Data density, like the track pitch, is related to the resolution of
the laser beam and so depends on the choice of infrared, red, or blue laser.
The encoding algorithm, including error correction, used for laser discs
complicates the relationship between the number of physical pits on the
disc surface and the number of bytes of information. Amazingly, a scratch
the width of a wide magic marker (2.5 mm) scraped across a CD will
not result in any loss of information or video/music quality. To achieve
this spectacular result encoding algorithms employ interleaving to distribute errors and parity to detect and correct them. As a rule, incorporating
error correcting information along with digital data on a CD increases the
space required for a given amount of information by about 25%.
Average values for the linear data density which can be used for this project are in the last column of Table 1.
Also, when measuring disc capacity in terms of data records, you may
assume that each record requires an average of 50 bytes of information
(this is typical for a phone record). Thus, for example, an infrared laser
CD can store a little over 2 phone records per millimeter on the innermost track.
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8 . D I G I TA L V I D E O C O M P R E S S I O N
Digital
video compression is essential for
disc storage of movies. Even with
medium resolution of 600 x 800 pixels and 256 colors at 30 frames per
second,
one second of raw video requires 14.4 megabytes of
data. At this data rate only about 8 minutes of video (with no sound)
would fit on one layer of a DVD—20 or more DVDs would be needed
for one movie.
The MPEG2 video/audio compression standard is used for DVD. Under
MPEG2, the data rate required to encode video varies with time. More
data is used to encode complex sequences (action on a moving background) and less for simple sequences (a small plane flying across a clear
sky). Typical fluctuations in the bit rate for encoding a movie are depicted in the following figure.
Figure 5:
VARIABLE BIT RATE FOR
MPEG2 DECODING
DVD players can process data at rates up to 1.25 megabytes per second.
By contrast, the byte rate used for digital satellite television (DSS) is fixed
at 0.75 megabytes per second—DVD picture quality is significantly higher. The standards for Digital TV (DTV) (which include high definition
TV) announced by the Federal Communications Commission at the end
of 1996 call for data rates of 4 times the rate of the first generation of
DVDs. Thus DVDs will offer video quality somewhere between satellite
digital TV and high definition TV.
The MPEG2 digital video compression used for DVD requires an average
of 0.62 megabytes per second to encode very high quality video along
with 5.1 channels of digital quality audio in 3 languages. This value of
0.62 megabytes per second can be used for estimating video capacity for
the different disc technologies under examination in this project.
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APPENDIX A:GLOSSARY OF TECHNICAL TERMS
Annular Region
A region bounded by two concentric circles. The area of such a region will be the difference between the areas of two discs.
Beam Waist
The diameter of the smallest circle to which a laser beam configuration can be focused.
This value is not zero due to diffraction and also due to imperfections in the mechanical parts and set-up.
Bit
The smallest unit of data, usually represented by the choice of either a 0 or 1 value.
Byte
A unit of data consisting of 8 bits, usually thought of as representing one character.
Constant Linear Velocity (CLV)
Used in reference to disc drives which vary the revolution speed from inner (shorter)
tracks to outer (longer) tracks to maintain a constant bit rate under the read head.
Constant Angular Velocity (CAV)
Used in reference to disc drives which operate at a fixed speed of rotation.
Cycle of a Spiral
The part of a spiral traced out during one revolution. A cycle is nearly a circle but the
two ends do not match up. There are more than 22,000 cycles in the data track of an
audio CD.
Diffraction
Bending or dispersion of a beam of light when passing by a sharp edge or through a
narrow aperture. Diffraction is due to wave interference and thus the extent of dispersion depends upon the wavelength or color of the light.
Linear Data Density
The quantity of data (typically in bytes) per unit length of a data track.
Linear Spiral
A spiral where the distance from the center is a linear function of the angle.
Megabit
A unit of data consisting of one million bits.
Megabyte
A unit of data consisting of one million bytes. Hard disc drives are usually rated in
terms of megabytes of data storage.
Micron
A tiny unit of length, one millionth of a meter. For example, one human hair is about
75 microns, a dust particle is 40 microns. A micron is symbolized by µ.
MPEG2
A video/audio coding and compression standard used for DVDs.
Random Access Time
The average time needed to jump from one section of a data stream to another.
Spiral
A geometric figure representing the path of a dot which moves radially outward as it
cycles around its center.
Track Pitch
Track pitch is the distance between successive cycles in a spiral track. The track pitch
is 1.6 micrometers for an audio CD.
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APPENDIX B:
NOTES FOR THE TEACHER
INTRODUCTION
The mathematics needed for this project is not daunting. The major difficulty is mastering a lot of detailed information and extracting an appropriate conceptualization of the problem and the relevant numerical values needed for the various requirements. The discussion of diffraction
and MPEG2 coding are for background information and are not needed
for the solution for the problem. The authors felt that some understanding of this background was valuable in its own right and, also, was essential for the authenticity of the project.
BRIEF CONCEPTUAL OUTLINE
The first step is to realize that a linear function in polar coordinates (a linear spiral) is the most appropriate mathematical representation of the spiral data track. The length is obtained using the standard arc length formula in polar coordinates. A trigonometric substitution is needed to evaluate
this integral. Approximations to this length can be based on conceptualizing the spiral as a family of concentric circles or else as a long thin strip
with thickness equal to the track pitch which is rolled into a coil.
Data capacity calculations depend on differences between CAV and CLV
drives. The easy case is CLV drives where the data capacity is track
length times the amount of data per unit length (linear data density).
Also, for CLV drives the size of the data area is established by convention
(essentially bigger is better in this case).
In the CAV case, the basic fact is that the amount of data is the same in
each cycle and this fixed amount depends only on the length of the
shortest spiral. Thus data capacity is the number of cycles or rings in the
track times the length of the shortest cycle times the linear data density.
The number of track cycles is the thickness of the annular data area
divided by the width of space between tracks (track pitch). Determining
the thickness of the annular data for a CAV drive is an optimization
problem solved quite easily with calculus (or more simply, with an
understanding of the parabolic shape of quadratic curves). The data area
for a CAV disc is annular with the inner radius equal to one-half the
outer radius (just like floppy discs).
Finally, data capacity in terms of data records and in terms of minutes of
video is obtained by dividing the data capacity in bytes by the values
given for either the number of bytes per record or the average number of
bytes per second of video.
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CLASSROOM DEMONSTRATIONS
Some aspects of this problem can be highlighted with classroom demonstrations.
DEMONSTRATION 1: A DIFFRACTION DEMONSTRATION
The physicist among the authors felt that diffraction and the modeling of
light as a wave phenomenon are concepts which are central to many
aspects of modern physics and technology and that these concepts are
largely missing from mathematics courses. As we worked out the details
of this project, the rest of the authors came to appreciate the validity of
this point of view. Unfortunately an understanding of diffraction is not
logically essential to the mathematical solution for this problem.
However, it is diffraction which limits the resolution of a focused laser
beam and thus the minimal size and packing of data pits on a CD. Also it
is diffraction which explains why laser wavelength is significant. The
non-physics authors were enlightened (and entertained) by seeing a simple diffraction demonstration, which is described in the next paragraph.
We encourage those teachers who use this module to contact the physics
department of their institution to borrow the equipment needed for their
own demonstration.
A laser source (for example a common red laser pointing device used by
speakers) and a slit micrometer are all that are needed for an interesting
demonstration which takes only 5 minutes of class time. With the beam
shining onto a wall through the slit, one sees that, at first, as the slit narrows the spot on the wall becomes narrower and sharper. However, as the
slit narrows further, the spot on the wall becomes fuzzy and expands. One
can see that there is a limit to how sharply the beam can be focused. The
light waves are bending around corners, i.e., diffraction is taking place.
DEMONSTRATION 2: DATA AREA OF A FLOPPY DISC
Floppy disc drives are constant angular velocity devices. Thus we expect
that floppy discs would be designed to have a data area with outer radius
2 times the inner radius as predicted by calculus. This can be verified by
simple measurements.
For a standard 3-inch floppy disc, the inner and outer radii of the data
area can be estimated by measuring the opening under the sliding metal
window cover. Approximate results are 4.2 cm outer radius and 2.0 cm
inner radius. This is not inconsistent with the theoretical 2 to 1 ratio.
Note that the actual data area will be somewhat smaller then the size of
the window opening. The actual data area does not seem to be visible on
the disc surface.
Measuring the faintly visible markings on the older 5-inch floppy discs,
the inner radius is approximately 3.2 cm and the outer radius is 6.4 cm.
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Interdisciplinary Lively Applications Project
INTERNET RESOURCES
Extensive materials on CD and DVD technology are available on the
internet, with most searches giving thousands of links. Five especially
informative links are given below. These URLs are listed here so that the
instructor can check that they are still valid before passing them onto
students.
Two substantial documents are available from C-Cube Microsystems, one
titled Digital Video Disc (http://www.c-cube.com/technology/dvd.html)
and the other titled MPEG Overview (http://www.c-cube.com/technology/mpeg.html).
Optibase Company maintains two particularly informative documents on
the web, one about digital video (http://www.optibase.com/dprimer.htm)
and a second about MPEG Video compression
(http://www.optibase.com/mprimer.htm)
Sony Corporation provides much general information about DVD technology at http://www.sel.sony.com/SEL/consumer/dvd/aboutdvd.html
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17
Red & Blue Laser CDs: How Much Data Can They Hold?
A P P E N D I X C : S A M P L E S O L U T I O N A N D A N A LY S I S
REQUIREMENT 1A: EXACT CALCULATION OF TRACK LENGTH
The simplest polar function (namely, linear or linear spiral) seems to be
the most appropriate functional model for the track:
r(θ ) = r1 +
p
θ, a ≤ θ ≤ b
2π
where r1 is the inner radius (22.5mm) and p is the track pitch. Note that
that since θ must rotate 2π radians for each cycle, the radius r(θ) increases by the track pitch p each complete cycle. Also note that the interval for
θ will be 0 ≤ θ ≤ 2π x R where R is the total number of cycles in the
track. The radius of the annular data region (outer radius minus the
inner radius) is 35.5mm so the basic relationship between the
track pitch p and the number of cycles or rings R is R = 35.5
p
where p is measured in millimeters. The values for p for the three laser
colors are obtained by multiplying the beam waist from Table 1 by 0.8 as
directed just following Table 1. Thus the final polar equation for the
spiral track is given by: r(θ ) = 22.5 +
p
35.5
θ, 0 ≤ θ ≤
× 2 π = 2 πR
2π
p
where the values of p are calculated as described above. In polar coordinates the element of arc length is given by ds2 = r2dθ 2 + dr2 so we get the
following integral formula for the track length Lp as a function of the
track pitch p:
2 πR
∫
Lp =
(
0
p
p
θ + 22.5)2 + ( )2 dθ .
2π
2π
Using the substitution u =
p
θ + 22.5 , we get
2π
Lp =
2π
p
u 2 + ( )2 du.
2π
p 22.5
58
∫
A trigonometric substitution then gives
Lp =
2π
p
© COPYRIGHT 1998 COMAP
58
u 2

p
1 p
p 
u + ( )2 + ( )2 ln  u + u 2 + ( )2  

2π
2 2π
2 π   22.5

 2
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This expression can now be evaluated to get the track length (in meters)
for different values of the track pitch p. The results are given in the next
table.
Laser Color
nm
Table 2:
TRACK LENGTH
CALCULATIONS FOR A
CLV DISC
Beam waist
microns
infrared (780)
red (640)
blue (410)
Pitch p
microns
2.000
0.925
0.400
1.60
0.74
0.32
Cycles R
number
Length Lp
mm
22,188
47,973
110,938
5,611,300
12,132,334
28,055,919
Length
Miles
3.49
7.54
17.44
The values in millimeters given in this table are correct (according to
Maple) to the nearest millimeter (this will be important for comparison
below). Dividing the length in millimeters by 1.609 X 106 gives the
length in miles.
REQUIREMENT 1B: ALTERNATE METHODS FOR ESTIMATING TRACK LENGTH
Three approximate methods will be given. Interestingly, all three methods
give the same approximate value Ap (depending on track pitch p) for the
approximation, namely, Ap = 2 π (22.5 +
where p is the track pitch and R =
35.5
35.5
35.5
) × R = 2 π (22.5 +
)×(
)
2
2
p
35.5
is the number of cycles.
p
Before presenting these methods, we first show the values of the approximation Ap as compared to the “exact” values Lp
Table 3:
EXACT AND
APPROXIMATE TRACK
LENGTH CALCULATIONS
Laser Color
Pitch p
Lp
Ap
Error
nm
microns
mm
mm
mm
infrared (780)
red (640)
blue (410)
1.60
0.74
0.32
5,611,300
12,132,335
28,055,919
5,611,179
12,132,279
28,055,895
121
56
24
Notice that the largest error is 121 mm (about 5 inches) out of 3.49
miles. The reasons for the errors being so tiny are the subject of requirement 1c.
The first two methods are based on modeling the spiral track as a series
of concentric rings and figuring the length as the sum of the circumferences of the rings.
Method A: Perhaps the easiest way to estimate the sum of the circumference of all of the concentric circular rings is to multiply the middle or
average ring circumference by the total number of rings. This gives
Approximate Track Length = 2π(22.5 +
35.5
35.5
35.5
) × R = 2π(22.5 +
)×(
) = Ap .
2
2
p
Note that the average circumference will be the middle circumference
since the circumference is a linear function of the radius.
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Red & Blue Laser CDs: How Much Data Can They Hold?
Method B: A second method for estimating the track length is by literally
adding up all of the circumferences of the concentric circles.
R−1
This gives Track Length ≈ ∑ 2 π (22.5 + p ⋅(n + 12 )).
n=0
N
This can be calculated with the formula
∑n =
n=0
N(N + 1)
.
2
Using the fact that Rp = 355, this second formula for track length also
ends up with Track Length ≈ Ap .
1
Notice that the expression ( n + 2 ) used above could reasonably be
replaced by just n or else ( n + 1) depending upon where the first of the
concentric rings is located.
The values obtained by these two modifications are, respectively,
(Ap + 35.5π × 10 −3 ) meters or (Ap − 35.5π × 10 −3 ) meters. The difference
between the two estimates and Ap is 35.5π ≈ 111.5 mm ≈ 4.4 inches.
Method C: A third method for approximating the track length was pointed out to the authors by David Carson. This method is based on the
observation that the total track length Lp times the pitch p will be approximately equal to the total area of the annular data region.
Annular Area = π (582 − 22.52 ) = 2 π
35.5
58 + 22.5
(58 − 22.5) = 2 π (22.5 +
)35.5.
2
2
From this and the above observation we get (using
Track Length ≈
Area
= Ap .
p
35.5
= R):
p
So this method gives the same value as the first two methods.
REQUIREMENT 1C: WHY ARE THE APPROXIMATIONS SO CLOSE?
It is curious that the approximation Ap is remarkably close to the
exact value. For instance, considering the standard red laser CD where
p = 1.6 x 10-3mm, the value of Ap is 5,611,179mm and the value from
the exact formula for Lp above is 5,611,299.5mm where both values are
calculated using Maple. So the difference is about 120 millimeters in 5
million millimeters (less than 5 inches out of 3.5 miles). The values are
even more accurate for the other values of p.
The accuracy of the estimates was surprising to the authors. Two explanations are given to see more clearly why this accuracy is to be expected—
one explanation involves estimating the arc length integral and the other
is by a simple analogy.
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Interdisciplinary Lively Applications Project
The first approach to investigating this further is to use an alternate
method to estimate the integral for the exact value obtained above:
58
Lp =
2π
p
u 2 + ( )2 du
2π
p 22.4
∫
We will use the substitution a =
p
2π
1
2
1
8
and the binomial expansion 1 + x = 1 + x − x 2 + h(x)
where h(x) will have higher powers of x and is alternating.
2π
p
1 2
1
1a 1  a 
a
u 2 + ( )2 =
u + a2 = u +
−
+ g( )
2π
a
a
2u 8  u 
u
p
3
This gives
where g(x) is a series whose lowest power is 5.
58
Notice that
1
udu = Ap ,
a
22.5
∫
58
1
58 
1a
du =  ln
a ≤ a, and

2
22.5 
2u
22.5
∫
58
3
a3  1
1 
1  a
du
=
−
≤ a3 .
∫22.5 8  u 
16  22.5 2 58 2 
Since the value of a =
p
≤ 10 −3 mm for the largest value
2π
of p (which is 0.0016), and especially since the additional terms are alternating, we expect that the exact value will not be far from Ap.
One final observation may give some further insight by analogy into why
the approximation is so close to the actual value. The approximations are
based on modeling the spiral as rings, i.e., the “slope” or pitch of the
track is ignored. The track (in the shortest case) is about 3.5 miles or
5,611,179 mm long. As the laser beam tracks this length the beam travels about 35.5mm from the inner to the outer tracks. By analogy we are
comparing the long side of a right triangle to the hypotenuse where the
long side is 5,611,179 and the other leg is 35.5. Note that this gives for
the length h of the hypotenuse h = 5,611,1792 + 35.52 = 5,611,179.00011229 .
Thus, since the slope of the track on a CD is so gradual, the diagonal and
“flat” lengths are virtually the same.
The reader may be familiar with the old problem which goes something
like this: “if a string is tied around the equator of the earth (a perfect
sphere), how much slack must be added so that an elephant can walk
under the string?” Only a fraction of an inch of slack is needed. Another
version of the problem is “if a mile-long piece of railroad track (attached
at both ends to a perfectly flat track bed) expands by one inch due to the
heat of the sun, can a horse walk under the track in the middle (assuming the new shape is an isosceles triangle with base 1 mile and with the
other two sides having length 1 mile and 1 inch)? In fact, the height of
the track in the middle is over 200 feet. Our problem here, in certain
respects, is like this: “if a wire is 5,611.179 mm long and one end is
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21
Red & Blue Laser CDs: How Much Data Can They Hold?
raised vertically by 35.5 mm, how much is the wire stretched?”
REQUIREMENT 2: OPTIMIZING DATA AREA FOR CAV DISCS
For a CAV disc, the optimal data area is an annular area with inner radius
one-half the outer radius. This can be verified by checking the data area
of a floppy disc. A mathematical argument goes as follows.
Let the variable r represent the inner radius of the annular data area.
Then the amount of data on the inside track would be 2 πr × k where k is
the linear data density (number of bytes of data per unit length). If the
D–r
outer radius of the disc is D, then the number of tracks is given by
p
where p is the track pitch (distance between tracks). Since the amount of
data on all tracks is the same, the total data capacity C(r)
is given by C(r) = 2 πr × k ×
D−r
= Kr(D − r).
p
This is quadratic in r. Either by drawing the graph and using properties
of parabolas or by setting the derivative to zero we can see that the
D
maximum of C( r ) will occur when r = as was claimed.
2
The effective track length for a CAV disc is simply the number of rings or
cycles times the circumference of the innermost ring:
Effect track length for CAV disc = R × 2 π × 29 =
29
1682 π
× 2 π × 29 =
p
p
The results for effective track lengths are given in the table below.
Laser Color
nm
Pitch p
micron
Rings
number
Effective Length
mm
Length
Miles
infrared (780)
1.60
18,125
3,302,599
2.05
red (640)
0.74
39,189
7,140,755
4.44
blue (410)
0.32
90,625
16,512,996
10.26
REQUIREMENT 3: DATA CAPACITY IN TERMS OF MINUTES OF VIDEO AND DATA
RECORDS
Knowing the track length (or effective track length in the case of
CAV discs) we can find the data capacity in megabytes by the simple
relationship
Data capacity (megabytes) = track length x linear data density
The capacity in minutes of video is the capacity in megabytes divided by
(0.62 megabytes/second x 60 seconds) = 37.2. The capacity in millions
of records (mega-records) is the capacity in megabytes divided by 50.
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Table 4:
EFFECTIVE TRACK
LENGTH FOR A CAV
DISC
22
Interdisciplinary Lively Applications Project
Final results for this report are summarized in the following two tables.
Laser Color
nm
Beam
micron
Pitch
micron
Rings
number
Length
mm
Length
Miles
Density
bytes/mm
Capacity
megabyte min video meg-records
infrared (780)
2.000
1.60
22,188
5,611,179
3.49
121
679.0
18
13.6
red (640)
0.925
0.74
47,973
12,132,279
7.54
387
4695.2
126
93.9
blue (410)
0.400
0.32
110,938
28,055,895 17.44
800
22444.7
603
448.9
Table 5:
CALCULATIONS FOR A CLV DISC
Laser Color
nm
Beam
micron
Pitch
micron
Rings
number
Length
mm
Length
Miles
Density
bytes/mm
Capacity
megabyte min video meg-records
infrared (780)
2.000
1.60
18,125
3,302,599
2.05
121
399.6
11
8.0
red (640)
0.925
0.74
39,189
7,140,755
4.44
387
2763.5
74
55.3
blue (410)
0.400
0.32
90,625
16,512,996 10.26
800
13210.4
355
264.2
Table 6 :
CALCULATIONS FOR A CAV DISC
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