Guide to Program
TEMPROFILE (5.3)

Temperature Profiling Simulation Software
from
MM Research, Inc.

February 2005

Copyright © 1981–2005 by MM Research, Inc. All rights reserved.

Guide to Program
TEMPROFILE (5.3)

For Windows 95, 98, 2000, NT, and XP Operating Systems

Information in this document is subject to change without notice and does not represent a commitment on
the part of MM Research, Inc.. The software described in this document is furnished under a license
agreement or nondisclosure agreement. The software may be used or copied only in accordance with theterms of the agreement. It is against the law to copy the software on any medium except as specificallyallowed in the license or nondisclosure agreement. The purchaser may make one copy of the software forbackup purposes. No part of this manual may be reproduced or transmitted in any form or by any means,electronic or mechanical, including photocopying, recording, or information recording and retrieval
systems, for any purpose other than the purchaser’s personal use, without the express written permission of
MM Research, Inc.

© 1981-2005 by MM Research, Inc.

All rights reserved.

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ii

Contents
Hard_key Installation and Setting up TEMPROFILE on the Hard Disk.....................................iv
1. Introduction.............................................................................................................................. 1
1.1. The Working Subdirectory................................................................................................ 1
1.2. Synopsis ............................................................................................................................ 2
1.3. Notation and the System of Units ..................................................................................... 2
2. Specifying the Multilayer Structure......................................................................................... 3
2.1. The Mesh Parameters........................................................................................................ 5
2.2. The System File ................................................................................................................ 6
3. Specifying the Laser Beam Profile, Pulse Shape, and Velocity .............................................. 7
3.1. Gaussian Beam.................................................................................................................. 7
3.2. User-defined Beam Intensity Profile................................................................................. 8
3.3. Laser Pulse Shape ............................................................................................................. 93.4. Velocity of the Beam ........................................................................................................11
4. Functional Units of TEMPROFILE .........................................................................................134.1. Computing Mark Size .......................................................................................................144.2. The PLOT Option .............................................................................................................14
5. Input and Output ......................................................................................................................15
6. Interactive and Non-interactive Sessions..................................................................................176.1. Structure of the COMMAND.DAT File ...............................................................................176.2. The $ Commands...............................................................................................................216.3. Running TEMPROFILE in the Non-interactive Mode .......................................................27
7. Test Run 1 ................................................................................................................................28
8. Test Run 2 ................................................................................................................................36
9. Test Run 3 .................................................................................................................................41
10. Test Run 4 ...............................................................................................................................47
References ................................................................................................................................51
Appendix A: Reprint of “Laser-induced Local Heating of Multilayers” ................................53
Appendix B: Reprint of “Laser-induced Local Heating of Moving Multilayer Media” .........65
Appendix C: Reprint of “Thermal Aspects of Magneto-optical Recording”...........................73iii

Hard_key Installation and Setting up TEMPROFILE on the Hard Disk
Your TEMPROFILE package contains an installation CD-ROM and a Rainbow
Sentinel SuperPro™ hard_key, together with the necessary software driverto operate the hard_key. The hard_key itself must be inserted in a USB port
or securely fastened to a parallel port on the back of your computer.
(Typically, this is the port that connects to a printer.) To activate thehard_key on your system, the INSTALL.EXE routine transfers the necessary
software driver(s) to your hard disk, then runs SETUP.EXE from the<RAINBOW> subdirectory located within TEMPROFILE’s home directory.
A USB hard_key should be inserted into an available USB port. To install aparallel port hard_key you must disconnect the printer cable (if any) fromthe back of your computer, mount the hard_key directly on the parallel port,tightly fasten the screws, then re-attach the printer cable (if any) to the backof the hard_key. To run TEMPROFILE under Windows® 95, 98, 2000, NT, or
XP, you must first install the program on your hard disk. This is done byfollowing four simple steps:
1. Insert the distribution CD in a CD-ROM drive, then switch to the CD’s
<Temprofile> directory.
2. Double-click on INSTALL.EXE. The installation routine prompts you forthe name of a home directory on the hard disk (the default is
C:\TEMPROFILE). All the files and directories on the CD-ROM are
subsequently transferred to the specified home directory, which will
eventually contain the subdirectories <PULSES>, <RIPS>, <RAINBOW>,<SYSTEMS>, and the files TEMPROFILE.EXE, README.DOC, and
LOGO.BMP.
3. The installation routine automatically installs the hard_key driver byrunning SETUP.EXE from the <RAINBOW> subdirectory. At this point you
may have to restart the computer to activate the hard_key driver.
4. Once the program and its hard_key are properly installed, switch to
TEMPROFILE’s home directory, then double-click on the program’s icon.
This is all you need to do to get the program running.
TEMPROFILE opens three windows on the monitor: The Dialogue Window

is where the program asks questions and expects to receive your answers.

iv

Help is automatically displayed in the Guiding Information window; thiswindow stays in the background and you must switch to it manually if you
need to look up its contents. The Graphics Window becomes active when
you invoke the plotting routine. Use the mouse-click or press Ctrl + Tab toswitch from one window to another.
Blank Windows. In some cases, when TEMPROFILE™ is installed underWindows 95, 98, 2000, NT, or XP, the Graphics and Guiding Information
Windows appear blank. This is caused by a setting of the desktop colors. To
solve the problem, follow these instructions: Click on the Start button. Clickon Settings, then on Control Panel. Double-click the Display icon, which
will open the Display Properties window, then click on Settings. There,change the setting from True Color to something else, such as High Color 16
bits. (Any setting other than True Color should work.) Then click the Apply
button, which will reset your computer. You should be able to use theGraphics and Guiding Information Windows now with the new settings.
Problems with Hard_key: The driver software is automatically loaded andactivated by the installation routine. However, in case of difficulty with the
hard_key, try the following steps:
1. Insert the installation disk in the CD-ROM drive. Switch to this drive andopen the <RAINBOW> directory. Double click on SETUP.EXE and wait asecond or two for the setup procedure to be completed. (Alternatively,
you may transfer the <RAINBOW> directory to your hard drive, then run
SETUP.EXE from the hard drive. In this way, in the unlikely event thatyou will need to reinstall the hard_key driver in the future, the driver
software will already reside on the hard disk, and the installation CD-ROM
will no longer be needed.)
2. Remove the Installation CD-ROM from its drive and keep it in a safe
place. You might need this disk again in the future to reinstall the
program in case of a system failure, or for installing the program on adifferent computer.
3. Restart your computer to activate the hard_key driver.
The hard_key is now activated on your system, and you may proceed to run
your protected application. If the difficulties persist, please read Readme.doc
residing within the <RAINBOW> subdirectory, or contact MM Research, Inc.
v

Contents of TEMPROFILE’s Subdirectories: The subdirectories <PULSES>
and <RIPS> contain examples of pulse shapes and radial intensity profiles ofthe laser beam. The user may construct his own laser pulses and/or intensity
profiles using these as templates. The <SYSTEMS> subdirectory contains
several complete systems (i.e., multilayer stack + laser beam profile + meshcharacteristics) created by TEMPROFILE; these can readily be imported by
the program during practice runs. Also, each time you run TEMPROFILE you
can create your own system file and save it to the <SYSTEMS> subdirectory.
Graphic Capabilities: TEMPROFILE’s graphic routine generates simple,
one-dimensional plots and saves them in the working directory as bitmap
files. These files can be read by Windows® utility programs, such as Word®
and Paint®, modified when necessary, and incorporated into documents. For
high-quality graphics, you may port the ASCII data-files created by
TEMPROFILE to other commercially available software. These data-files arecompatible with most graphics packages such as Matlab®, Origin®,
Corel Draw®, and Mathematica®.

viGuide to Program TEMPROFILE

1

1. Introduction: TEMPROFILE is a Fortran-based program for calculating
temperature profiles in multilayer stacks of dielectrics and metals, subject toillumination by a moving laser beam. Details of the analytical and numerical
techniques used in computing light absorption and heat diffusion in multilayers aswell as several application examples may be found in the references cited at the endof this manual. References 2 and 3, in particular, give a detailed description of thetheory behind the computations; for ease of reference, reprints of these articles areincluded in this manual as appendices A and B.
To launch the program from its home directory, double-click the mouse on the
program’s icon, Temprofile.exe, then answer all the questions in the Dialogue
Window on your terminal. The questions asked by TEMPROFILE are straightforwardand easy to answer. In case of difficulty, however, you may consult theaccompanying examples and Test_Runs for guidance.
The first message that you receive on your monitor is the following:
Units: Length = nanometer, Time = nanosecond, Power = milliwatt
In like manner, the program will continue to give you instructions as to what you
should be doing in each step, and asks for the parameter values needed in each stageof the calculations. The questions posed by TEMPROFILE appear in the Dialogue
Window, and the relevant information about these questions appears automaticallyin the Guiding Information window.
The data generated during a run will be saved to data-files within the working
subdirectory. Some of these files are simple text files that contain, for instance, arecord of the session, or descriptions of the multilayer stack, the laser beam, themesh, etc. Other files contain data for plotting curves. When these data-files are
given to the resident plotting routine within TEMPROFILE, the plots appear in theGraphics Window. Alternatively, one may hand these files over to other graphics
packages (e.g., Origin®, Matlab®, Mathematica®) for more elaborate renderings.
1.1. The Working Subdirectory. All data-files created during a session will berecorded within a working subdirectory, and all data-files needed by the program
must reside in the same subdirectory. The user is prompted at the outset to specify
the name of this working subdirectory. If the name is new, the program creates the2Guide to Program TEMPROFILE

subdirectory and makes it home directory for the entire session. If, on the other
hand, the name belongs to an existing subdirectory, the user is warned that some of
the resident files within that subdirectory might be subsequently renamed. The user
can then accept this option or specify a different working subdirectory.
1.2. Synopsis. TEMPROFILE calculates the distribution of temperature in a
multilayer stack illuminated by a normally-incident, monochromatic beam of light.
The stack (deposited on a flat substrate) may consist of any number of metal anddielectric layers up to MAXLAYER = 99. The incident beam, which can illuminate thestack either from the top surface or through the substrate, must have circularsymmetry but is otherwise arbitrary. Due to the linearity of the problem, theprogram first computes the response to a rectangular "impulse" having a fixed
POWER and a short duration τ. This "impulse-response" is then stored in a disk file,which may be retrieved subsequently for calculating thermal profiles bysuperposition.
The Compute Impulse Response option of the main menu allows the user tospecify the multilayer stack, mode of illumination, distribution of light intensity, andthe required mesh parameters. In the Generate Temperature Profile mode, adata-file containing a previously computed impulse-response is read into theprogram. The program then prompts the user for the temporal profile P(t) of the lightpulse, and the velocity V of the beam relative to the multilayer stack. At this point
the user may request one of several possible temperature profiles, namely, T(x),T(y), T(z), T(t), as well as various isotherms at fixed z.† The specified profiles are
saved to the hard disk within the working subdirectory, and the user is returned tothe main menu, where he can invoke the PLOT option to view the computed profiles.
1.3. Notation and the System of Units. The physical quantities encountered in
TEMPROFILE, their respective symbols, and their corresponding dimensional units
are listed in Table 1. The refractive index and the absorption coefficient are usuallycombined together to form the complex refractive index (n, k). In the notation usedin TEMPROFILE, the value of the absorption coefficient k must always be greaterthan or equal to zero (k ≥ 0). Each layer has a specific heat C and a thermal
conductivity K. The ratio K/C is known as the thermal diffusivity D. The parameterγ denotes the effective heat-loss-factor from the top surface of the stack.
†The stack’s layers are parallel to the XY-plane, the z-axis is the direction of the surface-normal
along which the beam propagates, t is the time, and T stands for temperature above ambient.

Guide to Program TEMPROFILE

3

QuantityTime
LengthPower
Temperature
Refractive index
Absorption coefficientSpecific heat
Thermal conductivity
Thermal diffusivity
Heat loss factor

Symboltr, x, y, z
P
T
n
k
C
K
D
γ

Units
nanosecond
nanometer
milliwatt
degrees Celsius
dimensionlessdimensionless
Joule/cm3/ºC
Joule/cm/sec/ºC
cm2/sec
1/cm

Table 1. Units of the physical quantities

2. Specifying the Multilayer Structure: Figure 1 shows the multilayer structure
within the Cartesian coordinate system XYZ. The multilayer stack consists of anarbitrary number of layers deposited on a flat substrate. (The current version of the
program is set up for a maximum of 99 layers, not counting the substrate). The
surface and all layer interfaces are flat and parallel to the XY-plane.
The light beam, which propagates along the z-axis, may be incident from the surface
side (TOP) or from the substrate side (BOTTOM). The substrate is layer 0, the layer
directly over the substrate is layer 1, and so on. Each layer is identified by its
thickness z (nm), complex refractive index (n, k), specific heat C (Joule/cm3/°C),
and thermal conductivity K (Joule/cm/sec/°C). The semi-infinite substrate issimilarly specified by its optical and thermal constants, although, for throughsubstrate incidence, the substrate’s absorption coefficient k must necessarily be
zero. The user is also prompted for a surface heat-loss-factor γ (1/cm), which is a
non-negative parameter. No heat escapes from the top of the stack when γ = 0. (Seeappendix A for a precise definition of the heat-loss-factor γ.)
TEMPROFILE assumes that the profile of the incident beam within the XY-plane iscircularly symmetric, and that the various layers of the stack are isotropic (bothoptically and thermally). The natural coordinate system for this problem, therefore,
is a cylindrical system in which Z is the axis of symmetry as well as the optical axisof the light beam (i.e., the axis along which the beam propagates), R is the radial
distance from Z, and φ is the azimuthal angle in the XY-plane of the stack.

4

Guide to Program TEMPROFILE

Figure 1. Multilayer stack consisting of N layers, coated on a semi-infinite substrate. The surfaceof the last layer is at Z = 0, while the interface between the substrate and layer 1 is at Z = z1 + z2 +··· + zN , where zi is the thickness of the i th layer. The incident beam has circular symmetry, andmoves with constant velocity V along the positive x-axis. The program allows the user to choose
the direction of incidence; thus the beam may enter through the substrate, or be incident on the
top surface of the multilayer stack, as indicated.

Guide to Program TEMPROFILE

5

2.1. The Mesh Parameters. The solution to the problem of light absorption andheat diffusion in a multilayer stack as addressed by TEMPROFILE takes advantage ofthe aforementioned circular symmetry. Circular symmetry enables analytical
integration of the relevant equations over φ, obviating thereby the need for numericalintegrations along this dimension. The discrete mesh used in these computations,
therefore, has only three dimensions along the radial (r), vertical (z), and temporal(t) directions; see Figure 2. Eliminating the azimuthal coordinate φ from theequations leads to substantial savings in the computation time.
In the radial direction the mesh size is Rmax and the step size is ∆R, leading to thenumber of pixels Imax = Rmax /∆R along this dimension. Similarly, in the verticaldirection the mesh size is Zmax and the step size is ∆Z, yielding Jmax = Zmax /∆Z pixels
in the Z direction. In the time domain the mesh size is tmax and the step size is τ, thechosen duration of the impulse (see section 3.3 for a description of τ). This leads tothe number of pixels Kmax = tmax /τ along the time axis. Upper bounds on the allowed
number of pixels are presently set as follows: (Imax, Jmax, Kmax) ≤ (2000, 2000, 2000).

Figure 2. The discrete mesh used in
TEMPROFILE calculations has three
dimensions along the vertical, radial, andtemporal directions. Since the multilayer isisotropic, and since the incident beam has
circular symmetry, the required spatial
coordinates are R and Z only. The mesh has
Imax cells in the R direction and Jmax cells in
the Z direction; the physical extent of themesh in these directions is Rmax and Zmax,
respectively. Time is discretized in steps of τ,with the number of cells along the temporalaxis being Kmax; the longest time for which
computations are performed is tmax.

Rmax and Zmax must be large enough to ensure negligible temperature rise at the meshboundaries for the time period of interest (i.e., 0 < t < tmax). Moreover,• ∆R must be small compared to the length-scale of variation of the beam's
intensity profile I(r); it must certainly be much less than the beam’s effective
radius r0 (see sections 3.1 and 3.2 for a definition of r0).

6

Guide to Program TEMPROFILE• ∆Z must be smaller than the thinnest layer of the stack. Also, to guarantee theexistence of a mesh point at each interface between adjacent layers, ∆Z must fit
an integer number of times into each layer’s thickness.
2.2. The System File. The structural parameters of the multilayer stack, theintensity profile of the incident beam, and the mesh parameters are initially enteredfrom the keyboard, and used in computing the impulse-response. However, theseparameters, collectively referred to as the system parameters, may be saved in adata-file within the working subdirectory, and imported to TEMPROFILE during
subsequent runs. Here is a message that informs the user of available options:
You may enter the system parameters directly from the keyboard or,alternatively, import them from a file such as XXXX.SYS, which must residewithin the working directory. Press RETURN for the keyboard option, or
else, enter the desired filename.
If you choose the keyboard option, you will be given the opportunity to save thedata in a file for future use. Here is the corresponding message:
You have the option of saving the system parameters (which you are about
to enter) in a file such as XXXX.SYS residing in the working directory. To
refuse this option press RETURN, otherwise enter the desired filename.
Note: Even if the option of saving the data is refused, TEMPROFILE automaticallysaves the data in the temporary file JUNK.SYS. Since this file may be overwritten
during subsequent runs without warning, if you intend to keep the file you should
consider renaming it or moving it to a different directory prior to starting a new run.

Guide to Program TEMPROFILE

7

3. Specifying the Laser Beam Profile, Pulse Shape, and Velocity. Incident on thestack (either from the TOP or from the BOTTOM) is a circularly symmetric,
monochromatic beam of light having vacuum wavelength λo. The radial profile ofthe beam's intensity distribution is allowed to be either Gaussian or user-defined. In
the first instance the user need only specify the 1/e radius of the Gaussian beam. In
the second case the entire intensity profile must be entered, point by point, either
from the keyboard or from a data-file residing within the working subdirectory.
The beam incident on the multilayer stack has a fixed spatial profile, but its power P(i.e., integrated intensity over the beam's cross-section in the XY-plane) can varywith time t. The beam is also allowed to move with constant velocity V over thestationary stack. Here it is generally assumed that the incident optical power is zeroprior to t = 0, the beam center coordinates at t = 0 are (x, y) = (0, 0), and the beam
moves along the positive x-axis.
3.1. Gaussian Beam. The circularly symmetric Gaussian beam has radius r0 at the
point where its intensity drops to 1/e of its peak value, that is, the beam’s intensity
profile in the XY-plane is proportional to exp [ – (r/r0)²] as shown in Figure 3(a). The
beam may be incident from the substrate side (BOTTOM), or it may illuminate thestructure from the surface side (TOP), as indicated in Figure 1. In either case thebeam diameter is assumed to be constant through the thickness of the multilayer
stack.‡
Being at the waist of a Gaussian beam, the incident beam has a uniform phase (i.e.,
flat wavefront), and remains more-or-less collimated over a distance of ± r02/λ0
along the z-axis. The 1/e amplitude radius of the beam is given by √2 r0 ≅ 1.4142 r0,
and its full-width at half-maximum intensity (FWHM) is given by (2√ln2) r0≅ 1.6651 r0. For accurate calculation of optical absorption within the stack, the total
multilayer thickness should not exceed the collimation range of the beam.
‡

Suppose the waist of the Gaussian beam is at the mid-point of the stack, that is, at
Z = ½ (z1 + z2 + ··· + zN). Then the range of values of Z within which the diameter of the beamremains within 10% of its waist diameter will be ≅ ± 1.44 r02/λ0. Thus, if the total thickness of thestack is less than this value, the approximation will be acceptable. Also note that TEMPROFILE
treats the incident beam as a single plane wave that is normally-incident on the stack, thus
ignoring its spatial frequency content. A more accurate method would have considered theoblique plane wave components of the incident beam, and would have calculated light absorption
within the multilayer for all such plane waves. The present approximation, however, is justifiable
for most problems of practical interest.

8

Guide to Program TEMPROFILE

Figure 3. (a) Spatial intensity distribution of the Gaussian laser beam throughout the multilayer
stack. This is a circularly symmetric Gaussian profile, having radius r0 at the 1/e (intensity) point.
(b) A typical time-dependence of the power of the laser beam, P(t). This piecewise linear function
is identified by the coordinates of its corner points, (ti, Pi).

3.2. User-defined Beam Intensity Profile. In this case the beam incident on themultilayer stack has a circularly symmetric profile defined by the radial intensity
distribution function I(r). The beam propagates along the z-axis, which is also its
axis of symmetry; its wavelength is λ0 and its maximum radius (as measured fromthe z-axis) is denoted by rs. The user is prompted for pairs of points (r, I) that, taken
together, define the function I(r). The first point must be at the beam center, r = 0,
and subsequent points must be entered in order of increasing r.
The radial profile is continuous and piecewise linear, starting at r = 0 and ending at
r = rs. (The maximum allowed number of points is presently set to 1000.) The
program continues to ask for input until r becomes equal to rs. It then computes aneffective radius r0 and the total power P0 corresponding to I(r). The effective radius
is similar to a Gaussian beam's radius at the 1/e intensity point. The intensity profile
will be automatically normalized by P0 to fix the total power of the beam at unity.

Guide to Program TEMPROFILE9

The radial intensity profile I(r) may be entered from the keyboard or imported froma data-file. The user will have to respond to the following query:
You may enter the radial intensity profile directly from the keyboard or,alternatively, import it from a file such as XXXX.RIP, which must reside within
the working directory. Press RETURN for the keyboard option, or else, enter
the desired filename.
When importing I(r) from the XXXX.RIP file, there is no need for specifying rs inadvance; the program stops accepting input when it reaches the end of the file.
If you choose the keyboard option, you will be given the opportunity to save thedata in a file for future use. Here is the corresponding message:
You now have the option to save the radial intensity profile (which you areabout to enter) in a file such as XXXX.RIP within the working directory. To
refuse this option press RETURN, otherwise enter the desired filename.
Note: Even if the option of saving the data is refused, TEMPROFILE automaticallysaves the data to the temporary file JUNK.RIP. Since this file may be overwritten
during subsequent runs without warning, if you intend to keep the file you should
consider renaming it or moving it to a different directory prior to starting a new run.
3.3. Laser Pulse Shape. The power of the laser is chosen by the user and can be an
arbitrary function of time, P( t ), as shown in Figure 3(b). The user has the option ofentering the laser pulse shape from the keyboard, or importing it from a file. The
following message provides the user with this option:
You may enter the laser pulse shape directly from the keyboard or,alternatively, import it from a file such as XXXX.PLS, which must residewithin the working directory. Press RETURN for the keyboard option, or
else, enter the desired filename.
If the keyboard option is chosen, the user will be given the opportunity to create adata-file for use in future runs. Here is the corresponding message:

10

Guide to Program TEMPROFILE

You now have the option to save the laser pulse shape (which you are about
to enter) in a file such as XXXX.PLS within the working directory. To refuse
this option press RETURN, otherwise enter the desired filename.
Note: Even if the option of saving the data is refused, TEMPROFILE automaticallysaves the data in the temporary file JUNK.PLS. Since this file may be overwritten
during subsequent runs without warning, if you intend to keep the file you should
consider renaming it or moving it to a different directory prior to starting a new run.
TEMPROFILE computes the temperature distribution within the multilayer stack in

two steps. In the first step the distribution resulting from a short, rectangular pulse(ideally an impulse) is computed and saved to a data-file, XXXX.IMP. This file willthus contain the “impulse-response” of the system. In the second step, the laser
pulse shape P(t) is considered to be a superposition of these short rectangularpulses, and the desired temperature profile is obtained by properly shifting andsuperposing the impulse-response function.
The user specifies the laser impulse by identifying its magnitude (POWER)§ andduration (τ), as shown in Figure 4. The duration τ must be short compared to thetime-scale of variation of the optical power P(t ) incident on the stack. The program
allows the user to break up τ into NSTEP sub-intervals, where NSTEP is a userdefined, integer-valued parameter. The discrete time-interval over which the
diffusion equation is numerically integrated is ∆t = τ /NSTEP. This integration stepsize must be short enough to ensure that changes in temperature during each such
step are relatively small. In other words, ∆t must be shorter than the characteristic
diffusion time of the most conductive layer of the stack. Although the heatabsorption and diffusion equations are solved in steps of ∆t = τ /NSTEP, the impulseresponse function is saved only for those instants of time that are integer-multiplesof τ. This feature of TEMPROFILE improves the numerical algorithm’s accuracy
without requiring an excessive amount of disk space for storing the computed
impulse-response function.
§

For all practical purposes POWER should be set to unity, although, when numerical accuracy
demands, one might consider using a different POWER for the impulse-response computation. To
ensure consistency, the program automatically scales the incident optical power by the magnitudeof this impulse.

Guide to Program TEMPROFILE

11

Figure 4. A narrow, rectangular pulse of laser light may be used in the computer simulations torepresent an impulse. This “impulse” has a duration of τ and a constant power level of POWER. The
width τ must be short as compared with the time scale over which P(t) has substantial variations.
POWER, however, is arbitrary, and may be fixed at any reasonable value (say, 1). Discrete
integration of the heat diffusion equation requires time steps ∆t that are small enough, such thatthe largest diffusion distance during a ∆t time interval would remain below the size of the cells of
the mesh in r-z space. For this reason, the user is given the option to divide τ in NSTEP equalintervals, and integrate the equations in time steps of ∆t = τ /NSTEP.

The laser pulse shape P(t) is continuous and piecewise linear, starting at the point(t = 0, P = 0) and ending at tmax = Kmax τ. This function is specified by entering pairsof numbers (t, P) corresponding to the corner points of P(t). Each time you enter thecoordinates of a corner point, you will be prompted for the next one. The process
ends when the entered value of t exceeds (or equals) tmax.
The corner points must be entered sequentially, starting at (t = 0, P = 0).** A corner
point entered in violation of the sequential order will be ignored. The laser pulse
shape in Figure 5, for example, may be specified by the corner points listed in Table
2. Note in particular that the point (0, 0) at the origin is missing from this Table(although its presence would have caused no problems either). Also, the last point,(100, 0), is necessary because it tells the program that between the end of the pulse(t = 18 nsec) and tmax (assumed here to be ≤ 100 nsec) P(t) is zero.
**The point at the origin, namely, (t = 0, P = 0), is automatically assumed to exist, even when it is
not explicitly specified.

12

Guide to Program TEMPROFILETime (nsec)
2
5
5
7
7
9
9
11
11
13
13
16
18
100

Figure 5. A typical laser pulse P(t). The corner
points of this pulse are listed in Table 2.

Power (mW)
5
5
10
10
5
5
10
10
5
5
3
3
0
0

Table 2. Corner coordinates ofthe laser pulse shown in Fig. 5.

3.4. Velocity of the Beam. The beam in TEMPROFILE is allowed to move with aconstant velocity V over the stationary multilayer stack. In general, the incident
optical power P(t) equals zero when t < 0, the beam center coordinates at t = 0 are(x, y) = (0, 0), and the beam moves along the positive x-axis at constant velocity.
For a given beam velocity V and beam radius r0, the mesh radius Rmax must be
chosen such that the beam does not leave the mesh (or even get close to its borders)
at t = tmax. Initially, of course, the beam is at the center of the mesh, but as time goeson, it gets closer to the mesh boundary. As a rule of thumb, the system parametersmust satisfy the relation Rmax2 > (V tmax + 3r0)2 + (3r0)2. If this rule is violated, theprogram issues the following warning:
Warning: Velocity too large for the current mesh size. Temperature
values at large (x, y) may be slightly in error.

Guide to Program TEMPROFILE13

4. Functional Units of TEMPROFILE: The program has three functional units, as
may be inferred by noting the available options in the program’s main menu:
Compute Impulse Response (COMP)
Generate Temperature Profile (TEMP)Plot Existing Data-files(PLOT)
Quit the Program(QUIT)
Upon choosing the first option, Compute Impulse Response, the thermal responseof a multilayer stack to a short, rectangular laser impulse is calculated. This impulseresponse is then saved to a binary data-file, XXXX.IMP, whose four-letter name XXXX
is user-selectable. To enable the program to compute the impulse-response, the user
must answer questions about the structure of the multilayer, its optical and thermal
constants, the incident beam profile, and the characteristics of the mesh.
The second option, Generate Temperature Profile, can be invoked only when
an impulse-response data-file happens to reside in the working subdirectory. In thiscase the user is prompted to provide the name of the impulse-response file, and to
specify the laser pulse shape P( t ) and the beam velocity V, as well as the parametersof the desired temperature profile(s). The program calculates the requested
temperature profile(s) by properly shifting and superimposing the impulse-response
function. As for the desired profiles, the available options are:Determine mark size (fixed z, temperature)
Isotherms in the XY-plane (fixed z, time)
Temperature versus x (fixed y, z, time)
Temperature versus y (fixed x, z, time)
Temperature versus z (fixed x, y, time)
Temperature versus time (fixed x, y, z)(MARK)(ISOS)(TVSX)(TVSY)(TVSZ)(TVST)The user may request the dimensions of a certain type of thermal mark (described insubsection 4.1 below), ask for one-dimensional profiles of temperature versus
x, y, z, or time, or request two-dimensional isotherms in planes parallel to XY.
Having chosen the type of curve to compute, the user must then answer additionalquestions regarding the parameter values for each curve. Attention must be paid at
this point to the filenames assigned to each profile; these ASCII files, which are
created in pairs, contain the horizontal and vertical coordinates of individual

14

Guide to Program TEMPROFILE

profiles. For example, if temperature-versus-time profiles are requested, the program
creates the files Tnn.DAT and TTnn.DAT, where nn are 2-digit numbers ranging from
01 to 99. Similarly, if isotherms are requested, the corresponding data-files will be
XInn.DAT and YInn.DAT. When finished, the PLOT option may be invoked to view
these profiles.
4.1. Computing Mark Size. A MARK is defined as the region in a given XY-planethat reaches above a certain threshold temperature, Tthreshold, under the influence ofthe laser beam. Consider the XY-plane located at a given distance z below themultilayer’s surface. This region of the stack may have the property that, when its
temperature reaches the critical value of Tthreshold, it undergoes a physicaltransformation. The region thus modified by the thermal process will be referred toas a MARK.
For the entire duration of the laser pulse (0 : tmax), at intervals of τ, TEMPROFILE
computes the isotherms at Tthreshold within the specified XY-plane. By definition, eachmark covers the (overlapping) areas within a number of successive isotherms. The
program then identifies the minimum and maximum values of x and y that boundsuch isotherms and, consequently, represent the boundaries of a single mark. If thelaser beam moves along the x-axis at the constant velocity V, it will be possible topulse the laser at desired intervals to create multiple marks in the x direction (at thegiven depth z of the stack).
TEMPROFILE proceeds to display the length (along x), width (along y), and center

coordinates of each and every mark thus formed by the laser beam during the timeinterval (0 : tmax). Also computed and displayed are the onset and freeze times, atwhich the rising and falling peak temperature within the specified XY-plane crosses
Tthreshold. At the end of these computations, TEMPROFILE creates (and saves to theworking directory) the data-files TIMEnn.DAT, TPEAKnn.DAT, and XPEAKnn.DAT,
which can be used to plot the time-dependence of the peak_temperature as well asthat of the peak_temperature’s x-coordinate. As usual, the index nn of these datafiles ranges from 01 to 99, with the value of nn being automatically incremented
each time the MARK option is selected.
4.2. The PLOT Option. The third functional unit of TEMPROFILE, the graphic
routine, enables the user to plot one or more curves on a set of XY coordinates
within a square frame. The input data-files are ASCII files XFILE.DAT and YFILE.DAT;
each filename may consist of up to 10 characters. The first plot is automaticallyscaled, but the user gets a chance to rescale the frame. (Only those data points thatfall within the frame will be displayed.) Once the user is satisfied with the scale ofGuide to Program TEMPROFILE

15

the frame, he will be given the option of overlaying more curves on the same plot.
For each new curve the user must specify the filenames XFILE.DAT, YFILE.DAT,
which contain the X- and Y-coordinates of the curve. There are no limits on thenumber of curves that can be simultaneously displayed, but the scale of the frame,
once set for the first curve, cannot be changed. The user selects the line-style andcolor of each curve by specifying a four-digit number SRGB in the range (0000 : 9999).
S is departure from solidity; thus S = 0 for solid, S = 1 for long dash, S = 2 forshorter dash, and so on. R signifies the strength of the color Red in the curve; thecolor is absent when R = 0 and fully present when R = 9. Similarly, G and B
represent the strengths of Green and Blue in the plot. The default value of SRGB is0000, yielding a solid black curve. At the end, the user will be given the opportunity
to save the screen in a bitmap file, FILE.BMP, where the filename is user-selectable.
5. Input and Output: At the end of each run of TEMPROFILE, all the input/output
information will be saved in a data-file called SESSION.DAT. The user is encouraged
to print this file in order to have a hardcopy record of the session. All input
parameters, whether entered from the keyboard or imported from a data-file, arerecorded in SESSION.DAT. The SESSION.DAT file also tells the user where the datafor plotting various curves are being stored.
Another data-file, called COMMAND.DAT, is created at the end of interactive (LIVE)sessions. The COMMAND.DAT file, which contains all the commands and parametervalues of the run, may be edited and used to control the flow of the program during
subsequent non-interactive (STEP or ZOOM) sessions.
Warning: Each new run of TEMPROFILE automatically renames theexisting SESSION.DAT and COMMAND.DAT files within the working
directory. The new filenames will be SESSION.nnn and COMMAND.nnn,
where nnn is the smallest available integer between 000 and 999.
At the end of a run, data for plotting each curve are stored in a pair of data-files:(Xnn.DAT, TXnn.DAT)(Ynn.DAT, TYnn.DAT)(Znn.DAT, TZnn.DAT)(Tnn.DAT, TTnn.DAT)(XInn.DAT, YInn.DAT)(TIMEnn.DAT, TPEAKnn.DAT)(TIMEnn.DAT, XPEAKnn.DAT)

for temperature versus x; T(x)for temperature versus y; T(y)for temperature versus z; T(z)for temperature versus time; T(t)for isotherms y(x)for peak temperature versus time, Tpeak(t)for Tpeak’s x-coordinate versus time, Xpeak(t)16Guide to Program TEMPROFILE

In each pair the first file contains the points on the horizontal axis, while the second
contains the points on the vertical axis. These curves may be displayed using either
the primitive plotting routine included with TEMPROFILE, or some other graphics
package. Figure 6 shows a collection of all the files that TEMPROFILE can generate.

COMMAND.DATSESSION.DATXXXX.IMP

XXXX.SYS

XXXX.RIP

XXXX.PLS

Xnn.DAT

TXnn.DAT

Ynn.DAT

TYnn.DAT

Znn.DAT

TZnn.DAT

Tnn.DAT

TTnn.DATXInn.DATYInn.DATTIMEnn.DAT

TPEAKnn.DAT

XPEAKnn.DAT

Figure 6. Diagram showing the files created by TEMPROFILE within its working directory. The
COMMAND.DAT file contains all commands and parameter values entered from the keyboard (orfrom data-files) during a LIVE session. A record of each session is kept in the SESSION.DAT file.
The impulse-response function is saved in (and retrieved from) the binary file XXXX.IMP. The text
file XXXX.SYS contains the structural parameters of the multilayer stack, the beam characteristics,
and the mesh parameters. The radial intensity profile of the laser beam is in the text file XXXX.RIP,
and the laser’s pulse shape is in the text file XXXX.PLS. ASCII files for plotting purposes are storedin pairs of files such as Xnn.DAT and TXnn.DAT, where nn is a 2-digit integer identifying individual
curves. The data points for the horizontal axis may also be in Ynn.DAT, Znn.DAT, or Tnn.DAT;
the corresponding vertical axis data-files are TYnn.DAT, TZnn.DAT, TTnn.DAT. The isotherms are
in XInn.DAT and YInn.DAT. When computing the mark size in a given XY-plane (MARK option),the program also computes the peak temperature and the x-coordinate of this peak as functions oftime. (The y-coordinate of the peak temperature is always zero, and the z-coordinate is fixed forthe given XY-plane.) The resulting data-files are TIMEnn.DAT, TPEAKnn.DAT, and XPEAKnn.DAT.

Guide to Program TEMPROFILE

17

6. Interactive and Non-interactive Sessions. When you run TEMPROFILE, it asks
in the beginning whether or not the session is going to be LIVE. If your answer ispositive, all subsequent commands and parameters must be entered from the
keyboard. If your answer is negative, TEMPROFILE assumes that all commands andparameters are stored in a file called COMMAND.DAT; it then proceeds to read thatfile and execute its commands.
Note: In the non-interactive mode you may choose to STEP throughCOMMAND.DAT or let the program run its course automatically (ZOOM). With
the STEP option the program will pause at the end of each screen and sounda tone. You must then press ENTER in order to resume program execution.
6.1. Structure of the COMMAND.DAT File. In the interactive (LIVE) mode, the file
COMMAND.DAT, a file which contains all the key-strokes entered during a session,
is automatically created. An example of COMMAND.DAT is shown in Fig. 7(a). The
title box of this file contains the syntax of the $ commands, which are described in
the next subsection. After the title box, there appear in columns 26-74 all the
key-strokes in the same sequential order in which they have been entered. The
guiding information appearing in columns 1-24 is intended to help the user, but is
not interpreted by TEMPROFILE. The colon (:) in column 25 separates the guiding
information from the actual key-strokes.
A number sign (#) is printed at the end of each line of COMMAND.DAT. You canassign any 2-digit number (00 - 99) to a given line by placing the number
immediately after # at the end of the line. These numbers are to be used inconjunction with the $ commands that you may choose to incorporate into
COMMAND.DAT. You can change the commands and/or the parameter values
appearing in COMMAND.DAT, so long as the new commands and/or parameters are
acceptable to TEMPROFILE. For example, in the COMMAND.DAT file of Fig. 7(a)you can change the value of 1.0E05 assigned to the Heat loss factor (GAMMA)
to 2.5E04 or to 3240.3 or to any other acceptable value, so long as you take carethat the colon (:) and the number sign (#) remain in place after such changes.
Figure 7(a). A COMMAND.DAT file created during a LIVE session. In the COMPUTE block, atrilayer stack is illuminated from the top by a Gaussian beam, and the impulse-response is saved in
AAAA.IMP. Next, in the TEMPROFILE block, the impulse response is read from AAAA.IMP, alaser pulse profile and a beam velocity are specified, then isotherms as well as temperature profiles
versus x, y, z, and t are created. The PLOT blocks that follow create plots of the various data files
generated in the preceding ISOS, TVSX, TVSY, TVSZ, and TVST command blocks.

18

Guide to Program TEMPROFILE@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ TEMPROFILE 5.3
File COMMAND.DAT @@#$@@ Date: 02/19/2005
Time: 13:07 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All lengths are in nanometers#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$#
Compute impulse-response: COMPUTE#!
#
Top/Bottom incidence: T#!
#
Substrate's index (n,k):1.500000
0.0000000#
Thermal constants (C,K):
2.000000
0.0500000#!
#
Number of layers: 3#!
#
Layer 01: Thickness,n,k:50.00000
2.1000000
5.3000000#
C,K:
1.850000
0.1000000#!
#
Layer 02: Thickness,n,k:120.0000
1.7600000
0.1000000#
C,K:
2.330000
0.0050000#!
#
Layer 03: Thickness,n,k:95.00000
1.5000000
0.0000000#
C,K:
1.700000
0.2000000#!
#
Heat loss factor (GAMMA):
1.0E05#!
#
Incident beam profile: Gaussian#
Lambda,R0:
650.0000
350.0000#!
#
Impulse: POWER,TAU,NSTEP:
1.000000
1.000000
5
#
!
#
Mesh:
RMAX,DLTAR:
6000.000
55.00000#
ZMAX,DLTAZ:
1000.000
8.000000#
TMAX:
50.00000#!
#
Save impulse-response: AAAA.IMP#$#
Temperature profile: TEMPROFILE#
Impulse-response file: AAAA.IMP#!
#
Point on pulse (t,P):
1.000000
2.000000#
Point on pulse (t,P):
2.000000
4.000000#
Point on pulse (t,P):
3.000000
6.000000#
Point on pulse (t,P):
30.00000
6.000000#
Point on pulse (t,P):
31.00000
4.000000#
Point on pulse (t,P):
32.00000
2.000000#
Point on pulse (t,P):
33.00000
0.000000#
Point on pulse (t,P):
50.00000
0.000000#!
#

Guide to Program TEMPROFILE
Beam velocity (V):19
2.500000$
Isotherms in XY-plane: ISOS
Z-coordinate of XY-plane:
150.0000
Time:
10.00000
Temperature:
55.00000
Time:
20.00000
Temperature:
150.0000
Time:
30.00000
Temperature:
250.0000$
Temperature versus X: TVSX
Z-coordinate:150.0000
Y,TIME:
0.000000
Y,TIME:
0.000000
Y,TIME:
100.0000$
Temperature versus Y: TVSY
Z-coordinate:150.0000
X,TIME:
0.000000
X,TIME:
0.000000
X,TIME:
0.000000$
Temperature versus Z: TVSZ
X,Y,TIME:
0.000000
X,Y,TIME:
0.000000
X,Y,TIME:
100.0000
X,Y,TIME:
300.0000$
Temperature versus Time: TVST
Z-coordinate:180.0000
X,Y:
0.000000
X,Y:
100.0000
X,Y:
300.0000$
Plot existing data-files: PLOT
X_Filename: T01.DATY_Filename: TT01.DAT
Curve type (SRGB): 0900!
Xmin,Xmax:-2.000000
Ymin,Ymax:
0.000000
Title of X-axis: time (ns)Title of Y-axis: Temperature (C)!
Plot another curve: Overlap
X_Filename: T02.DATY_Filename: TT02.DAT
Curve type (SRGB): 0090!
Plot another curve: Overlap
X_Filename: T03.DAT
Y_Filename: TT03.DAT
Curve type (SRGB): 0900!
Save graph (Filename): A001.BMP$
Plot existing data-files: PLOT
X_Filename: XI01.DATY_Filename: YI01.DAT
Curve type (SRGB): 0000!
Xmin,Xmax:-1000.000
Ymin,Ymax:-1000.000
Title of X-axis: X (nm)Title of Y-axis: Y (nm)10.00000
20.00000
30.00000

12.00000
24.00000
32.00000
0.000000
0.000000
200.0000
100.0000

0.000000
200.0000-100.0000

52.00000
350.0000

1000.000
1000.000

10.00000
20.00000
30.00000
25.00000

#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#

20

Guide to Program TEMPROFILE
!

#
#
#
#
#
!
#
Plot another curve: Overlap#
X_Filename: XI03.DAT#
Y_Filename: YI03.DAT#
Curve type (SRGB): 0090#!
#
Save graph (Filename): A002.BMP#$#
Plot existing data-files: PLOT#
X_Filename: Y01.DAT#Y_Filename: TY01.DAT#
Curve type (SRGB): 0000#!
#
Xmin,Xmax:-1200.000
1200.000#
Ymin,Ymax:
0.000000
350.0000#
Title of X-axis: Y (nm)#Title of Y-axis: Temperature (C)#!
#Plot another curve: Overlap#
X_Filename: Y02.DAT#Y_Filename: TY02.DAT#
Curve type (SRGB): 0900#!
#
Plot another curve: Overlap#
X_Filename: Y03.DAT#
Y_Filename: TY03.DAT#
Curve type (SRGB): 3592#!
#
Save graph (Filename): A003.BMP#$#
Plot existing data-files: PLOT#
X_Filename: Z01.DAT#Y_Filename: TZ01.DAT#
Curve type (SRGB): 0030#!
#
Xmin,Xmax:
0.000000
500.0000#
Ymin,Ymax:
0.000000
350.0000#
Title of X-axis: Z (nm)#Title of Y-axis: Temperature (C)#!
#Plot another curve: Overlap#
X_Filename: Z02.DAT#Y_Filename: TZ02.DAT#
Curve type (SRGB): 0910#!
#
Plot another curve: Overlap#
X_Filename: Z03.DAT#
Y_Filename: TZ03.DAT#
Curve type (SRGB): 0092#!
#
Plot another curve: Overlap#
X_Filename: Z04.DAT#
Y_Filename: TZ04.DAT#
Curve type (SRGB): 0090#!
#
Save graph (Filename): A004.BMP#$#
End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Plot another curve: Overlap
X_Filename: XI02.DATY_Filename: YI02.DAT
Curve type (SRGB): 0900

Guide to Program TEMPROFILE

21

6.2. The $ Commands: $ commands control the execution of TEMPROFILE in the
non-interactive mode. The syntax of all available $ commands appears in the title
box of COMMAND.DAT. These commands enable TEMPROFILE to operate in a loop
by allowing the user to modify specific items within COMMAND.DAT, and also by
allowing conditional jumps to marked positions within COMMAND.DAT. Some of
the lines appearing in COMMAND.DAT begin with a $ sign and end with a # sign, but
contain no information otherwise; these lines identify the end of one command blockand the beginning of the next. It is at such locations, namely, just before thebeginning of a command block, that $ commands may be inserted. One may insert
as many successive $ commands as is necessary at such locations. The $ commands
have a rigid format; the best way to use them is to copy them from the templates
provided in the title box, then substitute appropriate numerical values for thequestion marks (?) appearing within the template.
The $ commands generally target specific lines of COMMAND.DAT, or specific items
within these lines. A targeted line must be identified by a unique 2-digit number
placed immediately after the # sign at the end of the line. Within each line there canbe at most three items, identified as ITEM1, ITEM2 and ITEM3. A detailed
description of the $ commands follows.
i) The syntax of the INCREMENT command is$INCREMENT#??,ITEM?...BY: ??
Once a line in COMMAND.DAT has been labeled with a 2-digit number, it can beaddressed by the INCREMENT command. The ?? appearing after # in the command
syntax is the line number which contains the item to be incremented. Each targetedline could contain up to three items. The targeted item number (1, 2 or 3) must
replace ? immediately after ITEM in the command syntax. The amount by which thetargeted item is to be incremented replaces ?? after the colon (:). The increment
amount can be any arbitrary positive or negative real number.
ii) The syntax of the MULTIPLY command is$MULTIPLY#??,ITEM?....BY: ??
This command is similar to the INCREMENT command, except that the targeted item
is multiplied (rather than incremented) by the specified constant. Any line labeledwith a 2-digit number within COMMAND.DAT can be addressed by the MULTIPLY
command. The ?? appearing after # in the command syntax is the line number whichcontains the item to be modified by multiplication. Each targeted line could contain22

Guide to Program TEMPROFILE

up to three items. The targeted item number (1, 2 or 3) must replace ? immediately
after ITEM in the command syntax. The real-valued constant by which the targeted
item is to be multiplied replaces ?? after the colon; this constant can be any arbitrary
positive or negative number.
iii) The syntax of the MAKE command is$MAKE # ln,ITEM i ~ # lm,ITEM j:
The function of this command is to modify the value of an item on a given line of
COMMAND.DAT by an amount depending on the value of another item on the sameline or on a different line. The MAKE statement appears immediately after the $ sign;'ln' is a 2-digit line number in COMMAND.DAT which specifies the line targeted formodification; 'i' is a 1-digit number which specifies the targeted item within that line
(i = 1, 2, 3); ‘~’ is one of the ten operators +, −, *, /, =, E, L, S, C, T, which standfor addition, subtraction, multiplication, division, setting equal to, exponentialfunction, natural logarithm, sine, cosine, tangent. The 2-digit line number 'lm' andthe 1-digit item number 'j' specify the item within COMMAND.DAT that determines
the amount of change in the targeted item. In the process only the first item (i) is
modified; the second item (j) remains intact.
+
−
*
/
=
E
L
S
C
T

adds item j to item i.
subtracts item j from item i.
multiplies item j into item i.
divides item i by item j.
sets item i equal to item j.
replaces item i with the exponential of item j.
replaces item i with the natural logarithm of item j.
replaces item i with the sine of item j (angle in degrees).
replaces item i with the cosine of item j (angle in degrees).
replaces item i with the tangent of item j (angle in degrees).

iv) The syntax of the JUMP command is$JUMP#??_IF#??,ITEM?.GT.: ??
Upon reading the JUMP command, TEMPROFILE examines the current value of a
specified item within a specified line; if the item is greater than (or less than) a
certain amount, control will be transferred to a line also specified in the JUMP
command line; otherwise JUMP is ignored and the normal execution of the program
continues.

Guide to Program TEMPROFILE23

The destination address (i.e., the address to which control will be transferred)appears immediately after the JUMP command. The destination must be either a $command or the beginning of a command block (i.e., a line which begins with a $sign, ends with a # sign followed by a 2-digit number, and contains no information
otherwise). These are the only types of line to which a JUMP may be directed; inother words, TEMPROFILE cannot be instructed to JUMP to the middle of acommand block. The user must also ensure that the JUMP is logical and meaningful;for example, after the impulse-response is computed (in the COMPUTE block), one
should not skip the laser-pulse specification step (the TEMPROFILE block) and JUMP
to ISOS or TVSX or other blocks of this type. (The program needs to know the nameof an impulse-response file, the laser pulse shape, and the beam velocity, before it
can compute isotherms or profiles of temperature versus x, y, z, t.)
The line-number and the item within that line that serves as the condition parameter
for JUMP appear after IF in the command syntax. (Usually, this same item is also thetarget of an INCREMENT, MULTIPLY, or MAKE command, changing value with each
iteration.) The condition parameter is compared with a terminal value appearing
after the colon (:) in the JUMP command. The comparator .GT. stands for Greater
Than. Other comparators such as .LT. (Less Than), .GE. (Greater than or Equal to),and .LE. (Less than or Equal to) may also be used in this position.
Use extreme caution in choosing the condition parameter, the comparator, and theterminal value for the JUMP; a mistake in this command may either cause the JUMP
to be ignored, or it may result in an infinite number of jumps. Also make sure that
the JUMP destination is a valid starting point for a loop. Note in particular thatchanges in the impulse-response file (XXXX.IMP), laser pulse profile P(t), and beam
velocity V, will not be effective unless the corresponding TEMPROFILE block is
executed prior to the execution of ISOS, TVSX, TVSY, TVSZ, or TVST blocks.
v) The syntax of the TAGS command is$TAGS#??,ITEM?_#??,ITEM?:
Whenever this command is executed, the two targeted items are associated with twointernal variables, Tag1 and Tag2. Ideally, items used as tags should also be targetsof INCREMENT, MULTIPLY, or MAKE commands, which assume different values indifferent iterations.
Presently the $TAGS command does not perform any usefulfunctions; it will be exploited in future releases of TEMPROFILE.

24

Guide to Program TEMPROFILEThe TAGS command must specify exactly two items (no more and no less) within
COMMAND.DAT for association with Tag1 and Tag2. These items, however, may
have the same address or they may have different addresses. Tag1 and Tag2 are
dynamically associated with the targeted items, their values being updated wheneveran INCREMENT, MULTIPLY, or MAKE command changes the corresponding items.
The user may insert the TAGS command in more than one place within
COMMAND.DAT, thus changing the associations of Tag1 and Tag2 during program
execution. At the time when a particular output is being generated, however, thetags in effect will be the last ones invoked.
Example: Figure 7(b) shows a modified version of the COMMAND.DAT file ofFig. 7(a). Compared with the original file, we observe the following new features.
• The values of Heat_loss_factor (GAMMA) and Beam_velocity (V) have been
modified, the command blocks TVSX, TVSY, TVSZ, and TVST have been removed,
and all but two of the PLOT command blocks have been eliminated.
• Four of the lines have been marked as #10, #20, #25, and #30. These are thelines targeted by the $ commands described below.
• After the ISOS block we have inserted four $ commands. The first commandtargets line #20, ITEM1 (the substrate’s specific heat) for incrementing in theamount of 1.5. The second $ command multiplies ITEM3 of line #25 (i.e.,
absorption coefficient k of layer 2) by 2.5. The third command targets line #30
and decrements ITEM1 (z-coordinate of the xy-plane of isotherms) by 25. The
fourth command transfers control of TEMPROFILE to line #10 if the (recently
modified) value of the substrate’s specific heat happens to be ≤ 4.0.
Note that line #10 is a proper jump destination, since it begins with a $ sign, ends
with a # sign, and is empty otherwise; it is also a logical place to begin repeating thesimulation with modified parameters. Observe that in each iteration the impulseresponse file AAAA.IMP is overwritten, and that the new file is used in subsequent
TEMPROFILE calculations. Each isotherm is saved in the pair of files(XInn.DAT, YInn.DAT), where nn ranges from 01 to 03 in the first iteration, andfrom 04 to 06 in the second iteration. (Each type of temperature profile is saved in
numbered files whose indices range sequentially from nn = 01 to 99 automatically.)
The PLOT commands at the end of the COMMAND.DAT of Fig. 7(b) create two setsof isotherm plots, one for each iteration, saving them as A001.BMP and A002.BMP.

Guide to Program TEMPROFILE

25

Figure 7(b). A modified version of the COMMAND.DAT file of Fig.7(a), incorporating several$ commands. When used in a non-interactive mode (STEP or ZOOM), this COMMAND.DAT puts
TEMPROFILE through a loop that iterates through two sets of parameter values.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ TEMPROFILE 5.3
File COMMAND.DAT @@#$@@ Date: 02/19/2005
Time: 13:07 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All lengths are in nanometers#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$#10
Compute impulse-response: COMPUTE#!
#
Top/Bottom incidence: T#!
#
Substrate's index (n,k):1.500000
0.0000000#
Thermal constants (C,K):
2.000000
0.0500000#20!
#
Number of layers: 3#!
#
Layer 01: Thickness,n,k:50.00000
2.1000000
5.3000000#
C,K:
1.850000
0.1000000#!
#
Layer 02: Thickness,n,k:120.0000
1.7600000
0.1000000#25C,K:2.330000
0.0050000#!
#
Layer 03: Thickness,n,k:95.00000
1.5000000
0.0000000#
C,K:
1.700000
0.2000000#!
#
Heat loss factor (GAMMA):
23000.00#!
#
Incident beam profile: Gaussian#
Lambda,R0:
650.0000
350.0000#!
#
Impulse: POWER,TAU,NSTEP:
1.000000
1.000000
5
#
!
#
Mesh:
RMAX,DLTAR:
6000.000
55.00000#
ZMAX,DLTAZ:
1000.000
8.000000#
TMAX:
50.00000#!
#
Save impulse-response: AAAA.IMP#$#
Temperature profile: TEMPROFILE#
Impulse-response file: AAAA.IMP#!
#
Point on pulse (t,P):
1.000000
2.000000#
Point on pulse (t,P):
2.000000
4.000000#
Point on pulse (t,P):
3.000000
6.000000#
Point on pulse (t,P):
30.00000
6.000000#
Point on pulse (t,P):
31.00000
4.000000#

26

Guide to Program TEMPROFILE
Point on pulse (t,P):
Point on pulse (t,P):
Point on pulse (t,P):

32.00000
33.00000
50.00000

2.000000
0.000000
0.000000###!
#
Beam velocity (V):
1.250000#$#
Isotherms in XY-plane: ISOS#
Z-coordinate of XY-plane:
150.0000#30
Time:
10.00000#
Temperature:
55.00000#
Time:
20.00000#
Temperature:
150.0000#
Time:
30.00000#
Temperature:
250.0000#$#$INCREMENT#20,ITEM1...BY: 1.5#$MULTIPLY#25,ITEM3....BY: 2.5#$INCREMENT#30,ITEM1...BY: -25.0#$JUMP#10_IF#20,ITEM1.LE.: 4.0#$#
Plot existing data-files: PLOT#
X_Filename: XI01.DAT#Y_Filename: YI01.DAT#
Curve type (SRGB): 0000#!
#
Xmin,Xmax:-1000.000
1000.000#
Ymin,Ymax:-1000.000
1000.000#
Title of X-axis: X (nm)#Title of Y-axis: Y (nm)#!
#Plot another curve: Overlap#
X_Filename: XI02.DAT#Y_Filename: YI02.DAT#
Curve type (SRGB): 0900#!
#
Plot another curve: Overlap#
X_Filename: XI03.DAT#
Y_Filename: YI03.DAT#
Curve type (SRGB): 0090#!
#
Save graph (Filename): A001.BMP#$#
Plot existing data-files: PLOT#
X_Filename: XI04.DAT#
Y_Filename: YI04.DAT#
Curve type (SRGB): 0000#!
#
Xmin,Xmax:-1000.000
1000.000#
Ymin,Ymax:-1000.000
1000.000#
Title of X-axis: X (nm)#Title of Y-axis: Y (nm)#!
#Plot another curve: Overlap#
X_Filename: XI05.DAT#
Y_Filename: YI05.DAT#
Curve type (SRGB): 0900#!
#
Plot another curve: Overlap#
X_Filename: XI06.DAT#
Y_Filename: YI06.DAT#
Curve type (SRGB): 0090#!
#
Save graph (Filename): A002.BMP#$#
End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to Program TEMPROFILE

27

6.3. Running TEMPROFILE in the Non-interactive Mode. A COMMAND.DAT fileis usually created interactively, then opened with a text-editor (e.g., NotePad or
WordPad) for modification by the user. The modifications can be as simple aschanging a few of the numbers, or as extensive as cutting and pasting command
blocks from different files into one, adding $ commands, creating nested loops, etc.
Once a COMMAND.DAT file is ready, it should be placed within a subdirectory of
the home directory. The user then starts the program by double-clicking on
TEMPROFILE.EXE from the home directory, identifying as working directory thesubdirectory within which the desired COMMAND.DAT file resides, and instructing
TEMPROFILE to operate either in the STEP mode or in the ZOOM mode. The program
then moves automatically to the specified subdirectory, reads the COMMAND.DAT
file, and executes its commands sequentially.

28

Guide to Program TEMPROFILE

7. Test_Run 1. Simulated in this example is a quadrilayer structure consisting of aplastic substrate coated with a 32 nm-thick layer of aluminum, a 72 nm-thick layerof SiO2, a 12 nm-thick layer of TbFeCo, and another 100 nm-thick layer of SiO2.
The Gaussian laser beam is incident on the stack from the surface side, its 1/e radius
being 350 nm. The laser pulse shape is trapezoidal, having 60 nsec duration, with
10 nsec rise and fall times. The various simulation parameters are shown in theprintout of the SESSION.DAT file below. Calculated profiles of temperature versus
x, y, z, and time are shown in Figures 8(a-d), respectively. Figure 8(e) shows theisotherms in the TbFeCo layer.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ TEMPROFILE 5.3
File SESSION.DAT $$$$ Date: 02/19/2005
Time: 15:44 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
Units: Length = nanometer, Time = nanosecond, Power = milliwatt***************************************************************
Designated Working Subdirectory: <TR01>***************************************@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Setup for impulse-response computation@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Multilayer characteristics:***************************
Semi-infinite substrate:(n,k)=(
C=

1.46000,.00000)1.70000 K= .00200

Layer #

1:

Z=

32.0

(n,k)=(
C=

1.20000,
6.90000)
2.70000 K= 2.40000

Layer #

2:

Z=

72.0

(n,k)=(
C=

1.50000,.00000)2.00000 K= .01500

Layer #

3:

Z=

12.0

(n,k)=(
C=

3.10000,
3.50000)
2.60000 K= .16000

Layer #

4:

Z=

100.0

(n,k)=(
C=

1.50000,.00000)2.00000 K= .01500

Heat loss factor from top surface: GAMMA=.00E+00

Incident beam characteristics(Surface incidence):****************************************************
Radial intensity profile: Gaussian
LAMBDA=
632.8
R0=
350.0
POWER= 1.0000
TAU= 1.0000
NSTEP=
5****************************************************
Mesh characteristics:*********************
RMAX=
2000.
DLTAR=ZMAX=
400.
DLTAZ=
TMAX=
100.
TAU=

25.
4.
1.

IMAX=
JMAX=
KMAX=

80
100
100

Guide to Program TEMPROFILE

29

Characteristic parameters of the multilayer, theincident beam, and the mesh are saved in TR01.SYS*************************************************The impulse response function is saved in TR01.IMP**************************************************@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Generating Temperature Profile(s)@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@The impulse-response function is read from TR01.IMP***************************************************
Laser pulse shape:******************(Saved in TR01.PLS)

Time
****

Power
*****

10.00
50.00
60.00
100.00

2.00
2.00
.00
.00

Beam velocity =.00 meters/sec (along +X axis)************************************************
Profile(s) of temperature versus X at fixed Y,Z,T:**************************************************
Profile for Y=.00, Z=
108.00, Time=
X01.DAT and TX01.DAT

10.0000 is saved in

Profile for Y=
X02.DAT and TX02.DAT.00, Z=

108.00, Time=

20.0000 is saved in

Profile for Y=
X03.DAT and TX03.DAT.00, Z=

108.00, Time=

30.0000 is saved in

Profile for Y=
X04.DAT and TX04.DAT.00, Z=

108.00, Time=

40.0000 is saved in

Profile for Y=
X05.DAT and TX05.DAT.00, Z=

108.00, Time=

50.0000 is saved in

Profile(s) of temperature versus Y at fixed X,Z,T:**************************************************
Profile for X=.00, Z=
108.00 and Time=is saved in Y.DAT and TY01.DAT

10.0000

Profile for X=.00, Z=is saved in Y.DAT and TY02.DAT

108.00 and Time=

20.0000

Profile for X=.00, Z=is saved in Y.DAT and TY03.DAT

108.00 and Time=

30.0000

Profile for X=.00, Z=is saved in Y.DAT and TY04.DAT

108.00 and Time=

40.0000

Profile for X=.00, Z=is saved in Y.DAT and TY05.DAT

108.00 and Time=

50.0000

Profile(s) of temperature versus Z at fixed X,Y,T:**************************************************
Profile for X=.00, Y=.00, Time=
48.0000 is saved in
Z.DAT and TZ01.DAT
Profile for X=
Z.DAT and TZ02.DAT.00, Y=.00, Time=

60.0000 is saved in30

Guide to Program TEMPROFILE

Profile for X=
Z.DAT and TZ03.DAT.00, Y=.00, Time=

100.0000 is saved in

Profile(s) of temperature versus TIME at fixed X,Y,Z:*****************************************************
Profile for X=.00, Y=.00, Z=
108.00 is saved in
TIME.DAT and TT01.DAT
Profile for X=
175.00, Y=
TIME.DAT and TT02.DAT.00, Z=

108.00 is saved in

Profile for X=.00, Y=
TIME.DAT and TT03.DAT

350.00, Z=

108.00 is saved in

Profile for X=
525.00, Y=
TIME.DAT and TT04.DAT.00, Z=

108.00 is saved in

Profile for X=
700.00, Y=
TIME.DAT and TT05.DAT.00, Z=

108.00 is saved in

Profile for X=
875.00, Y=
TIME.DAT and TT06.DAT.00, Z=

108.00 is saved in

Isotherms in the XY plane at fixed Z and T:*******************************************
Isotherm at Z=
108.00, T=
50.00 is saved in XI01.DAT and YI01.DAT(Temperature= 193.00
MAX.TEMP= 194.13 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI02.DAT and YI02.DAT(Temperature= 180.00
MAX.TEMP= 194.13 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI03.DAT and YI03.DAT(Temperature= 160.00
MAX.TEMP= 194.13 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI04.DAT and YI04.DAT(Temperature= 140.00
MAX.TEMP= 194.13 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI05.DAT and YI05.DAT(Temperature= 120.00
MAX.TEMP= 194.13 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI06.DAT and YI06.DAT(Temperature= 100.00
MAX.TEMP= 194.13 at X=.00, Y=0)@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
TX01 versus X01 is plotted in Graphics Window
TX02 versus X02 is plotted in Graphics Window
TX03 versus X03 is plotted in Graphics Window
TX04 versus X04 is plotted in Graphics Window
TX05 versus X05 is plotted in Graphics Window
Graphics Window is saved to TVSX.BMP@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
TY01 versus Y is plotted in Graphics Window
TY02 versus Y is plotted in Graphics Window
TY03 versus Y is plotted in Graphics Window
TY04 versus Y is plotted in Graphics Window
TY05 versus Y is plotted in Graphics Window
Graphics Window is saved to TVSY.BMP

Guide to Program TEMPROFILE

31@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
TZ01 versus Z is plotted in Graphics Window
TZ02 versus Z is plotted in Graphics Window
TZ03 versus Z is plotted in Graphics Window
Graphics Window is saved to TVSZ.BMP@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
TT01 versus TIME is plotted in Graphics Window
TT02 versus TIME is plotted in Graphics Window
TT03 versus TIME is plotted in Graphics Window
TT04 versus TIME is plotted in Graphics Window
TT05 versus TIME is plotted in Graphics Window
TT06 versus TIME is plotted in Graphics Window
Graphics Window is saved to TVST.BMP@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
YI01 versus XI01 is plotted in Graphics Window
YI02 versus XI02 is plotted in Graphics Window
YI03 versus XI03 is plotted in Graphics Window
YI04 versus XI04 is plotted in Graphics Window
YI05 versus XI05 is plotted in Graphics Window
YI06 versus XI06 is plotted in Graphics Window
Graphics Window is saved to ISOS.BMP
End of session: TEMPROFILE terminated normally**********************************************

File SESSION.DAT created in Test_Run 1

32

Guide to Program TEMPROFILE

Temperature above ambient (degree C)

200

(a)

time = 50 ns
40 ns
30 ns

150

20 ns
100

10 ns
50

0
-1000

-750

-500

-250

0

250

500

750

1000

250

500

750

1000

X (nm)

Temperature above ambient (degree C)

200

(b)

time = 50 ns
40 ns
30 ns

150

20 ns
100

10 ns
50

0
-1000

-750

-500

-250

0

Y (nm)

Figure 8. (a,b) Plots of temperature versus x and y at z = 108 nm as computed in Test_Run 1.
Since the incident beam is circularly symmetric and does not move relative to the multilayer stack,the temperature profiles along X and Y are identical.

Guide to Program TEMPROFILE

200

33

(c)

Temperature above ambient (degree C)time = 48 ns

150
60 ns

100

50
100 ns

0
0

50

100

150

200

250

300

350

400

90

100

Z (nm)(d)200

Temperature above ambient (degree C)(x, y) = (0, 0)

150(x, y) = (175 nm, 0 )100(x, y) = (0, 350 nm )(x, y) = (525 nm, 0 )50(x, y) = (700 nm, 0)(x, y) = (875 nm, 0 )

0
0

10

20

30

40

50

60

70

80

time (nsec)

Figure 8. (c) Temperature versus z at (x, y) = (0, 0) at several instants of time. (d) Temperature
as function of time at z = 108 nm. Both sets of curves are computed in Test_Run 1.

34

Guide to Program TEMPROFILE

400

(e)

300
200

Y (nm)

100
0

-100
-200
-300
-400
-400

-300

-200

-100

0

100

200

300

400

X (nm)

Figure 8. (e) Isotherms in the XY-plane at z = 108 nm and time = 50 ns, as computed in
Test_Run 1. The hottest spot at the center has temperature T = 194.13ºC (above ambient). The
plotted isotherms, starting in the middle and moving outward, correspond to T = 193ºC, 180ºC,
160ºC, 140ºC, 120ºC and 100ºC.

The file TR01.PLS shown below contains the laser pulse shape. (The extension.PLS stands for “pulse.”) Pairs of points (t, P) at the corners of the piecewise linear
function P(t), initially entered from the keyboard, are saved in this file. The user canmodify the contents of this file with a text editor, and import it to the program
environment during subsequent runs of TEMPROFILE.
!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
@@ TEMPROFILE 5.3
File TR01.PLS @@#!
@@ Date: 02/19/2005
Time: 15:34 @@#!
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
#! Notes: Trapezoidal pulse shape of the laser beam. The rise and fall#!
times are 10 nsec each, and the total pulse width is 60 nsec.
#!
#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#
Point on pulse (t,P):
10.000000
2.0000000#
Point on pulse (t,P):
50.000000
2.0000000#
Point on pulse (t,P):
60.000000.00000000#
Point on pulse (t,P):
100.00000.00000000#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

File TR01.PLS created during Test_Run 1

Guide to Program TEMPROFILE

35

Shown below is the file TR01.SYS which contains the structural parameters of themultilayer, the laser beam profile, and the mesh parameters. (The filename extension.SYS stands for “system.”) This text file, which is originally created by
TEMPROFILE, can be modified by the user, and may be imported to the program
environment during subsequent runs.

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
@@ TEMPROFILE 5.3
File TR01.SYS @@#!
@@ Date: 02/19/2005
Time: 17:36 @@#!
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
#! Notes: Quadrilayer stack used in the surface-incidence configuration#!
of Test Run 1. The incident beam has a Gaussian profile.
#!
#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#$#
Top/Bottom incidence: T#$#
Substrate's index (n,k):1.4600000.00000000#
Thermal constants (C,K):
1.7000000
2.00000000E-03#$#
Number of layers: 4#$#
Layer 01: Thickness,n,k:32.000000
1.2000000
6.9000000#
C,K:
2.7000000
2.4000000#$#
Layer 02: Thickness,n,k:72.000000
1.5000000.00000000#
C,K:
2.0000000
1.50000000E-02#$#
Layer 03: Thickness,n,k:12.000000
3.1000000
3.5000000#
C,K:
2.6000000.16000000#$#
Layer 04: Thickness,n,k:100.00000
1.5000000.00000000#
C,K:
2.0000000
1.50000000E-02#$#
Heat loss factor (GAMMA):.00000000#$#
Incident beam profile: Gaussian#
Lambda,R0:
632.80000
350.00000#
Impulse: POWER,TAU,NSTEP:
1.0000000
1.0000000
5
#
$
#
Mesh:
RMAX,DLTAR:
2000.0000
25.000000#
ZMAX,DLTAZ:
400.00000
4.0000000#
TMAX:
100.00000#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

File TR01.SYS created in Test_Run 1

36

Guide to Program TEMPROFILE

8. Test Run 2. In this example a quadrilayer structure is simulated. The laser beam
is incident on the multilayer stack from the substrate side, and the various
parameters of the simulation are shown in the SESSION.DAT file below. Calculated
profiles of temperature versus z are shown in Figure 9(a). Isotherms in the XY-planeat z = 115 nm appear in Figure 9(b).
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ TEMPROFILE 5.3
File SESSION.DAT $$$$ Date: 02/19/2005
Time: 16:26 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
Units: Length = nanometer, Time = nanosecond, Power = milliwatt***************************************************************
Designated Working Subdirectory: <TR02>***************************************@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Setup for impulse-response computation@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Multilayer characteristics:***************************
Semi-infinite substrate:(n,k)=(
C=

1.53000,.00000)2.60000 K= .01000

LAYER #

1:

Z=

75.0

(n,k)=(
C=

2.00000,.00000)2.50000 K= .02000

LAYER #

2:

Z=

25.0

(n,k)=(
C=

3.67000,
3.86000)
3.50000 K= .50000

LAYER #

3:

Z=

60.0

(n,k)=(
C=

2.00000,.00000)2.50000 K= .02000

LAYER #

4:

Z=

40.0

(n,k)=(
C=

2.70000,
8.30000)
2.40000 K= 2.40000

Heat loss factor from top surface: GAMMA=.00E+00

Incident beam characteristics (Substrate incidence):****************************************************
Radial Intensity Profile: Gaussian
LAMBDA=
830.0
R0=
560.0
POWER= 1.0000
TAU= 1.0000
NSTEP=
5****************************************************
Mesh characteristics:*********************
RMAX=
6000.
DLTAR=ZMAX=
600.
DLTAZ=
TMAX=
100.
TAU=

50.
5.
1.

IMAX=
JMAX=
KMAX=

120
120
100

Characteristic parameters of the multilayer, theincident beam, and the mesh are saved in TR02.SYS*************************************************The impulse response function is saved in TR02.IMP**************************************************

Guide to Program TEMPROFILE37@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Generating Temperature Profile(s)@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@The impulse-response function is read from TR02.IMP***************************************************
Laser pulse shape:******************(Saved in TR02.PLS)

Time
****

Power
*****

.00
50.00
50.00
100.00

10.00
10.00
.00
.00

Beam velocity =.00 meters/sec (along +X axis)************************************************
Profile(s) of temperature versus Z at fixed X,Y,T:**************************************************
Profile for X=.00, Y=.00, Time=
10.0000 is saved in
Z.DAT and TZ01.DAT
Profile for X=
Z.DAT and TZ02.DAT.00, Y=.00, Time=

50.0000 is saved in

Profile for X=
Z.DAT and TZ03.DAT.00, Y=.00, Time=

60.0000 is saved in@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Generating Temperature Profile(s)@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@The impulse-response function is read from TR02.IMP***************************************************
Laser pulse shape:******************(Imported from TR02.LSR)

Time
****

Power
*****

.00
50.00
50.00
100.00

10.00
10.00
.00
.00

Beam velocity = 10.00 meters/sec (along +X axis)************************************************
Isotherms in the XY plane at fixed Z and T:*******************************************
Isotherm at Z=
115.00, T=
10.00 is saved in XI01.DAT and YI01.DAT(Temperature= 125.00
MAX.TEMP= 134.44 at X=
75.31, Y=0)
Isotherm at Z=
115.00, T=
20.00 is saved in XI02.DAT and YI02.DAT(Temperature= 125.00
MAX.TEMP= 174.06 at X=
149.06, Y=0)
Isotherm at Z=
115.00, T=
30.00 is saved in XI03.DAT and YI03.DAT(Temperature= 125.00
MAX.TEMP= 196.68 at X=
242.50, Y=0)
Isotherm at Z=
115.00, T=
40.00 is saved in XI04.DAT and YI04.DAT(Temperature= 125.00
MAX.TEMP= 211.75 at X=
342.50, Y=0)
Isotherm at Z=
115.00, T=
50.00 is saved in XI05.DAT and YI05.DAT(Temperature= 125.00
MAX.TEMP= 222.70 at X=
429.38, Y=0)

38

Guide to Program TEMPROFILE@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
TZ01 versus Z is plotted in Graphics Window
TZ02 versus Z is plotted in Graphics Window
TZ03 versus Z is plotted in Graphics Window
Graphics Window is saved to TVSZ.BMP@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
YI01 versus XI01 is plotted in Graphics Window
YI02 versus XI02 is plotted in Graphics Window
YI03 versus XI03 is plotted in Graphics Window
YI04 versus XI04 is plotted in Graphics Window
YI05 versus XI05 is plotted in Graphics Window
Graphics Window is saved to ISOS.BMP
End of session: TEMPROFILE terminated normally**********************************************

File SESSION.DAT created in Test_Run 2

The file TR02.PLS shown below was created during Test_Run 2, and contains thelaser pulse shape. Pairs of points (t, P) at the corners of the piecewise linear
function P(t), initially entered from the keyboard, are saved in this file. The user canmodify the contents of this file with a text editor, and import it to the program
environment during subsequent runs of TEMPROFILE.
!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
@@ TEMPROFILE 5.3
File TR02.PLS @@#!
@@ Date: 02/19/2005
Time: 16:26 @@#!
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
#! Notes: Rectangular laser pulse (Duration=50ns, Power=10mW)#!
#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#
Point on pulse (t,P):.00000000
10.000000#
Point on pulse (t,P):
50.000000
10.000000#
Point on pulse (t,P):
50.000000.00000000#
Point on pulse (t,P):
100.00000.00000000#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

File TR02.PLS created in Test Run 2

Guide to Program TEMPROFILE

39

Shown below is the file TR02.SYS, which contains the structural parameters of themultilayer, the laser beam profile, and the mesh parameters of Test_Run 2. This text
file, originally created by TEMPROFILE, can be modified by the user, and may be
imported to the program environment during subsequent runs.
!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
@@ TEMPROFILE 5.3
File TR02.SYS @@#!
@@ Date: 02/19/2005
Time: 16:26 @@#!
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
#! Notes: Quadrilayer stack used in the substrate-incident configuration#!
of Test Run 2. The incident beam has a Gaussian profile.
#!
#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#$#
Top/Bottom incidence: B#$#
Substrate's index (n,k):
1.5300000.00000000#
Thermal constants (C,K):
2.6000000
1.00000000E-02#$#
Number of layers: 4#$#
Layer 01: Thickness,n,k:75.000000
2.0000000.00000000#
C,K:
2.5000000
2.00000000E-02#$#
Layer 02: Thickness,n,k:25.000000
3.6700000
3.8600000#
C,K:
3.5000000.50000000#$#
Layer 03: Thickness,n,k:60.000000
2.0000000.00000000#
C,K:
2.5000000
2.00000000E-02#$#
Layer 04: Thickness,n,k:40.000000
2.7000000
8.3000000#
C,K:
2.4000000
2.4000000#$#
Heat loss factor (GAMMA):.00000000#$#
Incident beam profile: Gaussian#
Lambda,R0:
830.00000
560.00000#
Impulse: POWER,TAU,NSTEP:
1.0000000
1.0000000
5
#
$
#
Mesh:
RMAX,DLTAR:
6000.0000
50.000000#
ZMAX,DLTAZ:
600.00000
5.0000000#
TMAX:
100.00000#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

File TR02.SYS created in Test Run 2

40

Guide to Program TEMPROFILE

250

(a)

Temperature above ambient (degree C)time = 50 ns
200

150

100

60 ns
10 ns

50

0
0

100

200

300

400

500

600

20ns 30ns

40 50

Z (nm)
1000

(b)

800
600
400

Y (nm)

200
0time = 10ns

-200
-400
-600
Isotherms @ 125 degrees C
-800
-1000
-1000

-500

0

500

1000

X (nm)

Figure 9. (a) Profiles of temperature versus z at (x, y) = (0, 0) for several instants of time, as
computed in Test_Run 2. The beam is stationary with respect to the stack (i.e., V = 0).
(b) Isotherms in the XY-plane at z = 115 nm and at the fixed temperature T = 125ºC. The beam
moves at V = 10 m/s relative to the stack. The various isotherms, starting in the middle andmoving outward, belong to time = 10ns, 20ns, 30ns, 40ns, and 50ns.

Guide to Program TEMPROFILE

41

9. Test Run 3. In this example the quadrilayer stack is subject to a user-definedlaser beam profile that is incident from the surface side. The various parameters of
the simulation are shown in the SESSION.DAT file below. Calculated profiles oftemperature versus z are shown in Figure 10(a), and isotherms in the XY-plane atz = 108 nm appear in Figure 10(b).
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ TEMPROFILE 5.3
File SESSION.DAT $$$$ Date: 02/19/2005
Time: 17:32 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
Units: Length = nanometer, Time = nanosecond, Power = milliwatt***************************************************************
Designated Working Subdirectory: <TR03>***************************************@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Setup for impulse-response computation@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Multilayer characteristics:***************************(Imported from TR03.SYS)
Semi-infinite substrate:(n,k)=(
C=

1.46000,.00000)1.70000 K= .00200

Layer #

1:

Z=

32.0

(n,k)=(
C=

1.20000,
6.90000)
2.70000 K= 2.40000

Layer #

2:

Z=

72.0

(n,k)=(
C=

1.50000,.00000)2.00000 K= .01500

Layer #

3:

Z=

12.0

(n,k)=(
C=

3.10000,
3.50000)
2.60000 K= .16000

Layer #

4:

Z=

100.0

(n,k)=(
C=

1.50000,.00000)2.00000 K= .01500

Heat loss factor from top surface: GAMMA=.00E+00

Incident beam characteristics(Surface incidence):****************************************************
Radial intensity profile:RadiusIntensity****************************************.00
1.000000
50.00.972600
100.00.894800
150.00.778800
200.00.641200
250.00.499400
300.00.367900
350.00.256400
400.00.169000
450.00.105400
500.00.062200
550.00.034700
600.00.018300
650.00.009100
700.00.004300
750.00.001900

42

Guide to Program TEMPROFILE
800.00850.00900.001000.00.000800.000300.000100.000000

LAMBDA=
632.8
R0=
301.3
POWER= 3.0000
TAU= 1.0000
NSTEP=
5****************************************************
Mesh characteristics:*********************
RMAX=
2000.
DLTAR=ZMAX=
400.
DLTAZ=
TMAX=
100.
TAU=

25.
4.
1.

IMAX=
JMAX=
KMAX=

80
100
100

The impulse response function is saved in TR03.IMP**************************************************@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Generating Temperature Profile(s)@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@The impulse-response function is read from TR03.IMP***************************************************
Laser pulse shape:******************(Imported from TR03.PLS)

Time
****

Power
*****

10.00
50.00
60.00
100.00

2.00
2.00
.00
.00

Beam velocity =.00 meters/sec (along +X axis)************************************************
Profile(s) of temperature versus Z at fixed X,Y,T:**************************************************
Profile for X=.00, Y=.00, Time=
48.0000 is saved in
Z.DAT and TZ01.DAT
Profile for X=
Z.DAT and TZ02.DAT.00, Y=.00, Time=

60.0000 is saved in

Profile for X=
Z.DAT and TZ03.DAT.00, Y=.00, Time=

10.0000 is saved in

Profile for X=
Z.DAT and TZ04.DAT.00, Y=.00, Time=

100.0000 is saved in@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
TZ01 versus Z is plotted in Graphics Window
TZ02 versus Z is plotted in Graphics Window
TZ03 versus Z is plotted in Graphics Window
TZ04 versus Z is plotted in Graphics Window
Graphics Window is saved to TVSZ.BMP
Isotherms in the XY plane at fixed Z and T:*******************************************
Isotherm at Z=
108.00, T=
50.00 is saved in XI01.DAT and YI01.DAT(Temperature= 237.00
MAX.TEMP= 237.15 at X=.00, Y=0)

Guide to Program TEMPROFILE

43

Isotherm at Z=
108.00, T=
50.00 is saved in XI02.DAT and YI02.DAT(Temperature= 230.00
MAX.TEMP= 237.15 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI03.DAT and YI03.DAT(Temperature= 220.00
MAX.TEMP= 237.15 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI04.DAT and YI04.DAT(Temperature= 210.00
MAX.TEMP= 237.15 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI05.DAT and YI05.DAT(Temperature= 200.00
MAX.TEMP= 237.15 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI06.DAT and YI06.DAT(Temperature= 180.00
MAX.TEMP= 237.15 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI07.DAT and YI07.DAT(Temperature= 160.00
MAX.TEMP= 237.15 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI08.DAT and YI08.DAT(Temperature= 140.00
MAX.TEMP= 237.15 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI09.DAT and YI09.DAT(Temperature= 120.00
MAX.TEMP= 237.15 at X=.00, Y=0)
Isotherm at Z=
108.00, T=
50.00 is saved in XI10.DAT and YI10.DAT(Temperature= 100.00
MAX.TEMP= 237.15 at X=.00, Y=0)@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@
YI01 versus XI01 is plotted in Graphics Window
YI02 versus XI02 is plotted in Graphics Window
YI03 versus XI03 is plotted in Graphics Window
YI04 versus XI04 is plotted in Graphics Window
YI05 versus XI05 is plotted in Graphics Window
YI06 versus XI06 is plotted in Graphics Window
YI07 versus XI07 is plotted in Graphics Window
YI08 versus XI08 is plotted in Graphics Window
YI09 versus XI09 is plotted in Graphics Window
YI10 versus XI10 is plotted in Graphics Window
Graphics Window is saved to ISOS.BMP
End of session: TEMPROFILE terminated normally**********************************************

File SESSION.DAT created in Test_Run 3

44

Guide to Program TEMPROFILE

Shown below is the file TR03.SYS, which contains the structural parameters of themultilayer stack, the laser beam profile, and the mesh parameters of Test_Run 3.
!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
@@ TEMPROFILE 5.3
File TR03.SYS @@#!
@@ Date: 02/19/2005
Time: 17:35 @@#!
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
#! Notes: Quadrilayer stack used in Test Run 3. The incident beam profile #!
is user-defined.
#!
#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#$#
Top/Bottom incidence: T#$#
Substrate's index (n,k):1.4600000.00000000#
Thermal constants (C,K):
1.7000000
2.00000000E-03#$#
Number of layers: 4#$#
Layer 01: Thickness,n,k:32.000000
1.2000000
6.9000000#
C,K:
2.7000000
2.4000000#$#
Layer 02: Thickness,n,k:72.000000
1.5000000.00000000#
C,K:
2.0000000
1.50000000E-02#$#
Layer 03: Thickness,n,k:12.000000
3.1000000
3.5000000#
C,K:
2.6000000.16000000#$#
Layer 04: Thickness,n,k:100.00000
1.5000000.00000000#
C,K:
2.0000000
1.50000000E-02#$#
Heat loss factor (GAMMA):.00000000#$#
Incident Beam Profile: User-defined#
LAMBDA,RS:
632.80000
1000.0000#
Point on profile (r,I):.00000000
1.0000000#
Point on profile (r,I):
50.000000.97260000#
Point on profile (r,I):
100.00000.89480000#
Point on profile (r,I):
150.00000.77880000#
Point on profile (r,I):
200.00000.64120000#
Point on profile (r,I):
250.00000.49940000#
Point on profile (r,I):
300.00000.36790000#
Point on profile (r,I):
350.00000.25640000#
Point on profile (r,I):
400.00000.16900000#
Point on profile (r,I):
450.00000.10540000#
Point on profile (r,I):
500.00000
6.22000000E-02#
Point on profile (r,I):
550.00000
3.47000000E-02#
Point on profile (r,I):
600.00000
1.83000000E-02#
Point on profile (r,I):
650.00000
9.10000000E-03#
Point on profile (r,I):
700.00000
4.30000000E-03#
Point on profile (r,I):
750.00000
1.90000000E-03#
Point on profile (r,I):
800.00000
8.00000000E-04#
Point on profile (r,I):
850.00000
3.00000000E-04#
Point on profile (r,I):
900.00000
1.00000000E-04#
Point on profile (r,I):
1000.0000.00000000#
Impulse: POWER,TAU,NSTEP:
3.0000000
1.0000000
5
#
$
#
Mesh:
RMAX,DLTAR:
2000.0000
25.000000#
ZMAX,DLTAZ:
400.00000
4.0000000#
TMAX:
100.00000#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

Guide to Program TEMPROFILE

45

Shown below is the file GSSN.RIP, which contains the radial intensity profile (RIP)of the user-defined incident beam in Test_Run 3. The user can modify the contentsof this file with a text editor, then import it to the program environment during
subsequent runs of TEMPROFILE.
!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
@@ TEMPROFILE 5.3
File GSSN.RIP @@#!
@@ Date: 02/19/2005
Time: 17:35 @@#!
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
#! Notes: Gaussian intensity distribution with 1/e radius R0=300nm.
#!
#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#
Point on profile (r,I):.00000000
1.0000000#
Point on profile (r,I):
50.000000
0.9726000#
Point on profile (r,I):
100.00000
0.8948000#
Point on profile (r,I):
150.00000
0.7788000#
Point on profile (r,I):
200.00000
0.6412000#
Point on profile (r,I):
250.00000
0.4994000#
Point on profile (r,I):
300.00000
0.3679000#
Point on profile (r,I):
350.00000
0.2564000#
Point on profile (r,I):
400.00000
0.1690000#
Point on profile (r,I):
450.00000
0.1054000#
Point on profile (r,I):
500.00000
0.0622000#
Point on profile (r,I):
550.00000
0.0347000#
Point on profile (r,I):
600.00000
0.0183000#
Point on profile (r,I):
650.00000
0.0091000#
Point on profile (r,I):
700.00000
0.0043000#
Point on profile (r,I):
750.00000
0.0019000#
Point on profile (r,I):
800.00000
0.0008000#
Point on profile (r,I):
850.00000
0.0003000#
Point on profile (r,I):
900.00000
0.0001000#
Point on profile (r,I):
1000.0000
0.0000000#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

File GSSN.RIP created in Test_Run 3

The file TR03.PLS shown below was created during Test_Run 3, and contains thelaser pulse shape. Pairs of points (t, P) at the corners of the piecewise linear
function P(t), initially entered from the keyboard, are saved in this file.
!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
@@ TEMPROFILE 5.3
File TR01.PLS @@#!
@@ Date: 02/19/2005
Time: 17:44 @@#!
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#!
#! Notes: Trapezoidal pulse shape of the laser beam. The rise and fall#!
times are 10 nsec each, and the total pulse width is 60 nsec.
#!
#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#
Point on pulse (t,P):
10.000000
2.0000000#
Point on pulse (t,P):
50.000000
2.0000000#
Point on pulse (t,P):
60.000000.00000000#
Point on pulse (t,P):
100.00000.00000000#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

File TR03.PLS created in Test_Run 3

46

Guide to Program TEMPROFILE

Temperature above ambient (degree C)

250

(a)
time = 48 ns

200

60 ns

150

100
10 ns
50
100 ns
0
0

100

200

300

400

Z (nm)
400

(b)

300
200

Y (nm)

100
0

-100
-200
-300
-400
-400

-300

-200

-100

0

100

200

300

400

X (nm)

Figure 10. (a) Profiles of temperature versus z at (x, y) = (0, 0) computed at several instants oftime in Test_Run 3. (b) Isotherms in the XY-plane at z = 108 nm, time = 50 ns, computed in
Test_Run 3. The hottest spot at the center has temperature T = 237.15ºC (above ambient). The
plotted isotherms, starting in the middle and moving outward, correspond to T = 237ºC, 230ºC,
220ºC, 210ºC, 200ºC, 180ºC, 160ºC, 140ºC, 120ºC, and 100ºC.

Guide to Program TEMPROFILE

47

10. Test Run 4. In this example a trilayer stack is subject to a Gaussian laser beam(incident from the surface side) that moves at V = 5.0 m/s along the x-axis. During
the 60.0 ns time interval of the simulation, the laser is turned on and off three times,
creating three 10.0 ns pulses at equal intervals. Thus at the center of the second
layer, whose threshold temperature is 350°C (above ambient), three marks areformed. The various parameters of the simulation are shown in the COMMAND.DATand SESSION.DAT files below. Note in particular the MARK characteristics listed in
the SESSION.DAT. Figure 11 shows plots of the peak-temperature versus time andthe x-coordinate of the peak-temperature versus time in the XY-plane at z = 250 nm.
(a)

(b)

Figure 11. (a) Peak-temperature versus time and (b) x-coordinate of the peaktemperature versus time in the XY-plane at z = 250 nm.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ TEMPROFILE 5.3
File COMMAND.DAT @@#$@@ Date: 02/19/2005
Time: 17:53 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All lengths are in nanometers#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
$
#

48

Guide to Program TEMPROFILE

Compute impulse-response: COMPUTE!
Top/Bottom incidence: T!
Substrate's index (n,k):
1.560000
Thermal constants (C,K):
1.900000!
Number of layers: 3!
Layer 01: Thickness,n,k:50.00000
C,K:
1.650000!
Layer 02: Thickness,n,k:100.0000
C,K:
2.200000!
Layer 03: Thickness,n,k:200.0000
C,K:
1.450000!
Heat loss factor (GAMMA):
12500.00!
Incident beam profile: Gaussian
Lambda,R0:
405.0000!
Impulse: POWER,TAU,NSTEP:
1.000000!
Mesh:
RMAX,DLTAR:
2000.000
ZMAX,DLTAZ:
700.0000
TMAX:
60.00000$
Save impulse-response: AAAA.IMP$
Temperature profile: TEMPROFILE
Impulse-response file: AAAA.IMP!
Point on pulse (t,P):
1.000000
Point on pulse (t,P):
10.00000
Point on pulse (t,P):
11.00000
Point on pulse (t,P):
20.00000
Point on pulse (t,P):
21.00000
Point on pulse (t,P):
30.00000
Point on pulse (t,P):
31.00000
Point on pulse (t,P):
40.00000
Point on pulse (t,P):
41.00000
Point on pulse (t,P):
50.00000
Point on pulse (t,P):
51.00000
Point on pulse (t,P):
60.00000!
Beam velocity (V):
5.000000$
Determine mark size: MARK
Z,Threshold temperature:
250.0000$
Plot existing data-files: PLOT
X_Filename: TIME01.DATY_Filename: TPEAK01.DAT
Curve type (SRGB): 0000!
Xmin,Xmax:-2.000000
Ymin,Ymax:-10.00000
X-axis title: Time (ns)Y-axis title: PeakT (C)!
Save graph (Filename): PeakT.BMP$

0.000000
0.023000

2.100000
0.160000

5.900000

3.500000
0.068000

4.100000

1.730000
0.001000

0.000000

150.0000
1.000000
10.00000
2.000000

10.00000
10.00000
0.000000
0.000000
10.00000
10.00000
0.000000
0.000000
10.00000
10.00000
0.000000
0.000000

350.0000

62.00000
700.0000

10

#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#

Guide to Program TEMPROFILE

49

Plot existing data-files: PLOT#
X_Filename: TIME01.DAT#Y_Filename: XPEAK01.DAT#
Curve type (SRGB): 0000#!
#
Xmin,Xmax:-2.000000
62.00000#
Ymin,Ymax:-10.00000
250.0000#
X-axis title: Time (ns)#Y-axis title: PeakX (nm)#!
#Save graph (Filename): PeakX.BMP#$#
End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

File COMMAND.DAT created in Test_Run 4$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ TEMPROFILE 5.3
File SESSION.DAT $$$$ Date: 02/19/2005
Time: 22:57 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
Units: Length = nanometer, Time = nanosecond, Power = milliwatt***************************************************************
Designated Working Subdirectory: <TR04>***************************************@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Setup for impulse-response computation@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Multilayer characteristics:***************************
Semi-infinite substrate:(n,k)=(
C=

1.56000,
0.00000)
1.90000 K= 0.02300

LAYER #

1:

Z=

50.0

(n,k)=(
C=

2.10000,
5.90000)
1.65000 K= 0.16000

LAYER #

2:

Z=

100.0

(n,k)=(
C=

3.50000,
4.10000)
2.20000 K= 0.06800

LAYER #

3:

Z=

200.0

(n,k)=(
C=

1.73000,
0.00000)
1.45000 K= 0.00100

Heat loss factor from top surface: GAMMA=

0.13E+05

Incident beam characteristics(Surface incidence):****************************************************
Radial Intensity Profile: Gaussian
LAMBDA=
405.0
R0=
150.0
POWER= 1.0000
TAU= 1.0000
NSTEP= 10****************************************************
Mesh characteristics:*********************
RMAX=
2000.00DLTAR=
ZMAX=
700.00DLTAZ=
TMAX=
60.00
TAU=

10.00
2.00
1.00

IMAX=
JMAX=
KMAX=

200
350
60

The impulse response function is saved in AAAA.IMP**************************************************

50

Guide to Program TEMPROFILE@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Generating Temperature Profile(s)@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@The impulse-response function is read from AAAA.IMP***************************************************
Laser pulse shape:******************

Time
****
1.00
10.00
11.00
20.00
21.00
30.00
31.00
40.00
41.00
50.00
51.00
60.00

Power
*****
10.00
10.00
0.00
0.00
10.00
10.00
0.00
0.00
10.00
10.00
0.00
0.00

Beam velocity = 5.00 meters/sec (along +X axis)************************************************Mark characteristics in the XY-plane at fixed Z:************************************************
Plane of the Marks:
Z=
Threshold Temperature: TEMP=

250.00350.00Mark 01: Length=279.3, Width=
Onset Time=
4.0

279.3, Center (X,Y)=(40.6, 0.0)
Freeze Time= 12.0

Mark 02: Length=328.1, Width=
Onset Time=
23.0

327.5, Center (X,Y)=(135.7, 0.0)
Freeze Time= 33.0

Mark 03: Length=351.6, Width=
Onset Time=
43.0

349.6, Center (X,Y)=(233.7, 0.0)
Freeze Time= 53.0

Time-dependence of peak temperature and its location in XY-plane
at Z=
250.0 are saved in TIME01.DAT, TPEAK01.DAT and XPEAK01.DAT******************************************************************@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@TPEAK01 versus TIME01 is plotted in Graphics Window
Graphics Window is saved to PEAKT.BMP@@@@@@@@@@@@@@@@
Graphics Routine@@@@@@@@@@@@@@@@XPEAK01 versus TIME01 is plotted in Graphics Window
Graphics Window is saved to PEAKX.BMP
End of session: TEMPROFILE terminated normally**********************************************

File SESSION.DAT created in Test_Run 4

Guide to Program TEMPROFILE

51

References
1. M. Mansuripur, The Physical Principles of Magneto-optical Recording, Cambridge
University Press, London, 1995.
2. M. Mansuripur, G.A.N. Connell and J.W. Goodman, "Laser-Induced Local Heating ofMultilayers," Applied Optics, Vol.21, No.6, pp 1106-1114 (March 1982).
3. M. Mansuripur and G.A.N. Connell, "Laser-Induced Local Heating of Moving Multilayer
Media," Applied Optics, Vol.22, No.5, pp 666-670 (March 1983).
4. M. Mansuripur and G.A.N. Connell, "Thermal Aspects of Magneto-optical Recording,"
Journal of Applied Physics, Vol.54, No.9, pp 4794-4798 (September 1983).
5. M. Mansuripur, “Computer Modeling of Optical Storage media and Systems,” Proceedings of
SPIE, Vol. 1316, 70-80 (1990)
6. Y. C. Hsieh, M. Mansuripur, J. Volkmer and A. Brewen, “Measurement of the thermal
coefficients of nonreversible phase-change optical recording films,” Applied Optics, Vol. 36,
No. 4, pp 866-872 (February 1997).
7. M. Mansuripur, “Laser-induced Heating of Multilayers," Optics & Photonics News 10, 55-60,
May 1999.

52

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53

Appendix A
Reprint of the paper "Laser-induced Local Heating of Multilayers," by
M. Mansuripur, G.A.N. Connell and J.W. Goodman, Applied Optics,
Vol. 21, No.6, pp 1106-1114 (1982).

54

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61

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65

Appendix B
Reprint of the paper "Laser-induced Local Heating of Moving
Multilayer Media," by M. Mansuripur and G.A.N. Connell, Applied
Optics, Vol. 22, No.5, pp 666-670 (1983).

66

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67

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69

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Appendix C
Reprint of the paper "Thermal Aspects of Magneto-optical Recording,"
by M. Mansuripur and G.A.N. Connell, Journal of Applied Physics,
Vol. 54, No.9, pp 4794-4798 (1983).

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Guide to Program TEMPROFILEMM Research, Inc.
5748 N. Camino del CondeTucson, Arizona 85718

Laser-induced local heating of multilayers
Masud Mansuripur, G. A. Neville Connell, and Joseph W. Goodman

For a multilayer structure illuminated by a laser beam, absorption of optical energy in the absorptive layers

and the diffusion of the resultant heat throughout the structure are studied. Analytical and numerical procedures for this study are described, and, as a specific example, the profiles of temperature distribution during recording on a magnetooptical disk are presented. The technique is also expected to be of value for studies of thermal marking and laser annealing.

1.

layer, is propagating in the positive Z direction, and has

Introduction

Absorption of optical energy by a multilayer structure

and the consequent temperature rise within the structure have wide ranging applications. For example,
magnetooptical recording is achieved by heating a thinmagnetic film, which is typically overcoated with a thintransparent layer, with a narrow laser beam.",2 Thermal marking and laser annealing of silicon are otherexamples. We present in this paper, therefore, the tools

required for a numerical investigation of this class ofproblems. Notation and the general setup of the
problem are described in Sec. II, while details of analytical and numerical procedures are explained in Secs.
III and IV. Section V presents the numerical results
obtained for a case of practical interest, and the concluding section is devoted to some final remarks andconsiderations.
11. Notation and Sketch of the Problem

Figure 1 shows an N-layer structure which is illuminated on the surface by a Gaussian beam of light. The

coordinate system is cylindrical, and the surface of themultilayer is in the z = 0 plane. The structure extends
to infinity in all other directions. Starting at the substrate with k = 1, the layers are numbered in increasingorder. The kth layer has thickness Zk, refractive index
nk = n () - inlh,? specific heat Ck, and heat conductivity
Kk. The beam is stationary with respect to the multi-a Gaussian intensity profile with radius r at theexp(-1) point. The wavelengthand the instantaneous
power of the beam will be denoted by X0 and Po(t), respectively.
The temperature rise above ambient at point (r,z)
and time t is O(r,z,t)and it will be assumed that the rateof flow of heat from the surface is proportional to 0, i.e.,(O/9z)0(r,z = O,t) = yO(r,z = O,t), where the propor-

tionality constant y must be non-negative.
For a unit intensity incident plane wave,the averagerate of flow of optical energy through a plane perpendicular to the Z axis at z is denoted Y(z) and can be
determined from the electromagnetic wave theory.
Using Maxwell equations and the Poynting vector
theorem, we will derive a general expression for Y(z) inSec. III. The slope of Y(z) determines the rate of absorption of energy per unit volume at point z. As forthe diffusion of the heat thus generated, analytical solutions to the heat transfer equations do not exist, andwe must be content with numerical solutions. The
well-known implicit method for solving partial differential equations is briefly described in the Appendix,
and the necessaryinformation for the application of thistechnique to the present problem is provided in Sec.
IV.
111. Optics of Multilayers

A plane optical wave propagating in the positive Z
direction illuminates the surface of the multilayer in Fig.
1. The following Maxwell equations apply to theelectric and magnetic field components in the kth
Joseph Goodman is with Stanford

University, Department

ofElectrical Engineering,Stanford, California 94305;the other authors
are with Xerox Corporation, Palo Alto Research Center, 3333 Coyote
Hill Road, Palo Alto, California 94304.
Received 15 August 1981.
0003-6935/82/061106-09$01.00/0.
©1982 Optical Society of America.
1106

APPLIED OPTICS/ Vol. 21, No. 6 / 15 March 1982

layer 3 :I(d2/dZ 2 ) + [27rn(k)/Xo]2}Ek(Z) = 0;(1)

+ i(27r/Xo)Hk(Z) = 0.

(2)

(d/dZ)Ek(Z)The general solution to these equations can be
writtenEk(Z) = A

I((exp {i [27rn (k)/IO] (Z -+ AV exp {+i[27rn(k)/Xo] (z _

z4}

'

AVexp

{+i[2irn(k)/XoI(

(3)

~~N+1

-

E

sk)

-

it

N+1

r

-A
[2wn
exp
S+i ZZ

Hk(Z) = n(k)A?) exp l-i[27rn(k)/Xo]an Beam

z)

Ijl

N+1

(4)
ZN
_ , ZN-1

In the above equations ZN+1 = 0. Coefficients A1 andA2 must be determined in view of the continuity requirements at the boundaries. The continuity of H/E

ZN-2

is assured if the various values of A2 satisfy the following

recursive relation:
Ask) -

1[n(k) - n(k)]/[n(k) + n(k-1)] + Ak-')

1 + [n(k)- n(k-1)1/[n(k)+ n(k-)]JAl1)
X expt-i[4rn(k)/Xo]ZkJ;

iZ2

2 < k < N + 1.

Since the reflected component of light in the substrateis zero one can start with AV) = 0 and determine allother A2 from this equation. Note that AN+') is theZ1

amplitude reflectivity of the surface; nN+1 is the refractive index of air, through which light travels to reachthe multilayer, and ZN+1 = 0.
Once the coefficients A2 are calculated, one uses the-continuity of the electric field component to determine
the various A1 . From Eq. (3) one obtains

tion, illuminates the surface. The axis of symmetry for the beam isr = 0.

Aik) =(exp[-i[2rn(k+l)/Xo]zk+,1 + A?+l)exp{+i[27rn(k+1)/XIzk+1})
1 + Ak)xAk+')

1 <k <N.

independent of 0. Starting at time t = 0 a narrow, circularly symmetric, Gaussian beam of light propagatingin the z direction illuminates the surface. The incident
intensity distribution can be written

AIN+) is the incident light amplitude, and for light withI(r,t) = [Po(t)/(7rr')] exp[-(r/ro) 2 J,

unit intensity it is equal to \.
Other A 1's can be determined recursively from the above equation.
Having E and H determined, one invokes the
Poynting vector theorem3 to calculate the rate of flowof energy through the multilayer. The average amountof power per unit area that crosses the plane perpendicular to the Z axis at z can now be writtenY(z) = Re[l/ 2 E(z)H*(z)] = 1/2[np) (exp |-[47rn?)/oI-

2nhI)A)I|(Z

sin [47rnk) /Xo (Z -E

(6)where ro is the exp(-1) radius of the Gaussian beam andPo(t) is the instantaneous output power of the laser.
Denoting by O(r,z,t) the temperature rise above ambient and assuming that heat loss from the surface is
proportional to O(r,z = 0,t), one obtains the following
heat transfer equations4 :

zj)}-

L

|

Fig. 1. Multilayer in the cylindrical coordinate system (r,O,z). The
number of layers is N, and the surface is in the z = 0 plane. A circularly symmetric Gaussian beam of light, propagating in the z direc-

IA)

zj) +

2 exp
[A

)]

{+[47rnm/XoI Z
I

L

z)})
(5)

I,

where 4j[A ^)] is the phase angle of the complex number

Cn(o/9t)0(r,z,t) - KnV2O(rjz,t) = g(r,z,t);(7)A&)- If required, the rate of absorption of energy perunit volume (d/dz) Y(z) can be easily derived from Eq.

(a/z)0(r,z = O,t)= yO(rz = O,t);

(8)

(5).

O(rz = a,t) = O(r= -,z,t)

IV. Implicit Technique and the Problem of Heat
Transfer in Multilayers

O(r,z,t = 0) = f(r,z).

Consider Fig. 1 where a multilayer in the cylindricalcoordinate system (r,O,z) is shown. The number oflayers is N, and, starting at the substrate with n = 1, thelayers are numbered in increasing order. The initial
temperature distribution of the multilayer is f(r,z),

= 0;

(9)
(10)

O(r,z,t) and K(G/Oz)0(r,z,t) must be continuous at theboundaries between adjacent layers.
In Eqs. (7)-(10) Cn is the specific heat of the nthlayer, Kn is the heat conductivity of the nth layer, V2 =(02/3r2 ) + r-1(a/Or) + ( 2/0z2 ) is the Laplacian operator
15 March 1982 / Vol. 21, No. 6 / APPLIED OPTICS

1107

being that there are now two spatial coordinates rather
than one. Nevertheless, it is still possible to obtainstable solutions by separately applying the 1-D technique to each dimension. In the so-called alternating
direction-implicit technique,5 one divides the time stepfor circularly symmetric functions, g(r,z,t) is the power
delivered to the unit volume by the laser, y is a constantthat controls the rate of heat flow from the surface, andf(r,z) is the initial temperature distribution.
The implicit technique described in the Appendix
cannot be directly applied to Eqs. (7)-(10), the reason

At into two equal half steps, 1/2At each. In going from

Table 1. Basic Equationsof AlternatingDirection-ImplicitTechniquefor Problem of Heat Transfer In Multilayers

Part I: First Half Step (z Implicit, r Explicit)
iEo

J

BYWii1'1 + B
3,04+1/2
JmarJmx- = L 4

0+ AYO);/2 = A ,)(kl+1/2

1

3(

0

AV.) = [(Az) /(DN/t)]+ (1 + YAz)1(c)a0 < i < ima

Jmar

iOQ= f(iArjAz);
2

(b)

1

OS

4

2

jmax.. -

(a)

imax

0 <i S imax; 1 S j Smax0jk);++
1 = B3+ BY)ii

0i S imax;) j max

(d)

1

(e)

A0) = [(AZ)2/(DNAt)IoOk + 2(Az/Ar)2[1O0 - oOk] + [2AZ/(KNAt)]
X i1

-

2

exp[-('/ 2 Ar/r)2]i/[r(I/2Ar)

11[y(0) -Y(Q/ 2 Az)j[,fk+

Po(t)dt]

X l[(Az)2/(DNAt)]+ (1 + yAz)-'
AY"0) = [(Az)2/(DNAt)]iOk(f)2+ 1/2 (Az/Ar) 1[1 + 1/(2i)]i+Oko - 2iOo + [1 - 1/(2i)Ii-i01U

2

2+ [2Az/(KNt)](lexpl-[(i - 1/2)Ar/ro]21- exp[-[(i + /2)Ar/roI ]/[27i(Ar) ])X [Y(0) -Y('/2Az)] IIk

BW)= -1/ 2Dn [At/(Az) 2 ];2

X 1[(Az) /(DNAt)] + (1 + yAz)[-lPo(t)dt
1

fkAt

(g)

if j belongs to layer n
2
if j belongs to the interface b];

B) = -lDn/[1 + (Cn- 1/Cn)]j[At/(Az)2
if j belongs to layer n
B ) = 1 + Dn[At/(Az) ];2
B) = 1 + [Dn + (Cn- 1 /Cn)Dn- 1 ]/[1 + (Cn~-/Cn)Ii[At/(Az) I;2
1];
if j belongs to layer n2(Cn- 1 /Cn)][At/(Az) ];/[1+=-l(Cn-C/Cn)DnB2)
1(h)
if j belongs to the interface(iBY) = - / 2Dn[At/(Az)if ] belongs to the interface

BfPj'k) = oO4 + 2Dn[At/(Ar)2[1Oj - oO4] + [1/(CnAz)]
12
2- 1/2 )Az] - Y[(j + /2)AZ]}
X [11 - exp[-(/ 2 Ar/ro) ]1/[r(/ 2Ar) 1]l[(X

1Er(k+1/2)At
Po(t)dtj ;
k12

(J)

if j belongs to layer n

2
B3j~k = 004 + 21[Dn+ (Cn-1 /Cn)Dn-1 ]/[1 + (Cn-1/C0)Ii[At/(Ar) ](i04 - 0)22- /2)Az] - Y[(j + 1/]][Y[(jAr)
1/[7r(/
Ar/ro)
exp[-(/+ 12/[(Cn + Cn~-Az][112 )Az]]
22

rr (k+ l /2)At

X tJk

B3

1

if j belongs to the interface

Po(t)dtj;

2= iO4 + 1/2 Dn[At/(Ar) ]Jl1 + 1/(2i)]i+ 1 04 - 2iO + [1 - 1/(2i)]i-1Oj4
2
2
2+ 1/2 )Ar/roI D)/[2iri(Ar) ]]- /2)Ar/ro] 1-expl-[(i+ [1/(CnAz)][(exp-[(iX IY[(J - 1/2 )Az] - Y[(J + /2 )AzI

1J/'A

if j belongs to layer nPo(t)dtj;
2]
BV'ijk) = iOj + '/2 1[Dn + (Cn~-1Cn)Dn~-][1 + (C.-1/Cn)][At/(Ar)X |[1 + 1/(2i)]jijj0 - 20 + [1 - 1/(2i)]itij0; + 12/[(Cn + Cn-i)Azfl

22
X [(expl-[(i - /2 )Ar/ro]2 1- expl-[(i + 1/2 )Ar/ro] )/t27ri(Ar) ]]Y1(j - 1/2 )Az] - Yt(J+ 1/2)Az])
X[SJk1Po(t)dt

if j belongs to the interface;

Part II: Second Half Step (r Implicit, z Explicit)
oO4+1 = ASY)e10+

1

0 j s max0

+ Ap j);Bo,j)i-10j4+1 + BV-i)ij4+1 + B2' i)i+lo04+1= B36i'k);
km4+1= im 30+1/2 - 3(42204+1/2 -2

OJ' Available after the first step;

1

iimax-1;

maxo1);0i S imaX;

(a')

imax
0]s j smax

(b')

0 j

(d')

0

j

imax

(c')

continued
1108

APPLIED OPTICS/ Vol. 21, No. 6 / 15 March 1982

Table 1. Basic Equationsof AlternatingDirection-implicit Techniquefor Problem of Heat Transfer in Multilayers (continued)Aff) = [(IAr)2 /(2DNAt)I + 1}-1

(e')

2

if ] belongs to layer nA\°) = [(Ar) /(2DnAt)] + 1[-1;2A|°) = [(Ar) /(21[Dn+ (Cn- 1 /Cn)Dn-1]/[1 + (Cn_1/Cn)I}t)] + 11-1;
A T°')= {[(Ar)2/(2DNAt)]o0+0/2 + 1/2 (Ar/Az)2[o 1+'/

2

if j belongs to the interface+ -yz)O0k" 2 I

-(1

--

2
2
2
+ [(Ar) /(KNAzAt)I[1 - exp[-(/ 2 Ar/ro) 11/[7r(/ 2 Ar) ][Y(0) - Y(/2Az)](h+l1)Atk
2
Po(t)dtjj

X

X [(Ar) 2 /(2DNAt)] + 1}'(k+l/2)Qt

A(f')= [[(Ar)2/(2DnAt)IO00+1/2 + 1/4(Ar/Az)2(0(k+/ 2 - 200k+1/2+ 00 4 +1/2 )+ [(Ar)2/(2KnAzAt)](1- exp[-('/2Ar/ro)2 /[ir(1/2Ar)21)$Y[E( /2)AZ] Y[(J+ 1/2)AZ]1
X

r(k+ l)Qt

1

X [(Ar)2 /(2DnAt)

Po(t)dt

if ] belongs to layer n+ 11- ;

(k+1/2)At2A )= [(Ar)2/(21[Dn+ (Cn- 1/Cn)Dn-1]/[1 + (Cn-lCn )]^t)]004+1/2 + 1/4 (Ar/Az)

X [2(Cn-/Cn)Dn-1/[Dn + (Cn-1/Cn)Dn-]io0- 2o0e+1/2+ 2Dn/[Dn+ (Cn-1Cn)Dn-11}oei4-+ + (Ar) 2 /[(Kn + K-O)AZAt]}
2
1
2
X [1 - exp[-( / 2Ar/ro) 1W/[7r('/
2Ar) ]IY[(j - 1/2)Az] - Y[(j + /2)AZI

x(k+ l)Qt(k+1)At Po(t)dtl
J
J(k+l1/2)Qt[(Ar)2/(21[Dn+ (Cn-1/Cn)Dn-1 ]/[1 + (Cn-1/Cn)IAt)] + 11-1;Bo,°) = -/ 2 DN[At/(r)2

1[1 - (1/2i)]B.oj) = -1/ 2Dn[At/(r)2][1 - (1/2i)];

BOj) = -1/2 1[Dn + (Cn-1/Cn)Dn-1]/[12BV °) = 1 + DN[Atl(Ar)

2

(g')

if j belongs to layer n2+ (Cn-/Cn)][At/(Ar)][1 - (1/2i)];(h')
if j belongs to layer nB? i) = 1 + {[Dn+ (Cn_1 /Cn)Dn-1j/[1 + (C,,-.i/Cn)]i[At/(Ar)2 ];BY,.) = -/ 2 DN[At/(Ar)if j belongs to the interface]

B Vi) = 1 + Dn[At/(Ar) ];2

if j belongs to the interface

if j belongs to the interface(i')

1[1 + (1/2i)]2

if jbelongs to layer n

BYij) = -/ 2 Dn[At/(Ar) ][1 + (1/2i)];= -/ 2 1[Dn+ (Cn- 1 /Cn)Dn-1J/[1+ (Cn. 1/Cn)]j[At/(Ar)2 ][1 + (1/2i)];BY&.)

if j belongs to the interface

Avi'kl1
2

By °k) = o+1/2 + DN[At/(AZ)2][ekO+l/- (1 + yAZ),k0+1/2]+ [2/(CNAZ)]222X [(expl-[(i - 1/2)Ar/r0 ] 1 - expl-[(i + 1/2 )Ar/r0] j)/[2ri(Ar) ]]X [Y(0) - Y('/ 2Az)]
2po(t)dt
P(+l)otf(k+ 1/2)At1

Bi,j,k) - (0+1/ + /2 D[At/(AZ)
X [(expl-[(i- l/ 2)Ar/ro]
-

12

2

(i0,k+l/2

1-~~ ~ +'/2) Az{ MU - 12)AZI - Y[(7+-expl-[(i12)AZI

240'/

2

j')

2

+ t041/ ) + E1/(CnAz)I2+ 1/2 )Ar/ro] 2 1)/[27ri(Ar) ]]
+

tPOW&

if] belongsto layer n

B jk) = L0+1/2+ 1/21[Dn+ (Cn-1/Cn)Dn-1]/[1 + (Cn- 1 /Cn)][At/(Az)2 ]2-2jjIh0,+I
X (2t(Cn. 1 /Cn)Dn-1 /[Dn + (Cn.l/Cn)Dn-+ Cn-l)AZfl
j-+ ) + 12/[(Cn+ (Cn-llCn)Dn+2IDn/[Dn22
12

X [(expl-[(i - /2)Ar/ro] 1- expl-[(i + 1/2 )Ar/ro] j)/[27ri(Ar) ]](k+l)Atif]j belongs to the interface
Po(t)dt ;• MU[(- 1/2)AZ]- Y[J + 1/2 )AZI
ISfk+1/2)Ata In going from Eq. (9) to its discrete counterpart one could either select a large jmaxand set Ojma, = 0 or take a somewhat smaller jmaxand
extrapolate for 0 at the extreme edge of the grid. Equation (c) is the manifestation of the second approach.
b Grid must

include all the interfaces.
15 March 1982 / Vol. 21, No. 6 / APPLIED OPTICS

1109

-T

OVERCOAT(Glass, n-1.5)

1000A
120

-MAGNETIC FILM

7-CX--7(MnBi, n.3.1-3.5i)
T -- ,""I'll"....................................._INTERMEDIATE
LAYER

4

720 A

(Glass, n-1.5)- REFLECTOR(Aluminum, n 1.2- 6.9 i)

320 A

T

3
S
0
0.

2

SUBSTRATE

I(PMMA, n-1.46)
I

Fig. 2.

I

Structure of a quadrilayer magnetooptical

disk.
4
S
E

Table II.

Numerical Values for Optical and Thermal Parameters of the
QuadrilayerZ(A)

1. Substrate(PMMA)
2. Reflector

Refractive
heat C
index n (J/cm 3 /deg)(J/cm/deg/
sec)

X

1.46

1.7

0.002

320

1.2-6.9i

2.7

2.4

720

1.5

2.0

0.015

120

3.1-3.5i

2.6

0.16

1000

1.5

2.0

0.015

(aluminum)
3. Intermediate
layer (glass)
4. Magnetic Film
(MnBi)
5. Overocat(glass)

0.

2
1

Heat
conductivity
SpecificK

Thickness
Layer

3
0

20
Fig. 4.

30

40 50 60 70 80 90 1't(ns)Laser pulses used in the numerical calculations of tempera-ture distribution in the quadrilayer: (a) 60-nsec pulse with peak
power of 2 mW and total energy of 96 pJ used in calculations pertaining to Figs. 5-10; (b) 40-nsec pulse with peak power of 8/3 mW

and total energy of 64 pJ used in calculations pertaining to Fig. 11.

kAt to (k + /2 )At one treats r explicitly and z implicitly;
in going from (k + /2)At to (k + 1)At one treats z ex-

plicitly and r implicitly. (The order of r and z is unimportant and might be reversed.) The procedures foreach half step parallel those of the Appendix, and theequivalent of basic Eqs. (A5) is all that is required.

Table I represents these basic equations for the multilayer of Fig. 1. In this Table iO4stands for the approximation to temperature at r = ir,z = jAz, and t= kAt; D,, = K,,/C,, is diffusivity of the nth layer; and1.0

Y(-) is the rate of flow of energy through the multilayer,

which was discussed in the previous section.
V.

Numerical Results

In this section we consider as an example the problemof thermomagnetic writing on a magnetooptic disk.'Specifically, we apply the thermal analysis to the so-

Y (Z)called quadrilayer disk whose structure is shown in Fig.

2. A quadrilayer is an interference device that exhibits
large SNRs during magnetooptical readout and is thus

a preferred medium for magnetooptical recording.2 Weshow the profiles of temperature distribution in thequadrilayer when it is subjected to a narrow beam oflight and examine the effects of heat flow from thesurface on these profiles. We also give an example of
how energy can be used more efficiently in conjunction

with the recording process. The material parameters
used in this analysis are given in Table II. Throughout
OVERCOAT

FILM

LAYER

REFLECTOR SUBSTRATEZ (A)the section it is assumed that the beam has wavelength
X = 6328 A and radius r = 3500 A at the exp(-1)

point.
Fig. 3.

1110

Average rate of flow of optical energy through the quadrilayer
for a plane wave with unit intensity.
APPLIED OPTICS / Vol. 21, No. 6 / 15 March 1982

Figure 3 shows Y(z), the average rate of flow of opticalenergy through a plane perpendicular to the Z axis at

250

center begins with the decline of laser power but is delayed by the effects of diffusion at larger radii.
In reality, especially when the disk is rotating, convection losses at the front surface might be important.
To see what effect this has on the temperature profiles,

200

we choose a value of y = 105 cm-' that makes the convection losses be of the same order of magnitude as the
150
0

6
Z
01

250

100
200
\t-48

ns

50
0

150

c'J

t-60n

VI

0

500
I
OVERCOAT

1000

1500 2000

2500 3000

0
0

100

INTERMEDIATE
FILM
LAYER
REFLECTOR SUBSTRATE

Z(u)0Fig. 5. Quadrilayer's temperature (above ambient) vs z at the centerof the beam (r = 0) for several instants of time. At t = 0 the quadrilayer has been at ambient temperature. The laser pulse is shown in

50 _

\

t=100

Fig. 4(a), and the radius of the beam at the exp(-1) point is 3500A.
For these calculations the heat flow from the surface has been ne-

0 ... .. 1,, --,glected (i.e., y = 0).

4000

8000

12000

16000

r (AR)z for unit intensity incident light. Note in particularthat almost all the energy is absorbed by the magnetic
film (1000 A < z • 1120A) and that within the film theabsorption is quite uniform.
The heat generated in the absorbing layers diffusesin all directions and creates a complex pattern of temperature profiles. To begin, we assume that the heatflow from the surface is negligible (i.e., -y = 0). For thelaser pulse shown in Fig. 4(a) temperature profiles of thequadrilayer are plotted in Figs. 5, 6, and 7. Figure 5

shows temperature vs z at the center of the beam (r =0); the curves for three instants of time are shown. At
t = 48 nsec, while the laser is still on, the film has a hightemperature; some of its energy is escaping to theovercoat, but most of it is flowing rapidly to the highly
conductive aluminum layer. At t = 60 nsec, with theFig. 6. Radial temperature distribution of the magnetic film atseveral instants of time (conditions as in Fig. 5).

6
V

N
VI

0
0
0

0l

laser extinguished completely, the film is cooling down.

Again the aluminum layer and the substrate are absorbing most of the film's thermal energy. Finally at
t = 100 nsec, only 40 nsec after the laser is turned off,
the temperature is distributed uniformly throughout
the structure, and the situation is essentially back tonormal. Note that at all instances the film maintains
a uniform temperature throughout its thickness.
Figure 6 shows the radial temperature distribution

CD

0

10

20

30

40

50

60
t ns)

70

80

90 100in the film for the same three instants of time; the curves
are self-explanatory. In Fig. 7 we have plotted the timedependence of the film's temperature for several values

Fig. 7. Time dependence of the magnetic film's temperature atof r. Note that the temperature decay around the beamvarious radii (conditions same as in Fig. 5).
15 March 1982 / Vol. 21, No. 6 / APPLIED OPTICS

1111

heat lossesto the substrate. We believe, therefore, that
this is a reasonable upper limit for the value of -y. The
results, which are parallels of those in Figs. 5, 6, and 7,
are then shown in Figs. 8, 9, and 10. As expected, thetemperatures attained in this case are lower than those250

obtained previously, but the general features of thecurves have remained intact. We have also plotted inFig. 11 the time dependence of the film's temperaturefor several values of r when the narrow laser pulse of Fig.

4(b) is applied. Now in comparison with the results in
Fig. 10 it is observed that not only is less energy consumed by using the narrower pulse, but also higher
temperatures are achieved, because in the case of thenarrow pulse the local temperature buildup occurs before the accumulated heat gets a chance to diffuse to the200

neighboring regions.
In thermomagnetic recording, the region of themagnetic film whose temperature has risen above theCurie point becomes reverse magnetized in an appropriately applied magnetic field. For a typical material
with Curie temperature TC = 1300C,Fig. 7 shows that1500

a 50-nsec pulse with 2 mW of power is capable of writingN

a 1-,um diam spot when convection losses from thesurface are ignored. When they are taken into account100

0

the same result obviously requires more laser power, butaccording to Fig. 10, when y = 105 cm-', the required
power is increased only to 2.5 mW. We can, therefore,

50

conclude that for the quadrilayer structure convection
losses are unlikely to play a significant role in the write

process.
0

500

1000

1500

2000

2500

3000

Z(A)
Fig. 8. Quadrilayer's temperature (above ambient) vs z at the centerof the beam (r = 0) for several instants of time. At t = 0 the quadrilayer has been at ambient temperature. The laser pulse is shown in

Fig. 4(a), and the radius of the beam at the exp(-1) point is 3500 A.
For these calculations the escape of heat from the surface has beentaken into account by setting y = 105 cm- 1 .

Concluding Remarks

In this paper we described the analytical and numerical procedures for the study of laser-induced local
heating of multilayers. The optical aspect of the
problem was dealt with analytically, and we used
Maxwell equations to derive a general expression for therate of absorption of optical energy in different layersof the multilayer. The optical problem, however, was

250

250

200

200

0

6
V

6
V

VI.

N 150
VI

150

00

N
VI

0
0

00
-

100

100

CD

50

50

0

0

0

4000

8000

12000

16000

r(A)

t(ns)

Fig. 9. Radial temperature distribution of the magnetic film at

Fig. 10. Time dependence of the magnetic film's temperature atseveral instants of time (conditions same as in Fig. 8).

various radii (conditions same as in Fig. 8).

1112

APPLIED OPTICS/ Vol. 21, No. 6 / 15 March 1982

solved for plane incident waves, and the direct application of results to the case of narrow beams requires
some justification. To this end, the assumption thatthe depth of focusis larger than the multilayer thickness
has been implicit throughout our analysis.
The numerical solutions of the heat diffusion equation were obtained by the alternating direction implicit

technique. These solutions were found to be stable ina wide range of parameters, and the computation cost
was reasonably low. Finally, a specific case of magne-

tooptical recording on a quadrilayer disk was studied,and it was shown that with relatively low laser powers(-2 mW) and in rather short periods of time (50 nsec),
temperatures that are high enough for thermomagnetic
recording can be achieved.

250

200

Appendix: Explicit and Implicit Techniques for
Solving Partial Differential Equations

r-O
0

5

Consider the 1-D heat transfer equation4

r0.5ro

S

VI
N 150

[C(a/at) - K(a2/x 2 )]0(x,t) = g(x,t);0 S x <L; t 2 0, (Ala)

VI

0
0
0
0

with the following boundary conditions:(a/ax)0(x = 0,t) = yO(x = Ot);

100

0

0

10 20 30 40

50

60

70 80 90100t (ns)

Fig. 11. Time dependence of the magnetic film's temperature atvarious radii. At t = 0 the quadrilayer has been at ambient temperature.

The laser pulse is shown in Fig. 4(b), and the radius of thebeam at the exp(-1) point is 3500 A. For these calculations the escape of heat from the surface has been taken into account by setting-y = 105 cm-' (compare with Fig. 10).
QW(0,+'- 04)/At](Alc)

O(x,t = 0) = f(x).

(Ald)+ 0k4 )/(Ax)2 ] = [1/(AtAx)]

K[(O,+1 -20&-

O(x = L,t) = 0;

In the above equation O(x,t) is the temperature rise
above ambient at point x and instant t; C is the specificheat; K is the heat conductivity; g(x,t) is the power
delivered to the medium per unit length; -y is a constantthat controls the rate of heat flow at x = 0; and f(x) is
the temperature distribution at t = 0.
A numerical solution to Eqs. (Al) requires the replacement of the continuous space-time domain witha discrete set of points. Let us define
as the approximation to the temperature at (xt) = (Ax,kAt),
where k 0and 0 <j ima = L/Ax. One way to approximate Eqs. (Al) with difference equations is towrite__________________0[r (k+1)Atd(j+1/2)Ax
1
kAt
Jd1/2x g(x,t)dx;-Kf[2-O)/At](Alb)<jmax-1;k20

1]

(A2a)

x)OI/(x)2}0= [2/(AtAx)]-2(1 +[r(k+)At

X fk1tr /2Ax1
dt g
g(x,t)dx ;hAt

O'm,= =0;k

OJ =fjAx);

0

k

(A2b)

S

0

(A2c)

0

j <•Imax(A2d)

The following rearrangement of Eqs. (A2) shows how

the solution at time k + I can be directly derived fromthe solution at time k (diffusivity D is substituted for

K/C):
OJ5=f(jAx);
=

J ,

-kl=

O'+'
=0
Jmakx

0 < j imax<

+ [DAWt/( x)

2

[2O

(A3a)
-

2(1 + -y\x)Ok] + [2/(CAx)]l + [D/t/(Ax) 2 [O+1 - 2
k0

J

+

[SkAt5_I + [1/(CAx)]f()
J_
JkAt

dt

dt 5

Jfm1/2)Axg(xt)dx;g(x,t)dx

I

k

0

1<

(A3b)

< ma -1; k

o

0

(A3c)
(A3d)

15 March 1982 / Vol. 21, No. 6 / APPLIED OPTICS

1113

The explicit technique above for solving partial differential equations has several shortcomings, the mostimportant being the heavy dependence of 9k+1 on Ok.
The error, which is the difference between the solutions

A) = -BY)/[BV) + BY)AV`)],(A6b)A)= [Byk) - B+)AV-1)/[B)+ B)AV`)].

(A6c)

We are now in a position to calculate Ok by the fol-of Eqs. (Al) and (A2), is magnified at each stage of
calculation by a factor of [DAt/(Ax)2 ]. To ensure
convergence it has been shown that [DAt/(Ax)2 ] mustbe less than 1/2. In practice, this constrains At to verysmall values, resulting in long computation times andlowing procedure:(1) set k = 0;(2) calculate BY),BV),BY),Bank) for 1 < i < imax1 from Eqs. (A5g)-(A5j);(3) calculate AV),A&7)'for
0 < < imax -1 from Eqs.

excessive round-off errors. 6(A5e),(A5f) and (A6b),(A6c);(4) calculate 0 1 for 0 2• < imax from Eqs. (A5c)and (A6a);

The implicit technique, which we shall shortly describe, overcomes the aforementioned problems.6 In
this technique one starts with the following approximation to Eqs. (Al):(5) set k = k + I and repeat the procedure.

2
C[(04+ - O)/At] - K[(0,++ - 20+1 + Ok+i)/(AX)] = [/(AtAX)1r(h+)Atp (+1/2)kt
dt(_ 112, g(x,t)dx
hAt
S1 2)AxX(A4a)2
k[2O^+-2(1+ YAX)O(+l]/(Ax)1 = [2/(AtAx)]

K

C[(@O+l- O)/At]-

1 <j <jmax-I;k 0(k+ 1)AtX

X IIk~

hAt

04jm
J.

=

;

k

0jQ= f UAx);

dt So

1/2Ax

JO

g~x,t~uxi;

k 0

(A4b)
(A4c)

0

0•j,-ijrnaxi..

(A4d)

These equations differ from Eqs. (A2) in that c2 0/0x2
is now evaluated at time k + 1 rather than k. The recursive technique of solving Eqs. (A4) begins with thefollowingrearrangement:= A\)Oh+l + A&°);
BY,')04+'+ BV)O+' + BY')O+

Byk);

-

k > 0

1 < j < jm

(4+m = ;

k

-

(A5a)
1; k 2 0
(A5bj

0

(A5c)
(A5d)

J=x

in whichAi') = 1[(Ax)2 /(2DAt) + (1 + yAx)1'-(Ae)

A20)= {[(Ax)2/(2DAt)10o + [AxI(KAt)]
X II(h+1)At

p1/2A

dt

2

(A5g)

With the implicit technique stable solutions to thepartial differential Eq. (Al) can be obtained. At andAx must be small enough to make the discrete approximation to the continuous problem reasonable;
unlike the explicit technique no other restrictions
exist.

(Ah)

References

g(x,t)dx

SO
X [(Ax)2 /(2DAt)] + (1 + yAx)}-',kAt

BY) = -[DAt/(Ax)

2

1,
2

By) = 1 + 2[DAt/(Ax) 1,
BY) = -[DAt/(Ax)

2

],

(A5f)

(A5i)

BWk) = oh + [1/(CAx)]xS(h+1)At

JkAt

dt

5

(j+1/2)Axg(x,t)dx * (A5j)

J u-1/2)I

From Eqs. (A5a) and (A5b) and with the aid of induction we obtain the general equation
J=A

k+1 + A)-0< j < jm-1,(A6a)where the coefficients for j = 0 are given by Eqs. (A5e)and (A5f), while for 1 S j < imax - 1 they can be ob-tained recursively from1114

APPLIED OPTICS / Vol. 21, No. 6 / 15 March 1982

1. D. Chen, G. N. Otto, and F. M. Schmit, IEEE Trans. Magn.
MAG-9, 66 (1973).
2. M. Mansuripur, G. A. N. Connell, and J. W. Goodman, "Signal andNoise in Magneto-Optical Readout," to be published in J. Appl.
Phys. May 1982.
3. J. D. Jackson, Classical Electrodynamics(Wiley, New York,1967).

4. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids(Oxford U. P., London, 1954).
5. G. Birkhoff, R. S. Varga, and D. Young, "Alternation

Direction

Implicit Methods," in Advances in Computers, F. L. Alt and M.
Rubinoff, Eds. (Academic, New York, 1962).

6. W. F. Ames, Numerical Methods for Partial Differential Equations (Barnes & Noble, New York, 1969).

Laser-induced local heating of moving multilayer media
Masud Mansuripur and G. A. N. Connell

Earlier work on the local heating of stationary multilayer structures by focused laser light has been extended
to deal with nonstationary situations. The numerical procedures described here are therefore applicableto many important technologies including optical recording, thermal marking, and laser annealing. We
demonstrate this in two examples, namely, the effects of readout intensity on the readout signal from a quadrilayer magnetooptic disk and the writing threshold for ablative materials in single-layer and three-layer
structures.

1. Introduction

Local heating of thin films by focused laser beams is
becoming increasingly important in technology, withlaser annealing of silicon and thermal marking perhaps
the most obvious examples. In a recent paperl we
presented a technique for calculating the temperaturerise in a stationary multilayer medium that occurs whena laser beam, focused to a Gaussian spot, is incident on
it. While this approach provides very useful results fora large variety of problems, it is inadequate for cases inwhich the motion of the medium (or laser spot) is nolonger small. This occurs when the relative distancetraveled by the medium during the laser pulse is comparable to or greater than the diameter of the laser spot,
a condition often met with in practical situations. In
Sec. II, therefore, we show how our earlier work can beextended to allow the calculation of the temperature
profiles in time and space for moving media. The
method can then be applied to numerous new problems,
and we select here as examples an exact application inmagnetooptic recording in Sec. III and an approximate
one in nonerasable optical recording in Sec. IV. Finally,a brief summary of the results is presented in Sec. V.
II. Temperature Distribution in Moving Media by
Superposition of Static Distributions

The problem of light absorption and heat diffusion
in multilayer media exposed to a stationary beam oflight with circular symmetry has been studied in aprevious paper.1 In that study we invoked the Poynting
vector theorem2 to calculate the amount of energy ab-

The authors are with Xerox Palo Alto Research Center, 3333 Coyote
Hill Road, Palo Alto, California 94304.
Received 13 August 1982.
0003-6935/83/050666-05$01.00/0.
© 1983 Optical Society of America.
666

APPLIED OPTICS / Vol. 22, No. 5 / 1 March 1983

sorbed by the structure. We then applied the numerical method known as the alternating direction implicit
techniques to solve the differential equations thatgovern the processes of heat diffusion throughout themedia and heat loss from the surface. In this way, wewere able to determine the temperature distribution
throughout the media as a function of time for an arbitrary laser pulse of total instantaneous power P(t).
To extend the results of this calculation to moving
media, consider a cylindrical coordinate system centered on the axis of the beam and with origin at thesurface of the medium. The positive Z axis enters themedium and thus the point (xo,yozo) occurs at a radial
distance (XI + yl)1/ 2 from the beam axis and at a depthzO below the surface of the structure. Assuming thatthe media is moving with a constant linear speed V in
the positive X direction, we can then approximate its
motion by a discrete (stepping motorlike) motion inwhich the media jumps a distance VAt in the positive
X direction at the end of each time interval At. During
each At, however, the media is stationary and the static
techniques is applicable to the problem. Moreover, forthe specific case of temperature-independent opticaland thermal parameters, the absorption and diffusion
equations become linear, and the moving media problem may be solved by superposition of the static solutions at progressive time intervals. This is the basis forwhat follows.
Using our static technique we calculate, therefore,
the temperature distribution O(r,z,t) that develops
when a narrow stationary pulse with Gaussian intensity,(rr')'
tI 0

exp[-(r/ro)2 ] 0 < t < At,otherwise,

falls on a multilayer medium. Then to determine thetemperature T(xoyozoNAt) for any integer N that
occurs in response to the laser pulse P(t), we calculate
the set of numbers PI where............
................ .
.................

111-.7,loi-.

. .../ .. -..
/,./. 1./ 111 11
11
. /1 ./,

Fig. 1. Schematic of a quadrilayer magnetooptic disk. The thicknesses of the reflector (hatched), intermediate layer, magnetic film
(shaded), and overlayer are given in the text.
PI = (/A~t)

rn_1A P(t)dtn At

J1 (n-

1 _ n <N

1)At

represents the average strength of the pulse in the nthinterval. We also calculate the sequence of numbers{rn} wherern = [xo - (N - n + 1/2)VAt] 2 + yl/ 2< n <N

represents the radial distance between (x0 ,yozo) andthe beam axis during the nth interval. Finally we add
the individual responses to obtain the total response,
namely,T(xo,yo,zo,NAt) =N

PO[rnzo,(N - n + 1)At].

n=1

Since (r,z,t) is originally calculated only for a discrete set of points, it might be necessary to use interpolation to obtain its intermediate values. Thus in ther direction for the calculations of the following sections
we have used linear interpolation, but interpolation in
the Z and t directions has been avoided by choosing
appropriate values for z and At.
The results can be presented in numerous ways: as
isotherms at fixed time, as specific isotherms at different times, as the temperature along a specific line atvarious times, etc. In the following examples we shallsee that each of these can be useful under different
circumstances.
Ill. Thermal Effects of the Read Beam on the
Readout Signal inMagnetooptic Recording
In magnetooptic disk storage, 4 information is stored
in the form of small reverse-magnetized domains on amagnetic surface with perpendicular magnetization.
Retrieval of data from the rotating disk is then achievedby means of a cw laser beam which is focused to form asmall spot, comparable in size with the magnetic domains on the disk surface. The polar Kerr magnetooptic effect, 5 which can be briefly described as thechange in the state of polarization of light on reflection
from a perpendicularly magnetized medium, is thephenomenon most commonly used in readout. The
Kerr effect is proportional to magnetization, and since
the magnetization is in general a decreasing function oftemperature, the read signal is expected to increase
sublinearly with laser power as a result of heating of themedium by the read beam. Therefore, in this section
we apply our moving media technique to determine thetemperature distribution in a magnetooptic disk during
readout as the first step in calculating this readout signal behavior.
Figure 1 shows a multilayer structure that has beenshown to have favorable properties for application inmagnetooptic recording.4 Here, a thin magnetic film
is sandwiched between two dielectric layers and deposited on a glass substrate with an aluminum reflecting
surface to provide an interference enhanced SNR during readout. The thicknesses of the different layers for
optimum performance at X= 840 nm are 60, 60, 20, and
100 nm for the aluminum reflector, SiO intermediate
layer, magnetic film, and SiO overlayer, respectively.
The optical and thermal parameters of this structure
are shown in Table I. In the thermal calculations below,
it was assumed that the beam is focused to a Gaussian
spot with e-1 radius (ro) of 500 nm and that the disk ismoving with the linear velocity of 20 m/sec with respect
to the beam. Since the temperature above ambient is
proportional to the total power in the beam, the rise intemperature will be given in units of C/mW that stands
for degrees centigrade per milliwatt of laser power applied.
Figure 2 shows steady-state contours of constanttemperature in the magnetic film with the central
contour representing the hottest spot on the film at
40.60C/mW. The second contour corresponds to 90%of the maximum temperature, the third to 80%, and soon. In this figure the laser is assumed to be located at
the origin of the coordinate system and the disk ismoving in the positive X direction. Note that the
maximum temperature does not occur at the beam
center, and that due to the radial diffusion effects, thetemperature distribution is slightly wider than the laser
intensity profile.
For comparison, we have shown similar isotherms forthe surface of the multilayer and for the aluminum layer

Table 1. Optical and Thermal Parameters of Materials Used in Examples

Material
Glass
PMMA
SiO
Aluminum
Magnetic film
Tellurium

Refractive index
450 nm
840 nm1.5
1.5
2.0
0.5 + 5.Oi
3.3 + 3.9i

1.5
1.5
2.0
2 + 7.1i
3.67 + 3.85i-

Specific heat(J/cm 3/deg)

Heat conductivity(J/cm/deg/sec)

2.0
2.0
2.0
2.7
3.2
1.5

0.015
0.0015
0.015
2.4
0.4
0.015

1 March 1983 / Vol. 22, No. 5 / APPLIED OPTICS

667

1000t

U

z

0

t-

Q

-1000 '

-1000

0

-1000

1000

DISTANCE (nm)Fig. 2. Steady-state isotherms in the magnetic film when the disk
moves at 20 m/sec from left to right relative to the beam. The central
spot at 40.60 C/mW is the hottest spot in the film, and each successive
contour represents a temperature that is lowered by 10% of thismaximum.

in Figs. 3 and 4, respectively. The maximum temperature on the surface is 37.9°C/mW which is very close
to the maximum in the magnetic film itself. This ispartly due to the fact that we have ignored effects ofheat loss from the surface in this calculation and partly
due to the low thermal conductivity of the overcoat
material. The latter is also responsible for the lack ofappreciable broadening of profiles on the surface compared with those of the magnetic film. In contrast, thetemperature profile in the aluminum layer, which hasa high thermal conductivity, is broadened tremendously, thus resulting in a maximum temperature thatis only 12.90C/mW.
These results provide the basis for evaluating thedrop in magnetization and the subsequent effect on theread signal from the device. While the details of thiscalculation are not within the scope of this paper, theutility of the approach for selecting the most efficientmagnetooptic structure for use in practical situations
must nevertheless be apparent. Such calculations will
be published elsewhere.
IV. Recording Threshold inMultilayer Ablative Media
When a film of a material such as bismuth or tellurium is heated locally by a short pulse from a focused laser
beam, there is a threshold laser power above which
ablation occurs. 6 In this process, material from within
the melted portion of the film is pulled by surface tension into a toroidal rim to create a hole with dimensions
comparable with the original melt. We have shown bycomparison of static measurements and calculations
that the center of the melt at threshold reaches a temperature close to the boiling point of the material.7 In
practical situations, it is evidently advantageous to incorporate a thin film of ablative material in a three-layer
structure either to increase the recording sensitivityof the film or to protect the film from environmental
attack. It then becomes unclear whether the changes
in recording threshold observed are caused solely by thenew, more complicated conditions of energy absorption
and heat flow or whether in the latter case a change is668

APPLIED OPTICS / Vol. 22, No. 5 / 1 March 1983

0
DISTANCE (nm)1000

Fig. 3. Steady-state isotherms at the surface of the overlayer when
the disk moves at 20 m/sec from left to right relative to the beam. The
central spot at 37.90 C/mW is the hottest spot on the surface, and eachsuccessive contour represents a temperature that is lowered by 10%of this maximum.

E
C.S

z
I-

-1000

0
DISTANCE (OminFig. 4. Steady-state isotherms in the aluminum reflector layer when
the disk moves at 20 m/sec from left to right relative to the beam. The
central spot at 12.90 C/mW is the hottest spot in the layer, and eachsuccessive contour represents a temperature that is lowered by 10%of this maximum.

also induced in the threshold temperature of the film
itself by entrapment. The answers to these questions
provide important information about the nature of thenucleation stage of the ablation process, and while thiscan be obtained from static measurements and calculations, we show here that similar information is obtained from dynamic results. Moreover, much is also
learned about the development of the recorded spot in
realistic recording situations.
We take for our example the ablative thresholdcharacteristics of tellurium films in single-layer andthree-layer configurations. The situations examined
are shown in Fig. 5. The single-layer medium consistsof a 15-nm thick tellurium film deposited on a polymethylmethacrylate (PMMA) substrate, while thethree-layer media, either regular or inverted, employ a
100-nm thick aluminum film, a 75-nm thick PMMA
film, and a 7.5-nm thick tellurium layer on PMMA
substrate. Except for the regular three-layer medium,
which required direct top-surface illumination, the

Ii

(a)

I

(b)

(c}

Fig. 5. Schematic representations of (a) single-layer, (b) three-layer,and (c)inverted three-layer ablative structures. (The thin ablative
material is shaded and the reflector layer is hatched.)

W 800

.600

ablative thresholds were obtained using through-substrate illumination as the media scanned at 20 m/sec
past a focused helium-cadmium laser with X= 450 nm.
The e-1 radius of the laser spot was 500 nm, and theduration of the laser pulse was 100 nsec (including
10-nsec rise and fall times). The threshold power so
obtained for each of the media configurations is given
in Table II. The optical and thermal parameters required for the calculations are shown in Table I. For
the purposes of this comparative study, it was not necessary to include the heat of fusion of tellurium nor thetemperature dependence of the thermal constants
themselves, since their omission leads to the same systematic error in the estimate of the temperature rise ineach case. Moreover, the temperature reached in thesteady state for a given irradiation power is independentof the exclusion or inclusion of the heat of fusion and isin error only inasmuch as the optical and thermal parameters change very much with temperature. Thus,
while the calculated transient temperature behavior toa large extent and the equilibrium temperature to asmall extent will be in error, they contain some useful
information including answers to the previously posedquestions.
Calculated results at threshold power for the singlelayer medium are shown in Fig. 6. From these curves
that represent the temperature along the track at various times, several conclusions can be extracted. First,
the steady state is reached in -100 nsec and the maximum temperature rise is close to 10000C, confirmingthat hole nucleation occurs at a temperature close to theboiling point of tellurium. Second, the temperaturerises to its maximum value -50 nsec into the pulse,
implying that shorter pulses might provide better resolution without loss in sensitivity. Finally, we note thatthe envelope of maximum temperature along the trackis considerably broader than the laser spot itself,
Table 11. Measured and Calculated Threshold Results for Tellurium
Ablative MediaMedia structure

Measuredthreshold power(mW)

Calculated
maximum
temperaturerise (C)Single layer
Three layer (regular)
Three layer (inverted)

5
3
5

960
950
965

(b(

400

200

01000

a

0

1000
DISTANCE(nm)

2000

3000

Fig. 6. Temperature rise in the single-layer tellurium film vs distancealong the track at (a) 10, (b) 30, (c) 60, (d) 90, and (e) 110 nsec after
the 5-mW pulse is switched on. (The dotted line indicates that there
is no laser irradiation at this time.) The distance is measured fromthe point at which the pulse is turned on.

suggesting that elongated holes might occur in thesewriting conditions. Since the hole opens to the edge ofthe tellurium melt, we obtain the actual aspect ratio of
the written hole by examining the envelope of isotherms
that correspond to the melting point of tellurium.
These results and photomicrographs of tracks written
in similar conditions are shown in Fig. 7. It is seen thatthe calculated aspect ratio of -2 is in good agreementwith the observed data.
Equivalent results for the three-layer media configurations have also been obtained. In each case themaximum temperature rise at threshold power is close
to 1000'C, as indicated in Table II, even for the inverted
three-layer structure. Thus encapsulation affects theablation threshold by adding extra thermal capacity andnot by affecting the nucleation process through entrapment.
In Fig. 8 the temperature rise at various positions
along a track is shown vs time for the single-layer
structure. The maximum temperature is now seen to
occur at different times according to position, andtemperatures near 10000C are registered between 0.5
and 1.5 m at times ranging from 30 to 100 nsec into thepulse. This is a particularly visual representation of themachining action of the laser pulse. A qualitative understanding of these results can be obtained by recognizing that for times less than 2ro/V the medium may
be considered stationary. Consequently, in our example the temperature rise induced is approximately
equivalent to the outcome of static irradiation for 50
nsec with two laser spots of radius 500 nm on centers1000 nm apart and staggered in time by 50 nsec.
1 March 1983 / Vol. 22, No. 5 / APPLIED OPTICS

669

u9
z

L

uE
0

600

c-

cc
2 600

400

1000
DISTANCE (nin)

b)

C)

C)

400

600
800
TIME (Wmc)

e

200
Ca)(f

Fig. 7. 400'C isotherms in the single-layer tellurium film at 14,20,
30, 60,90,100, and 120 nsec after the 5-mW pulse is turned on. (Thedotted lines indicate that there is no laser irradiation at this time.)
The envelope of these isotherms represents the hole ablated in thetellurium film. This may therefore be compared with the photomicrographs of holes written in similar conditions shown in the insert.
(The track shown was written considerably above threshold to aid in
reproduction in the journal. Tracks at threshold have the same
aspect ratio and each hole is 2000 nm long, in agreement withcalculations.)

0200

Thanks are due to David Treves for many helpful
discussions on the subject.
References
1. M. Mansuripur, G. A. N. Connell, and J. W. Goodman, Appl. Opt.
21, 1106 (1982).
2. J. D. Jackson, Classical Electrodynamics (Wiley, New York,1967).
3. G. Birkhoff, R. S. Varga, and D. Young, "Alternation Direction
Implicit Methods," in Advances in Computers, F. L. Alt and M.
Rubinoff, Eds. (Academic, New York, 1962).
4. M. Mansuripur, G. A. N. Connell, and J. W. Goodman, J. Appl.
Phys. 53, 4485 (1982).
5. P. S. Pershan, J. Appl. Phys. 38, 1482 (1967).
6. D. Maydan, Bell Syst. Tech. J. 50, 1761 (1971).
7. G. A. N. Connell, unpublished.
8. A. E. Bell and F. W. Spong, IEEE J. Quantum Electron. QE-14,
487 (1978).
APPLIED OPTICS / Vol. 22, No. 5 / 1 March 1983

1200

Fig. 8. Temperature rise of the single-layer tellurium film vs timeat (a) -500, (b) 500, (c) 1000, (d) 1500, (e) 2000, and (f) 2500 nm along
the track. The distance is measured from the point at which the
5-mW pulse is turned on.

V. Summary
We have demonstrated the utility of a technique forcalculating temperature profiles in moving multilayer
media under irradiation by a laser beam for two cases:
magnetooptic readout from a quadrilayer structure andablative recording on single-layer and three-layer
structures. The effects of media motion and pulse
length are apparent in these examples, and how eachaffects the permissible laser power in the case of readout
or required laser power in the case of writing can bereadily worked out. We believe that this kind of information is of value in other situations of current interest.

670

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