Progress Report to National Aeronautics and Space Administration (NASA)

Greenbelt, MD 20071

NAG 5-526

Theoretical Studies of Resonance Enhanced Stimulated Raman Scattering

(RESRS) of Frequency Doubled Alexandrite Laser Wavelength in Cesium

Vapor

Principal Investigator

Nabil M. Lawandy

Associate Professor of Engineering and Physics

Brown University

Providence, Rhode Island 02912

July-December, 1987

(KASA-CB-182371). TEEGBEIICAl STUDIES OP N88-23.8.41

BSSONANCE- ENHANCED SlIBOLATEJ) EABAN

SCATTERING (SESIS) OF FEEQUEKCY DOUBLED

AIEXAHBEITE LASIE HAVElENGlEiIK CESIOH .VAPOB UnclaS

Progress Bepprt«-Jul. -Dec. 1S87 (Brown G3/25 01^2330
Progress Report/NASA -1- NAG 5-526

Introduction

The third phase of research on this project will focus on the propagation and

energy extraction of the pump and SERS beams in a variety of configurations including oscillator structures. In order to address these questions a numerical code capable

of allowing for saturation and full transverse beam evolution is required. The proposal

which is forthcoming, will describe this effort in detail. However, as we do not want

to lose continuity on the research we have begun to solve this problem as this time.

The method we propose is based on a discretized propagation energy extraction

model which uses a Kirchoff integral propagator coupled to the three level Raman

model we have already developed. The model will have the resolution required by

diffraction limits and will use the previous density matrix results in the adiabatic following limit. Owing to its large computational requirements, such a code must be

implemented on a vector array processor such as the Cyber facility at the John von

Neumann Center for Scientific Computing at Princeton University.

We have already begun testing one code on the Cyber by using previously understood two-level laser models as guidelines for interpreting the results. We have implemented two tests: 1) the evolution of modes in a passive resonator (Fox & Li), and 2)

the evolution of a stable state of the adiabatically eliminated laser equations. These

1

 '

results which show mode shapes and diffraction losses for the first case and relaxation

oscillations for the second one are shown in figures la and Ib, respectively. Finally,

in order to clarify the computing methodology that we must use in order to exploit the

Cyber's computational speed we have measured the time it takes to perform both of

the computations previously mentioned to run on the Cyber and VAX 730. The laser
Progress Report/NASA -2- NAG 5-526

steady state problem required approximately 5 hours to run on the VAX and 30

seconds to run on the Cyber. This incredible time reduction along with the way in

which our problem lends itself to parallel computation, we feel, will result in some

very exciting results on the SERS problem as well as other problems in laser physics.

Attached is also a short description of our current "laser" model (CAVITY.FOR)

and a flow chart (figures 2a - 2d) of the test computations.
Spatial Mode Profile

F.N. = 2, No gain

1.00

0.67

=> 0.33

c

0)

0.00

0.00 0.33 0.67 1.00

Mirror Position

All losses in this case are due to diffraction

Fig. la
9.00

c

35

CO

*-•

!5

L.

a:

i_

a>

o

Q.

6.00

3.00

0.00

Poiuer us. Pass Number

G = 2. Csi = 1. F.N. = 2

All losses in this case are due to diffraction

1.00 14.00

Pass Number

27.00 40.00

Ficr. lb
Notes on CAVITY.FOR

CAVITY is a Fortran program running on the John Von Neumann Computing Center

Cyber 205 Supercomputer. The program calculates the growth of modes in a Laser or

Maser cavity by iterating Green function solutions for the electromagnetic fields from one

mirror to the next, and then letting the fields jnteract with a "gain disk" at each mirror using

2 level rate equations.

There are several unusual features to this program, mostly stemming from the vector

processing nature of the Cyber 205. In Cyber Fortran, arrays can be manipulated as

objects using new data structures called "descriptors". To simplify the flowcharts

descriptors and the arrays they reference have been identified. The routines that initialize

variables and output data have also not been included for the sake of brevity.

In the formulation of the Green function problem the Fresnel number limit has been used.

The Fresnel number becomes important where (a/b)**2 is small*, and reduces the number

of calculations for the integral. In the case where the mirrors are highly symmetric, the

number of calculations can be reduced by a factor of M, where M is the number of

gridpoints on the mirror( approx. 10000). This reduces what would normally be a "Msquared" calculation to a calculation somewhere on the order of "M" (with some uncertainty

due to the strange way vectors are handled by the Cyber).

Times for running programs on the Cyber are on the order of 30 sec. per iteration, using

100x100 grids. The number of points that must be used for the problem is proportional to

the Fresnel number. The IBM PC AT, by comparison, took several hours per iteration for

40X40 grids. Times for both these machines can be improved by efficient algorithm

design, but by doing so one loses much of the clarity of the original program.

1

Fox, A.G., and Li,T.,"Resonant Modes in a Maser Interferometer",Bell Syst.

Tech. Jou., Vol. 40(1961)
Main Routine

Call Initial

Input Nit

1. Initializes variables for

the program.

2. Nit: No. of passes

L

-^

Loop: I = 1, Nit

j

Call Fields

\

Call Invers

\

Call Output

v

End

3. Solves for EM field

4. Solves for population

inversion

5. Prints (or plots) out

field values.

Fia. 2a
Subroutine Fields

Find] (N + l)th field from Nth field

using Green function method!

I

Common all respective arrays,

variables:

G. Darray. Ul. U2, Xinv. N

Ul - Ul • e*p( G • Xinv )

Loop: I . I, N

' Loop: J - 1, N

Call RRange( Darray, NewD.I.J)

U2(I.J) = Q8SDOT(U1, NewD)

U2(U) - U2(I,J)*exp( ikB)«k»DAMPI

1. G : gain-length product

of the medium

Xinv : Population inversion

Darray : "Distance product" array

U1.U2: fields on respective mirrors

N: No. of mesh points

2. Reaaranges the distance

product array so that

it corresponds to the

actual distances from the

(I,J)th element on one mirror

to all elements on the other

mirror.

3. Q8SDOT: Routine on the

Cyber 205 taking the

dot product of 2 vectors.

4. The last 2 boxes replace

an integral by a discrete sum.

k: wave vector

B: distance between mirrors

DA: differential area

End Subroutine

Fig. 2b
Subroutine RRange

Dl. D2,1. J

1. Dl, D2:Old, new distance product

arrays

I, J : Index of mirror element

Common N

Dimension Index(N.N) ,J1(N,N)..J2(N,N)

I

Loop: II - 1, N

J1(1:N, II) = Abs(Il-I ) + 1

I2(1:N. 11) . Abs(tl-I) * N

i

D2 = Q8VGATHR(Dl,Index)

End Subroutine

2. N : No. of points

Index : Index for D2

3. This somewhat opaque

code is necessary as the

Cyber stores arrays as length

N**2 vectors.

12 - Transpose( J2 )

Index - Jl + J2

4. Q8VGATHR: Vector command on the

Cyber, accomplishing

D2( I) = Dl(Index(OV

I- 1. N

Fig. 2C
Subroutine Inuers

Solves for (N+l)th inversion

using Nth inversion, fields

1

Common variables, arrays:

Csi, Xinv. Uold, Unew

I

Fold = Uold"2

Pnew « Unew "2

Dummy «• (Fold + Pnew)/4

XI - 1/Csi - Dummy - \ri

X2 «. 1/Csi -i- Dummy + 1/2

Xinv . Xinv'(Xl/X2) + 1/X2

1

1. Csi: Medium relaxation time

over cavity rountrip time.

Xinv: inversion of the medium.

Uold : Nth field

Unew:(N+l)th field

Uold, Unew NxN arrays

2. Fold, Pnew: Intensity values

for the rate equations

3. This formula is a

discretization of a 2-level

rate equation.

End Subroutine

Fiq. 2d