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Self-focusing of laser beams in a non-circular fiber
WAVE MOTION 8 (1986) 341-348 NORTH-HOLLAND
SELF-FOCUSING
341
OF LASER BEAMS IN A NON-CIRCULAR FIBER
L.Y. SHIH Division of Electrical Engineering, National Research Council, Ottawa, Canada KlA OR8
Received
29 November
1985
This paper deals with self-focusing of a high-intensity laser beam in a non-circular fiber. Weak nonlinearity is assumed for the energy-dependent dielectric. It is assumed that the electric field lines form concentric loops about the longitudinal axis. Non-orthogonal cylindrical coordinates are introduced based on the assumption that a fiber with circular cross-section has been deformed, while the cross-sectional area remained constant. Numerical examples are given for the particular case of an elliptic fiber.
1. Introduction
Nonlinear electromagnetic phenomena occurring in the optical region are usually associated with high-intensity laser beams. Self-focusing of light has fascinated many researchers in the past. It is typical of the sort of nonlinear wave propagation which depends critically on the transverse profile of the beam. Most of the previous work on the self-focusing of an optical beam in a nonlinear fiber has concentrated on the circular cross-section. In these papers, axial symmetry and weak nonlinearity were assumed. Chiao et al. studied the self-trapping of a cylindrical beam with the assumption of circular polarization [l]. Abakarov et al. considered the cylindrical beam of a different type, by assuming rotationally symmetric filaments in which the electric field lines form concentric circles about the longitudinal axis [2]. Using the spinor representation of electromagnetic fields in cylindrical coordinates, Hillion and Quinnez obtained quasiplane-wave solutions describing the self-trapping of optical beams [3]. Kelly introduced the concept of a self-focusing length, and the numerical solution for a cylindrically symmetrical beam has been obtained with an equi-phase Gaussian intensity 0165-2125/86/$3.50
1986, Elsevier Science Publishers
profile given for the input beam [4]. Talanov considered the cylindrical case of self-focusing in an isotropic nonlinear dielectric E [5]. He discussed features of the paraxial beam in such a medium in the case of weak nonlinearity of E. In the present paper, we examine the selffocusing phenomenon for a non-circular fiber. Numerical examples are given for an elliptical fiber as a particular case. Non-orthogonal cylindrical coordinates are introduced based on the assumption that the fiber cross-section represents a deformation from a circular fiber while the crosssectional area remained constant. Self-trapping as a limiting phenomenon is studied in Section 4. Solutions for self-focusing under the paraxial approximation are discussed in Section 5.
2. Self-focusing for non-circular fibers For fibers with a small uniform non-circular general noncross-section, we introduce orthogonal cylindrical coordinates, say (x’, x2, z), where z denotes the longitudinal coordinate, x1 characterizes the configuration of the cross-section with x1 = 0 at the center and x1 = constant on the fiber surface, and the plane X1X2 is perpendicular
B.V. (North-Holland)
L. Y. Shih / Self-focusing of laser beams
342
to the Z-axis. In this coordinate system the metric tensor is expressed as
&j =
g11
g12
0
g12
gz2
0
[ 0
0
be obtained by substituting expression (2.2) into equation (2.8): -~o~=k,2eZxV[V+ZxG)]-~]GI]2G.
*
(2.1)
(2.9)
1I
We consider transverse dielectric fiber
waves propagating
E = G(x’, x2, f; t) exp[i(bz
- @)I,
in a
(2.2) ;(N2)+$V.[(NV+)N]=0
G = (a,,/ e2)“‘A exp(i#),
(2.3)
where A and (cr are dimensionless functions of (x1, x2, 5, t), and I is the reduced longitudinal coordinate defined as z - (~,,/k,,)t. The dielectric coefficient of the fiber is given as E=E,,(w)+E~(w)~E~~.
(2.4)
It is well known that propagation of optical pulses in a dielectric fiber is governed by the nonlinear wave equation VxVxE++E)=O.
If Q represents the z-component
With expression (2.3), eq. (2.9) may be split into two coupled equations for the conjugate amplitude defined as N = eZx A and the phase $:
(2.5)
of the curl E, then
Q=V.(Exe,)
0.6)
while V x ( Qez) = e, x V[V * (e, x E)].
(2.7)
(2.10)
0
and
=$V(V
s N)+N2.
(2.11)
0
Equation (2.10) may be interpreted as the conservation law of beam intensity. It is analogous to the continuity equation for an it-rotational flow of compressible fluid, where A2 corresponds to the fluid density and (oo/ ki) V, + corresponds to the flow velocity u. It reveals that the beam intensity decays only when Vll, - Vx’> 0. Equation (2.11) is of a form known as the Hamilton-Jacobi equation. Applying the gradient operator to eq. (2.11), we write ~+(U.v)n=v(U~-Uo),
(2.12)
Thus, the wave equation becomes where Uo denotes the diffraction term a2E pso a2E 2-z catz az
u D =__I2
@o 2N-W-N),
(
~ZN 0
(2.13)
>
and Ur denotes the self-focusing term For e2 CCe0 as in most practical applications {e.g. Nd:glass laser (A = 1.06 microns) in Si02’ glass fiber, Ed/ a0 = 3 x 1O-26cgs-esu}, we investigate weakly nonlinear waves where G is assumed to be a slowly varying function. Similar to the case of a circular fiber [6], an approximate equation may
(2.14) It should be pointed out that although only the transverse field is considered in this paper, due to dyadic nature of the susceptibility tensor there exists a longitudinal component of the electric
L. Y. Shih / Self-jocusing qf laser beams
polarization. Substituting expressions (2.2) and (2.3) into the Maxwell’s equation for div 0, the imaginary term yields A-V+=O.
343
The metric tensor may be evaluated expressed as
(2.15)
Equation (2.15) indicates that V+ is perpendicular to A.
g,, = cos n( 1 -I-cos* 4 tan2n),
(3.6)
g,, = p* cos T( 1+ sin* 4 tan* n),
(3.7)
g,, = -p sin Q,cos 4 sin n tan 7,
(3.8)
where the determinant g=g11g22-g12-p
3. Coordinate system for elliptic fibers From
In this paper, non-circular fibers of arbitrary cross-section are considered. Solutions and numerical examples will be treated in detail for the particular case of elliptical fibers. For this purpose, we shall first choose a suitable coordinate system for elliptical fibers. Let us assume x1 to be concentric ellipses p(x, y) of the same eccentricity as expressed by sin 7. Suppose that p is defined as the square root of the product of a semi-major axis and a semi-minor axis, then p*=x*cos
n+y*secn.
tan C$= y/(x cos n). The transformation as
*_
of g, is 2.
(3.9)
eqs. (3.6)-(3.8), it is easy to show that a -g,, ‘34
=2g,,=
a j-gg12
=
-- ;
;g2*. (3.11)
&T**lP - Plzll,
and p(cos
ml+g22lp=
7+sec
77).
(3.12)
Let us introduce two functions of 4, which will be used in Section 4, as follows:
(3.1)
We may imagine the elliptical fiber as an elastic deformation of the circular fiber, caused by a symmetric compression in Y-direction. This results in an elongation in X-direction by a factor of Jseol and a contraction in Y-direction by a factor of G,b ut t h e cross-sectional area of the fiber remains unchanged. For this reason confocal ellipses and hyperbolas will not be considered as a good choice for our coordinate system. Let x2 be 4, corresponding to radial lines of the circular fiber. We have
[7] and
a(+) =
&‘lP,
P(d)
=
-g,*/&*.
(3.13)
By eqs. (3.9)-(3.12), we obtain da ,,=P,
(3.14)
dS --3 G=(Y -(y,
(3.15)
$=
(3.16)
-(1+3*-4)p,
a-*+p*=gl*,
cu2+g11=cos n+sec
(3.2)
(3.17) n.
(3.18)
of coordinates may be written 4. Self-trapping as limiting case
x = p cos I# sec”‘q,
(3.3)
y = p sin 4 cos”*n.
(3.4)
If v denotes the angle of Vp, the normal to an ellipse, then tan v = tan C$set 7.
(3.5)
The situation when U, = U, is called self-trapping. Equating expressions (2.13) and (2.14), we obtain a partial differential equation of parabolic
type N.V(V - N)+k;N4=0.
(4.1)
L. X Shih / Self-focusing of laser beams
344
Chiao et al. considered the case of circular fibers, and no analytic solution has been found even with the assumption of circular polarization [ 11. In view of the boundary conditions E = 0 at the center and E- Vx’ = 0 on the surface of a fiber, for small cross-sections one may assume A’ to be negligibly small. Considering elliptical fibers, we write from expressions (A.3) and (3.13) N’ = -aA,
N2 = -PA/p.
(4.2)
Substituting expressions (4.2) into eq. (4.1) with the aid of expression (AS), and making use of eqs. (3.13)-(3.17), we obtain
where F = bb,,A, 5 = p/ bO, and b0 denotes the initial value of p for the laser beam. The parameter p=secnfor+=Owhilep=cosnfor+=$n.The variables F and 5 are so chosen to yield dimensionless moderate numerical values. Note that p = 1 corresponds to the case of circular fibers [2]. Equation (4.6) may be solved numerically subject to the boundary conditions that F vanishes at both 6 = 0 and 5,. We choose n = $7~and 5, = 5. In Fig. 1 F is plotted versus 6: the solid curve represents the variation along the major axis, while the dashed curve represents that along the minor axis.
+ k;A3 = 0.
It was found that no solution exists under the assumption of elliptic polarization A(p, 4) = a(+)R(p), where R is a function of p only to be determined from eq. (4.3). It is apparent that the solution is symmetric to both principal axes of the elliptical cross-section, namely the major axis and the minor axis. We are not interested in the general solution of eq. (4.3) for an arbitrary value of $. To have a picture of the self-trapping phenomenon, we would rather examine the solutions at these particular orthogonal directions. Along the major axis where 4 = 0, eq. (4.3) is reduced to
P/b, Fig. 1. Self-trapping of laser beam in an elliptic fiber with v = 7r/4. Numerical solutions of equation (4.6): the solid curve represents the solution on the major axis, while the dashed curve represents the solution on the minor axis.
d2A
cos qy+sec
5. Solution for elliptic fibers
dp while along the minor axis where & = &T, eq. (4.3) becomes d2A
set qz+cos dp
d A q- +kiA3=0. dp 0 P
(4.5)
Equations (4.4) and (4.5) are of the same form
$ (+;)
+pFLo,
(4.6)
To determine evolution of the wave front in an elliptical fiber we must solve the coupled eqs. (2.10) and (2.11). As mentioned in Section 2, eq. (2.11) has the same form as the Hamilton-Jacobi equation, which leads to Euler’s equation (2.12) for the irrotational flow of a compressible inviscid fluid, with the enthalpy defined as U = U, - UF. However, in our case according to eqs. (2.13) and
L..Y. Shih / Self-focusing of laser beams
(2.14)
(5.1) is known only if A(p, 4, t) is known, but A can only be obtained by solving the coupled eqs. (2.10) and (2.11). Let us turn to the same case as considered in Section 4, namely, that A has no component in p direction. We obtained +*V(V*N)
345
because at one single point there cannot exist a’ vector of finite magnitude but with many different directions. As an approximation, one may assume a certain functional form for F(& t) such that each ray follows a trajectory with p/b(t) = po/ bo, where the subscript ‘0’ indicates the initial conditions. This approximation is found to be reasonable as long as ray bending during focusing is insignificant. Let us define A = b(t)/ bo, and assume that the central part of the beam retains its Gaussian profile as it propagates, but the beam size varies with time: i.e., for p” b(t) (5.6)
(5.2) Equation (2.15) implies that JI is independent of 4. Thus we have ‘1 =
-k,:
ap,
u*=o,
(5.3)
where the subscripts indicate the components of a contravariant vector. In the elliptic coordinates, eq. (2.11) may be expressed as
wow 1 --+-(au,)2=-u. k:at
(5.4)
2
which is known as the paraxial or aberrationless approximation. Expression (5.6) possesses singular derivatives along the longitudinal axis. Apart from this singularity, a$,/85 = 0 may be described as the condition on the longitudinal axis. Considering again the solutions along two principal axes, eqs. (5.4) and (5.5) with expressions (5.2) and (5.6) are reduced to a* 2p~=pF*+H-u2
(5.7)
where the function H(A, p) is defined as
Similarly, with the aid of eq. (A.12), eq. (2.10) may be written as
[’
1+p2
=4---A
A2
p2 ,f2’
(5.8)
and ~0.
(5.5) ~-($+2~)u=2p(2-~)$
To avoid extremely small and extremely large values in our numerical solutions, we introduce a time scale of ( kQo)‘/wo and a length scale of bo. Following eq. (5.3) the velocity in new scales may be expressed as u1= a$/ag. Instead of A and p, we shall use F and 5 as defined in Section 4. As p approaches zero the ellipse shrinks to a point. Thus F must be zero on the longitudinal axis
(5.9)
Let us first solve eq. (5.9) for u
u-*’ exp(-u2) -!PA’ =2
U(2 - UZ)rPZ exp(-u2)
du+ C(t, p), (5.10)
L. K Shih / Seijljbcusing of laser beams
346
where u is defined as s/A = p/b for convenience, and C( t, p) is the constant of integration. Properties of the solution in the form of eq. (5.10) can hardly be visualized. For this reason we expand the integral through repeated integration by parts, and eventually arrive at an expression valid for pz1: v=
C(t,p)uP2exp(uZ)+u+(3+p2)u
PA’
xc
k=O
(2U2)k (1 -P2)(3 -P2) * * * (2k+ 1 -p’>’ (5.11)
which vanishes on the longitudinal axis. For convenience let us write the right-hand side of eq. (5.11) as C(t)X(u)+ Y(u). It may be shown from eq. (5.10) that the ratio -Y/X has a minimum value C* which occurs at u = &. For the example with 7) =&r, C* = 8.842 on the major axis and C* = -4.396 on the minor axis. Under normal conditions u and A’ are of the same sign, thus the right-hand side of eq. (5.11) must be positive. This leads to the conclusion that C > C*. Equation (5.11) may be integrated with respect to u to yield the wave phase
We shall examine the effect of F,, appearing in the profile (5.6), on the self-focusing of a beam. As we know, pF2+ H represents the net effect of U,- U,,, which must be positive for a beam to focus. By expression (5.8), it yields the condition for self-focusing F~>e”*[p(l+u-2)+(1-u2)/p].
(5.15)
As depicted in Fig. 2 for 77=&T, the areas above the curves are the regions for self-focusing, where the solid curve corresponds to the major axis and the dashed curve to the minor axis. As an illustration, Fo= 4 may be considered as a reasonable value.
~upZe”‘S,(u,p)
1
+$+
Mu, PI u,
where
(5.12)
0.8
1.2
\
1.6
,
2
Fig. 2. Limiting conditions for self-focusing of laser beam in an elliptic fiber with 1)= ?r/4. Gaussian profile is assumed. The solid curve represents the condition on the major axis, while the dashed curve represents the condition on the minor axis.
(2U*)*k
k=l (3+P2)(5+P2)
0.4
P/b
2u2 &=1---_-I 3+P
+C
8 8I
0
* - * (4k+ 1 +p’)
6. Cooeluding remarks
2u2 4k+3+p2
>
(5.13)
and
s,=]+?$ (2u2)k
xck=l(k+1)(3-p2)(5-p2)+2k+1-p’)’ (5.14)
To study self-focusing of the laser beam in a non-circular fiber, a pair of generalized equations has been derived in this paper: one represents the conservation law of beam intensity, while the other is of the form known as the Hamilton-Jacobi equation. Instead of A, it was discovered that N is a better variable for analytical purposes.
L. Y. Shih / Self-focusingof laser beams
The condition of axial symmetry cannot be assumed for the case of non-circular fibers. Nonorthogonal cylindrical coordinates have been chosen based on the assumption that the fiber cross-section has been deformed from the circular fiber while the cross-sectional area remained constant. The particular case of an elliptical fiber has been treated in detail. The solutions are symmetric with respect to two principal planes. Numerical solutions along the major and minor axes have been depicted graphically.
Following eq. expressed as
347
(A.4),
the
Laplacian
may
be
(A.6) Curl. The curl of vector A may be expressed in tensor notation as g
-l/2eijk_ a; (gk,A”).
(A-7)
Define the vector F = e, x V[V * (ez x A)], which appears in eqs. (2.7)-(2.9) as part of the expression for curl curl A.
Appendix. Formulas for non-orthogonal coordinate system
-i$ (gl2A’+ g22A2)
To aid readers with the derivations in this paper, some useful formulas for a non-orthogonal coordinate system are collected below. Illustrations are given for the particular system of cylindrical coordinates as described in Section 2, with the metric tensor defined in eq. (2.1). Vector product. Define C = A x B. In tensor notation we write C’ = g-l/2e’ikAjBk.
(g,,A’+ g12A2)
,
64.8) F2 = g-‘/2
_5
g’/2 {
[
-5
$2(g,tA’+ gtA2) kt,A1+g22A2) .
(A.1)
(A.9)
(A.2)
Conjugate metric tensor. The contravariant tensor conjugate to the metric tensor gii defined by expression (2.1) is of the form of
Using the relation & = g,A”,
-3
in cylindrical coordinates the vector N defined by eq. (2.14) may be expressed in tensor notation as
(A.lO)
N1= -g-“2(g,2A1 + g2,A2), (A.3)
N2 = g-1’2(g,,A1+g,2A2).
Thus the quantity lV$)2 may be expressed as Divergence. V . A = g-l12 3
(g1/2Ai)_
(A.4)
Thus the expression N* V(V *N) appearing in eqs. (2.13) and (4.1) may be written in cylindrical coordinates in tensor notation as g-‘12
-& (g”‘N’)
I .
(A.9
g&!w=~(zty+~(~) --- a* a* g ax1ax2’
_2g12
(A.ll)
The expression (u - V)Q appears in eq. (2.10) where u is defined as the gradient of a scalar potential while Q is a scalar quantity. It may be
L.. Y. Shih / SelfIocusing
348
of laser beams
except
expressed as
(A.15) (A.12) The expression (v - V)v appears in eq. (2.12) where u is defined as the gradient of a sclar potential. In contravariant form it may be expressed as while
References
gjkVi,jVk
aVi
Ly
Vi,j=~-
i I
j
va.
(A.13)
I
The Christoffel symbol of the second kind is defined as {
i”j} +@!+?ap).
(A.14)
In terms of the elliptic coordinates defined in Section 3, we found CJ!
{ i j
I
=o
[l] R.Y. Chiao, E. Garmire and C.H. Townes, “Self-trapping of optical beams”, Phys. Rev. Len. 13,479 (1964) (Erratum: 14, iO56 (1965)]. PI D.I. Abakarov, A.A. Akopyan and S.I. Pekar, “Contribution to the theory of self focusing of light in nonlinearly polarizing media”, Soviet Phys. JETP 25, 303 (1967). [31 P. Hillion and S. Quinnez, “Analysis of self-trapping using the spinor wave equation”, Opt. Engrg. 24, 290 (1985) [41 P.L. Kelley, “Self-focusing of optical beams”, Phys. Rev. Lett. J5, 1005 (1965). 151V.I. Talanov, “Self focusing of wave beams in nonlinear media”, JEEP Len. 2, 138 (1965). 161Y.R. Shen, The Principles of Nonlinear Optics, Wiley, New York (1984) 307-313. . . ._ . [7] I.S. Sokolnikoff, Tensor Analysis: Theory and Appflcatlons to Geometry and Mechanics of Continua, Wiley, New York, 2nd ed. (1964).
SELF-FOCUSING
341
OF LASER BEAMS IN A NON-CIRCULAR FIBER
L.Y. SHIH Division of Electrical Engineering, National Research Council, Ottawa, Canada KlA OR8
Received
29 November
1985
This paper deals with self-focusing of a high-intensity laser beam in a non-circular fiber. Weak nonlinearity is assumed for the energy-dependent dielectric. It is assumed that the electric field lines form concentric loops about the longitudinal axis. Non-orthogonal cylindrical coordinates are introduced based on the assumption that a fiber with circular cross-section has been deformed, while the cross-sectional area remained constant. Numerical examples are given for the particular case of an elliptic fiber.
1. Introduction
Nonlinear electromagnetic phenomena occurring in the optical region are usually associated with high-intensity laser beams. Self-focusing of light has fascinated many researchers in the past. It is typical of the sort of nonlinear wave propagation which depends critically on the transverse profile of the beam. Most of the previous work on the self-focusing of an optical beam in a nonlinear fiber has concentrated on the circular cross-section. In these papers, axial symmetry and weak nonlinearity were assumed. Chiao et al. studied the self-trapping of a cylindrical beam with the assumption of circular polarization [l]. Abakarov et al. considered the cylindrical beam of a different type, by assuming rotationally symmetric filaments in which the electric field lines form concentric circles about the longitudinal axis [2]. Using the spinor representation of electromagnetic fields in cylindrical coordinates, Hillion and Quinnez obtained quasiplane-wave solutions describing the self-trapping of optical beams [3]. Kelly introduced the concept of a self-focusing length, and the numerical solution for a cylindrically symmetrical beam has been obtained with an equi-phase Gaussian intensity 0165-2125/86/$3.50
1986, Elsevier Science Publishers
profile given for the input beam [4]. Talanov considered the cylindrical case of self-focusing in an isotropic nonlinear dielectric E [5]. He discussed features of the paraxial beam in such a medium in the case of weak nonlinearity of E. In the present paper, we examine the selffocusing phenomenon for a non-circular fiber. Numerical examples are given for an elliptical fiber as a particular case. Non-orthogonal cylindrical coordinates are introduced based on the assumption that the fiber cross-section represents a deformation from a circular fiber while the crosssectional area remained constant. Self-trapping as a limiting phenomenon is studied in Section 4. Solutions for self-focusing under the paraxial approximation are discussed in Section 5.
2. Self-focusing for non-circular fibers For fibers with a small uniform non-circular general noncross-section, we introduce orthogonal cylindrical coordinates, say (x’, x2, z), where z denotes the longitudinal coordinate, x1 characterizes the configuration of the cross-section with x1 = 0 at the center and x1 = constant on the fiber surface, and the plane X1X2 is perpendicular
B.V. (North-Holland)
L. Y. Shih / Self-focusing of laser beams
342
to the Z-axis. In this coordinate system the metric tensor is expressed as
&j =
g11
g12
0
g12
gz2
0
[ 0
0
be obtained by substituting expression (2.2) into equation (2.8): -~o~=k,2eZxV[V+ZxG)]-~]GI]2G.
*
(2.1)
(2.9)
1I
We consider transverse dielectric fiber
waves propagating
E = G(x’, x2, f; t) exp[i(bz
- @)I,
in a
(2.2) ;(N2)+$V.[(NV+)N]=0
G = (a,,/ e2)“‘A exp(i#),
(2.3)
where A and (cr are dimensionless functions of (x1, x2, 5, t), and I is the reduced longitudinal coordinate defined as z - (~,,/k,,)t. The dielectric coefficient of the fiber is given as E=E,,(w)+E~(w)~E~~.
(2.4)
It is well known that propagation of optical pulses in a dielectric fiber is governed by the nonlinear wave equation VxVxE++E)=O.
If Q represents the z-component
With expression (2.3), eq. (2.9) may be split into two coupled equations for the conjugate amplitude defined as N = eZx A and the phase $:
(2.5)
of the curl E, then
Q=V.(Exe,)
0.6)
while V x ( Qez) = e, x V[V * (e, x E)].
(2.7)
(2.10)
0
and
=$V(V
s N)+N2.
(2.11)
0
Equation (2.10) may be interpreted as the conservation law of beam intensity. It is analogous to the continuity equation for an it-rotational flow of compressible fluid, where A2 corresponds to the fluid density and (oo/ ki) V, + corresponds to the flow velocity u. It reveals that the beam intensity decays only when Vll, - Vx’> 0. Equation (2.11) is of a form known as the Hamilton-Jacobi equation. Applying the gradient operator to eq. (2.11), we write ~+(U.v)n=v(U~-Uo),
(2.12)
Thus, the wave equation becomes where Uo denotes the diffraction term a2E pso a2E 2-z catz az
u D =__I2
@o 2N-W-N),
(
~ZN 0
(2.13)
>
and Ur denotes the self-focusing term For e2 CCe0 as in most practical applications {e.g. Nd:glass laser (A = 1.06 microns) in Si02’ glass fiber, Ed/ a0 = 3 x 1O-26cgs-esu}, we investigate weakly nonlinear waves where G is assumed to be a slowly varying function. Similar to the case of a circular fiber [6], an approximate equation may
(2.14) It should be pointed out that although only the transverse field is considered in this paper, due to dyadic nature of the susceptibility tensor there exists a longitudinal component of the electric
L. Y. Shih / Self-jocusing qf laser beams
polarization. Substituting expressions (2.2) and (2.3) into the Maxwell’s equation for div 0, the imaginary term yields A-V+=O.
343
The metric tensor may be evaluated expressed as
(2.15)
Equation (2.15) indicates that V+ is perpendicular to A.
g,, = cos n( 1 -I-cos* 4 tan2n),
(3.6)
g,, = p* cos T( 1+ sin* 4 tan* n),
(3.7)
g,, = -p sin Q,cos 4 sin n tan 7,
(3.8)
where the determinant g=g11g22-g12-p
3. Coordinate system for elliptic fibers From
In this paper, non-circular fibers of arbitrary cross-section are considered. Solutions and numerical examples will be treated in detail for the particular case of elliptical fibers. For this purpose, we shall first choose a suitable coordinate system for elliptical fibers. Let us assume x1 to be concentric ellipses p(x, y) of the same eccentricity as expressed by sin 7. Suppose that p is defined as the square root of the product of a semi-major axis and a semi-minor axis, then p*=x*cos
n+y*secn.
tan C$= y/(x cos n). The transformation as
*_
of g, is 2.
(3.9)
eqs. (3.6)-(3.8), it is easy to show that a -g,, ‘34
=2g,,=
a j-gg12
=
-- ;
;g2*. (3.11)
&T**lP - Plzll,
and p(cos
ml+g22lp=
7+sec
77).
(3.12)
Let us introduce two functions of 4, which will be used in Section 4, as follows:
(3.1)
We may imagine the elliptical fiber as an elastic deformation of the circular fiber, caused by a symmetric compression in Y-direction. This results in an elongation in X-direction by a factor of Jseol and a contraction in Y-direction by a factor of G,b ut t h e cross-sectional area of the fiber remains unchanged. For this reason confocal ellipses and hyperbolas will not be considered as a good choice for our coordinate system. Let x2 be 4, corresponding to radial lines of the circular fiber. We have
[7] and
a(+) =
&‘lP,
P(d)
=
-g,*/&*.
(3.13)
By eqs. (3.9)-(3.12), we obtain da ,,=P,
(3.14)
dS --3 G=(Y -(y,
(3.15)
$=
(3.16)
-(1+3*-4)p,
a-*+p*=gl*,
cu2+g11=cos n+sec
(3.2)
(3.17) n.
(3.18)
of coordinates may be written 4. Self-trapping as limiting case
x = p cos I# sec”‘q,
(3.3)
y = p sin 4 cos”*n.
(3.4)
If v denotes the angle of Vp, the normal to an ellipse, then tan v = tan C$set 7.
(3.5)
The situation when U, = U, is called self-trapping. Equating expressions (2.13) and (2.14), we obtain a partial differential equation of parabolic
type N.V(V - N)+k;N4=0.
(4.1)
L. X Shih / Self-focusing of laser beams
344
Chiao et al. considered the case of circular fibers, and no analytic solution has been found even with the assumption of circular polarization [ 11. In view of the boundary conditions E = 0 at the center and E- Vx’ = 0 on the surface of a fiber, for small cross-sections one may assume A’ to be negligibly small. Considering elliptical fibers, we write from expressions (A.3) and (3.13) N’ = -aA,
N2 = -PA/p.
(4.2)
Substituting expressions (4.2) into eq. (4.1) with the aid of expression (AS), and making use of eqs. (3.13)-(3.17), we obtain
where F = bb,,A, 5 = p/ bO, and b0 denotes the initial value of p for the laser beam. The parameter p=secnfor+=Owhilep=cosnfor+=$n.The variables F and 5 are so chosen to yield dimensionless moderate numerical values. Note that p = 1 corresponds to the case of circular fibers [2]. Equation (4.6) may be solved numerically subject to the boundary conditions that F vanishes at both 6 = 0 and 5,. We choose n = $7~and 5, = 5. In Fig. 1 F is plotted versus 6: the solid curve represents the variation along the major axis, while the dashed curve represents that along the minor axis.
+ k;A3 = 0.
It was found that no solution exists under the assumption of elliptic polarization A(p, 4) = a(+)R(p), where R is a function of p only to be determined from eq. (4.3). It is apparent that the solution is symmetric to both principal axes of the elliptical cross-section, namely the major axis and the minor axis. We are not interested in the general solution of eq. (4.3) for an arbitrary value of $. To have a picture of the self-trapping phenomenon, we would rather examine the solutions at these particular orthogonal directions. Along the major axis where 4 = 0, eq. (4.3) is reduced to
P/b, Fig. 1. Self-trapping of laser beam in an elliptic fiber with v = 7r/4. Numerical solutions of equation (4.6): the solid curve represents the solution on the major axis, while the dashed curve represents the solution on the minor axis.
d2A
cos qy+sec
5. Solution for elliptic fibers
dp while along the minor axis where & = &T, eq. (4.3) becomes d2A
set qz+cos dp
d A q- +kiA3=0. dp 0 P
(4.5)
Equations (4.4) and (4.5) are of the same form
$ (+;)
+pFLo,
(4.6)
To determine evolution of the wave front in an elliptical fiber we must solve the coupled eqs. (2.10) and (2.11). As mentioned in Section 2, eq. (2.11) has the same form as the Hamilton-Jacobi equation, which leads to Euler’s equation (2.12) for the irrotational flow of a compressible inviscid fluid, with the enthalpy defined as U = U, - UF. However, in our case according to eqs. (2.13) and
L..Y. Shih / Self-focusing of laser beams
(2.14)
(5.1) is known only if A(p, 4, t) is known, but A can only be obtained by solving the coupled eqs. (2.10) and (2.11). Let us turn to the same case as considered in Section 4, namely, that A has no component in p direction. We obtained +*V(V*N)
345
because at one single point there cannot exist a’ vector of finite magnitude but with many different directions. As an approximation, one may assume a certain functional form for F(& t) such that each ray follows a trajectory with p/b(t) = po/ bo, where the subscript ‘0’ indicates the initial conditions. This approximation is found to be reasonable as long as ray bending during focusing is insignificant. Let us define A = b(t)/ bo, and assume that the central part of the beam retains its Gaussian profile as it propagates, but the beam size varies with time: i.e., for p” b(t) (5.6)
(5.2) Equation (2.15) implies that JI is independent of 4. Thus we have ‘1 =
-k,:
ap,
u*=o,
(5.3)
where the subscripts indicate the components of a contravariant vector. In the elliptic coordinates, eq. (2.11) may be expressed as
wow 1 --+-(au,)2=-u. k:at
(5.4)
2
which is known as the paraxial or aberrationless approximation. Expression (5.6) possesses singular derivatives along the longitudinal axis. Apart from this singularity, a$,/85 = 0 may be described as the condition on the longitudinal axis. Considering again the solutions along two principal axes, eqs. (5.4) and (5.5) with expressions (5.2) and (5.6) are reduced to a* 2p~=pF*+H-u2
(5.7)
where the function H(A, p) is defined as
Similarly, with the aid of eq. (A.12), eq. (2.10) may be written as
[’
1+p2
=4---A
A2
p2 ,f2’
(5.8)
and ~0.
(5.5) ~-($+2~)u=2p(2-~)$
To avoid extremely small and extremely large values in our numerical solutions, we introduce a time scale of ( kQo)‘/wo and a length scale of bo. Following eq. (5.3) the velocity in new scales may be expressed as u1= a$/ag. Instead of A and p, we shall use F and 5 as defined in Section 4. As p approaches zero the ellipse shrinks to a point. Thus F must be zero on the longitudinal axis
(5.9)
Let us first solve eq. (5.9) for u
u-*’ exp(-u2) -!PA’ =2
U(2 - UZ)rPZ exp(-u2)
du+ C(t, p), (5.10)
L. K Shih / Seijljbcusing of laser beams
346
where u is defined as s/A = p/b for convenience, and C( t, p) is the constant of integration. Properties of the solution in the form of eq. (5.10) can hardly be visualized. For this reason we expand the integral through repeated integration by parts, and eventually arrive at an expression valid for pz1: v=
C(t,p)uP2exp(uZ)+u+(3+p2)u
PA’
xc
k=O
(2U2)k (1 -P2)(3 -P2) * * * (2k+ 1 -p’>’ (5.11)
which vanishes on the longitudinal axis. For convenience let us write the right-hand side of eq. (5.11) as C(t)X(u)+ Y(u). It may be shown from eq. (5.10) that the ratio -Y/X has a minimum value C* which occurs at u = &. For the example with 7) =&r, C* = 8.842 on the major axis and C* = -4.396 on the minor axis. Under normal conditions u and A’ are of the same sign, thus the right-hand side of eq. (5.11) must be positive. This leads to the conclusion that C > C*. Equation (5.11) may be integrated with respect to u to yield the wave phase
We shall examine the effect of F,, appearing in the profile (5.6), on the self-focusing of a beam. As we know, pF2+ H represents the net effect of U,- U,,, which must be positive for a beam to focus. By expression (5.8), it yields the condition for self-focusing F~>e”*[p(l+u-2)+(1-u2)/p].
(5.15)
As depicted in Fig. 2 for 77=&T, the areas above the curves are the regions for self-focusing, where the solid curve corresponds to the major axis and the dashed curve to the minor axis. As an illustration, Fo= 4 may be considered as a reasonable value.
~upZe”‘S,(u,p)
1
+$+
Mu, PI u,
where
(5.12)
0.8
1.2
\
1.6
,
2
Fig. 2. Limiting conditions for self-focusing of laser beam in an elliptic fiber with 1)= ?r/4. Gaussian profile is assumed. The solid curve represents the condition on the major axis, while the dashed curve represents the condition on the minor axis.
(2U*)*k
k=l (3+P2)(5+P2)
0.4
P/b
2u2 &=1---_-I 3+P
+C
8 8I
0
* - * (4k+ 1 +p’)
6. Cooeluding remarks
2u2 4k+3+p2
>
(5.13)
and
s,=]+?$ (2u2)k
xck=l(k+1)(3-p2)(5-p2)+2k+1-p’)’ (5.14)
To study self-focusing of the laser beam in a non-circular fiber, a pair of generalized equations has been derived in this paper: one represents the conservation law of beam intensity, while the other is of the form known as the Hamilton-Jacobi equation. Instead of A, it was discovered that N is a better variable for analytical purposes.
L. Y. Shih / Self-focusingof laser beams
The condition of axial symmetry cannot be assumed for the case of non-circular fibers. Nonorthogonal cylindrical coordinates have been chosen based on the assumption that the fiber cross-section has been deformed from the circular fiber while the cross-sectional area remained constant. The particular case of an elliptical fiber has been treated in detail. The solutions are symmetric with respect to two principal planes. Numerical solutions along the major and minor axes have been depicted graphically.
Following eq. expressed as
347
(A.4),
the
Laplacian
may
be
(A.6) Curl. The curl of vector A may be expressed in tensor notation as g
-l/2eijk_ a; (gk,A”).
(A-7)
Define the vector F = e, x V[V * (ez x A)], which appears in eqs. (2.7)-(2.9) as part of the expression for curl curl A.
Appendix. Formulas for non-orthogonal coordinate system
-i$ (gl2A’+ g22A2)
To aid readers with the derivations in this paper, some useful formulas for a non-orthogonal coordinate system are collected below. Illustrations are given for the particular system of cylindrical coordinates as described in Section 2, with the metric tensor defined in eq. (2.1). Vector product. Define C = A x B. In tensor notation we write C’ = g-l/2e’ikAjBk.
(g,,A’+ g12A2)
,
64.8) F2 = g-‘/2
_5
g’/2 {
[
-5
$2(g,tA’+ gtA2) kt,A1+g22A2) .
(A.1)
(A.9)
(A.2)
Conjugate metric tensor. The contravariant tensor conjugate to the metric tensor gii defined by expression (2.1) is of the form of
Using the relation & = g,A”,
-3
in cylindrical coordinates the vector N defined by eq. (2.14) may be expressed in tensor notation as
(A.lO)
N1= -g-“2(g,2A1 + g2,A2), (A.3)
N2 = g-1’2(g,,A1+g,2A2).
Thus the quantity lV$)2 may be expressed as Divergence. V . A = g-l12 3
(g1/2Ai)_
(A.4)
Thus the expression N* V(V *N) appearing in eqs. (2.13) and (4.1) may be written in cylindrical coordinates in tensor notation as g-‘12
-& (g”‘N’)
I .
(A.9
g&!w=~(zty+~(~) --- a* a* g ax1ax2’
_2g12
(A.ll)
The expression (u - V)Q appears in eq. (2.10) where u is defined as the gradient of a scalar potential while Q is a scalar quantity. It may be
L.. Y. Shih / SelfIocusing
348
of laser beams
except
expressed as
(A.15) (A.12) The expression (v - V)v appears in eq. (2.12) where u is defined as the gradient of a sclar potential. In contravariant form it may be expressed as while
References
gjkVi,jVk
aVi
Ly
Vi,j=~-
i I
j
va.
(A.13)
I
The Christoffel symbol of the second kind is defined as {
i”j} +@!+?ap).
(A.14)
In terms of the elliptic coordinates defined in Section 3, we found CJ!
{ i j
I
=o
[l] R.Y. Chiao, E. Garmire and C.H. Townes, “Self-trapping of optical beams”, Phys. Rev. Len. 13,479 (1964) (Erratum: 14, iO56 (1965)]. PI D.I. Abakarov, A.A. Akopyan and S.I. Pekar, “Contribution to the theory of self focusing of light in nonlinearly polarizing media”, Soviet Phys. JETP 25, 303 (1967). [31 P. Hillion and S. Quinnez, “Analysis of self-trapping using the spinor wave equation”, Opt. Engrg. 24, 290 (1985) [41 P.L. Kelley, “Self-focusing of optical beams”, Phys. Rev. Lett. J5, 1005 (1965). 151V.I. Talanov, “Self focusing of wave beams in nonlinear media”, JEEP Len. 2, 138 (1965). 161Y.R. Shen, The Principles of Nonlinear Optics, Wiley, New York (1984) 307-313. . . ._ . [7] I.S. Sokolnikoff, Tensor Analysis: Theory and Appflcatlons to Geometry and Mechanics of Continua, Wiley, New York, 2nd ed. (1964).