£ 2
Due to the increase in intermodal dispersion of a temporal pulse as the dielectric difference between core and cladding increases,
practical optical fibres employ small dielectric differences to maximise
the information carrying capacity. This condition can be stated as
£ - £
A = — --- - = sin2 6 « 1 , (20)
£ i
where £j,£2 are defined in fig. 2, and 0^ is the acceptance angle of the
waveguide.
Equation (20) allows sin 0^ to be approximated by
sin 0 ~ 0 (21)
c c
Using this physical constraint, equation (20), in equations (8) - (17) for the amplitude of the field components and eigenvalue euqation, introduces considerable simplification of the expressions for the modal fields.
Since the longitudinal propagation constant, 3/ must satisfy
k2 = k 0£2^ < 3 < k x = k 0£ ^ 2 (22)
for a bound mode, the weakly guiding fibre approximation, equation (20), has a clear physical interpretation using geometric optics. By defining an angle 0^ to the z-axis as in fig. 1.5 such that
3 = k, cos 0 , (23)
1 z
we see that bound modes of the weakly guiding fibre correspond to wave vectors with
0 < 0 « 1 ; (24)
z c
and if we interpret the direction of the wave vector as the ray direction of a geometric optics ray, the "bound rays" are paraxially
directed. Thus, the weakly guiding approximation is also known as the
paraxial ray approximation.
In the practical optical fibre y 1 = y2 as well as E l/ c 2 ~ 1*
Perturbing the eigenvalue equation about A £=0, we readily see that the
where U 2 (r) ^0 = + Kq , i„ V w i t h a . 3 f ' h '1 »' 1 1
i
w 2 H (1) H £ r=P (25) (26)and K 0 from equation (11) is
using the paraxial approximation of 3/k2 ~ 1/
l V 2 (p) P U 2 (p) W 2
(27)
In conventional parlance for the step-index fibre, the modes corresponding to the + signs are referred to as the HE^ and EH^ modes with the upper sign for the HE modes and the lower for the EH modes. This convention, established by Beam et at. [24] , refers to the dominant axial component of the electromagnetic field. The EH^ fields are those for which the magnitude of the z-component of the electric field is greater than the corresponding component of the magnetic and vice versa for the HE^ modes.
In the weakly guiding fibre approximation equation (16) or (17) for the inter-relationship of the magnitudes of two polarisations defined in equations (6) and (7) become
B £ = ± i . (28)
In the following analysis the upper sign is taken as referring to the HE modes and the lower sign for the EH modes following the
extension of the nomenclature of Beam to weakly guiding dielectric w a v e guides [7], From equation (28) it can be seen that the raison d ’etre for the nomenclature becomes shallow for the weakly guiding fibre.
Upon substitution of equation (28) into equations (1) - (4) via (6) and (7), using the paraxial approximation, the modal electromagnetic fields for the weakly guiding fibre become (excluding the common factor exp{i (cot - 3z - £(f>) }) ,
E z
E r
and the complex magnetic field in becomes
A AF A (r) (29)
W r)
(30)i iE, (31)
ie travelling wave form, viz. (e'U *)
(32)
where the radial variation of the transverse fields G Q (r) is defined by
Gl<r>
k i (- -5- 1+F' ( r ) --- F 0 (r)£ (33)(The superscript + on G ^(r) refer to the HE and EH modes respectively.) k x is the wavenumber at the axis of the waveguide, defined in equation
(22). We have consistently used the weak guidance approximation throughout equations (29) - (33).
F^(r) in equation (33), used to represent the field in both the core and cladding regions, is defined by
F £ (r) = f£(r) : r < p f
f 1
- J
*„ ( 1 )
iWr : r > P * H (1)| Ä ' r=p l P J (34)Furthermore, upon investigation of equations (30) - (33) we find that the transverse field components are related by
re.
S
t%
z x E m
~ ~T (35)
where z is the unit vector in the z-direction (along the fibre axis and the subscript T refers to the transverse components.
In the weakly guiding dielectric fibre, the bound mode field components which are dominant are the transverse fields. However, our
derivation of the field components expresses the transverse fields in terms of the longitudinal field. It would be preferable to have the components expressed in terms of the transverse fields. Such a study has been used by Kurtz and Streifer [22] for more complicated
_+
longitudinal field solutions. Substituting G^(r) into the radial wave equation, equation (1.27), (or by using equations (1.3-6)) results in a
_+
similar equation to equation (33) for f^(r) in terms of G^(r) and we find (ignoring terms in d£/dr and higher order)
V r)
1 l+
[G +l 1 £+
1 gt 1 *1r
r (36)Using equation (36) together with equation (33), the transverse wave function G^(r) is found to satisfy the second order differential equation 1_ _d_ r dr d ' r — + l drj , 2, %
a
+ i)V r) - -TT-
0 , (37)again ignoring all terms in de/dr and higher.
Comparing equation (37) with the radial wave equation, equation (l. 27), G~(r) satisfies an identical equation with "£", the azimuthal mode number replaced by " (£+1)". G^(r) is known as the
"scalar optical field" [22], which from equation (37) can now be written as G £ (r) N £+l f £ + l (r'^£) m u(1) M £ + l H £+l iWr l P for r < p for r > p (38)
where are both constants which must be chosen so transverse fields satisfy the boundary conditions at r = p. find that the T h u s , we where £+1 £+1 - l £+1 P $ f (r) P___ (1) (iW) r=P (39) (40)
Using the recurrence relations for cylinder functions [17] and equation (34) , within the weak guidance approximation, can
written as
M
£+1 *£ > ° ' (41)
where is defined in equation (40) . Using equation (38) in (36) we find that the radial variations of the longitudinal fields can now be written as
N _ f __ N
f £ (r'^£} = “ \ ± f £ + l (r'^£} r f£ + l (r' V j ' (42)
where we have explicitly included the dependence of the functions, f^(r), on 3^, the eigenvalue, so that equation (42) is not confused with
ordinary recurrence relations for the functions.
Using the above relationships between the longitudinal and transverse field components we can now write the modal electromagnetic fields defined in equations (29) - (34) in a simpler form. In the fields presented below, we have extended the analysis to include the modes with azimuthal mode number £, defined in the exp{-i£(j)} variation of the
fields, to include £ < 0, so that now the modal fields in the waveguide i*
can be written as, for r < p
£ Ä I I + 1 1 fi. ] + 1 r |s,| + l <p)j x [ Ä. I -1 -i£(j) fini(r) e-i£cj) <\> A |£| + l [ f |£| + 1 (r)/ f |£| + l (P)y and i ^ E £ <t>
and the complex magnetic field, when written in this azimuthal travelling wave form satisfies
(43)
(44)
(45)
The modulus signs on the subscripts of the functions fß(r) etc. is necessary, since only £ 2 appears in the radial wave equation, equation
(1.27). From now on however, for the sake of conciseness, we will omit these signs from the order of the function.
H = (46)
where we have normalised the transverse electric field to unity at the
[h e]
core-cladding boundary, r = p. As usual the + signs refer to the < ? modes respectively. A^-^, the power normalisation coefficient is related to A^ defined in equations (29) - (34) by
A£+l = N£+l A£ '
where is defined by equations (39) and (41). The sign | | denotes the modulus of the number inside. In the cladding r > p, the modal fields are and E z M . f iw (1) *■ U I + l \ k 2P P I r lW - PJ -i£cb e -i£cj) e (47) (48)
with all other field components obtained from equations (47)-(48) by application of the relations of equations (45)-(46). These fields are correct to ignoring all terms in de/dr. Higher order corrections to these fields and the eigenvalue equation can be developed using the formulation of Appendix 2B.
In this travelling wave form we can observe that these results agree with those derived by Marcuse for the step-index weakly guiding fibre. The standing wave forms, viz. cos £<f), sin £(j) forms, are obtained simply by addition and subtraction of the two travelling wave modal f i e l d s .
Either by direct substitution of equations (33) and (42) into the eigenvalue equation, equation (25), or by the observation from (42), we can show that all boundary conditions can be satisfied by the
condition
£+1 £+1 r=p
Hm (iw)
where as usual the ' indicates d/dr.
This simple form of the eigenvalue equation for the modes with azimuthal mode number £, allows easy computation of the cutoff values of a mode, exactly analogous to Suitzer's treatment for the cutoff values for the step-index fibre modes. It is immediately apparent from
equation (49), that the HE^ and EH^ modes satisfy identical eigenvalue equations so that the degeneracy of these mode pairs is the same as in the step-index weakly guiding fibre.
Equation (49) is identical to the eigenvalue equation derived by the "scalar" approximation recently used by Arnaud [24] for step-
index waveguides, or from the analysis of graded-index fibres by Okoshi and Okamoto [25], and more recently Hashimoto et al. for cladded
parabolic profiles [26].
To conclude our discussion on the modes of the weakly guiding fibre, we must determine the formulae for the normalisation coefficient A^_i (defined below equation (46)), defined to be the arbitrary constant that yields unit power propagating in the mode with azimuthal mode
number &.
The conventional power orthogonality between electromagnetic modes propagating in the same axial direction, wit h fields defined by
h } and {je t ,h , in a dissipationless medium [5] is X/iTi %j rn X/ rQ
• z dA (50)
where the 6's are the usual Krönecker delta and A is the infinite
00
cross-sectional plane. (The additional subscripts m and m' refer to the mth (or m'th) solution of the eigenvalue equation for the mode with azimuthal mode number £ (or £').)
Using equation (35) we can now write equation (50) as
(51)
becomes
1+ 1, m.
- 2
fe 1
t i ^2
II 2
dA , (52)
which upon conversion of the integral to cylindrical polar co-ordinates can be written as
- 2
£+1, in
£ h