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Lecture 8: Light Guiding Structures
Above, in Lecture 5, we investigated the total intrinsic reflection (TIR) phenomenon from the intersection of two medium. Let us now, using results obtain there to analyse the light waveguiding in such guiding structures, as a slab and a cylindrical fiber.
8.1. Light Waveguiding in the Dielectric Slab
First of all we consider a symmetrical dielectric structure shown in Fig. 8.1. Here we have an infinite (in width and length) dielectric slab of refractive index , sandwiched between two other infinite slabs, each of refractive index . As we will see, all analytical results obtained in Lecture 7 for different guiding structures using the guiding wave modes presentation can be obtained by use the light ray concept.
Using the Cartesian axes defined in the figure let us consider a light ray starting at the origin of the coordinate system and propagating within the first slab at an angle . If is greater than the critical angle introduced in Lecture 5, as was shown there, the light will bounce the first medium by means of a series of TIRs at the boundaries with the other two slabs, as is shown in Fig. 8.1. Since the wave is thus confind to the first slab it is said to be “guided” by a structure, which is also called a “waveguide”. Let us firstly consider TE guided light wave, that is, a lineary polarized wave for which the electric field component is normal to the plane of incidence (-polarization) as sketched in Fig. 8.1.The electric field of the incidence ray represented by symbol “i” in the picture can be written according to analysis carried out in Lectures 5 and 7 as:
(8.1)
where is an amplitude of the incident light wave. In the same manner we can write the ray field reflected from the first boundary between two slabs (see Fig. 8.1) taking into account the ray phase change at TIR for TE-polarized wave, :
(8.2)
These two waves will be superimposed on each other and will thus interfere. The interference pattern is obtained by adding them:
(8.3)
This is a ray propagating in the z-direction with wavenumber , and it is amplitude modulated in the x-direction according to .
According to symmetry of the problem, the intensity of the ray must be the same at each of the two boundaries (see Fig. 8.1). This requires that it is the same for x=0 and x=2a , i.e.,
(8.4)
or
(8.5)
where m is an integer. Since depends only on (according to Fresnel’s formulas), this angle can have only certain discrete values if the interference pattern is to remain constant along the length of the slab (fiber). Each interference pattern is characterized by a value m which provides a corresponding value for . The allowed interference patterns are corresponds to the “modes” of the waveguide (see Lecture 7), and then can be called the “modes” of optical fibers.
If we now turn to the propagation of the light wave along the quide structure (i.e., down the z-axis), we see from (8.3) that this is characterized by wavenumber, say
(8.6)
Furthermore, since the TIR condition requires that (see Lecture 5), it follows that
(8.7)
so that longitudinal wavenumber always lies between those of the two media (slabs). The transverse wave number is now can be defined as
(8.8)
If, for convenience, we also define a parameter p where
(8.9)
we discover that we can rewrite the “transverse resonance” condition (8.5) into more convenient form:
(8.10a)
for the TE mode propagation (with-polarization) and
(8.10b)
for TM mode propagation(with -polarization). We can use equations (8.10a) and (8.10b) to characterize the modes for any given slab geometry. The solutions of equations can be separated into odd and even types depending on whether m is odd or even. For odd m we have
(8.11a)
and for even m
(8.11b)
Taken, say, m to be even, we can write (8.10a), for example, in the form
(8.12)
Now from the definitions of q from (8.8) and p from (8.9) it is clear seen that
(8.13)
If we now take rectangular axes ap and aq, the latter relation between p and q transforms into a circle of radius (Fig. 8.2). If, on the same axes, we also plot the function , equation (8.12) is satisfied at all points of itersection between two functions (the left-side of (8.12) must be equal to the right-side of (8.13), see Fig. 8.2). In the same manner we could obtain the same relationships for odd m. Therefore these points provide the values of which correspond to the allowed modes of the guide described in Lecture 7. Having determined a value of for a given k, we can now obtain from (8.6): .
Finally, can be determined as a function of k for a given integer m. The resulting curves presented in Fig. 8.2 is an other form of presentation of “dispersive” properties of guiding wave, which were examined in more detail in Lecture 7 by use Fig. 7.5 for guiding TE and TM modes. From mentioned there and above is clear that the number of possible guiding modes (the corresponding interference patterns of reflected and incident rays) depends upon the values of the guiding structure parameters. But as was shown in Lecture 7, there will always be at least one mode solution. The same conclusion is seen from Fig. 8.2: the circle will always intersect the tangent curve even for a vanishingly small circle radius. If there is to be just one solution then Fig. 8.2 shows that the radius of the circle u must be less than , i.e.
(8.14)
or
(8.15)
The last equation (8.15) is the single-mode condition for this symmetrical slab waveguide. It represents an important case, since the existance of just one mode in a waveguide considerably simplifies the behavior of the power flux transport within it. If we remember, the same results were obtained in Lecture 7 both for TE and TM modes in rectangular and cylindrical (circle) guiding structures (TE , TM and TE, respectively).
8.2. Light Waveguiding in the Optical Fiber
Let us now consider the cylindrical dielectric structure as shown in Fig. 8.3. This is just the geometry of the optical fiber, where the central region is konwn as the “core” and the outer region as the “cladding”. In this case the same basic principles as for the dielectric slab, but the circular rather than planar symmetry changes the mathematics. We use solution of Maxwell’s equation in the cylindrical coordinates both for the coaxial cable and the circular waveguide, where we deal mostly with guiding modes rather than the ray concept [1-3]. Using the same approach, as in Lecture 7 for the axial cable, we finally can obtain solutions for the field of rays through the modified Bessel functions of first and second order, J(qr) and K(pr), where q and p are determined by (8.8) and (8.9), respectively, which can be rewritten as
, (8.16)
Thus the solution in the core is
(8.17)
and a similar solution for the cladding:
(8.18)
where l is the azimuth integer.
Again, as in the case of a slab, we can determined for a fiber the allowable values for p, q, and by imposing the boundary conditions at r=a. The result is a relationship which provides the versus k or “dispersion” curves shown in Fig. 8.4. The full mathematical approach is very complicated, and we use so-called “weakly guiding” approximation. This makes use of the fact that if the ray’s angle of incidence on the boundary must be very large if TIR is to occur. The ray must bounce down the core almost at grazing incidence. This means that the wave is very nearly a transverse wave, with very small z-components. By neglecting the longitudinal components , a considering simplification of the analysis results (see Fig. 8.5). Finally we obtain the same transverse waves propagation, as in the coaxial cable. Since the wave within fiber is considered to be transverse, the solution can be resolved conveniently into two linearly polarized components, just as for free-space propagation. The modes are thus called “linearly polarized” (LP) modes [1-3]. All solutions obtained above relate directly to the optical fiber. The latter has just the cylindrical geometry, and if for a typical fiber we have that than the “weakly guiding” approximation is valid. Some of the low order “LP modes” of light intensity distribution are shown in Fig. 8.6. Together with their polarizations and values for the azimuth integer l. There are two possible LP optical fiber modes: LP (l=0) and LP (l=1). For cylindrical geometry the single-mode condition is (instead (8.15)) [1-3]:
(8.19)
Bibliography
[1] Adams, M. J. An Introduction to Optical Waveguides, Wiley, New York, 1981.
[2] Elliott, R. S., An Introduction ro Guided Waves and Microwave Circuits, Prentice Hall, New Jersey, 1993.
[3] Optical Fiber Sensors: Principles and Components. Ed. by J. Dakin and B. Culshaw, Arthech House, Boston-London, 1988.
[4] Jackson, J. D., Classical Electrodynamics, New York: John Wiley & Sons, 1962.
[5] Chew, W. C., Waves and Fields in Inhomogeneous Media, New York: IEEE Press, 1995.
Additional Material (from Topic 5):
As was stated un Topic 5, the behaviour of Fresnel’s formulas (5.1)-(5.2) depends on boundary conditions. Thus if the fields are to be continuous across the boundary, as required by Maxwell’s equations, there must be a field disturbance of some kind in the second media (see Fig. 5.1). To investigate this disturbance we can use Fresnel’s formulas. We, first of all rewrite . For we can present by use some additional function , which can be more than unit. If so, . Hence we can write the field component in the second medium to vary as (for nonmagnetic materials )
(8.20a)
or
(8.20b)
This formula represents a ray travelling in the x-direction in the second medium (that is, parallel to the boundary) with amplitude decreasing exponentially in the z-direction (at right angles to the boundary). The rate at which the amplitude decreases with z can be wtitten
(8.21)
where is a wavelength of the light in the second medium. As seen, the wave attenuate significantly (~) over distances z of about .
Example: At the glass-air interface, the critical angle of is . For a light in the glass incident on the glass-air boundary at we find that . Hence the amplitude of the wave in the second medium is reduced by a factor of in a distance of only one wavelength, which is of the order of .
Even though the wave is propagating in the second medium, it transports no light energy in a direction normal to the boundary. All the light is totally internally reflected (TIR) at the boundary. The fields, which exist in the second medium give a Poynting vector which averages zero in this direction over one oscillation period of the light wave. The guiding effect is based on TIR phenomenon: all energy transport occurs along the boundary of two media after TIR, without any penetration of light energy inside the intersection.
Moreover, according to (5.1)-(5.2) and (5.6), we can note that the totally internal reflected (TIR) wave undergoes a phase change which depends on both the angle of incidence and the field polarization. This directly follows from derivations of Fresnel’s equation (5.1)-(5.2). Namely, for TM wave (i.e., -polarization) from (5.2) for and from above mentioned it follows that
(8.22)
This complex number provides the phase change on TIR as where for TM wave (-polarization) we have
(8.23a)
and for TE wave (-polarization)
(8.23b)
We note also that there is a close relationship between the light wave phase changes at TIR for both kinds of waves, that is,
(8.24)
and that
(8.25)
The variations of phase changes,, and their difference, , as a function of the incident angle are shown in Fig. 5.1. It is clear that the polarization state of light undergoing TIR will changed as a result of the differential phase change . By choosing appropriately and perhaps using two TIRs, it is possible to produce any required final polarization state from any given initial state. It is interesting to note that the reflected ray in TIR appears to originate from a point, which is displaced along the boundary from the point of incidence. This is consistent with the incident ray’s being reflected from a parallel plane which lies a short distance within the second boundary (see Fig. 5.2). This view is also consistent with the observed phase shift, which is now regarded as being due to the extra optical path traveled by the ray. The displacement is known as the Goos-Hanchen effect and provides an entirely consistent alternative expantion of TIR [4].