Finally, the conventional slot waveguide is connected to a strip waveguide through an s-shape strip- to-slot mode converter. Figure 3(f) shows the measured transmission spectrum of a200μmlong1DPCslot waveguide filledwith EOpolymer.Aclear bandedge canbeobserved near 1550 nm. The proposed slot waveguide consists of a vertical slot structure placed above a silicon slab, with a low- index buffer layer between them to construct a T-shaped slot region. Such a T-shaped slot waveguide could be realized by using similar fabrication methods of the conventional slot waveguides.

1. Optical Waveguide Mode
2. Slot Waveguide Mode Converter
3. Slot Waveguide Model
4. Slot Waveguide Mode Profile

Waveguide Tutorial Includes:
Waveguide basicsWaveguide modesWaveguide impedance & matchingWaveguide cut-off-frequencyWaveguide flangesWaveguide junctionsWaveguide bendsFlexible waveguideWaveguide types & sizes

Electromagnetic waves can travel along waveguides using a number of different modes.

The different waveguide modes have different properties and therefore it is necessary to ensure that the correct mode for any waveguide is excited and others are suppressed as far as possible, if they are even able to be supported.

Waveguide modes

Looking at waveguide theory it is possible it calculate there are a number of formats in which an electromagnetic wave can propagate within the waveguide. These different types of waves correspond to the different elements within an electromagnetic wave.

• TE mode: This waveguide mode is dependent upon the transverse electric waves, also sometimes called H waves, characterised by the fact that the electric vector (E) being always perpendicular to the direction of propagation.
• TM mode: Transverse magnetic waves, also called E waves are characterised by the fact that the magnetic vector (H vector) is always perpendicular to the direction of propagation.
• TEM mode: The Transverse electromagnetic wave cannot be propagated within a waveguide, but is included for completeness. It is the mode that is commonly used within coaxial and open wire feeders. The TEM wave is characterised by the fact that both the electric vector (E vector) and the magnetic vector (H vector) are perpendicular to the direction of propagation.

Text about the different types of waveguide modes often indicates the TE and TM modes with integers after them: TE_m,n. The numerals M and N are always integers that can take on separate values from 0 or 1 to infinity. These indicate the wave modes within the waveguide.

Only a limited number of different m, n modes can be propagated along a waveguide dependent upon the waveguide dimensions and format.

Optical Waveguide Mode

For each waveguide mode there is a definite lower frequency limit. This is known as the cut-off frequency. Below this frequency no signals can propagate along the waveguide. As a result the waveguide can be seen as a high pass filter.

It is possible for many waveguide modes to propagate along a waveguide. The number of possible modes for a given size of waveguide increases with the frequency. It is also worth noting that there is only one possible mode, called the dominant mode for the lowest frequency that can be transmitted. It is the dominant mode in the waveguide that is normally used.

It should be remembered, that even though waveguide theory is expressed in terms of fields and waves, the wall of the waveguide conducts current. For many calculations it is assumed to be a perfect conductor. In reality this is not the case, and some losses are introduced as a result, although they are comparatively small.

Rules of thumb

There are a number of rules of thumb and common points that may be used when dealing with waveguide modes.

• For rectangular waveguides, the TE₁₀ mode of propagation is the lowest mode that is supported.
• For rectangular waveguides, the width, i.e. the widest internal dimension of the cross section, determines the lower cut-off frequency and is equal to 1/2 wavelength of the lower cut-off frequency.
• For rectangular waveguides, the TE₀₁ mode occurs when the height equals 1/2 wavelength of the cut-off frequency.
• For rectangular waveguides, the TE₂₀, occurs when the width equals one wavelength of the lower cut-off frequency.

Waveguide propagation constant

A quantity known as the propagation constant is denoted by the Greek letter gamma, γ. The waveguide propagation constant defines the phase and amplitude of each component or waveguide mode for the wave as it propagates along the waveguide. The factor for each component of the wave can be expressed by:

Where:
z = direction of propagation
ω = angular frequency, i.e. 2 π x frequency

It can be seen that if propagation constant, γ_m,n is real, the phase of each component is constant, and in this case the amplitude decreases exponentially as z increases. In this case no significant propagation takes place and the frequency used for the calculation is below the waveguide cut-off frequency.

It is actually found in this case that a small degree of propagation does occur, but as the levels of attenuation are very high, the signal only travels for a very small distance. As the results are very predictable, a short length of waveguide used below its cut-off frequency can be used as an attenuator with known attenuation.

The alternative case occurs when the propagation constant, γ_m,n is imaginary. Here it is found that the amplitude of each component remains constant, but the phase varies with the distance z. This means that propagation occurs within the waveguide.

The value of γ_m,n is contains purely imaginary when there is a totally lossless system. As in reality some loss always occurs, the propagation constant, γ_m,n will contain both real and imaginary parts, α_m,n and β_m,n respectively.

Accordingly it will be found that:

This waveguide theory and the waveguide equations are true for any waveguide regardless of whether they are rectangular or circular.

It can be seen that the different waveguide modes propagate along the waveguide in different ways. As a result it is important to understand what he available waveguide modes are and to ensure that only the required one is used.

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Metal pipe waveguides are often used to guide electromagnetic waves.The most common waveguides have rectangular cross-sections and so arewell suited for the exploration of electrodynamic fields that dependon three dimensions. Although we confine ourselves to a rectangularcross-section and hence Cartesian coordinates, the classification ofwaveguide modes and the general approach used here are equallyapplicable to other geometries, for example to waveguidesof circular cross-section.

The parallel plate system considered in the previous threesections illustrates much of what can be expected in pipe waveguides.However, unlike the parallel plates, which can support TEM modes aswell as higher-order TE modes and TM modes, the pipe cannot transmit aTEM mode. From the parallel plate system, we expect that a waveguidewill support propagating modes only if the frequency is highenough to make the greater interior cross-sectional dimension of thepipe greater than a free space half-wavelength. Thus, we will findthat a guide having a larger dimension greater than 5 cm wouldtypically be used to guide energy having a frequency of 3 GHz.

We found it convenient to classify two-dimensional fields astransverse magnetic (TM) or transverse electric (TE) according to whether E or H was transverse to the direction ofpropagation (or decay). Here, where we deal with three-dimensionalfields, it will be convenient to classify fields according to whetherthey have E or H transverse to the axial direction ofthe guide. This classification is used regardless of thecross-sectional geometry of the pipe. We choose again the ycoordinate as the axis of the guide, as shown in Fig. 13.4.1. If wefocus on solutions to Maxwell's equations taking the form

then all of the other complex amplitude field components can bewritten in terms of the complex amplitudes of these axial fields,H_y and E_y. This can be seen from substituting fields having theform of (1) and (2) into the transverse components of Ampère'slaw, (12.0.8),

and into the transverse components of Faraday's law, (12.0.9),

If we take _y and _y as specified, (3) and(6) constitute two algebraic equations in the unknowns _xand _z. Thus, they can be solved for these components.Similarly, _x and _z follow from (4) and (5).

We have found that the three-dimensional fields are asuperposition of those associated with E_y (so that the magneticfield is transverse to the guide axis ), the TM fields, and those dueto H_y, the TE modes. The axial field components now play the roleof 'potentials' from which the other field components can be derived.

We can use the y components of the laws of Ampère andFaraday together with Gauss' law and the divergence law for H toshow that the axial complex amplitudes _y and _ysatisfy the two-dimensional Helmholtz equations.

TM Modes(H_y = 0):

where

and

TE Modes(E_y = 0):

where

These relations also follow from substitution of (1) and (2) into they components of (13.0.2) and (13.0.1).

The solutions to (11) and (12) must satisfy boundary conditions onthe perfectly conducting walls. Because E_y is parallel to theperfectly conducting walls, it must be zero there.

TM Modes:

The boundary condition on H_y follows from (9) and (10), whichexpress _x and _z in terms of _y. On the walls at x = 0 and x = a, _z = 0. On the wallsat z = 0, z = w, _x = 0. Therefore, from (9) and(10) we obtain

TE Modes:

The derivative of _y with respect to a coordinateperpendicular to the boundary must be zero.

The solution to the Helmholtz equation, (11) or (12), follows apattern that is familiar from that used for Laplace's equation in Sec.5.4. Either of the complex amplitudes representing the axial fields isrepresented by a product solution.

Substitution into (11) or (12) and separation of variables then gives

where

Solutions that satisfy the TM boundary conditions, (13), are then

TM Modes:

so that

When either m or n is zero, the field is zero, and thus m and nmust be equal to an integer equal to or greater than one. For a givenfrequency and mode number (m, n), the wave number k_y isfound by using (19) in the definition of p associated with (11)

with

Thus, the TM solutions are

For the TE modes, (14) provides the boundary conditions, and we areled to the solutions

TE Modes:

Substitution of _m and _n into (17) therefore gives

The wave number k_y is obtained using this eigenvalue in thedefinition of q associated with (12). With the understanding thateither m or n can now be zero, the expression is the same as thatfor the TM modes, (20). However, both m and n cannot be zero. Ifthey were, it follows from (22) that the axial H would be uniformover any given cross-section of the guide. The integral of Faraday'slaw over the cross-section of the guide, with the enclosing contour Cadjacent to the perfectly conducting boundaries as shown in Fig.13.4.2, requires that

where A is the cross-sectional area of the guide. Because the contouron the left is adjacent to the perfectly conducting boundaries, theline integral of E must be zero. It follows that for the m =0, n = 0 mode, H_y = 0. If there were such a mode, it wouldhave both E and H transverse to the guide axis. We willshow in Sec. 14.2, where TEM modes are considered in general, thatTEM modes cannot exist within a perfectly conducting pipe.

Even though the dispersion equations for the TM and TE modes onlydiffer in the allowed lowest values of (m, n), the field distributionsof these modes are very different.

The superposition of TE modes gives

where m n 0. The frequency at which a given modeswitches from evanescence to propagation is an important parameter.This cutoff frequency follows from (20) as

TM Modes:

TE Modes:

Rearranging this expression gives the normalized cutoff frequency asfunctions of the aspect ratio a/w of the guide.

These normalized cutoff frequencies are shown as functions of w/ain Fig. 13.4.3.

The numbering of the modes is standardized. The dimension w ischosen as w a, and the first index m gives the variation ofthe field along a. The TE₁₀ mode then has the lowest cutofffrequency and is called the dominant mode. All other modeshave higher cutoff frequencies (except, of course, in the case of thesquare cross-section for which TE₀₁ has the same cutofffrequency). Guides are usually designed so that at the frequency ofoperation only the dominant mode is propagating, while allhigher-order modes are 'cutoff.'

In general, an excitation of the guide at a cross-section y =constant excites all waveguide modes. The modes with cutoff frequencies higher than the frequencyof excitation decay away from the source. Only the dominant mode hasa sinusoidal dependence upon y and thus possesses fields thatare periodic in y and 'dominate' the field pattern far away fromthe source, at distances larger than the transverse dimensions of thewaveguide.

Example 13.4.1. TE₁₀ Standing Wave Fields

The section of rectangular guide shown in Fig. 13.4.4 is excitedsomewhere to the right of y = 0 and shorted by a conducting plate in theplane y = 0. We presume that the frequency is above the cutofffrequency for the TE₁₀ mode and that a > w as shown. Thefrequency of excitation is chosen to be below the cutoff frequencyfor all higher order modes and the source is far away from y = 0(i.e., at y a). The field in the guide is then that of theTE₁₀ mode. Thus, H_y is given by (25) with m = 1 andn = 0. What is the space-time dependence of the standing wavesthat result from having shorted the guide?

Because of the short, E_z (x, y = 0, z) = 0. In order to relate thecoefficients C⁺₁₀ and C^-₁₀, we must determine _zfrom _y as given by (25) using (10)

and because _z = 0 at the short, it follows that

Slot Waveguide Mode Converter

so that

and this is the only component of the electric field in this mode. We can now use (29) to evaluate (25).

In using (7) to evaluate the other component of H, remember thatin the C⁺_mn term of (25), k_y = _mn, while in theC^-_mn term, k_y = -_mn.

To sketch these fields in the neighborhood of the short anddeduce the associated surface charge and current densities, considerC⁺₁₀ to be real. The j in (31) and (32) shows that H_x andH_y are 90 degrees out of phase with the electricfield. Thus, in the field sketches of Fig. 13.4.4, E andH are shown at different instants of time, say E when t = and H when t = /2. The surfacecharge density is where E_z terminates and originates on the upper and lowerwalls. The surface current density can be inferred from Ampère'scontinuity condition. The temporal oscillations of these fieldsshould be pictured with H equal to zero when E peaks, andwith E equal to zero when H peaks. At planes spaced bymultiples of a half-wavelength along the y axis, E is alwayszero.

The following demonstration illustrates how a movable probe designedto couple to the electric field is introduced into a waveguide withminimal disturbance of the wall currents.

Demonstration 13.4.1. Probing the TE₁₀Mode.

A waveguide slotted line is shown in Fig. 13.4.5. Here theline is shorted at y = 0 and excited at the right. The probeused to excite the guide is of the capacitive type, positioned sothat charges induced on its tip couple to the lines of electric fieldshown in Fig. 13.4.4. This electrical coupling is an alternative tothe magnetic coupling used for the TE mode in Demonstration 13.3.2.

Slot Waveguide Model

The y dependence of the field pattern is detected in theapparatus shown in Fig. 13.4.5 by means of a second capacitiveelectrode introduced through a slot so that it can be moved in they direction and not perturb the field, i.e., the wall is cut along the lines of the surface current K. From the sketch of K given in Fig. 13.4.4, it can be seen that K is inthe y direction along the center line of the guide.

Slot Waveguide Mode Profile

The probe can be used to measure the wavelength 2 /k_y of the standing waves by measuring the distance between nulls in theoutput signal (between nulls in E_z). With the frequency somewhatbelow the cutoff of the TE₁₀ mode, the spatial decay away fromthe source of the evanescent wave also can be detected.