US2738468A - Wave guide filters - Google Patents

Description

' March 13, 1956 w. A. MILLER 2,738,468

WAVEGUIDE FILTERS Filed July 24, 1950 I 2 Sheets-Sheet 1 2& 1-9- We 0 5 04M! 0 t/ p o 0}/- kl, 04 La I 4 I 6 1 1 a 72m;

United States Patent M WAVE GUIDE FILTERS William A. Miller, Place, Y.-,' assigiio i" to Radio Corporation of America, a corporationof Delaware Application July 24, 1950, Serial No. 175,625

The terminal years of the term of the patent to be granted has been disclaimed 1 Claim. (Cl. 333-73) This invention relates to microwave guide filters and particularly to such filters consisting of a series of waveguide sections.

More particularly, the filter of this invention consists of a series of waveguide sections connected together in a predetermined sequence and at predetermined angles, the guide sections having predetermined indexes of refractions and having predetermined lengths.

in my copending application Serial No. 166,350, filed June 6, 1950, and'in my copending application Serial No. 166,050, filed June 3, 1950, there were disclosed filters for a microwave system in which interference of other than the pass-band frequencies was accomplished by assembling together a series of waveguide sections having different indexes of refraction. The waves entered and passed through these filters at zero angle of incidence, that is, the angle between the direction of propagation and the normal to the junctions (hereinafter referred to as surfaces) between waveguide sections, was zero degree.

In the embodiment of the present invention, the filter consists of a plurality of waveguide sections connected together and arranged symmetrically in both directions from the center guide section. The filter is, therefore, bidirectional. The outer or end guide sections are connected to their contiguous intermediate guide sections at a critical angle. This angle is determined such that there would be total reflection of transmitted waves in the filter at the surfaces between the outer guide section and its adjoining intermediate guide section, except for the predetermined index of refraction of the intermediate guide sections. In other words, total reflection of the transmitted waves at the end intermediate surfaces is frustrated or prevented by the intermediate sections having predetermined indexes of refraction in relation to the indexes of refraction of the outer guide sections for the physical angle between the outer and intermediate guide sections. The index of refraction and the length of the center guide section is such that, in combination with the outer and intermediate sections, interference of waves .of other than the band-pass frequencies is accomplished.

The principal object of the invention is to provide a microwave filter in which a narrow pass-band is obtained by causing interference between waves of other than the pass-band frequencies.

Another object of the invention is to provide a filter of high discrimination in which the transmitted waves are passed into one section of the filter at a critical angle.

Another object of the invention is to provide a filter of high discrimination in which assembled sections of waveguides of critical dimensions are connected together at critical angles.

Another object of the invention is to provide a filter of high discrimination in which the waves are fed into one section of the filter at an angle of incidence greater than zero degree.

Another object of the invention is to provide a filter of high discrimination in which waves within the filter would be totally reflected except for a frustrating guide section of the filter.

Other objects and advantages of the invention will be apparent from the following detailed description made with reference to the' accompanying drawings in which:

Fig. 1 is a simplified or line diagram of waveguides of the filter of this invention, showing the reflection of an' electromagnetic wave by a surface separating two waveguide sections having different indexes of reflection;

Fig. 2 is a line diagram showing" the transmission of an electromagnetic wave through three joined waveguide sections having different indexes of refraction;

Fig. 3 is a line diagram showing the transmission of an electromagnetic wave through five waveguide sections having indexes of refraction of n2, n1, no, In, and m, respectively;

Fig. 4 is a line diagram showing the transmission and some of the reflections occurring when an electromagnetic wave enters and traverses five waveguide sections having indexes of retraction of n2, 111, no, In, and nz, respectively;

Fig. 5 is a cross-section of one end of a frustrated total reflection waveguide filter;

Fig. 6 is a graph showing the relation of the index of refraction of a waveguide section to the ratio of (1) the wavelength of a wave in the guide to (2) the critical wavelength of the guide;

Fig. 7 is a top viewof a frustrated total reflection waveguide filter;

Fig. 8 is a front view of the frustrated total reflection filter in Fig. 7;

Fig. 9 is a view in and 8; and

Fig. 10 is a graph showing the relation between the relative power transmitted through the filter in Fig. 9 to the wavelength of the waves propagated through the filter.

Referring to Fig. l, 0 and 1 are two waveguide sections having indexes of refraction of no and m, respectively, and separated by surface 0-1. Four (4) is the ray-tracer of an electromagnetic wave transmitted through section 1, striking surface 01 at point 5 and being reflected as wave 6. For this reflecting condition to exist, in must be greater than no and the sine of the angle of incidence, 0 (the angle between the direction of propagation and the normal 7 to the surface 01 at the point of incidence 5) must be equal to or greater than the ratio of the smaller to the larger index of refraction. As the index of refraction is defined as the ratio of (l) the velocity of the electromagnetic Wave in air (taken as unity) to (2)' the group velocity of the wave in the waveguide section, no equals 1/ V0 and n1 equals 1/ V1, where V0 and V1 are the velocities of propagation of the wave in sections 0 and 1, respectively. For a reflecting condition at point 5, therefore, V0 must be greater than V1.

Referring to Fig. 2, waveguide section 0 is separated from section 1 by surface 0-1 and section 2 is separated from section 1 by surface 2-1. By directing the Wave 4 at surface 2-1 at an angle of incidence 02 and by introducing between section 2 and section 0, respectively, a section 1 of a proper length and index of refraction, wave 4 will enter section 1 at' angle of refraction 61, be transmitted through section 1 at the velocity V1, enter section 0 at an angle of refraction of B0 and be transmitted through section 0 at a velocity of V0. For such a condition, V1 must be greater than V2 and V2 must be greater than Vo.

It follows from the preceding that for other than the certain values of angles of incidence, lengths and indexes of refraction of the waveguide sections total reflection of electromagnetic waves may be made to occur or, for

perspective of the filter in Figs. 7

certain values, as aforesaid, transmission through the Patented Mar. 13, 19 56 3 sections may occur. The three sections, therefore, accomplish frustrated total reflection.

To apply these conditions to provide a filter effective in either direction of propagation through the filter, five sections are assembled, as in Fig. 3, in the order of indexes of refraction of 21,, n,, 11,, n,, and n respectively, with corresponding velocities of propagation of V2, V1, V0, V1, and V2, respectively. To constitute a frustrated total reflection filter, V must be less than V2 and V2 must be less than V1- In applying the principles of this invention to the construction of a filter, the angle of incidence 02 is made sufliciently large to ensure total reflection at the surface 21 if the index of refraction of section 1 is unity, that is, total reflection would occur in the filter at surface 21 if section 1 is a waveguide the index of refraction of which is unity. If n it, and n. are the indexes of refraction of waveguides relative to free space, then for total reflection,

which is merely a statement of the law of total reflection, and, by Snells law of refraction,

sin 02 1 /n where "s is the index of refraction of the 5th section relative to free space) and A is the wavelength of the radiation in free space.

To design a filter for a particular waveguide system, then, it is necessary, first to ensure an angle of incidence 02 such that total reflection will occur at surface 21 when the index of refraction of section 1 is unity and then change the index of refraction of section 1 by changing its dimensions to frustrate total reflection at this angle of incidence. As waveguides may be constructed over a wide range of indexes of refraction, 02 may be determined by construction considerations such as, for example, the mechanical arrangement of sections in relation to sections of a waveguide system.

There are, however, limits that are imposed on the range of selection of values of 02, which may be expressed as It is apparent that if the left half of inequality (4) is not met, total reflection will not occur in the absence of a frustrating section 1 and if the right half of the inequality (4) is not met, total reflection will occur at the surface 2-1 even in the presence of the frustrating section 1. It is, therefore, concluded that l n n, (5

and since n must be greater than 11,, total reflection at the surface 1-0 cannot occur, regardless of any real assumed value of 01 and 00 must be less than 01.

Frustrated total reflection filter In the filter arrangement in Fig. 3, the transmitted wave only is shown. There will, of course be reflections at the several surfaces, which phenomena make the arrangement respond as an interference filter. In the interests of simplicity, the reflections occurring (Fig. 4) will be considered in only one direction of transmission. As the reflections at the lower surface 2-1 and the lower surface 1-0 are minor in strength and effect, they may be neglected. Likewise, the reflections transmitted through the lower section 2 may be neglected and only the reflections that throw back energy into section 0, at 76 both the upper and lower surfaces 1-0, will be considered.

If R is the coefiicient of reflection and To is the coefiicient of transmission at a surface between two sections, and zero absorption is assumed,

If 1r/J is the phase difference between (1) a wave transmitted from section 1 (lower) through medium 0 and into section 1 (upper) and (2) the same wave except reflected by upper surface 1-0, to the lower surface 1-4), to and through the upper surface 10 and into section 1 (upper), the intensity of the radiation (1) transmitted per unit incident radiation through such surfaces is given by the equation T, T, +4R site (11)) (7) Equation 8 (see Max Born, Optik, Springer 1933, 123), however, applies to parallel surfaces when the angle of incidence is zero. As the sections under consideration have different indexes of refraction, and the angle of incidence cannot be set equal to the angle of refraction, Equation 8 becomes where Rll-l is the coefiicient of reflection at the lower or upper surface 0-1, 6, is the phase advance on a single passage of the wave through the 0 section and A is the phase shift of the wave incident in the 0 section upon reflection from one of surfaces 01.

The phase advance 6,, on a single passage of a wave through the section may be defined by the equation 5,=21rn, cos 0, (10) where d5 is the perpendicular equivalent length of the 5th section.

It will be noted from Equation 9 that the value of (I) is a maximum when where m is an integer. The values of A and Ro-1 may, therefore, be calculated as follows:

Reflection coefficient (R) for a multi-section filter The relation between index of refraction n and the angle of incidence 0 for a particular (5th) refracting section (ts) may be defined as it, cos 0, t,'

K, cos 0.-1 (13) Where its is defined in Equation 3.

The effective index ofrefraction of each of the s layers (Ts) would, therefore, be

I 1 --1 T,+1 tan 5. (14) The reflection coefiicient of a multi-section filter may then be found by determining from the known values of t +1 and fi found from Equations 13 and 10, respectively. The values of T3 for successively decreasing subscripts are then derived until T1 is found. From this value, the value of R may be calculated, using Equation 12.

To evaluate R for a multi-section filter in which the electric vector is parallel to the plane of incidence, Equation 13 is written as lc, cos 0 (16) and the values obtained are substituted in Equations 14 and 15 to obtain the value of R under these conditions. It will be noted that Equation 16 is not the reciprocal of Equation 13.

Obtaining the value of R for a frustrated total reflection filter The first step in evaluating the reflection coeflicient R for a frustrated total reflection filter is to determine the coefficient of reflection at the boundary surfaces of the interfering waveguide 0. It is apparent that the value of R at only one boundary surface and for only one direction of propagation need be obtained.

In Fig. 5, which is the waveguide equivalent of the line diagram Fig. 2, wave 4 is transmitted into the waveguide 8 at an angle 00 and passes through frustrating waveguide section 9 of index of refraction 11,, the bound ary surface between the two guide sections being indicated by line 10. After traversing guide section 9, the wave is transmitted into guide section 11, which is connected to guide section 9 at an angle of 92 degrees.

From Equation 14, the value of T1 for the surface 2-1 is and the value of 61 in Equation 10 become imaginary and may be defined as (19a) 19b) (190) T1 then becomes T z't (-il +tanh 'y) 1 l-it tanh 'y The value of T1 may be substituted in Equation 12 to obtain the value of R0-1.

Evaluation of A In deriving the value of A, it may be assumed that total reflection occurs. The error introduced by this assumption is very slight for a narrow band filter such as this invention provides. The assumption may be stated as tan A=l (21) or T1=it,' and From Equation 22 and by the laws involved in phase change upon reflection,

Calculation of the dimensions of a frustrated total reflection filter In the practical application of the principles of this invention to filters, certain values and conditions are set and determined by the waveguide system into which the filter is to be included. For example, let the wavelength to be transmitted in the guide system be 1 cm. and the radiation be of the TEM mode. The index of refraction of waveguides may be determined from the graph in Fig. 6, for selected ratios of (1) the wavelength in the guide to (2) the cutoff wavelength. Hence, the dimensions of the waveguide may be selected for a mean value of n,, for example, n =1.5.

From Equation 1, with a value of n,=l.5, 62 is greater or equal to 42 degrees. Selecting a reasonable value of 11 for the frustrating guide section of 11 equals 1.25, from Equation 4, 02 is equal to or less than 56 degrees. The value of 02 equals 50 degrees is thus found to be satisfactory.

The value of the index of refraction n of section 0 must be somewhat greater than n but, from a consideration of the trend of the curve in Fig. 2, a value of n =2.50 is selected to avoid high dispersion in this section.

The value of k is determined by For 0 =50, sin 0,=0.766, and cos 0,=0.643.

From Equation 2,

sin 62 and cos 0 =0.886 From Equation 10 and setting the length of the 0 section equal to 1.11

6 =4.9551r and from Equation 11 4.9951r+A=51r and A=0.0451r From Equation 23 t,'=-tan 4 or From the Inequality (4), the radiation propagated in the direction of the arrow in Fig. 5, the right part of Inequality (4) must be used for the angle of incidence in the frustrating guide section, and

sin 0 7 cos 0,, therefore equals:

cos 0,- =i(1.04l) =0.2i From Equation 10 t, is obtained from Equation 13 Applying the determined values to Equation 20 Substituting this value of T1 in Equation 12,

and R0-1 =0.895.

From the data hereinbefore determined, the dimensions of the filter and its performance may be calculated. The dimensions of waveguide sections in Fig. 7 (a top view of the filter) and in Fig. 8 (a side elevation of the filter) corresponding to the reference characters are stated or calculated as follows:

From Fig. 6, with 110:2.50, M/ \=0.92 and as 101:1 cm. and \c=1.09 cm., Wo, therefore, equals 0.545 cm. Similarly for values n and n w equals 0.78 cm. and w, equals 0.87 cm.

d, and d were calculated on the basis of their radiation length and, therefore, their linear length is found by multiplying the values of d, and d by their respective indexes of refraction.

d,'=0.605.1.25=0.756 cm. and

d =1.11.250=2.79 cm.

Internal resistors 12 are provided in the frustrating section 1 to dampen any spurious cavity modes. The

length l of the filter is not critical, if it is several waveof refraction. As filters in microwave systems are energized by radiations of only slightly wider band of frequencies than the band-widths of the filters, i, may be assumed to be a constant, that is, the waveguide dispersion may be neglected. On the other hand, 6 =k/ which is not a constant.

The relative power transmitted in relation to the Wavelength in ems. has been plotted in Fig. 9 using the equation 1 1+325 sin (1.76+13.9/)\) For a pass-band at 1 cm., the pass-bands closest to the 1.0 cm. pass-band, as calculated from Equation 21, are located at 1:082 cm. and \=1.3 cm. A filter such as disclosed in my pending application Serial No. 166,050, filed June 3, 1950, will provide more than ample isolation for the pass-band centered at 1 cm.

There has thus been disclosed herein a bidirectional, interference type of waveguide filter consisting of waveguide sections in series of predetermined indexes of refraction and lengths, the end sections of which are connected to intermediate sections at predetermined angles to produce at the joint surfaces of these sections, frustrated total reflection and accompanied interference filtering.

What is claimed is:

A microwave guide filter for selectively transmitting microwave energy at a predetermined frequency comprising, a first and central waveguide section having a longitudinal axis and an index of refraction of n at said frequency, second and third waveguide sections connected to opposite ends of said first waveguide section each having indexes of refraction 11 at said frequency and their longitudinal axes parallel to the axis of said first waveguide section, means within said second and third waveguide sections for damping spurious modes within said second and third waveguide sections, and fourth and fifth waveguide sections connected to the outermost ends of said second and third waveguide sections each of said fourth and fifth waveguide sections having indexes of refraction n at said frequency, said fourth and fifth waveguide sections being connected to said second and third waveguide sections, respectively, with the longitudinal axes of said fourth and fifth waveguide sections inclined with respect to the longitudinal axes of said second and third waveguide sections at an angle 0,. where s sin 0,5 "2 "2 References Cited in the file of this patent UNITED STATES PATENTS 2,129,712 Southworth Sept. 13, 1938 2,142,138 Llewellyn Jan. 3, 1939 2,261,130 Applegate Nov. 4,1941 2,407,911 Tonks Sept. 17,1946 2,436,828 Ring Mar. 2, 1948 2,531,437 Johnson Nov. 28, 1950 2,576,186 Malter Nov. 27, 1951 2,601,806 Turner July 1, 1952 2,623,121 Loveridge Dec. 23, 1952

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