Penetration of VLF Radio Waves through the Ionosphere By Ken-ichi MAEDA and Hiroshi OYA Kyoto University, Kyoto, Japan (Read May 24; Received November 25, 1962) Abstract The rate of energy penetration of VLF (10Kc) radio waves through a sharp ionospheric boundary under a constant magnetic field is calculated and the directions of-the ray and the wave normal at the boundary are discussed. For the case in which the VLF radio waves go up into the ionosphere from free space below, a rigorous calculation is performed using Bremmer's method and the existence of a quasi-brewster's angle is ascertained. In the case in which the waves come downwards from above, an approximate method is set up for the calculation and it is shown that the VLF waves can penetrate down to the earth's surface, only when the wave normal lies in a narrow band of angle near the normal of the boundary. In this case the ray direction of the penetrating waves is close to the magnetic field. 1. Introduction It is evident that the whistler waves must pass through the ionosphere twice and the VLF emissions such as the hiss and the chorus must pass through once, for their reception on the earth's surface. It is important to disclose the aspect of penetration of the VLF radio waves through the ionospheric boundary. As for the reflection of the VLF radio waves from the ionosphere, a number of analyses have been performed assuming a sharp plane boundary model by Budden (1951), Wait and Perry (1956), and Johler and Walters (1960). Budden set up a Q-L approximation method for the VLF radio waves and pointed out the existence of a quasi-brewster's angle. In this paper the penetration coefficient of the VLF radio waves through the ionosphere will be analysed. A sharp plane boundary which separates the ionosphere from the free space below is assumed. At first, a case in which the VLF radio waves penetrate into the ionosphere from free space is treated using the theory worked out by Bremmer (1949). A rigorous calculation which is performed by an electronic computer shows the existence of a quasi-brewster's angle. Next, a case in which the VLF radio waves are transmitted downwards across the boundary from anisotropic dispersive medium into the free space is treated and the rate of energy penetration through the ionospheric boundary is calculated by an ap- (151)
152 K. MAEDA and H. OYA proximate method. For both cases the ray direction and the wave normal direction are calculated by an approximate method neglecting the collision. 2. Theory of Radio Wave Penetration into the Ionosphere from Free Space 2.1 Theory Fig. 1 shows the Cartesian coordinates in which the xy-plane is the boundary between the anisotropic dispersive medium (upper, z>0) and ing equation, after normalizing its amplitude. (1) c is the velocity of light, and the matrix of the vectors is defined as follows. (2) field (ExI or exi) polarized perpendicularly to the incident plane has been excluded, because this wave component is reflected almost perfectly at any angle of incidence. The reflected wave (R) which propagates in free space has the form, (3) Tee is the reflection coefficient of the wave, the electric field of which is polarized in the incident plane and Tern is that of the wave, the magnetic field of which is polarized in the incident plane. The latter wave is generated at the reflection, as the ionosphere is anisotropic. As for the penetrated wave, a complete set of the fields (see Fig. 1) can be simply fixed on the boundary, by the following orthogonal transformation. (4)
Penetration of VLF Radio Waves through the Ionosphere 153 (5) (6) (7a) then and next (7b) (7c) H (or H) denotes the constant magnetic field in the dispersive medium. If (8) and (9a) (9b) (10) (11)
154 K. MAEDA and H. OYA (12) At the boundary, the two equations, and (13) are set up. The second equation of (13) is known as Snell's relation. Then from (10), equation is derived. (14) The above is rewritten in the form of a quartic in x with complex coefficient, namely, (15) and The quartic equation (15) is equivalent to the Booker's quartic (1936), which contains the unknown, q, that is related to x as follows, Equation (15) can be solved by algebraic method by means of an electronic computer. conditions that the tangential components of e and h in the two media are equal at z=0, lead to four linear equations of the four unknowns, A0, Ae, Tee and Tern. Namely, the boundary conditions are (16) The suffix T is used to express the tangential component of each vector in (x, y, z) system. Taking notice of equation (5), we can write
Penetration of VLF Radio Waves through the Ionosphere 155 (17) As for the magnetic field a similar transformation can be applied. After all, the following formulae are required at the boundary. For the electric field: (18) For the magnetic field: (19) The notation is used to express that the quantities belonging to the two modes, (P0) and (Pe), are summed up all together. The vector equations (18) and (19) yield four linear equations of the four unknowns A0, Aef Tee and Tem. After having solved these equations, the rate of energy penetration Su is given by the following equation, (20) * denotes the complex conjugate. 2.2 Numerical result Numerical calculation has been made on the assumption that the electron density is 105cm-3, and the wave at a frequency of 10kc propagates in the north-south direction. Fig. 2 shows the absolute values and phase angles of the reflection coefficients Tee and Tern. The both coefficients are rewritten, are the phase angles of the coefficients. The curves, (a) to (i) correspond to the cases remarked in the figure caption. In general, when the incident angle is smaller than the quasi-brewster's angle. The range of small reflection coefficient increases with
156 K. MAEUA and H. OVA Fig. 2. Absolute value and phase angle of the complex reflection coefficient. which is the reflection coefficient of the wave, magnetically polarized in the incident plane, can take place even if that component is not included in the incident wave. This is the similar result pointed out by Budden. Fig. 3. Rate of energy penetration versus incident angle.
Penetration of VLF Radio Waves through the ionosphere 157 corresponding to the cases mentioned in the figure caption. When the incident angle of a case in which the collision frequency is equal to 107sec-1 (curve d). The rate increases up to its maximum value of about 80% at the quasi-brewster's angle. As the upper side of the sharp boundary is an anisotropic medium, it may be noticed that the aspect of penetration in the case of incidence from south to north differs from that in the case of incidence from north to south. For VLF radio waves, however, the reflection coefficient is nearly symmetrical for the incidences form both directions. Therefore, we can use Fig. 2 and Fig. 3 for both directions of incidence. 3. Theory of Radio Wave Penetration into Free Space from the Ionosphere 3.1 Acceptance angle for penetration Fig. 4 shows the Cartesian coordinates used in this case, in which the incident wave (I) in yz-plane lies in the dispersive medium. The incident wave is expressed by (21 All quantities are expressed by the same form same time the reflected waves (R) which propagate in the plasma are also expressed by sign of exponential term in (8). (22) Pee is the intensity factor of the penetrated wave electrically polarized in the incident plane. Pem is that of the penetrated wave polarized magnetically in the incident plane.
158 K. MAEDA and H. OYA In the case of penetration downward to free space, we can apply Q-L approximation (24) becomes a real angle between the wave normal vector, wi and the constant magnetic vector, H. For the same reason mentioned above, we can write, from (22) a relation for the incident wave (I), namely, (26) As shown in Fig. 4 the following relation is obtained: (27) Then from (26) and (27) the following equation is written. Namely (28) from (28), the following, (29) for the efficient penetration of the wave (30) It can be found that the VLF radio waves coming downwards can penetrate across the ionosphere only under the limited condition that the incident angle of the down-
Penetration of VLF Radio Waves through the Ionosphere 159 3.2 Determination of the orthogonal transformation (31) Then, following (7), (KImn) is determined. (32) (33) Using the method similar to that applied to the incident wave, KRe32 is required to satisfy the following relation. KRe32=-e. (34) Then, following equation (7), (KRemn) is written as follows. (35) to the reflected ordinary mode wave, (R0), is written, (36) So, from equations (22), (32), and (36), KR032=-ie, (37) and, following the method used above, (38) 3.3 Rate of energy penetration In the preceding subsection 3.2, we found out the form of orthogonal transformation
160 K. MAEDA and H. OYA for each wave, I, R0, and Re. Then if we determine the full set of the electric and system, and then, the boundary conditions yield the coefficients of penetration, Pee and Pem. (39) The boundary condition is written for the electric field, (40) For the magnetic field, (41) From (40) and (41) the four linear equations are derived, (42) In the process solving equation (40) and (41), the higher order terms of a are Finally we can obtain the penetration coefficient, (43) And if we choose AI so as to make the normal component of Poynting vector of
Penetration of VLF Radio Waves through the Ionosphere 161 the incident waves to be equal to cost, the rate of energy penetration is defined as follows, (44) 3.5 Numerical result 3.5.1 Acceptance factor for penetration which is defined as the ratio of the acceptance solid angle to that of uniform penetration, (45) Some results of calculated K are shown in Fig. 5 in which the solid curves are plotted for the frequency, 10kc and the dotted curves for the frequency, 2kc. It may be noticed that K changes from 10-4 10kc and this change from daytime to night condition is comparable with the diurnal change in the ionospheric attenuation of about 20db which was given by Kimura (1961). 3.5.2 Penetration coefficient The equation (43) shows that the quantities, Pee and Pem depend upon the choice of the value, AI, the in equation (21). Especially, when we take A1=1, let the values of Pee and Pem be written as Pee and Pem respectively. The physical meaning of the quantities, Pee and Pem will be explained in the following. It must be born in mind that for mathematical sim- plicity, we have normalized the intensity of the electric and magnetic fields. To find out actual intensity of the fields of the penetrated wave, let us define the following field intensity,
162 K. MAEDA and H. OYA between equations (21) and (23) we can write the actual intensity of the penetrated wave as follows, (47) Fig. 6. Relations of polarization and intensity of incident and penetrated waves. The aspect of polarization and the intensity of the fields of incident and penetrated waves are shown in Fig. 6. To examine the quantitative change in the field intensities we must define the penetration coefficients for the electric and magnetic fields separately because the intensities E0 and H0 are quite different and are related only by the intrinsic impedance of the medium. We may define the four penetration coefficients. The first of them is the penetration coefficient of the electric field of the penetrated wave which is polarized parallel in the incident plane, that is, Pee. The second is the penetration coefficient of the electric field of the wave polarized perpendicularly to the incident plane, that is, Pem. The third is the coefficient of the magnetic field of the wave polarized perpendicularly For the four coefficients we can use the following expression. (48) the fact that even if the path of the penetrated wave is extremely oblique, for example of the boundary, z (see Fig. 4) and hence x and y components of the field of the incident wave (I-wave) do not change so much, however oblique the path of the penetrated wave may be. As a result, to meet the boundary condition, that is, the equality of the tangential components of the fields in both sides of the boundary, the field intensity
Penetration of VLF Radio Waves through the Ionosphere 163 number. Then we can use the following equality for the phase angles of the four penetration coefficients, namely, (49)
164 K. MAEDA and H. OYA Fig. 9. Phase angle of the penetration coefficient for the electric and magnetic fields versus penetrated
Penetration of VLF Radio Waves through the Ionosphere 165 (50) SIz is z-component of the Poynting vector of the incident wave. Fig. 10 shows penetration, 0.05. In general, when the direction of the penetrated wave becomes nearly the wave penetrated into free space with this large rate hits the ionosphere again without reaching to the earth's surface. Assuming the bight of the ionospheric lower boundary to be equal to 100km, the largest penetration angle of the ray which can the most part of the energy of downcoming waves is transmitted back again into the ionosphere. The magnitude of Poynting vector of a wave shows the mean energy density which is defined as the energy flow per unit surface perpendicular to the ray direction. We can find the following relation for SP, which is defined as the energy density of the penetrated wave, From Figs. 7 and 8, it can be noticed that the energy density is concentrated in case (51) 4. Ray Direction 4.1 Theory In an anisotropic dispersive medium, the ray direction does not generally coincide with the wave normal and it is important to determine the ray direction of penetrated wave. Taking the real part of a complex Poynting vector, we can find the ray direction, Namely (52) the real part of a complex quantity. Two cases are considered, (i) the VLF radio wave from the free space is penetrated into the ionosphere across the sharp boundary, and (ii) the wave from the ionosphere is penetrated into free space. In the first case, mentioned above, an approximate method is adopted rather than the rigorous calculation which was considered for the computation of the rate of energy penetration Su Fig. 11 shows the relationship between the ray direction and the wave
166 K. MAEDA and H. OYA normal in the presence of a constant magnetic field, taking the incident plane in y-z plane. The first equation of (24) is used instead it follows that (53) From the Shell's law, (54) Applying the same Q-L approximation for equation (9a) and putting, for convenience, APe =1, the field of penetrated wave is expressed by (55) From equation (52), the ray directional vector is found (56) which coincides with the wave normal. Then the angle between the ray direction and (57) This formula is identical with the one obtained by Storey (1953) for the ray direction of VLF radio waves in an anisotropic dispersive medium. (58) (59)
Penetration of VLF Radio Waves through the ionosphere 167 (66) (61) Then, In the second case in which the wave from the ionosphere is transmitted into free space, equation (39) is used. In this equation, suffix a is replaced by I for the incident wave and replaced by Re for the reflected wave. Fig. 12 shows the relation of the wave normal and ray direction in the presence of a constant magnetic field. In the similar way as the first case, the from equations (22) and (39). Namely, (62) (63) (64) Summerizing the above results concerning the ray directional angle of wave, (a) (b) (c) (65) Equation (65) shows that the ray directions are reversible at a such sharp boundary which separates the anisotropic medium above and free space below. For detail illustration of these relations, it is convenient to consider a case in which wave hits the boundary in the direction Fig. 13. The ray directions, for the incident, penetrated and reflected waves at the boundary. Fig. 13. Equations (a) and (b) in (65) show
168 K. MAEDA and H. OVA that a wave propagating in free space and entering into the boundary in the direction respect to the normal of the boundary. The ray directions of penetrated, incident, and reflected waves at the boundary by the approximate method using the following function, (66) 4.2 Numerical result
Penetration of VLF Radio Waves through the ionosphere 169
170 K. MAEUA and H. QYA
Penetration o f VLF Radio Waves through the Ionosphere 171 5. Summery There is a certain favourable direction in which the VLF radio waves can penetrate into the ionosphere from free space. The range of an arrowed angle is very narrow and it may be called quasi-brewster's angle which was pointed out by Budden (1951). from the vertical at 10 kc and the character of the penetration is nearly symmetrical with respect to the normal of the boundary. After the VLF wave is penetrated, the wave normal is nearly in the normal of the boundary. The ray direction, on the other hand, bends towards the magnetic field following the Storey's relation. The ray direction of the penetrated wave for various incident angles, however, does not deviate so much For the VLF waves penetrating downwards from the ionosphere into free space, the reflected wave turns back, with a very small angle from the incident ray direction, owing to the existence of the magnetic field. The allowed penetrated wave into free space is confined within a very narrow range of incident angle near the normal of the boundary, which may be called the acceptance angle for penetration. The width of this acceptance angle is inversely proportional to the electron density. It is noted that the role of this acceptance angle is as important as the effect of collisional attenuation through the ionosphere for the reception of the VLF radio waves. The favourable incident ray direction for the penetration into free space is almost fixed in the direction near the magnetic line of force. The allowed margine of the ray direction is very small, for an efficient penetration. When the wave normal lies within the acceptance angle, the rate of energy penetra- oblique penetration, the energy density of the penetrated wave is highly concentrated, comparing with the penetration directed towards normal of the boundary. Although the present treatment has been made for a simplified case such as a sharp boundary, the conclusion obtained in this paper will reveal essential features of the penetration of VLF radio waves through the ionosphere. Acknowledgements The authors wish to express their sincere thanks to Prof. T. Obayashi, Dr. I. Kimura, Dr. T. Takakura, Dr. J. Outsu and Mr. K. Sakurai for their valuable discussions on this work. References Booker, H. G., 1935, Proc. Roy. Soc. A., 150, 267. Bremmer, H., 1949, Terrestrial Radio Waves (Elsevier Publishing Co., New York). Budden, K. G., 1951, Phil. Mag., 2, 42, 833. Wait, J. R. and Perry, L. B., 1957, J. Geophys. Research, 43, 62. Johler, R. and Walters, L. C., 1960, Jour. of Research, N. B. S., 64D, 269. Kimura, I., 1961, The propagation of Electromagnetic Waves in the Ionosphere and Exosphere (Doctoral dissertation of Kyoto University, Japan.).