Pointsource response

Here we derive formally the response RPS'(t) of a two-telescope system to a point source.

Wavefront samples

The two telescopes sample the incoming electromagnetic wave at two places in the plane wavefront. The two telescopes are configured to select the same component of the incoming (vector) E field (the same linear polarization); thus the measured scalar components E1 and E2 can be used to describe the electric fields at telescopes 1 and 2. These fields may be written as

E2 = E0 cos(wt — 0) (Electric field; tel. 2) (7.7)

where m is the radio angular frequency and 0 is the phase delay in radians that corresponds to the path-length difference nk in Fig. 5 a, where n, the equivalent number of wavelengths, need not be an integer. The phase delay may be written with the aid of (3) which is generalized to allow non-integer n so that 9n ^ 9(t), nk B

0(t) = 2n— = 2nn = 2n— cos 9(t) (Phase delay; radians) (7.8)

The earth's rotation causes the angle 9(t) to vary with time, thereby continually shifting the phase 0(t) of E1 relative to E2. This leads to beating of the two waves as they move into and out of phase with one another. The beat frequency is the frequency with which the visibility lines pass over the point source, or the frequency of the slow modulation of the response function R'(t) shown in Fig. 2d.

The interference can be viewed in another way: the earth's rotation causes one telescope to steadily move closer to (or farther from) the source compared to the other. The telescope getting closer (relative to the other) detects the wavefront at shorter time intervals, i.e., at a slightly higher frequency; this is the Doppler effect. The signals from the two telescopes thus have slightly different frequencies, and this leads to the observed beats.

Summed waves

Electric fields are additive; two fields at a given position and time yield a net field obtained by vector addition. After detection of the two wave samples from our two telescopes, the fields could be added by bringing them together with coaxial cables. In our case (same polarizations), one may add the scalar components algebraically to obtain the net E field, from (6) and (7),

E = E0 cos Mt + E0 cos(Mt — 0) (Summed EM waves) (7.9)

The sum of these two cosine functions may be rewritten, with the aid of the appropriate trigonometric relations, as a product of cosines that separate out the phase 0 dependence, j? oj? ( , 0(t)\ 0(t) mm

This function describes a high frequency oscillation at radio frequency w, which is modulated by a slowly varying cosine function, cos (0/2), which forces the oscillations to zero amplitude at 0 = n, 3n, 5n,... radians (Fig. 6a). The slower oscillations are due to the waves going into and out of phase with one another as the earth rotation continuously repositions the telescopes. As noted above, the radio and beat frequencies will actually differ by a large factor, 106 or more.

The power in an electromagnetic wave is proportional to the square of the summed electric field,

E2 = 4 Eg cos2 ^wt — cos2 0- (Proportional to (7.11)

power in wave)

which is illustrated in Fig. 6b. Since the cos2(0/2) term may be written as (1 + cos 0)/2, this modulation has a sinusoidal shape as the phase delay 0 changes, but it is offset from zero so it never goes negative.

The large difference in the radio and fringe frequencies allows one to carry out a series of time averages on the function (11), each over a time long compared to the period of the rapid radio-frequency oscillations but short compared to the period of the oscillations of 0(t). This will remove the rapid radio-frequency oscillations from the signal. This type of averaging is called a low-pass filter because it lets low frequencies pass through but blocks the higher frequencies. The output of the low-pass filter follows from the fact that the average of a cos2 function over an integral number of cycles is 1/2. Invoking this average for the rapid cos2[wt — (0 /2)] term, we have

O O 1 1 + cos 0(t) o E2p = 4E022-= E02[1 + cos 0 (t)] (7.12)

(After low-pass filter)

This function, plotted in Fig. 6c, has the form shown previously in Fig. 2d.

The energy flux density (W/m2) in the wave is the magnitude of the Poynting vector F = E x B/^0, where B is the magnetic field, underlined here so as not to confuse it with baseline B used in this chapter. Since Maxwell's equations tell

2E02Hv~' "ps

Eo2 2

Figure 7.6. Interference of electromagnetic waves E\ = Eo cos mt and E2 = Eo cos(mt - 0). The rapid modulations represent the high frequency m of the electromagnetic waves, or sometimes 2m, while the slow oscillations are due to the changing relative phase 0 between the two interfering signals as the earth rotates. (a)-(c) The waves are added, squared, and then passed through a low-pass filter. (d)-(e) The waves are multiplied and then sent through the low-pass filter. These plots are qualitative sketches.

Figure 7.6. Interference of electromagnetic waves E\ = Eo cos mt and E2 = Eo cos(mt - 0). The rapid modulations represent the high frequency m of the electromagnetic waves, or sometimes 2m, while the slow oscillations are due to the changing relative phase 0 between the two interfering signals as the earth rotates. (a)-(c) The waves are added, squared, and then passed through a low-pass filter. (d)-(e) The waves are multiplied and then sent through the low-pass filter. These plots are qualitative sketches.

us that E is perpendicular to B and |B| = |E|/c (in SI units),1 the magnitude of the Poynting vector becomes2

ioc ioc . x density)

This expression is also known as the response RPS'(t) to a point source,

RPS'(t) =| F"| a E2[1 + cos 0(t)] (Summed point-source (7.14)

response)

Note that we define R' to be the response after low-pass filtering. Each peak in the function (13) or (14) indicates a change of the path-length difference to the source by one wavelength X.

Multiplied waves

The same variation can be obtained if the waves are multiplied rather than added. To motivate this, take the square of the summed waves (9) directly,

E2 = E2[cos2 Mt + 2 cos Mt cos(Mt — 0) + cos2(Mt — 0)] (7.16)

The first and last terms on the right side are separately the fluxes of the two signals; the time-average value of each is the constant E02/2. Thus the second term contains all the interference information; it is known as the interference term.

The interference term is simply twice the product of the two waves (6) and (7). The product

E1E2 = eo2 cos Mt cos(Mt — 0) (Product of waves) (7.17)

may be rewritten to isolate the variation of 0(t), again with the aid of trigonometry relations,

Here the rapid radio-frequency term cos(2Mt — 0) is summed with the slow interference term cos 0 (Fig. 6d). A low-pass filter averages the rapid oscillations to zero while the beat term is not affected appreciably (Fig. 6e). The point-source response function thus becomes

1 We use | V | for the magnitude of the vector V, that is, the root of the vector dot product (V • V)1/2.

2 The symbol S is usually used for the Poynting vector, but we reserve it for a related quantity, spectral flux density (Wm—2 Hz—1); see Section 8.2.

Rps(t) a (EiE2)lp = cos 0(t) (Multiplicative (7.19)

point-source response function)

This expression is the multiplicative response to a point source after the two signals have passed through a multiplier followed by a low-pass filter. Only the beat-oscillation term remains. The basic temporal variation is essentially identical to the summation case except for a constant offset and different additive and multiplicative factors; see (14) and Fig. 6c.

The function RPS(t) oscillates about zero rather than always being positive; we use R without the prime to signify that the average is zero. The amplitude of the response is again proportional to and hence to the energy flux in the wave (W/m2). The phase shift 0(t) may again be expressed in terms of the observatory parameters (8), to yield

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