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Production of seemingly doppler-shifted radiation by time- and space-dependent radiant-energy sources
1. Quant. Specrrosc. Radiar. Transfer. Vol. 10, pp. 831-833. Pqamon
Press 1970. Printed in Great Britain
NOTE
PRODUCTION OF SEEMINGLY DOPPLER-SHIFTED RADIATION BY TIME- AND SPACE-DEPENDENT RADIANT-ENERGY SOURCES*
s. s. PJZNNEX Institute for Pure and Applied Physical Sciences and Department of the Aerospace and Mechanical Engineering Sciences, University of California, San Diego, La Jolla, California 92037 (Received 11 January 1970)
Ahalract-An optical arrangement is sketched which will produce monochromatic beams with frequencies that may be interpreted as corresponding to Doppler-shifted radiation. Thus the existence of monochromatic, Doppler-shifted radiation does not necessarily imply the presence of moving media. Doppler-shifted radiation may, in particular, be produced by appropriately-tailored, time- and space-dependent, “intensity” (i.e. power density) fluctuations viewed under suitably chosen conditions. Hence the presumption that the observation ‘of Doppler-shifted radiation is associated with moving media of welldefmed velocities is not unique.
OUTLINE
OF THEORETICAL
CONSIDERATIONS
IT IS easy to demonstrate(‘) an intimate connection between Young’s modified two-slit interference experiment and the Doppler frequency-shift. Thus it may be shown that it is possible to derive Doppler’s law from the time-dependent intensity variations in interference patterns produced in schlieren interferometry with laser radiation on moving density gradients. (r) An obvious corollary to this derivation is the statement that frequency shifts may be produced by appropriately viewed, fluctuating radiant-intensity fields. In other words, the observation of a (small) frequency shift for monochromatic radiation is not uniquely related to the physical presence of moving media. This conclusion may have cosmological implications relating to the customary interpretation of observed Doppler shifts as produced by moving sources in astrophysical observations. We fist note a requirement of reversibility, namely, a fluctuating intensity field produced by interfering beams must regenerate these same beams when it is produced by other means and viewed through the same optical system that produces it when it is formed by interfering monochromatic beams with slightly different frequencies. The requisite experimental arrangement is sketched in Fig. 1. * This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U.S. Army Research OtIlce-Durham under Contract DA-31-142-ARO-D-257. 831
s. s. PENNER
832
PLANE
OF
RADIATION SOURCES
COLLIMATING
LENS
PLANE OF OBSERVATION SLtTS
/ /
/
/
FIG. 1. Schematic diagram showing an optical arrangement in which time- and space-varying radiant intensities in the “plane of radiation sources” produce monochromatic radiation in the horizontal direction and at an angle tI with respect to the horizontal direction. The horizontal beam and the beam observed at the angle 8 act like Doppler-shifted beams with respect to each other,
corresponding to an apparent velocity component u sin 0 = c(Av/v) along the O-direction for sufficiently low frequencies of the fluctuating intensity components in the “plane of the radiation sources”.
The intensity Z(t, z) of radiation at a height z above the horizontal axis and at time t in the “plane of radiation sources” is prescribed as follows : Z(t, z) - I,(& z) = A y4z cos[27r(v, - v,)t + 27rsz/(c/v,)x]
(1)
where zh(t, z) = A: cos2(27rv,t+cp,)+A~
cosz(27rv,t+cp2)+A,A2
cos[27r(v, +v,)t+(cp,
+(pz)], (2)
A1 and A, are amplitude factors, qpl and cp2 are phase angles such that cpi -q2 N 27rsz/ (c/v,)x, v1 >- (vi -v,), v2 >> (vi - v2), z/x << 1 ; in practice, Av/v, w 10-l to low4 where for visible radiation. The high-frequency component Av = (v,-v,)andv, N 6~10’~sec-’ I,,(&z) acts as pseudo-steady background noise to the observed radiant intensity Z(t, z). It is apparent from equation (1) that Z(t, ze - z) - I&, z,, - z) = A,A,[cos 27r(v, - v2)t + 2n(z,-z)s/(c/v,)x], where we choose z0 to correspond to an intensity maximum in the low-frequency component at times t = n/(vr - vz) for
n = 1,2, . . . ;
thus zo = (c/v1 )x/s so
that Z(t, zo) - I&, zo) = 4.42 cos 27r(v, - vz)t
(3)
and Z(t, z. &-mzo) - Z,(t, z. * mz,) = A ,A, cos 27r(v, - v,)t
for
m = 1, 2, . . . .
(W
Production of seemingly Doppler-shifted
radiation by time- and space-dependent
radiant-energy sources
833
What are the frequencies of monochromatic beams observed in the horizontal direction (0 = 0) and at the small angle 0 (which is uniquely determined by the distance s and the focal lensfof the lens)? These frequencies must be such that they produce the intensity pattern described by equations (1)-(3a) when they illuminate the “plane of radiation sources.” Therefore, they are monochromatic beams of frequencies v1 and v2 in the horizontal and O-directions, respectively, with amplitudes A 1 cos(27rv,t + qr) and AZ cos(27rv,t + (p2), respectively. A stationary observer viewing the “plane of radiation sources” through the specified optical system at the angle 8 with respect to the horizontal direction will conclude that he sees a monochromatic beam with frequency v2. If he is familiar from laboratory experiments with the frequency vr as corresponding to a known spectroscopic transition, then he may conclude that the observed frequency v2 corresponds to a Doppler shift in the direction of observation which is produced by a velocity component u
sin 8 = CAV/V N c(vl - v,)/vi.
(4)
An obvious procedure (I) for obtaining the desired intensity distribution in the “plane of radiation-sources” involves interference betw.een laser beams with slightly different frequencies after passage through a suitable double-slit arrangement. Dr. J. Harris, in commenting on this Note, has made the following observation: “For the case of a single source of radiation of the blackbody type, the requirements for producing a red shift are apparent. If the original spectrum is N(f) and the shifted spectrum is N(f+ Af), then the spectral filter which is required to produce the shift is ITI
= W-+
~fYW-)~
where T is the complex amplitude transmittance of the filter. The fact that the shift is observed in terms of radiance values shows that we only care about the modulus squared of this filter. This makes it clear that there are, in general, an infinite number of complex-amplitude transmittance filters which will produce the desired effect. If we visualize this family of filters in terms of temporal weighting functions, then we have knowledge of only the autocorrelation function of the temporal filter (by virtue of Fourier-transforming IT12). For the case of the spectrum shift, the Fourier shift theorem tells us that a linear phase shift with time is a filter which satisfies the requirement. A linear phase shift with time can, of course, be produced by linear motion. There is, in principle, an infinite family of temporal filters which would have the same autocorrelation function as the linear-phase-shift-with-time filter. Perhaps this family could be explored to see if other members have physically significant counterparts or mechanizations.” REFERENCES 1. S. S. PENNER, W. DAVID~Rand F. BIEN,“Determination of Interference Patterns (or of the Doppler frequency shift) from Velocity Measurements of Intensity Maxima in Schlieren Interferometry with Laser Radiation,” unpublished studies (1969).
Press 1970. Printed in Great Britain
NOTE
PRODUCTION OF SEEMINGLY DOPPLER-SHIFTED RADIATION BY TIME- AND SPACE-DEPENDENT RADIANT-ENERGY SOURCES*
s. s. PJZNNEX Institute for Pure and Applied Physical Sciences and Department of the Aerospace and Mechanical Engineering Sciences, University of California, San Diego, La Jolla, California 92037 (Received 11 January 1970)
Ahalract-An optical arrangement is sketched which will produce monochromatic beams with frequencies that may be interpreted as corresponding to Doppler-shifted radiation. Thus the existence of monochromatic, Doppler-shifted radiation does not necessarily imply the presence of moving media. Doppler-shifted radiation may, in particular, be produced by appropriately-tailored, time- and space-dependent, “intensity” (i.e. power density) fluctuations viewed under suitably chosen conditions. Hence the presumption that the observation ‘of Doppler-shifted radiation is associated with moving media of welldefmed velocities is not unique.
OUTLINE
OF THEORETICAL
CONSIDERATIONS
IT IS easy to demonstrate(‘) an intimate connection between Young’s modified two-slit interference experiment and the Doppler frequency-shift. Thus it may be shown that it is possible to derive Doppler’s law from the time-dependent intensity variations in interference patterns produced in schlieren interferometry with laser radiation on moving density gradients. (r) An obvious corollary to this derivation is the statement that frequency shifts may be produced by appropriately viewed, fluctuating radiant-intensity fields. In other words, the observation of a (small) frequency shift for monochromatic radiation is not uniquely related to the physical presence of moving media. This conclusion may have cosmological implications relating to the customary interpretation of observed Doppler shifts as produced by moving sources in astrophysical observations. We fist note a requirement of reversibility, namely, a fluctuating intensity field produced by interfering beams must regenerate these same beams when it is produced by other means and viewed through the same optical system that produces it when it is formed by interfering monochromatic beams with slightly different frequencies. The requisite experimental arrangement is sketched in Fig. 1. * This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U.S. Army Research OtIlce-Durham under Contract DA-31-142-ARO-D-257. 831
s. s. PENNER
832
PLANE
OF
RADIATION SOURCES
COLLIMATING
LENS
PLANE OF OBSERVATION SLtTS
/ /
/
/
FIG. 1. Schematic diagram showing an optical arrangement in which time- and space-varying radiant intensities in the “plane of radiation sources” produce monochromatic radiation in the horizontal direction and at an angle tI with respect to the horizontal direction. The horizontal beam and the beam observed at the angle 8 act like Doppler-shifted beams with respect to each other,
corresponding to an apparent velocity component u sin 0 = c(Av/v) along the O-direction for sufficiently low frequencies of the fluctuating intensity components in the “plane of the radiation sources”.
The intensity Z(t, z) of radiation at a height z above the horizontal axis and at time t in the “plane of radiation sources” is prescribed as follows : Z(t, z) - I,(& z) = A y4z cos[27r(v, - v,)t + 27rsz/(c/v,)x]
(1)
where zh(t, z) = A: cos2(27rv,t+cp,)+A~
cosz(27rv,t+cp2)+A,A2
cos[27r(v, +v,)t+(cp,
+(pz)], (2)
A1 and A, are amplitude factors, qpl and cp2 are phase angles such that cpi -q2 N 27rsz/ (c/v,)x, v1 >- (vi -v,), v2 >> (vi - v2), z/x << 1 ; in practice, Av/v, w 10-l to low4 where for visible radiation. The high-frequency component Av = (v,-v,)andv, N 6~10’~sec-’ I,,(&z) acts as pseudo-steady background noise to the observed radiant intensity Z(t, z). It is apparent from equation (1) that Z(t, ze - z) - I&, z,, - z) = A,A,[cos 27r(v, - v2)t + 2n(z,-z)s/(c/v,)x], where we choose z0 to correspond to an intensity maximum in the low-frequency component at times t = n/(vr - vz) for
n = 1,2, . . . ;
thus zo = (c/v1 )x/s so
that Z(t, zo) - I&, zo) = 4.42 cos 27r(v, - vz)t
(3)
and Z(t, z. &-mzo) - Z,(t, z. * mz,) = A ,A, cos 27r(v, - v,)t
for
m = 1, 2, . . . .
(W
Production of seemingly Doppler-shifted
radiation by time- and space-dependent
radiant-energy sources
833
What are the frequencies of monochromatic beams observed in the horizontal direction (0 = 0) and at the small angle 0 (which is uniquely determined by the distance s and the focal lensfof the lens)? These frequencies must be such that they produce the intensity pattern described by equations (1)-(3a) when they illuminate the “plane of radiation sources.” Therefore, they are monochromatic beams of frequencies v1 and v2 in the horizontal and O-directions, respectively, with amplitudes A 1 cos(27rv,t + qr) and AZ cos(27rv,t + (p2), respectively. A stationary observer viewing the “plane of radiation sources” through the specified optical system at the angle 8 with respect to the horizontal direction will conclude that he sees a monochromatic beam with frequency v2. If he is familiar from laboratory experiments with the frequency vr as corresponding to a known spectroscopic transition, then he may conclude that the observed frequency v2 corresponds to a Doppler shift in the direction of observation which is produced by a velocity component u
sin 8 = CAV/V N c(vl - v,)/vi.
(4)
An obvious procedure (I) for obtaining the desired intensity distribution in the “plane of radiation-sources” involves interference betw.een laser beams with slightly different frequencies after passage through a suitable double-slit arrangement. Dr. J. Harris, in commenting on this Note, has made the following observation: “For the case of a single source of radiation of the blackbody type, the requirements for producing a red shift are apparent. If the original spectrum is N(f) and the shifted spectrum is N(f+ Af), then the spectral filter which is required to produce the shift is ITI
= W-+
~fYW-)~
where T is the complex amplitude transmittance of the filter. The fact that the shift is observed in terms of radiance values shows that we only care about the modulus squared of this filter. This makes it clear that there are, in general, an infinite number of complex-amplitude transmittance filters which will produce the desired effect. If we visualize this family of filters in terms of temporal weighting functions, then we have knowledge of only the autocorrelation function of the temporal filter (by virtue of Fourier-transforming IT12). For the case of the spectrum shift, the Fourier shift theorem tells us that a linear phase shift with time is a filter which satisfies the requirement. A linear phase shift with time can, of course, be produced by linear motion. There is, in principle, an infinite family of temporal filters which would have the same autocorrelation function as the linear-phase-shift-with-time filter. Perhaps this family could be explored to see if other members have physically significant counterparts or mechanizations.” REFERENCES 1. S. S. PENNER, W. DAVID~Rand F. BIEN,“Determination of Interference Patterns (or of the Doppler frequency shift) from Velocity Measurements of Intensity Maxima in Schlieren Interferometry with Laser Radiation,” unpublished studies (1969).