Reply to Comments on: Analysis of Atmospheric Laser Doppler Velocimeters W. M. Farmer and D. B. Brayton ARO, Inc., Arnold Air Force Station, Tennessee 37389. Received 20 July 1972. A number of significant questions have been raised concerning the basic properties of laser velocimeter systems, and it seems that some further discussion of these points is worthwhile. The statements in our paper that Owens seems to object to are (1) optical signal power—not signal amplitude, adds as the √n for 2978 APPLIED OPTICS / Vol. 11, No. 12 / December 1972 RB systems, and (2) the S/N power ratios we calculated repre- sent an optimum for any laser velocimeter system used in the atmosphere. Owens has asserted that the reason the AC optical signal power adds as the √N is that the scattered electromagnetic fields for n particles add randomly as the √n; therefore, identical results from the two different approaches are coincidental. Such a claim ap- pears to rest on interesting assumptions that we feel are yet to be verified satisfactorily. Indeed, until these questions can be satis- factorily answered, it is not precisely clear exactly what the statistical addition of n should be, nor is it clear whether our re- sults are merely coincidental with the result obtained by summa- tion of the scattered electromagnetic field with phases defined independently of the phase of each field relative to the field of the local oscillator. Consider Owen's diagram. As the illustration shows, an interference pattern exists that consists of alternate bright and dark interference fringes that are planes parallel to the bisector of the incident K vectors. On the center line of a dark fringe, no photons exist to define the electromagnetic field, and one may say it is nonexistent. A more classical approach would be to say that the phases of the fields are such that they cancel, and the net observable intensity is zero. The question arises as to whether a particle with a diameter much less than a fringe spacing can scatter light when centered on a dark fringe. For DS systems, it can be demonstrated that for paraxial front or backscatter, the linear superposition of the two scatter fields accurately reflects the illuminating fringe intensity distribution. 1 This could be true, however, due to the optical symmetry involved in the ob- servation. (Identical portions of the Mie lobes scattered from each beam are observed.) If the signal is due to the linear super- position of the two single beam Mie scatter lobes, which do not reflect the phases of the illuminating beams, an observation point could conceivably exist where a large scatter amplitude for one beam would exist but not for the other beam, in which case, scattered light could be observed when the particle was centered on a dark fringe, independent of the fact that the other beam exists. Such assumptions are questionable at best and lead to the seemingly paradoxical situation of obtaining scattered in- tensity when the illumination intensity is zero. We believe that a complete analysis will show that the scattered intensity re- flects where the scattering took place, i.e., no light is scattered for a point particle on a dark fringe. If this is the case, it would appear that the scattered electromagnetic fields mixed with the local oscillator must reflect the fact that they are spatially modu- lated in the probe volume. Hence, phases of the scattered fields cannot be chosen randomly á priori as though the illuminating field were spatially uniform. Therefore, Owen's claim that our result coincidentally gives the correct answer for RB system ap- pears to rest on an incomplete analysis of the problem. Argu- ments using the fringe model, which account for the spatial vari- ations in the probe volume illuminating intensity, appear to offer the most direct solution to the problem if our answer to the pre- vious question is correct, while Owen's approach of á priori sum- mation of the scattered fields with phases defined independently of the local oscillator would be applicable if a particle does scatter when it is in the dark. We await further light on this question. In objecting to our assertion about the superiority of the DS system-S/N in atmospheric work, Owens is correct in stating that direct comparison of the two systems is difficult due to differ- ent geometries in the various optical systems. However, in forming his comparative conclusions on the relative merits of the two systems, he has overlooked a number of significant facts. A primary oversight is that the solid collection angle for DS sys- tems is not limited by radiation alignment which allows as much scattered light to be collected as possible or desirable, indepen- dent of the solid angle subtended by the illuminating intensity. 2 - 3
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Reply to Comments on: Analysis of Atmospheric Laser Doppler Velocimeters W. M. Farmer and D. B. Brayton
ARO, Inc., Arnold Air Force Station, Tennessee 37389. Received 20 July 1972.
A number of significant questions have been raised concerning the basic properties of laser velocimeter systems, and it seems that some further discussion of these points is worthwhile. The statements in our paper that Owens seems to object to are (1)
optical signal power—not signal amplitude, adds as the √n for
RB systems, and (2) the S/N power ratios we calculated represent an optimum for any laser velocimeter system used in the atmosphere.
Owens has asserted that the reason the AC optical signal power adds as the √N is that the scattered electromagnetic fields for n particles add randomly as the √n; therefore, identical results from the two different approaches are coincidental. Such a claim appears to rest on interesting assumptions that we feel are yet to be verified satisfactorily. Indeed, until these questions can be satisfactorily answered, it is not precisely clear exactly what the statistical addition of n should be, nor is it clear whether our results are merely coincidental with the result obtained by summation of the scattered electromagnetic field with phases defined independently of the phase of each field relative to the field of the local oscillator.
Consider Owen's diagram. As the illustration shows, an interference pattern exists that consists of alternate bright and dark interference fringes that are planes parallel to the bisector of the incident K vectors. On the center line of a dark fringe, no photons exist to define the electromagnetic field, and one may say it is nonexistent. A more classical approach would be to say that the phases of the fields are such that they cancel, and the net observable intensity is zero. The question arises as to whether a particle with a diameter much less than a fringe spacing can scatter light when centered on a dark fringe. For DS systems, it can be demonstrated that for paraxial front or backscatter, the linear superposition of the two scatter fields accurately reflects the illuminating fringe intensity distribution.1 This could be true, however, due to the optical symmetry involved in the observation. (Identical portions of the Mie lobes scattered from each beam are observed.) If the signal is due to the linear superposition of the two single beam Mie scatter lobes, which do not reflect the phases of the illuminating beams, an observation point could conceivably exist where a large scatter amplitude for one beam would exist but not for the other beam, in which case, scattered light could be observed when the particle was centered on a dark fringe, independent of the fact that the other beam exists. Such assumptions are questionable at best and lead to the seemingly paradoxical situation of obtaining scattered intensity when the illumination intensity is zero. We believe that a complete analysis will show that the scattered intensity reflects where the scattering took place, i.e., no light is scattered for a point particle on a dark fringe. If this is the case, it would appear that the scattered electromagnetic fields mixed with the local oscillator must reflect the fact that they are spatially modulated in the probe volume. Hence, phases of the scattered fields cannot be chosen randomly á priori as though the illuminating field were spatially uniform. Therefore, Owen's claim that our result coincidentally gives the correct answer for RB system appears to rest on an incomplete analysis of the problem. Arguments using the fringe model, which account for the spatial variations in the probe volume illuminating intensity, appear to offer the most direct solution to the problem if our answer to the previous question is correct, while Owen's approach of á priori summation of the scattered fields with phases defined independently of the local oscillator would be applicable if a particle does scatter when it is in the dark. We await further light on this question.
In objecting to our assertion about the superiority of the DS system-S/N in atmospheric work, Owens is correct in stating that direct comparison of the two systems is difficult due to different geometries in the various optical systems. However, in forming his comparative conclusions on the relative merits of the two systems, he has overlooked a number of significant facts. A primary oversight is that the solid collection angle for DS systems is not limited by radiation alignment which allows as much scattered light to be collected as possible or desirable, independent of the solid angle subtended by the illuminating intensity.2-3
To compare the two systems with identical solid collection angles does not take full advantage of the inherent characteristics of the DS system. If a small solid collection angle RB system is compared with a large solid collection angle DS system, the RB system requires large n in the probe volume to remain competitive in S/N with the DS system. Obviously, there is a point where n will beeome sufficiently large tha t the R B system S/N will surpass that of the DS system and would then be more desirable. However, long range atmospheric R B backscatter systems are even further collection aperture limited due to atmospheric turbulence. This is a severe problem for visible wavelengths and is generally overcome by using light in the far ir. In doing so, scattering efficiency is decreased as is spatial resolution. DS systems are not collection aperture limited by turbulence in the same sense as R B systems since the light is premixed before it is scattered. As long as the individual beam separation in the DS transmitter is less than a turbulence correlation length, turbulence effects will be minimal.4
We have not observed signals from large n for naturally occurring aerosols for transmitter ranges less than 30-50 m and with probe volumes of about 7.3-15.7 cc. While there are many hundreds of particles per cc greater than 0.1 µm in the Junge size distribution describing natural aerosols, the observed real time signals in DS systems have been due to a few large single particles present in the distribution passing one at a time through probe volume. I t is not clear what portion of the particle size distribution is observed by R B systems, since S/N for there systems is poor5
and is not many times tha t of the DS system as Owens has suggested. Perhaps the difficulty arises in attempting to decide exactly what constitutes large n for RB systems and how biased the systems are toward signals from single large particle systems. In any case, for experiments performed thus far, it has been observed tha t even under the most artificial seeding conditions, R B systems do not give proportionately higher S/N ratios than DS systems as predicted by Owens or by calculations made by anyone else.5–7 This suggests tha t the theoretical calculation for the RB, S/N is either incomplete, or misapplied, or tha t experimental comparison with the theory awaits for more precise experiments.
Owens is correct in asserting tha t increasing the local oscillator strength does increase the absolute magnitude of the signal while decreasing the fringe visibility. This can also adversely affect the signal in a numbers of ways. The most obvious effect is tha t the ac/dc signal level decreases, requiring increased dynamic range for the detector and signal processing electronics. I t can be shown tha t if the signal is from particles whose sizes are comparable to a fringe spacing, cases exist where nearly as much light is scattered from the dark fringes as from the bright fringes. This reduces the scattered ac/dc power to a value much less than that determined by the relative visibility of the illuminating fringes.1 This means tha t the illuminating fringe visibility should be kept as high as possible for maximum values of ac/dc if large particles are observed. I t follows tha t increasing the local oscillator strength for increased conversion gain could act as a detriment to signal detection.
Furthermore, Owens seems to have misinterpreted Eq. (29) from which it can be shown that for the assumptions made, S/N for DS system is independent of n. Ps in Eq. (29) is the gross scattered power from the probe volume incident on the detector. Therefore, when n is such tha t the scattered light power is greater than tha t due to stray background light, S/N is independent of n.
The main claim for superior RB system S/N in atmospheric work seems to rest on the increase of S/N with n. We will agree that at some long range (not yet reported), where the probe volume is large, RB systems may out perform DS systems. However, it has been our observation that at a range of the magnitude probably required for this to occur, the probe volume would be so large tha t the velocity distribution of the particles contributing
to a collective observation and turbulence effects would cause the S/N for the ensemble to decrease. This would probably limit velocity observations to those from single particles.
The primary objection Owens has raised to statements in our paper appear to rest on misinterpretations of these statements, a lack of experimental experience with self-aligning DS systems, and on questions tha t are yet to be resolved. Resolution of these questions awaits theoretical comparison with carefully controlled quantitative experiments.
References 1. W. M. Farmer, Appl. Opt., 11, Nov. (1972). 2. D. B. Brayton, " A Laser Doppler Shift, Velocity Meter with
Self Aligning Optics," EOSD Conference, New York, New York (September 1969).
3. W. T. Mayo, "Laser Doppler Flowmeters—A Spectral Analysis," Ph.D. Thesis, School of Electrical Engr., Georgia Institute of Technology (1969) pp. 26-27.
4. E . B. Treacy, Instrumentation In The Aerospace Industry (Instrument Society of America, Pittsburgh, 1971), p. 17.
5. R. M. Huffaker, Appl. Opt. 9, 1026 (1970). 6. M. K. Mazumder and D. L. Wankum, Appl. Opt. 9, 633 (1.770). 7. D. B . Brayton and W. H. Goethert, Trans. ISA 10, 40 (1971)
