Measuring properties of superposed light beams carrying different frequencies

When two electromagnetic fields of different frequencies are physically superposed, the linear superposition equation implies that the fields readjust themselves into a new mean frequency whose common amplitude undulates at half their difference frequency. Neither of these mathematical frequencies are measurable quantities. We present a set of experiments underscoring that optical fields do not interfere with each other or modify themselves into a new frequency even when they are physically superposed. The multi-frequency interference effects are manifest only in materials with broad absorption bands as their constituent diploes attempt to respond collectively and simultaneously to all the optical frequencies of the superposed fields. Interference is causal and real since the dipoles carry out the operation of summation dictated by their quantum mechanical properties. 2003 Optical Society of America OCIS codes: (260.3160) Interference; (999.9999) Interference, multi-frequency; (140.4050) mode locked laser. References and links 1. P. A. M. Dirac, The principles of quantum mechanics, 4 ed., (Clarendon Press, Oxford, 1958). 2. L. Mandel, “Interpretation of Instantaneous Frequencies,” Am. J. Phys., 42, 840-846 (1974). 3. A. E. Siegman, Lasers, (University Science Books, 1986). 4. A. T. Forrester et al, “Photo-electric Mixing of Incoherent Light,” Phys. Rev. 99, 1691 (1955). 5. Lenore McMackin and David G. Voelz, “Multiple wavelength heterodyne array interferometry,” Opt. Express 1, 332-337 (1997). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-1-11-332 6. E. Gelmini, U. Minoni, and F. Docchio, “Tunable, double-wavelength heterodyne detection interferometer for absolute-distance measurements,” Opt. Lett. 19, 213-215 (1994). 7. R. Dandliker, R. Thalmann, and D. Prongue, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339-341 (1988). 8. P. Zemanek and J. Lazar, “Use of the beat frequency between two modes for frequency stabilization of internal-mirror lasers,” Appl. Opt. 33, 6333-6339 (1994). 9. R. Balhorn, F. Lebowsky, and D. Ullrich, “Beat frequency between two axial modes of a He-Ne laser with internal mirrors and its dependence on cavity Q,” Appl. Opt. 14, 2955-2959 (1975). 10. Katuo Seta and Tadanao O’ishi, “Distance meter utilizing the intermode beat of a He-Ne laser,” Appl. Opt. 29, 354-359 (1990). 11. Shuko Yokoyama, Tsutomu Araki, and Norihito Suzuki, “Intermode beat stabilized laser with frequency pulling,” Appl. Opt. 33, 358-363 (1994). 12. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley and Sons, 1991). 13. E. Hecht, Optics, 3 ed., (Adision-Wesley, 1998). 14. Jan Max Walter Kruger, “External cavity diode laser (ECDL),” (2001), http://hubble.physik.unikonstanz.de/jkrueger/thesis/node5.html. 15. Ole Trinks and Walter H. Beck, “Application of a diode-laser absorption technique with the D2 transition of atomic Rb for hypersonic flow-field measurements,” Appl. Opt. 37, 7070-7075 (1998). 16. Albert A. Michelson, Studies in Optics, (University of Chicago Press, Phoenix Edition, 1968). 17. A. G. Zajonc, The Quantum Challenge, (Jones and Bartlett, 1997). (C) 2003 OSA 21 April 2003 / Vol. 11, No. 8 / OPTICS EXPRESS 944 #2210 $15.00 US Received March 11, 2003; Revised April 15, 2003 18. A. van der Merwe and A. Garuccio, Waves and Particles in Light and Matter, (Plenum, 1994). 19. Max Tegmark, “Parallel Universes”, Scientific American, p.46, May 2003. 20. C. Roychoudhuri, J. M. Siqueiros and E. Landgrave, “Concepts of spectroscopy of pulsed light”, pp. 87-94, in Optics in Four Dimensions-1980, Ed. M. A. Machado and L. Narducci, American Institute of Physics, 1981.


Introduction
Dirac states in his book [1] that a photon interferes only with itself, interference between different photons never occurs.The purpose of this experimental paper is to clarify the fundamental concepts behind the observable effects when electromagnetic fields of different coherent frequencies are literally superposed, with a self-consistent, semi-classical argument that does not require the concept of a photon.
We define the frequency of the EM field as that frequency at which the electric vector oscillates and what Mandel [2] defines as the instantaneous frequency and some others call carrier frequency.Both the traditional multi-frequency laser mode locking [3] and twofrequency beat [4] phenomena fall under the category of superposition of light beams carrying different optical frequencies.Our key objective is to give qualified support to the established view that light beams of different frequencies do not interfere by themselves.However, they do impose the interference effects on the excitability of dipoles in the materials used as sensing tools.EM fields, by themselves, do not produce the mode-locked pulses or beat signals; the materials do by virtue of their excitation and absorption characteristics in response to the presence of multi-frequency fields.However, while the beat signal does not exist independently (outside the detector), the mode locked pulses, once generated, are real, propagating time pulses of the EM field.Beat signals are now used routinely as a precision tool for measuring surface quality, distance, frequency of light beams, stabilizing a laser frequency, etc. [5][6][7][8][9][10][11]. Mode locking, of course, has become the most important tool for controlling laser energy that facilitates applications from delicate, living gene manipulation to ablating hardest materials on the Earth.
Let us start with the generalized equation representing a superposition of N periodically separated optical frequencies, without any arbitrary phases.This represents a simple mode locked laser [3], where ν is the mean of the N periodic frequencies and δν is the periodic laser mode spacing.
Let us now consider the conceptually simplest case of superposition of two different coherent frequencies [12,13], or the conditions for locking two modes (N = 2): where, the mean frequency, ν = (ν 1 +ν 2 )/2 and half the difference frequency, ν m = (ν 1 -ν 2 )/2.The basic question is whether the EM fields of different frequencies, when physically superposed, carries out the "summation" operation of the left-hand-side of Eqs. ( 1) and ( 2) and transform themselves into the right-hand-side as a new E-vector oscillating at the mean frequency of ν whose amplitude modulates at a frequency, δν/2 as a "grating" function for Eq. 1, and at ν m as a cosine function for Eq.(2).From the viewpoint of an experimentalist, the question can be rephrased as whether ν and δν/2 (or, ν and ν m ) are real and measurable.Accordingly, we have carried out the following four experiments to measure the properties of superposed light beams carrying two coherent, but different frequencies represented by Eq. (2).
First, let us explain the overall experimental set up shown in Fig. 1.We have chosen a tunable, external cavity, very narrow line, CW 780 nm laser that can be made to resonate to any one of the Rb-resonant fluorescence lines.This laser line is sent through a 2GHz acoustooptic modulator (AOM) to introduce 2 GHz Doppler shift in the optical frequency.The AOM gives back the original, undeviated laser beam ν L and a diffracted, Doppler shifted beam ν S .The beams ν L , ν S , and (ν L +ν S ) are then separately sent through three separate Rb-vapor tubes (shaded yellow box).This superposed beam (ν L +ν S ) can also be thought of as the simplest but synthetic "mode locking" in a two-mode laser.

Experiment 1. A high-speed detector observes beat signal (ν L -ν S ).
The superposed beams were directed onto a high-speed detector and we registered the beat signal (ν L -ν S ), (i) directly by a high-speed, photoconductive detector and an oscilloscope (HP Model 86100A) and also (ii) by the same detector and an electronic spectrum analyzer.The beat signal vanished when any one of the two beams was blocked before the beam combining beam splitter.This straightforward result is well understood since the time of Forrester [4].But for extra clarity, let us recall the basics.Equation (2) does not represent anything directly observable or measurable.Please, note that the detected beat frequency is (ν L -ν S ), and not (ν Lν S )/2, as indicated by Eq. 2. The measurable quantity is the energy absorbed by the dipoles in the detector during interaction with the superposed fields.It is proportional to the time (or ergodic ensemble) average of the square of the electric field amplitude or, <|E(t)| 2 > [12,13]: It is instructive to note the significance and convenience of complex representation.The RHS of Eq. (3).could have been arrived at by simple square modulus, had we written Eq.( 2) in complex representation, as in Eq. ( 1).For a high-speed photo-conductor, <|E(t)| 2 > is the probability of the rate of excitation of electrons from the valence to the conduction band, where, ∆E bands represents the maximum energy differential between the neighboring, excitable bands of the material, and ν p represents all possible E-vectors that can simultaneously induce dipole oscillations in the material [see Fig. 2(a)].This is also not a new understanding.All semi-classical approaches to light-matter interaction follow this route.However, the value of this paper comes from our underscoring that matter has to carry out the operation of summation locally for the superposition effects to be manifest.Photons or EM fields do not have to have any non-local knowledge of the apparatus facilitating the superposition.
During the photoconductive detection process, electrons are excited from the valence band to the conduction band of a fast solid-state detector.If more than one coherent optical frequency, but of same polarization, is incident on the detector, the dipoles try to oscillate collectively and simultaneously at all the allowed frequencies ν p , effectively carrying out the summation operation given by the Eqs.( 1) and (2).This leads to time varying dipole excitation and consequent time varying energy exchange between the EM field and the detector.This time varying photoconductivity oscillates at the beat frequency given by Eq. ( 3).It is to be noted that the beat oscillation did not exist in the free EM field as demonstrated by the high-resolution spectroscopy experiments.
In the case of Rb gas, although the atoms collectively have a Doppler broadened energy band of about 1.5GHz, the energy levels of each independent atom are very sharp [see Fig. 2(b) and Fig. 3(a)].Thus to excite any Rb-atom dipole, the incident light beam must have a precisely matching optical frequency ν Rb (albeit Doppler shifted) so that the energy difference between the sharp atomic levels is exactly ∆E = hν Rb .Rb atomic dipoles cannot respond to the mean frequency of Eq. ( 2) even when it arithmetically matches to ν Rb = (ν L +ν S )/2, because EM fields did not interfere by themselves to generate this new, mean frequency.
It is very instructive to discuss the case of laser mode locking of Eq. ( 1) here, because it provides both contradiction and conceptual harmony.There is a contradiction because, unlike the beat signal, mode locked pulses do exist as time pulses of EM fields.There is conceptual harmony because the pulses are generated due to the superposition effect in the material (modulated gain medium, saturable absorber (SA), etc.), and not by the free fields themselves.(See Siegman [3] for various types of mode locking techniques.)The two-level band diagram for a photo-conductor, as depicted in Fig. 2(a), can also represent the dipoles of an SA, except that the excited electrons in the upper band rapidly relaxes back to the lower band.It functions like a time domain, on-off gate by responding to the in-phase (mode locked), superposed light carrying all the frequencies ν p and absorbing energy from the field given by the time averaged, square modulus of Eq. (1).Spontaneous emission in a laser cavity starts opening the SA gate to the cavity mirror, allowing weak feedback to establish the longitudinal modes of the cavity.The desired cascading effects start due to a fortunate, statistical phase matching of the complex amplitudes of different longitudinal modes (frequencies) in the SA, making it more transparent and enhancing the gain further.Even though the field superposition is linear in the amplitudes, the energy exchange is probabilistic and quadratic <|E(t)| 2 >, creating a natural preference for in-phase oscillation (higher gain due to quadratically higher transparency of the SA).Our model predicts that an appropriate spectral analysis of truly mode locked pulses should always reveal the existence of all the cavity longitudinal modes allowed by the gain envelope, since the EM fields do not rearrange themselves to a new, mean E-vector frequency.

Discussion
Light does not interfere with light!In any linear, transparent medium, light beams cross each other without modifying their frequency, or redistributing their energy (see p. 738 of Ref. [12]).We have only re-validated this through a set of four experiments.However, many important and subtle interpretations can be derived in the process of finding conceptual continuity amongst these experiments.We will briefly underscore only three of them here.(i) Ascribing the effects of interference exclusively on the field entities will naturally perpetuate conceptual conflicts, like "non locality of interference", "delayed choice", "quantum many worlds", etc., elaborated in the Refs.[17][18][19].Instead of interpreting a dark fringe (in time and/or space) due to non-arrival of photons (assumption -fields interfere by themselves), we should recognize that local dipoles could not be excited (and hence, could not absorb energy) when the resultant amplitude of the superposed fields in the vicinity of the detecting dipole is zero.Thus, we claim that interference is causal and local.(ii) The concepts of Fourier synthesis and decomposition, represented by Eqs. ( 1) and ( 2), going from left-to-right and from right-to-left, respectively, are mathematical embodiment of the principle of superposition.Unfortunately, extra confusion comes in because the solutions for Maxwell's equations and the basis functions for Fourier analysis are both sinusoidal.This paper has focused on Fourier synthesis, going from left-to-right of Eqs. ( 1) and ( 2).We have demonstrated that these equations, while mathematically right, do not represent reality of nature generally, but only conditionally, when the right dipoles interact with the fields.Consequently, by mathematical self-consistency argument, Fourier decomposition (going from right-to-left) cannot also be a generalized principle of physics.In other words, Fourier transformed mathematical frequencies of an amplitude modulated light beam could not represent real E-vectors in any linear medium.This point has been separately elaborated in Ref. [20].Only excited dipoles can generate new EM frequencies.Finally, (iii) we must caution against indiscriminate assignment of observed quantum properties of material dipoles onto the EM fields, because fields do not operate on each other, only dipoles do.Further developments will be presented in the follow up papers.

Fig. 1 .
Fig. 1.The schematic diagram shows a set up for four different experiments.A tunable 780nm source at the left generates ν L .The AOM generates a Doppler shifted line ν S .The three Rb cells, in a shaded yellow box, separately receive ν L , ν S , and superposed (ν L , ν S ) beams.The superposed beam is then further analyzed by optical Fabry-Perot (F-P) and electronic spectrum analyzers (ESA) using high speed detector and monitored by oscilloscopes (Osc.).