There is considerable interest in measuring the time for an electromagnetic wave to propagate through a medium, which may be a vacuum, free space, or some medium with properties intermediate to those of free space and vacuum. It is convenient to work in the Fourier domain, so that a spatial phase profile s(x,y) is given by s=∫ℝ2F(kx,ky)e−j2πk⋅rdr and may be represented in Cartesian coordinates by a phase (φ(r,t)=∫ℝ2F(kr)e−jω(k)t) associated with a complex valued amplitude function (A(r,t)=F(kr)e−jω(k)t). Assuming that the wave propagates in the z direction, then the time-averaged Poynting vector S is given by S=(1/2)Re{ρA*(r,t)∇φ(r,t)}. If one knows all of the field components for the region under consideration, then it is possible to calculate the time-averaged Poynting vector, from which information on the phase velocity can be calculated. It is therefore of interest to develop a means to probe the field components.
Because of its potential for providing a simple means of measurement, there is substantial interest in developing an optical system that measures the phase profile of a beam of electromagnetic radiation.
Several methods are known for determining the phase profile of electromagnetic radiation. These include interferometric means in which a beam of electromagnetic radiation is divided into two portions and the portions are caused to travel in opposite directions. A reference wave front phase distribution is then calculated by combining the information gathered in each direction. The phase distribution can then be subtracted from the original phase distribution to produce a signal indicative of a wave front phase distribution. Because it relies on an interferometric method of measurement, this system cannot be used to measure the phase profile of a beam of electromagnetic radiation where such radiation is being used in an optical system, where it might otherwise disturb the reference wave front phase distribution. An interferometric system must also be calibrated with a known distribution to enable phase detection.
Another technique uses a single probe beam that is propagated along the z direction through an interaction region, and is then combined with the beam of interest. The resulting interference pattern is measured to determine the phase profile. This technique suffers from the use of a probe beam that is spatially extended, so that the phase measurement is not localized.
There is a need for an improved optical method for determining the time-averaged Poynting vector of a phase profile associated with a beam of electromagnetic radiation.
3. Statement of the Problem
The invention addresses the need for an improved optical method for determining the time-averaged Poynting vector of a phase profile associated with a beam of electromagnetic radiation. The beam is divided into a plurality of phase-differentiated beams, each of which propagates along a path of substantially equal phase and different path length, and an interference pattern is measured at a plurality of locations in the interaction region. The pattern produced by the probe beam is correlated with the pattern produced by each phase-differentiated beam. The phase of each phase-differentiated beam is measured by examining the spatial profile of the amplitude of the detected probe beam.
The determination of the phase profile of the probe beam relies on the well known fact that the path length from the reference frame of an electromagnetic wave to a detector, which may be called an observation point, corresponds to a position in the observed wave that is associated with an angle of propagation. A relationship between the position of an observation point and the distance measured from that point to a source of the electromagnetic radiation is generally given by d=c1×A2, where A is an angle measured from a reference axis. A relationship between a position in a beam of radiation and a position in the plane perpendicular to the propagation axis of the beam can be given by r=c2×α, where r is a distance measured from the axis and α is an angle measured from a reference axis. In this relationship c1 and c2 are factors of wavelength. These relationships hold for a source of electromagnetic radiation that provides a plane wave. In the present description, (θ) is used as a phase difference between adjacent beam paths of the phase-differentiated beams. For a narrow angle,
θ = 2 π λ Δ l ( t ) ( 1 ) where Δl(t) is an optical path length difference between adjacent phase-differentiated beams that is different for each phase-differentiated beam, λ is the wavelength of the radiation, and θ is measured from a reference axis. From the definition of the path length difference, Δl(t) can be expressed as follows:
∫ 0 L E r ( z ) 2 ⅆ z = L E r ( L ) 2 ( 2 ) where z is an axis along which the beam propagates. L is the path length of the phase-differentiated beam, and Er(L) is an amplitude of the electric field at a reference point in the phase-differentiated beam. By substituting Eq. (2) into Eq. (1), it is found that
θ = 2 π λ { E r ( L ) E r ( 0 ) } ( L 0 L 1 ) ( 3 ) In Eq. (3) L0 is an initial distance from the reference point to a point of interest, and L1 is an optical path length difference between adjacent beams in the phase-differentiated beams, which may be made very large. By substituting Eq. (2) into Eq. (1), it is found that θ=2π{(Eer(L0))2/Eer(L1)}.
From Eq. (3) it is understood that when the reference point is at the center of the phase-differentiated beam, θ=2π and the electric field has a constant amplitude. Consequently, the phase of the electric field varies as a function of z. It follows from the above that if a correlation coefficient is defined asρ=Eer(z)eiθ, (4)where ρ is the correlation coefficient, Eer(z) is the electric field at a reference point in the phase-differentiated beam, and eiθ is the unit amplitude sinusoidal function, then a value of the correlation coefficient ρ will be maximized or minimized depending on whether the reference point is ahead or behind a source point of the electromagnetic radiation as z varies, and whether the phase difference θ is equal to zero or π.
Since the phase-differentiated beams are formed from one electromagnetic beam, an interference pattern results at a plurality of locations in the interaction region, and the pattern produced by the probe beam is correlated with each pattern produced by the phase-differentiated beams
