Interference of diffraction and transition radiation and its application as a beam divergence diagnostic

We have observed the interference of optical diffraction radiation (ODR) and optical transition radiation (OTR) produced by the interaction of a relativistic electron beam with a micromesh foil and a mirror. The production of forward directed ODR from electrons passing through the holes and wires of the mesh and their separate interactions with backward OTR from the mirror are analyzed with the help of a simulation code. By careful choice of the micromesh properties, mesh-mirror spacing, observation wavelength and filter band pass, the interference of the ODR produced from the unperturbed electrons passing through the open spaces of the mesh and OTR from the mirror are observable above a broad incoherent background from interaction of the heavily scattered electrons passing through the mesh wires. These interferences (ODTRI) are sensitive to the beam divergence and can be used to directly diagnose this parameter. We compare experimental divergence values obtained using ODTRI, conventional OTRI, for the case when front foil scattering is negligible, and computed values obtained from transport code calculations and multiple screen beam size measurements. We obtain good agreement in all cases.


Introduction
The term 'diffraction radiation' is commonly used to describe the radiation produced when a charged particle moving at a constant velocity passes near, but does not intercept, a material whose dielectric constant differs from the medium in which the particle is traveling [1]. This radiation is caused by a rapid change in the induced polarization of the impacted medium caused by the transiting particle. It is the polarization current that radiates. The radiation can be observed in the far field (Fraunhofer zone) or the near field (wave or Fresnel zone). The far field spectral-angular properties of DR are similar to those of transition radiation (TR), which is produced by a charge particle passing through a solid boundary between two media with different dielectric constants, but has some distinguishing features [2]. Like TR, the spectralangular distribution of DR is altered by the angular distribution of the beam particles and thus it can be used to diagnose the beam's divergence and mean trajectory angle.
However, unlike TR, the spectral-angular distribution of DR is also a function of the beam size and its intensity is a function of the proximity of the charged particle to the impacted medium.
The relevant parameter which governs the intensity of DR produced is the socalled radiation impact parameter, a = γλ/2π; here γ is the Lorentz factor of the moving charge and λ is the observed wavelength. The parameter a is a measure of the degree of fall-off of the radial component of electric field of the moving particle [3]. Significant DR is produced when a l ≤ , the distance of closest approach of the particle to the impacted medium, e.g. the edges of circular aperture or slit through which the beam traveling in a vacuum passes. The radiation impact parameter is also relevant to the production of transition radiation, since when a r ≥ , the size of the radiating medium, diffraction effects from the edges of the radiator are significant [4,5]. These effects include cutoffs in the spectral density at low frequency for a finite size solid radiator and at high frequency for an aperture, as well as modulations (fringes) in the angular distribution of the radiation.
TR from a finite size screen and DR from an aperture are closely related complementary effects. In fact, Babinet's principle applies to the radiation fields; i.e. TR from a finite size radiator is equal to the difference of TR from an infinite plane and DR from a complementary aperture [6][7][8]. In this sense there is no formal distinction between the two radiation phenomena and we will refer to both TR from a finite size screen and DR from an aperture as diffraction radiation, when the relevant size of the radiator or aperture is less than or of the order of the radiation impact parameter. Our interest in this paper is the investigation of incoherent (λ << beam dimensions), far field, optical (λ = 400-700 nm) diffraction radiation (ODR) from beams with moderate energies (i.e. 10-100 MeV), where the radiation impact parameter a is in the range 10-100 μm.
Theoretical investigations have shown that the far field angular distribution (AD) of ODR can be used as a non-interceptive beam size and divergence diagnostic for relativistic beams [6,9,10]. Experiments have verified that the far field AD from a single screen can be used to measure the beam size for a low divergence beam [11]. 'Near field imaging', a term which is somewhat loosely used to describe imaging the spatial distribution of DR at the source to elicit information about beam position and size has also been investigated theoretically [12,13]. Recently an experimental study of 'near field imaging' of ODR from a single metal edge has been reported [14].
In an earlier study [7] we showed computationally how optical diffractiontransition radiation interferometry (ODTRI) could be used to measure the divergence of moderate energy (10-100 MeV) electron beams. This technique uses a device similar to a conventional two foil OTR interferometer [15]. However, in an ODTR interferometer the first foil is replaced by a micromesh [16] whose cell dimensions are comparable to the radiation impact parameter for visible wavelengths but much smaller than the beam radius (100's to 1000's of microns).
A general schematic of a reflection ODTR interferometer is shown in Figure 1.
The diagram shows the production of ODR from unscattered (u) electrons passing through the holes of the micromesh and ODR from scattered (s) electrons passing through the wires of the mesh. These forward directed ODR components reflect and interfere with backward OTR generated by the beam impinging on the mirror itself.
While both ODR components from the mesh are due to a changing induced polarization current on the metal in wires we can consider them to be independent effects.
The total ODR intensity from unperturbed electrons passing through the holes is comparable to that produced by electrons passing through the wires when the transparency of the mesh is about 50% [7]. However, by properly choosing the atomic number and thickness of the mesh material one can take advantage of electron scattering in the wires to wash out the interference of ODR from the scattered component and backward OTR from the mirror produced by this component. The scattered contribution to the observed radiation pattern then forms a smooth background and by proper choice of the optical band pass, the fringes from the ODR from unperturbed electrons passing through the holes can be made visible above this background. The visibility of these fringes is sensitive to the unperturbed beam divergence. In [17] we presented preliminary results of an rms beam divergence measurement made using ODTRI. In this paper we present detailed results, analysis and comparisons of divergences obtained using three different techniques: ODTRI, OTRI and multiple screens-transport code calculations. We report measurements of both vertical and horizontal components of the divergence on two different electron beam accelerators with beam energies 50 and 100 MeV, respectively. We also present a more detailed explanation of the model employed in the simulation code we have developed to calculate ODTRI than previously given in ref. [17]. We also provide a detailed explanation of how we use the simulation code results to fit the data. The excellent agreement between all these measurements and calculations firmly establishes ODTRI as a viable new technique for the measurement of beam divergence for moderate energy relativistic electron beams. In addition, ODTRI has a distinct advantage over conventional OTR interferometry, which is subject to the limitation that only divergences comparable to or exceeding the rms scattering angle in the primary foil can be measured; no such limitation is present with ODTRI.

OTR Interferometry
The performance of a conventional OTR two-foil interferometer can be evaluated from the expression for the far field spectral-angular distribution of backward reflected radiation observed in the detection plane. This plane is perpendicular to the direction of specular reflection (for backward reflected radiation) or to the direction of the average beam velocity (for forward radiation). While in reality the radiation expands as a spherical wave in the far field, measurement of the radiation in the detection plane, which is tangent to the spherical wave front, is a good approximation for small angles of observation measured from the tangent point, i.e. z = 0. k β , Ê is the electric field of the radiation, , ⊥ E , are the and ⊥ components of Ê with respect to the observation plane and ζ is the observation or scan angle in the observation plane measured from the direction of β . Note that the observation plane can be oriented arbitrarily in the detector plane and, in general, it does not pass through the direction β nor through the z axis. Note also that β is not generally collinear with the z axis.
The far field spectral-angular density for interference OTR measured in the observation plane is given by: where ω is the frequency, Ω is the solid angle subtended by the source at the detector plane, 2 , r ⊥ are the , ⊥ Fresnel reflection coefficients of the foil, which are both approximately unity for a highly conductive metallic surface, , I ( ) ⊥ ζ are the single foil OTR intensities polarized parallel and perpendicular to the plane of observation and Ψ is the phase difference between the radiations generated at the two foils [15].
The intensities = λ π γ + θ is the coherence or 'formation length', defined as the distance over which the field of the electron and the co-moving radiation photon differ in phase by 1 radian.
For all inter foil distances the radiation from two foils will interfere. However, the number of interferences per angular interval increases as the interfoil spacing and angle of observation increase. The visibility of these interferences is a function of the divergence and energy spread of the beam which typically are fractions of 1/γ for high quality beams. However, we have shown [21] that if the energy spread is smaller than the normalized divergence of the beam (i.e. ΔE/E << γσ), which is the case for our experimental conditions for all angles, the divergence effect dominates and hence the fringe visibility becomes a diagnostic for this quantity. The sensitivity of the interferometer to a given range of divergence can be optimized by adjustment of the interfoil spacing and the band pass of the measurement.

Description of Simulation Code used to Calculate ODR and ODTRI
In a conventional OTR interferometer forward OTR from a solid foil reflects and interferes with backward OTR from the mirror, where both the forward TR from the first foil and the backward TR from the mirror each has the form given above in Eqs. (2,3,5).
However, when the first foil is a mesh, the radiation is ODR from two distinguishable sources: 1) the beam electrons passing through the holes of the mesh and 2) the beam electrons passing through the solid wires separating the holes. Each of these ODR components interferes with two corresponding OTR components generated from the unscattered and scattered beam electrons emerging from the mesh interacting with the mirror. Thus the total intensity observed is composed of four contributions, two ODR components from the mesh and two corresponding OTR components from the mirror.
Analytic expressions for , ODR I ⊥ from the mesh similar to , OTR I ⊥ given above in Eqs.
(2,3) are not available. Hence we have developed a simulation code to calculate them. In addition our code computes the angular convolution of these components with a two dimension distribution of beam trajectory angles represented by one or more 2D Gaussian distributions each with a width representing , x y σ , the rms x and y divergences of the corresponding beam component. For OTRI such a convolution can be directly applied to the analytic forms for the OTR intensities components given above by Eqs. (2,3). For ODTRI the convolutions are incorporated into the ODR simulation code. The latter simulation code results should agree in the OTRI calculations in the limit of zero mesh cell size (i.e. continuous foil limit). We have use this limit as well as other checks [see ref . 7] to establish the validity of the simulation code.

Calculation of DR and TR from the two foils of the interferometer
Our simulation code calculates the angular distribution of the intensity of ODR produced by an electron beam passing through two parallel foils which are separated by the distance d measured along the direction of the beam velocity. In the analysis and experiments described in this paper the foils are tilted by an angle ν = 45 O . In the code we neglect the longitudinal component of the electric field of the electron. This simplifying approximation is valid for high energies (E > 50 MeV) even if the foils are tilted with respect to the electron beam velocity. We assume that the mesh perforations are perfectly symmetric rectangular holes with width h , which are evenly and symmetrically distributed on the foil with period p. The foil structure is represented as sum of translations of the unit cell shown in Figure 3., which shows a portion of the perforated foil projected onto a plane normal to the mean beam velocity. We refer to this plane as the source plane. The perforations are shown as white rectangles and a single perforation and its surrounding solid area (unit cell) is shown as two concentric rectangles. Radiation from the first and second foils is calculated assuming that the forward and backward radiations are symmetric about the surface of the tilted foil. If the size of the foil is large and the beam cross section is much larger than the period of the perforations p, the beam density profile varies very slowly over the cell period p and is considered to be constant over each cell.

Calculation of ODR Intensities for a single particle within cell of the mesh
For simplicity, the formulas presented below describe forward directed radiation from both foils of the interferometer considering the z axis to be directed forward and the detection plane is normal to this axis. There is no loss of generality in this approach because forward and backward specularly reflected radiations are mirror symmetric.
Following the picture introduced in the above paragraph, we introduce Cartesian coordinates , , x y z to describe the mesh perforations and the coordinates , We introduce various observation planes, which are normal to the source plane.
The horizontal observation plane is defined to be coplanar to (x, z ); the vertical observation plane is defined to be coplanar to ( y , z ); and we use cylindrical coordinates In local cylindrical coordinates , , r z ϕ ′ ′ ′ the longitudinal Fourier components with respect to time of the electric and magnetic fields of a relativistic electron in free space can be written as: Note that the electric field has only a radial component, the magnetic field has only an azimuthal component and that both fields are azimuthally symmetric about the z axis. Fourier components of fields of the electron can be interpreted as waves propagating along with the moving electron whose field is concentrated within a radius Now consider an electron which is incident on or emerges from the surface of perfect conductor. Inside the conductor the total field equals zero because the perfect conductor "screens out" all fields. This means that the conductor can be modeled as a region where, in addition to the fields of the electron, there are "primary induced" electric E i and magnetic B i fields with amplitudes equal and opposite in sign to the fields of the electron, i.e. Ei ≡ -Ee and Bi ≡ -Be at all points in the conductor including the surface.
Additionally we assume that these primary induced fields are " non radiative" inside this region. As a result we can consider the metallic boundary as a surface S′ with a known distribution of electric and magnetic source fields.
We assume that the induced surface fields radiate into the vacuum and that the field radiated into free space can be found using the Huygens-Fresnel principle. For example, the components of the electric field parallel and perpendicular to the ϕ plane at the observation point R are given by: where , , r z ϕ ′ ′ ′ are coordinates of the surface element / cos dS r dr dϕ is the modulus of the wave vector ( , , ) x y z k k k k = , ω is the frequency of the radiation, c is the speed of light in vacuum, At large distances from the radiator where r R R ′ Δ = − . Thus the angular distribution of the radiation is determined by the term , , which gives the radiation field produced from the area S′ in the direction k . The spectral energy density at the observation point averaged over the period of oscillation of the field is given by: where ds is an elementary surface normal to k at distance R .
In the far field zone (radiation zone) the energy spectral density per unit solid angle dΩ in the direction k ( later in this paper referred to as the intensity of radiation) is: which corresponds to the peak of OTR intensity at normal incidence, and S ⊥ ′ is a large solid area of integration normal to the particle trajectory.
The angular distribution of DR depends strongly on the size of integration area, the coordinates of the particles, the distribution of the holes in the foil and the angle of inclination of the foil. Note also that, in general, the perpendicular intensity DR I ⊥ is not zero even if V is parallel to the observation plane. In contrast to DR, the angular distribution of TR is independent of the coordinates, the spatial distribution of the particles and the angle ϕ and TR I ⊥ = 0, when V is parallel to the observation plane.
In our model a small deviation of the trajectory angle of an electron from the z axis corresponds to a small deviation of the tilt angle of the foil from the angle We have found that for small angular deviations, i.e. 5 / 0.05 rad ν γ Δ ≤ ≈ the angular pattern of the radiation produced from any particle in the unit cell is practically unaffected by the deviation angle (i.e. the intensity changes less than few percent in the worst case). We conclude that the pattern of radiation of an electron deflected from the z axis by a small deviation angle is centered about the deviation angle with the same distribution as that of an undeflected electron about its trajectory angle. This situation is well known for TR, i.e the centroid of the far field radiation pattern "follows" the angle of trajectory of electron for forward TR and the specular reflection angle for backward (reflected) TR.

Observations in the Detection Plane
In addition to the angular coordinates , As it is shown above the center of the radiation pattern of any electron interacting with the mesh follows the direction of the trajectory of the particle. Mathematically, this means that the distribution of intensity produced by the particle with trajectory e θ can be written as:  (8) and (10).

Total radiation from two parallel foils
In the interferometer the particle passes through two foils: (1) a perforated mesh and (2) a solid foil, producing DR and TR respectively. Using the variables , , , , where π ΔΨ = when the forward radiation from the first foil is reflected and interferes with the backward radiation from the second foil. In the limit , 0 The radiation produced by the scattered S or unscattered U fractions of the beam is a summation of radiations produced by all the particles from each beam fraction.
The parallel component can be written as: , ,, , are summation terms collinear to the direction of the trajectory of the particle, with index i representing a particular beam particle with coordinates i e r ( , The perpendicular component of the intensity is calculated in the same manner as described above: Note that the perpendicular components do not contain an interference phase term because the radiation intensity from the solid foil is TR and, as such, does not have a perpendicular component. The total radiation from the two interferometer foils produced by all particles of S or U fraction with trajectory e θ is then: In practice these summations shown in Eqs. (18,19) are

Computing the effect of beam divergence
The effect of beam divergence on the intensities computed above is performed by means of a two dimensional angular convolution. In order to perform this convolution it is necessarily to know the intensity produced by all particles of each beam fraction (scattered and unscattered) with trajectory angle e θ at the observation point k θ , as well as the distribution of trajectory angles of the beam electrons.
We model the distribution of electron trajectory angles as a sum of up to three individual Gaussian components. For instance in the case of the mesh the wires substantially scatter electrons up to few mrads completely "hiding" the original angular distribution which is usually a fraction of one mrad. The scattered portion of the beam evidently has wider angular distribution than the unperturbed beam passing through the holes and is represented as a single wide Gaussian component. However, the angular distribution of the "unperturbed" unscattered portion of the beam is more complex and cannot be represented by a single Gaussian distribution. We have modeled this situation by splitting the U fraction of the beam into two individual Gaussian components, the minimum number required for our fits. Note that the zero angle of the total distribution is the same before and after scattering and is the same for all components.
In angular coordinates and using the small angle approximation, the multi-Gaussian component beam can be presented as: where n is the number of Gaussian components including scattered and unscattered portions, xn σ , yn σ are standard angular deviations and n A are normalization constants.
In this paper, the numbers n = 1 and 2 designate the 1st and 2nd components of the unscattered beam and n = 3 designates the single scattered component.
The radiation produced by the n th component at the observation point , which is the OTRI limit. In our analysis of OTRI data, the angular convolution for TR interferometer is performed using Eq. (26) and two Gaussian components to represent the unscattered beam distribution.

Additional Convolutions
To account for variations of the beam energy and the observation wavelength, we can optionally perform separate convolutions or averages over these variables as well as angular convolution. To account for finite band of observed wavelengths, we assume that the spectral characteristic of the band pass filter is rectangular and perform an convolution of the intensity scan over the filter transmission function. To account for possible variations in beam energy we can perform a convolution of the scanned intensity over energy under the assumption that the energy distribution has a cosine distribution.
Optional forms for the energy variation are Gaussian or rectangular distributions.

Unpolarized OTRI and ODTRI
In the experiments described below unpolarized OTRI and ODTRI images are used, i.e. no polarizer was used. The reason for this is that analysis shows that the use of a polarizer does not give any advantage when a 2D angular convolution code is used to evaluate the data.

Demonstration of Simulation Code Results
In Figure 5. we present results of simulation code calculations of the sum of ODR contributions from the beam electrons which pass through the holes of the mesh (unscattered beam) and the sum of ODR contributions from the electrons passing through the mesh wires (scattered beam), for a 5 micron thick, 750 lines per inch copper mesh, with 25 μm square holes and a cell period p = 33 μm (55% transparency). The percentage of the ODR intensities from unscattered and scattered beam components is about 10% of the total radiation. The OTR generated at the mirror from the scattered and unscattered components are 55% and 45% respectively, in accordance with the mesh transparency.
The beam energy used in the calculations shown in Figure 5. reduce the visibility of the interference fringes. This effect is the basis of beam divergence diagnostics with both OTRI and ODTRI when the energy spread is smaller than the normalized divergence which is the case in the present study.
The interference term for OTRI and ODTRI is the same since this term depends only on the relative phase of the radiation from the first foil and the mirror. Figure 6.
shows the interferences generated from the individual ODR intensity components  In this situation the fringes due to the unscattered (unperturbed) beam component are made visible above the smooth (incoherent) scattered beam contribution. The two components add to form the black curve in Figure 6. The fringe visibility is affected by the inherent (unperturbed) beam divergence, which for illustration is σ = 0.5 mrad. The wavelength chosen for the calculation is 650 nm with a delta function band pass.

Description of the Experimental Setup
The beam energies used in our experiments are 50 and 95 MeV. The setups for both experiments are essentially the same. Both employ optical trains which accept and maintain an angular field of view of approximately 10/γ and transport the light to cameras positioned away from the foil-mirror position to reduce the x-ray background. A schematic of the optics used at the BNL/ATF is given in Figure 7. for illustration. Details of the experimental setup of the NPS 95 MeV experiment have been previously described in [19]. For each experiment, care must be taken to insure that the second camera is focused on the plane of the mirror, i.e. the OTR radiator/reflector. This is done with the help of a graticule target, whose surface is coplanar with the mirror.
To image the far field angular distribution, i.e. the OTR or ODTR interference pattern we used Apogee Instruments Inc., 16 bit, Peltier cooled, high QE, low noise CCD cameras each of which is equipped with an electronically controlled mechanical shutter; this allows the CCD to integrate the light produced from multiple electron beam pulses.
A model Alta E47+ was used at ATF and a model AP230E was used at NPS. A second less sensitive RS 170, 8-bit CCD camera (Cohu 4912 or GBC-CCTV 500E) was used to monitor the beam's spatial distribution.
The first lens shown in Figure 7, which has a focal length f1=44cm, is placed 47 cm from the mirror. In the focal plane of this lens an image of the far field angular distribution (AD) appears. A second lens whose focal length f 2 = 20cm placed at 168 cm from the mirror is used to re-image the AD in the image plane of camera C2. The object and images distances for C2 are 77 and 28 cm respectively. A light baffle is used to prevent direct reflection from the first foil or mesh from entering the optical path.
Identical interferometers but with different foil-mirror spacings: d = 37 and 47 mm, for the 50 and 95 MeV beams respectively, were used in the experiments.
A photograph of the target ladder housing the interferometers is shown in Figure   8. This apparatus was mounted on a stepper motor driven, 6-inch linear actuator. One of four positions (components) of the ladder could be placed into the beam: 1) a graticule, used to determine the magnification of the system; 2) an aluminized Silicon mirror cut from a 0.5 mm wafer, used alone for alignment of the optics with an upstream laser; 3) the OTR interferometer, consisting the mirror and a 0.7 micron thick foil of 99.5% pure aluminum mounted on a stretcher ring (the apparatus seen on the right hand side of  . The foilmirror spacings were determined from calculations to produce an optimal number of interferences for the beam energy and expected range of divergence for each beam. The optimal spacings and band passes for these two situations were determined prior to the experiment by applying the results of computer code runs for both OTRI and ODTRI interferences.
At NPS a pair of quadrupole magnets up stream of the target chamber were used to magnetically focus the beam to either an x or y waist condition at the mirror position.
The waist condition was confirmed by observing the maximum sharpness of the higher order interference fringes [20]. At a beam waist the visibility of the observed OTR or ODTR interference fringes in the x (horizontal) or y (vertical) directions is a measure of the corresponding x or y rms beam divergence. Thus, together with knowledge of the rms (x or y) size at the corresponding (x or y ) waist obtained from the spatial image of the beam and the corresponding rms divergence obtained from the interference pattern, the rms x and y beam emittance could be determined.

Results and Analysis
Data Fitting Procedure To fit the scanned data to simulation code scans we compare the data to a family of theoretical curves each obtained for a particular set of beam parameters: divergence, energy, energy spread and fractional weight, when more than one beam component is required, and interferometer foil spacing. The goal is to achieve the best set of parameters which simultaneously provides a 'best fit' to both the horizontal and vertical sector averaged data scans.
The goodness of the fit of measured ( ) E θ and calculated ( ) T θ scans is determined first qualitatively by eye and then more rigorously using the following procedure. The essence of this procedure is to scale the data scan by a constant A until θ θ θ θ θ is the experimentally measured intensity distribution, scaled by A , an arbitrary amplitude,and T(θ) is the line scan calculated from the simulation code The goal is to achieve the best set of parameters which simultaneously gives the best similarity for both the horizontal and vertical scans. The "best fit" occurs when we have found a set of beam parameters, interferometer parameters and values of min D and min A for which the eye judges that the best similarity is achieved. Note that in the ideal case D min = 0.
Adjustment of the parameters used to fit the experimental scan data to simulation code or theoretical calculations is performed in the following way: 1) parameters of the interferometer and expected (starting) parameters of the beam, i.e. foil separation d, filter pass band, electron energy band, parallel σ p and normal σ n angular divergences, angular interval for fitting are inputted.
2) theoretical/simulation code calculations are performed for both horizontal and vertical scans and the theoretical and renormalized experimental curves are plotted.
3) a comparison of experimental and theoretical curves and adjustment of the input parameters is made to achieve the best similarity between theory and experiment simultaneously for both horizontal and vertical scans, i.e. by minimizing the rms deviation function D.

Example of Data Fitting Procedure
To illustrate the procedure employed to fit the sector averaged line scans used in all our analyses of OTRI and ODTRI patterns, we will use an OTRI from NPS as an example. The OTR interference pattern for the NPS beam focused to a y (vertical ) beam waist is shown in Figure 9. The measured rms y size of the beam at the y waist condition is ~ 1mm. The OTRI pattern was obtained by exposing the far field CCD camera for 45 seconds; the picture is taken with an optical filter band pass filter in place ( 650nm, 70 nm FWHM band pass). The colored sectors overlaying the image in Figure 9. show the angular regions used to average the intensity at each radius measured from the center of the pattern.  A variational analysis the interference phase term in Eq. (1) shows that the effect of divergence on the fringe visibility is proportional to the fringe order so that the effect of increasing the divergence is seen as a reduced fringe visibility for the higher order fringes first [21]. The higher order fringes are better fit by single Gaussian with σ = 0.6 mrad but the lower order fringes are not fit well with this same function; this is evidence that the real beam angle distribution is not well represented by a single Gaussian.
To improve the fit to the data we have introduced a second 2D Gaussian function in addition to the primary Gaussian to model the distribution of electron angles. The fractional amplitudes and rms widths of both Gaussians were adjusted to provide the best fit (dot dashed blue line) to the data (solid red line) shown in Figure 11. The primary distribution fraction is weighted by 0.75 and its rms width, σ = 0.6 mrad.
The effect of the primary distribution (Comp1) on the OTRI fringes is shown by the dashed black curve. The effect of the secondary distribution (Comp2) is shown by the dot-dashed green line. The total effect of the two components is represented by the dotdashed blue line. As is seen from Figure 11. the overall fit to the data is excellent with the  Figure 11.

NPS Data Fits
An ODTRI pattern at the y waist NPS beam condition obtained with an integration time of 60 seconds and the same band pass filter used for the OTRI is shown in Figure 12.    A comparison of the other ODTRI and OTRI fit parameters is provided in Table   1. There is a slight difference in the spacing, 36.5 mm for the ODTRI vs. 37.2 mm for the OTRI, which is most likely due to a small difference in orientation of the two foils with respect to the beam direction due to rotational wobble in the linear drive. Previous analysis [15,21] has shown that the position of the fringes is a sensitive function of the beam energy and spacing but that the visibility of the fringes is primarily affected by the divergence, when the energy spread of the beam is small in comparison to the normalized divergence.   Note the apparent offset of the colored sectors from horizontal, which is due to a slight rotation of the mirror with respect to the optical axis. This offset is observable and known from the diffraction pattern of the laser, which follows the same optical path as the ODTR.    . This is also verified by code calculations which show that an energy spread of up to 2% has little effect on the fringe visibility.

Discussion
We have examined the possible causes of the inferred bimodal distributions and consequent two beam divergences obtained from our fits to the ODTRI and OTRI data.
These are listed and analyzed below.

Energy Spread
The energy spread of the ATF beam was monitored during the experiment and is less than 0.5%. Both variational analysis of the interference terms in Eq. 1. [21] and our convolution codes show that this spread is too small to be responsible for the observed fringe blurring; i.e. the energy spread would have to be 16 times larger (8%) to show the effect observed at ATF. The energy spread at NPS is higher than ATF, i.e. a few percent.
However, the divergence of the NPS beam is also higher. Both variational analysis and computer convolution calculations show that the energy spread is not sufficient to significantly affect the observed visibility.

Beam Instabilities
There are several types of beam stabilities that could be present and possibly affect our results. These include: a. Jitter in the beam position This type of jitter has no effect on the far field angular distributions of OTRI and ODTRI.
b. Random jitter in the trajectory angle of the beam This effect would combined with the effect of the beam core divergence resulting in as a single Gaussian distribution whose full width would be calculable from quadrature addition of the FWHMs of the components, one related to the inherent temperature of the beam, the other to a jitter of the trajectory angle would be seen as a single Gaussian with a wider rms width: . A single Gaussian with width σ tot would have a predictable effect on the fringe visibility. Since the fringes cannot be fit with a single Gaussian distribution, a random jitter effect must be ruled out.
c. Non-random jitter of the trajectory angle of the beam This effect would appear as a distinct non Gaussian distribution and possibly explain the need for a second distinct component in addition to the core beam angular distribution. Since we did not continuously monitor the beam position during the experiment and acquire the data that would allow a statistical calculation of the jitter, we cannot rule out this possibility. We therefore conclude that a nonrandom instability in the beam position, during the long image integration times, is a possible explanation for the observed two Gaussian component fit. A future experiment will be needed to test this possibility.

Conclusions
We have obtained nearly identical divergences and beam fractions using two independent measurements, i.e. OTRI and ODTRI. The divergence obtained agree wel with those obtained by independent multiple screen measurement-transport code calculations. The analysis of the OTRI and ODTRI data are very different. OTRI analysis uses a direct convolution of an exact analytic expression to obtain a fringe pattern, which is then fit to the data. The analysis and fitting of ODTRI, on the other hand, is much more complicated and requires a simulation code to perform. It is very unlikely that these two independent calculations and fits to data would produce nearly identical results that both agree with the independent multiple screen analysis. We conclude that ODTRI and OTRI are indeed being affected by the same, real physical effect and that the simulation code and data fitting procedures we have employed are correct and consistent.
We emphasize that we have not set out to prove that electron beams, under certain operating conditions can show bimodal distributions -which is the conclusion of our Since the ODTRI fringes are sensitive to the unperturbed beam, which passes through the holes of a micromesh foil, ODTRI overcomes the lower limit on measurable divergence present in a conventional OTR interferometer, i.e. scattering in the solid first foil. Hence ODTRI can be used to diagnose lower emittance and/or lower energy beams.
Our demonstration of ODTRI as a useful rms divergence diagnostic indicates that it may be possible to use ODTRI to make localized beam divergence and trajectory angle measurements, i.e. within the beam's spatial distribution, and therefore to produce a transverse phase space map of the beam. Optical phase space mapping (OPSM) has already been demonstrated using OTRI [19,20]. Presuming the problem of the lower intensity yield of ODR compared to OTR can be overcome, e.g. by increasing the integration time or beam current, OPSM using ODTRI should be straightforward.