tag:blogger.com,1999:blog-42674962072900970692024-03-07T19:25:18.983-08:00Astronomical Optical Interferometry TelescopeFunkhttp://www.blogger.com/profile/13086037268464989921noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-4267496207290097069.post-52062632356524624122009-11-01T09:21:00.000-08:002009-11-01T09:24:49.146-08:00Astronomical Optical Interferometry<h1>Astronomical Optical Interferometry</h1> <h2>A Literature Review by Bob Tubbs</h2> <h2>St John's College Cambridge</h2><span style="font-family:Arial;font-size:130%;"><b><p>Abstract</p></b></span> <span style="font-family:Arial;font-size:100%;"> <p>This report documents the development of optical interferometry and provides a physical explanation of the processes involved. It is based upon scientific papers published over the last 150 years, and I have included references to the ones which are most relevant. The reader is assumed to have an understanding of modern optical theory up to undergraduate level - References 28 and 29 give explanations at a more basic level. The formation of images from interferometric measurements is discussed and several example images are included.</p> </span> <span style="font-family:Arial;font-size:130%;"><b><p>Introduction</p></b></span> <span style="font-family:Arial;font-size:100%;"> <p>Fizeau first suggested that optical interferometry might be used for the measurement of stellar diameters at the Academie des Sciences in 1867. The short wavelength of light and the absence of sensitive calibrated detectors precluded more sophisticated interferometric measurements in the optical spectrum for over a century. After the Second World War most researchers instead turned to the radio spectrum, where macroscopic wavelengths and electronic detection greatly simplified the measurement of interferometric quantities. Modern computers, lasers, optical detectors and the data processing techniques developed for radio interferometry have recently enabled astronomers to produce high resolution images with optical arrays. At present only a few optical interferometer arrays are capable of image formation but many more are planned or under construction. The basic principles underlying the operation of optical interferometers have not changed, so I begin with a look at some of the earliest instruments.</p> <p>Notes: </p><ul><li>Superscript numbers 1) link to the References section of this report and relate to relevant reference numbers.</li><li>All unusual symbols are presented as GIF images.</li></ul> For a more detailed description of astronomical optical interferometry I would recommend this <a href="http://www.astro.lsa.umich.edu/%7Emonnier/Publications/ROP2003_final.pdf">review article</a> by John Monnier (68 pages). </span>Funkhttp://www.blogger.com/profile/13086037268464989921noreply@blogger.com0tag:blogger.com,1999:blog-4267496207290097069.post-40810226666879850982009-11-01T09:20:00.002-08:002009-11-01T09:21:15.055-08:00Early Optical Interferometry<span style="font-family:Arial;font-size:130%;"> </span>The American physicist A. A. Michelson demonstrated the practicability of measuring light sources using optical interferometry<sup><a href="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/refer.html#2">2</a></sup> in 1890 with the experimental apparatus shown in Figure 1.<span style="font-family:Arial;font-size:100%;"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image23.gif" width="614" height="739" /></td></tr> <tr><td width="100%"><b><i>Figure 1 - Michelson?s experimental apparatus</i></b></td></tr> </tbody></table> <p>Various masks were placed in front of incoherent light sources, acting as "artificial stars" for the experiment. Light from a distant artificial star passed through slits <i>O</i> and <i>O'</i> and was then focused by a lens of focal length <i>y</i> to form an image on the screen. In a mathematical analysis of this experiment it is easier to first consider a monochromatic point source at <i>Q</i> on the optic axis. Spherical wavefronts will radiate from the source reaching slits <i>O</i> and <i>O'</i> simultaneously. Light passing through slit <i>O</i> will interfere with light passing through slit <i>O'</i> forming intensity fringes on the screen either side of point <i>P</i>. The optical path length from <i>Q</i> to point <i>P</i> on the screen is the same for rays travelling through either slit. This will not be the general case for light rays travelling to an arbitrary point on the screen from <i>Q</i>. The difference in optical path length between light rays travelling via slit <i>O</i> and those travelling via <i>O'</i> will then be <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image24.gif" width="20" height="37" /> to a first approximation, where <i>v</i> is the co-ordinate on the screen shown in Figure 1. When light rays from the two slits are combined on the screen they will interfere producing intensity proportional to <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image25.gif" width="64" height="42" />, where <i>k</i> is the wavenumber defined as <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image26.gif" width="69" height="37" />. Light rays from a point source offset from <i>Q</i> by an angle <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> as shown in Figure 1 give light intensity on the screen proportional to <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image27.gif" width="96" height="45" />. An extended incoherent source placed at Q can be considered as a distribution of many such point sources. A chromatic source can be considered as the superposition of many monochromatic sources of different frequency. The intensity observed on the screen will be the sum of the intensities produced by each point on the source.</p> <p>Michelson was not able to make quantitative measurements of the visibility of interference fringes on the screen but did make measurements of the slit separation <i>x</i> which gave minimum fringe visibility. The size of the artificial star can be calculated from this measurement provided its shape and distance are known. With modern photodiode detectors it is possible to make accurate intensity measurements and hence calculate fringe visibilities. The viewing screen is replaced by four light intensity detectors as shown in Figure 2. Detector 1 is positioned so that the optical path lengths from the detector to slit <i>O</i> and from the detector to slit <i>O'</i> are equal. Detector 2 is positioned so that the optical path lengths to <i>O</i> and <i>O'</i> differ by a <sup>1</sup>/<sub>4</sub> of the mean wavelength. For detectors 3 and 4 the path differences are <sup>1</sup>/<sub>2</sub> of a wavelength and <sup>3</sup>/<sub>4</sub> of a wavelength respectively. If <b><i>A</i></b> is the complex amplitude of the light arriving at detector 1 along the path through slit <i>O</i>, the amplitude of the light arriving via slit <i>O'</i> will be <b><i>A</i></b>exp[-<i>i<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> kx</i>], giving a total amplitude of <b><i>A</i></b>+<b><i>A</i></b>exp[-<i>i<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> kx</i>]. The intensity at detector 1 will be:</p><dir> <dir> <p align="JUSTIFY"> <span style="font-size:85%;"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image28.gif" width="188" height="76" /></span></p></dir> </dir> <p>Similarly if <b><i>A</i></b> is the amplitude of the light arriving at detector 2 along the path through slit <i>O</i>, the intensity at the detector will be: </p><dir> <dir> <p align="JUSTIFY"><span style="font-size:85%;"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image29.gif" width="161" height="70" /> </span></p></dir> </dir> <p>For detector 3:</p><dir> <dir> <p align="JUSTIFY"> <span style="font-size:85%;"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image30.gif" width="150" height="54" /></span></p></dir> </dir> <p> For detector 4:</p><dir> <dir> <p align="JUSTIFY"> <span style="font-size:85%;"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image31.gif" width="166" height="70" /></span></p></dir> </dir> <p>I have defined the complex fringe intensity <b><i>I</i></b> as (<i>I</i><sub>1</sub>-<i>I</i><sub>3</sub>)+<i>i</i>(<i>I</i><sub>2</sub>-<i>I</i><sub>4</sub>) where <i>I</i><sub>1</sub> to <i>I</i><sub>4</sub> are the intensities shown above, and <i>i</i> is <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image32.gif" width="29" height="20" />. In the case of the point source shown in Figure 2 </p><dir> <dir> <b><i> </i></b><p><b><i><span style="font-size:85%;">I</span></i></b><span style="font-size:85%;">=4<b><i>AA</i></b><sup>*</sup>(cos[<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> <i>kx</i>]+<i>i</i>sin[<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> <i>kx</i>])</span></p> <p><span style="font-size:85%;">=4<b><i>AA</i></b><sup>*</sup>exp[<i>i<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> kx</i>]</span></p></dir></dir> <p> </p> <p> </p> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr> <td valign="BOTTOM" width="50%" align="CENTER"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image33.gif" width="306" height="637" /></td> <td valign="BOTTOM" width="50%" align="CENTER"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image34.gif" width="382" height="759" /></td></tr> <tr><td valign="TOP" width="50%" align="CENTER"><b><i><span style="font-family:Arial;"> Figure 2 - Visibility measurement</span></i></b></td> <td valign="TOP" width="50%" align="CENTER"><b><i><span style="font-family:Arial;"> Figure 3 - Alternative optical arrangement</span> </i></b></td></tr> </tbody></table> </span><span style="font-family:Arial;font-size:100%;"> <p>As the complex intensity <b><i>I</i></b> is a linear combination of intensities, the complex intensity of an extended incoherent source can be calculated by summing the contributions from each point on the source. The amplitude <b><i>A</i></b>(<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ) of the light received from points between <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> and <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> +d<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> on the source will be dependent on the source brightness distribution <i>B</i> (<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ) in the following manner:</p><dir> <dir> <p> <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image35.gif" width="105" height="30" /> (assuming <i>d<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> </i> is small)</p></dir> </dir> <p>The complex intensity for light received between <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> and <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> +d<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> will be <b><i>I</i></b>(<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> )=4<i>B</i>(<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> )exp[<i>i<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> kx</i>] d<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> . Integrating over all <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> gives:</p><dir> <dir> <p align="JUSTIFY"> <span style="font-size:85%;"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image36.gif" width="117" height="68" /></span></p></dir> </dir> <p>If the variable <i>u</i> is defined as <i>u</i>=<i>kx</i>, then <b><i>I</i></b><i><sub>TOTAL</sub></i> is proportional to the Fourier transform of the one dimensional source brightness distribution B(<img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ) with respect to <i>u</i>. If this Fourier transform is normalised to have a total intensity of unity we obtain the complex visibility:</p><dir> <dir> <p align="JUSTIFY"><span style="font-size:85%;"> <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image37.gif" width="177" height="46" /></span></p></dir> </dir> <p>Michelson did not have sensitive electronic detectors so his measurements relied on human eyesight. He succeeded in calculating the diameters of Jupiter's satellites <sup><a href="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/refer.html#3">3</a></sup> using an aperture mask with two slits of adjustable separation placed over the objective of a 12-inch telescope. He measured the slit separations at which the fringes were least visible, and calculated the diameters of the satellites by assuming them to be circular disks with uniform illumination. His results agreed well with visual estimations of the satellite diameters which had been made using large optical telescopes.</p> <p>With the optical arrangement of Figure 2 a large objective lens or mirror is required for measurements with large slit separations and much of the light that passes through the slits in the aperture mask is wasted. Figure 3 shows an alternative optical arrangement which uses separate optical elements for the two beams. The incident light is from a distant point source at angle <img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> . Light entering each of the slits is split into four equal beams which are then directed to the detectors. The path differences between rays travelling through <i>O</i> and <i>O'</i> to each of the detectors are the same as in Figure 2, but in this arrangement all the light entering the apparatus is used efficiently. In practice glass blocks might produce reflections within the apparatus and would probably not be used. Instead, the appropriate difference in optical path length from the detectors to each of the slits could be produced by careful adjustment of the mirror positions. By varying the optical path length of one of the beams it is possible to calculate the complex visibility with just one detector. As the optical path length is varied the interference fringes will be scanned past the detector. The amplitude and phase of the intensity variations at the detector will be linearly related to the amplitude and phase of the complex visibility. In most modern interferometers the intensity variation with time is Fourier transformed to give an amplitude and phase for the complex visibility.</p> <p>In 1891 Michelson <sup><a href="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/refer.html#4">4</a></sup> discussed the possibility of obtaining information about the brightness distribution within a source from interferometric measurements. He conceded that this was not practicable as it would require accurate measurements of fringe visibility at many different slit separations. Over the next sixty years most of the work on optical interferometry concentrated instead on the measurement of stellar diameters and the separation of binary stars<sup><a href="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/refer.html#5">5</a></sup>. In 1920 A. A. Michelson and F. G. Pease<sup><a href="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/refer.html#6">6</a></sup> constructed a <a href="http://web.archive.org/web/20050516083547/http://www.mtwilson.edu/Tour/100inch/20ftI/">separate-element Michelson stellar interferometer</a> as shown in Figure 4. The separation of the siderostat mirrors was equivalent to the slit separation in his earlier interferometers. Separations of over 20ft were possible, enabling measurements of the diameters of several large stars to be performed. An <a href="http://web.archive.org/web/20050516083547/http://www.mtwilson.edu/Tour/Michelson/50ftI/">interferometer with a 50ft siderostat separation</a> <sup><a href="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/refer.html#7">7</a></sup> was built in 1930, with mirrors attached to 9 tons of steel girderwork on the front of a 40 inch optical telescope. Very few astronomical measurements were made with this instrument due to the difficulty of operating it. With both of these interferometers atmospheric fluctuations produced phase variations which caused the fringes to "shimmer", making observation extremely difficult. R. Hanbury Brown<sup><a href="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/refer.html#8">8</a></sup> estimated that atmospheric fluctuations may have led to errors of between ten and twenty percent in Michelson and Pease's stellar diameter calculations. Hanbury Brown produced more accurate measurements using an intensity interferometer in Navarra <sup><a href="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/refer.html#8">8</a></sup>. Intensity interferometers look at the statistical relationship between the intensities at two separated detectors observing a distant source. Quantum mechanics suggests that this is related to the amplitude of the complex visibility function, allowing measurements of visibility with large detector separations. Unfortunately the phase of the complex visibility cannot be determined, and accurate visibility amplitudes can only be calculated for bright astronomical sources.</p> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20050516083547/http://www.geocities.com/CapeCanaveral/2309/image38.gif" width="349" height="361" /></td></tr> <tr><td width="100%"><i><b> Figure 4 - Simple separate element interferometer</b></i></td></tr></tbody></table></span>Funkhttp://www.blogger.com/profile/13086037268464989921noreply@blogger.com0tag:blogger.com,1999:blog-4267496207290097069.post-33241256294283217162009-11-01T09:20:00.001-08:002009-11-01T09:20:37.520-08:00Development of Radio Interferometry<span style="font-family:Arial;font-size:130%;"> </span><span style="font-family:Arial;font-size:100%;">Much of the early work in interferometric imaging was done by radio astronomers. Cosmic radio emissions were discovered in the 1930s<sup><a href="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/refer.html#9">9</a></sup> and radio interferometry developed after the Second World War. In 1946 Ryle and Vonberg<sup><a href="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/refer.html#10">10</a></sup> constructed a radio analogue of the Michelson interferometer and soon located a number of new cosmic radio sources. The signals from two radio antennas were added electronically to produce interference. Ryle and Vonberg's telescope used the rotation of the Earth to scan the sky in one dimension. Fringe visibilities could be calculated from the variation of intensity with time. Later interferometers included a variable delay between one of the antennas and the detector as shown in Figure 5.</span> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image39.gif" width="501" height="790" /></td></tr> <tr><td width="100%"><b><i>Figure 5 - Radio interferometer</i></b> </td></tr> </tbody></table> <span style="font-family:Arial;font-size:100%;">In Figure 5 radio waves from a source at an angle <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> to the vertical must travel a distance <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> <i>l</i> further in order to reach the left-hand antenna. These signals are thus delayed relative to the signals received at the right hand antenna by a time <i>c<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> l</i>=<i>ca</i>sin[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ] where <i>c</i> is the speed of the radio waves. The signal from the right hand antenna must be delayed artificially by the same length of time for constructive interference to occur. Interference fringes will be produced by sources with angles in a small range either side of <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> determined by the coherence time of the radio source. Altering the delay time <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> <i>t</i> varies the angle <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> at which a source will produce interference fringes. It should be noted that the effective baseline of this interferometer will be given by the projection of the telescope positions onto a plane perpendicular to the source direction. The length of the effective baseline, shown at the bottom of Figure 5, will be <i>x</i>=<i>a</i>cos<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> where <i>a</i> is the actual telescope separation.</span> <p><span style="font-family:Arial;font-size:100%;">An interferometer constructed from two antennas with separation variable in one direction can only provide information about the sky brightness distribution in one dimension. However, a two dimensional map of the sky can be produced if the separation vector is varied in two dimensions. In Figure 6 the separation between two radio antennas is described by the vector (<i>a</i>,<i>b</i>) constructed from two cartesian co-ordinates. The position of the source in the sky is described using the angles <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> in the plane of the <i>a</i> axis and <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> in the plane of the <i>b</i> axis. As in Figure 5, the effective baseline (<i>x</i>,<i>y</i>) will be the projection of the separation vector onto a plane perpendicular to the source direction: (<i>x</i>,<i>y</i>)=(<i>a</i>cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ],<i>b</i>cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> ]). Measurements of complex visibility are usually plotted in the Fourier transform plane of the sky brightness distribution using the dimensionless variables <i>u</i> conjugate to angle <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> and <i>v</i> conjugate to angle <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> . These can be calculated as <i>u</i>=<i>kx</i> and <i>v</i>=<i>ky</i>, where <i>k</i> is the wavenumber of the radio source defined as <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image40.gif" width="101" height="38" />. Either the phase of signals from the left-hand antenna can be measured relative to the those from the right hand antenna, or the phase of the signals from the right-hand antenna can be measured relative to those from the left. A measurement of complex visibility for an antenna separation (<i>a</i>,<i>b</i>) can thus provide values of the complex visibility function at two points in the <i>u</i>-<i>v</i> plane:</span></p><dir> <dir> <span style="font-family:Arial;font-size:100%;"><span style="font-size:85%;"><p> (<i>u</i>,<i>v</i>)=(<i>kx</i>,<i>ky</i>)=(<i>ak</i>cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ],<i>bk</i>cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> ]) and (<i>u</i>,<i>v</i>)=(-<i>kx</i>,-<i>ky</i>)=(-<i>ak</i>cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ],-<i>bk</i>cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> ])</p> </span></span><p><span style="font-family:Arial;font-size:100%;"> </span></p></dir> </dir> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image41.gif" width="553" height="385" /></td></tr> <tr><td width="100%"><b><i>Figure 6 - The telescope separation vector (a,b)</i></b> </td></tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;">In order to produce a perfect map of the sky brightness distribution the complex visibility would have to be known for all points in the <i>u</i>-<i>v</i> plane (Fourier transform plane). The complex visibility must be known at all points in a <i>n</i>×<i>m</i> rectangular array in the <i>u</i>-<i>v</i> plane for a portion of the sky to be mapped with resolution equivalent to <i>n</i>×<i>m</i> pixels. The radio source brightness distribution <i>B</i>(<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ,<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> ) is reconstructed by Fourier transforming the array of complex visibility measurements. Figure 7 shows a typical cosmic radio source with brightness distribution <i>B</i>(<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ,<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> ). Fourier transforming a 40×40 array of complex visibility measurements in the <i>u</i>-<i>v</i> plane gives a relatively accurate model of the source brightness distribution, as shown in Figure 8. Figure 9 shows the cruder model formed from a 9×9 array of complex visibility measurements.</span></p> <table width="100%" border="0" cellpadding="4" cellspacing="0"> <tbody><tr><td valign="BOTTOM" width="33%" align="CENTER"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image42.gif" width="160" height="160" /></td> <td valign="BOTTOM" width="33%" align="CENTER"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image43.gif" width="160" height="160" /></td> <td valign="BOTTOM" width="33%" align="CENTER"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image44.gif" width="160" height="160" /></td></tr> <tr><td valign="TOP" width="33%" align="CENTER"> <i><b>Figure 7 - Source brightness distribution</b></i></td><td valign="TOP" width="33%" align="CENTER"><i><b>Figure 8 - brightness distribution with 40x40 Fourier components</b></i></td><td valign="TOP" width="33%" align="CENTER"><i><b>Figure 9 - brightness distribution with 9x9 components</b></i></td></tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;"> </span></p> <p><span style="font-family:Arial;font-size:100%;"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image45.gif" width="53" height="54" /><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image46.gif" width="192" height="27" /></span></p> <p><span style="font-family:Arial;font-size:100%;"><i><b>Axes and brightness key</b></i></span></p><span style="font-family:Arial;font-size:100%;"> </span><p><span style="font-family:Arial;font-size:100%;">For direct measurement of the complex visibility at a rectangular array of points in the <i>u</i>-<i>v</i> plane a large number of different baselines is required. The cost of radio antennas soon led astronomers to try and find methods for calculating the complex visibility throughout the <i>u</i>-<i>v</i> plane using measurements from only a small number of antennas. The most important of these is the Earth rotation aperture synthesis technique.</span></p> <p><span style="font-family:Arial;font-size:100%;">If an interferometer is constructed from two antennas with a separation which is not parallel to the Earth's axis of rotation, the effective baseline of the interferometer will rotate. Figure 10 shows an interferometer in the northern hemisphere with antennas located at <i>A</i> and <i>B</i>. During the day antenna <i>A</i> will move to <i>A'</i> and then <i>A''</i> whilst <i>B</i> moves to <i>B'</i> and <i>B''</i>. Only the relative positions of the two antennas are relevant when constructing a map of complex visibility in the <i>u</i>-<i>v</i> plane. To an irrotational observer standing beside antenna <i>A</i>, antenna <i>B</i> would appear to rotate in a circle, and vice-versa. In a twelve hour period the complex visibility can be measured at all points on an ellipse in the <i>u</i>-<i>v</i> plane. If one of the antennas is mobile, the antenna separation can be altered every day so as to measure complex visibilities in a different part of the <i>u</i>-<i>v</i> plane. A mathematical function which approximates the complex visibility is created by interpolation from the measurements made. This can then be Fourier transformed to give an approximation to the source brightness distribution.</span></p> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/sphere.jpg" width="238" height="265" /></td></tr> <tr><td width="100%"><i><b>Figure 10 - Rotation of the Earth</b> </i></td></tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;">Information about the fine structural detail of a radio source is found at large values of <i>u</i> and <i>v</i> due to the reciprocal nature of the Fourier transform plane. In order to produce a radio map of high angular resolution it is therefore necessary to measure fringe visibilities over very long baselines. The radio signal received at an antenna cannot be sent further than a few tens of kilometers by electrical cable due to the signal loss incurred. Electronic amplification en route introduces delays and distortion to the signal. The most effective method for measuring the complex visibility for <a href="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/nm_astro.html#vlbi">very long baseline interferometry (VLBI)</a> is to first record the signals received by each antenna along with timing signals from a local atomic clock. The recorded signals from each antenna can then be sent to a laboratory where they are replayed to produce interference. Figure 11 shows the received signals from three antennas being recorded onto magnetic tapes along with timing signals from local atomic clocks. From these tapes the complex visibility can be calculated at six points in the <i>u</i>-<i>v</i> plane corresponding to the antenna separations <i>a</i><sub>1</sub>, -<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, - <i>a</i><sub>2</sub>, <i>a</i><sub>3</sub> and -<i>a</i><sub>3</sub> in Figure 11.</span></p> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image48.gif" width="691" height="369" /></td></tr> <tr><td width="100%"><i><b>Figure 11 - Recording radio signals for very long baseline interferometry</b> </i></td></tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;">Each antenna will be a different distance from the radio source, and as with the short baseline radio interferometer (Figure 5) the delays incurred by the extra distance to one antenna must be added artificially to the signals received at each of the other antennas. The approximate delay required can be calculated from the geometry of the problem. The tapes are played back in synchronous using the recorded signals from the atomic clocks as time references, as shown in Figure 12. If the position of the antennas is not known to sufficient accuracy or atmospheric effects are significant, fine adjustments to the delays must be made until interference fringes are detected. If the signal from antenna <i>A</i> is taken as the reference, inaccuracies in the delay will lead to errors <span style="font-family:Symbol;">e</span> <i><sub>B</sub></i> and <span style="font-family:Symbol;">e</span> <i><sub>C</sub></i> in the phases of the signals from tapes <i>B</i> and <i>C</i> respectively. As a result of these errors the phase of the complex visibility cannot be measured with a very long baseline interferometer.</span></p> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><br /></td></tr> <tr><td width="100%"><i><b>Figure 12 - Visibility measurements in very long baseline interferometry</b> </i></td></tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;">The phase of the complex visibility depends on the symmetry of the source brightness distribution. Any brightness distribution B(<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ,<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> ) can be written as the sum of a symmetric component and an anti-symmetric component <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image51.gif" width="137" height="37" />. The symmetric component B<sub>S</sub> of the brightness distribution only contributes to the real part of the complex visibility, while B<sub>A</sub> only contributes to the imaginary part. To demonstrate the dependence of the phase of the complex visibility on the symmetry of the source I separated the 9×9 array of complex visibility used to produce Figure 9 into real and imaginary parts. Figure 13 was produced using only the real component of the visibility, with the imaginary component set to zero. As the phase of the complex visibility is zero throughout the u-v plane the image is symmetric about its centre. In Figure 14 the real component was removed instead, giving an anti-symmetric image. As the phase of each complex visibility measurement cannot be determined with a very long baseline the symmetry of the corresponding contribution to the source brightness distributions is not known.</span></p> <table width="480"> <tbody><tr><td valign="BOTTOM" width="35%" align="CENTER"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image52.gif" width="159" height="160" /></td> <td valign="BOTTOM" width="35%" align="CENTER"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image53.gif" width="160" height="160" hspace="8" /></td><td valign="BOTTOM" width="30%" align="CENTER"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/image54.gif" width="125" height="116" /></td></tr> <tr><td valign="TOP" width="35%" align="CENTER"><b><i>Figures 13 - Symmetric components</i></b></td><td valign="TOP" width="35%" align="CENTER"><b><i>Figure 14 - Anti-symmetric components</i></b></td><td width="30%"> </td> </tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;">R. C. Jennison developed a novel technique for obtaining information about visibility phases when delay errors are present, using an observable called the closure phase. Although his initial laboratory measurements of closure phase had been done at optical wavelengths, he foresaw greater potential for his technique in radio interferometry. In 1958<sup><a href="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/refer.html#11">11</a></sup> he demonstrated its effectiveness with a radio interferometer, but it only became widely used for long baseline radio interferometry in 1974<sup><a href="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/refer.html#12">12</a></sup>. A minimum of three antennas are required. I will initially look at the simplest case, with three antennas in a line separated by the distances <i>a</i><sub>1</sub> and <i>a</i><sub>2</sub> shown in Figure 11. The radio signals received are recorded onto magnetic tapes and sent to a laboratory as described above. The effective baselines for a source at an angle <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> will be <i>x</i><sub>1</sub>=<i>a</i><sub>1</sub>cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ], <i>x</i><sub>2</sub>=<i>a</i><sub>2</sub>cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ] and <i>x</i><sub>3</sub>=(<i>a</i><sub>1</sub>+<i>a</i><sub>2</sub>)cos[<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> ]. The phases of the complex visibility of the radio source corresponding to baselines <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub> and <i>x</i><sub>3</sub> I will call <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>1</sub>, <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>2</sub> and <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>3</sub> respectively. The phase of interference fringes on each baseline will contain errors resulting from <span style="font-family:Symbol;">e</span> <i><sub>B</sub></i> and <span style="font-family:Symbol;">e</span> <i><sub>C</sub></i> in the signal phases. The measured phases for baselines <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub> and <i>x</i><sub>3</sub>, denoted <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub>1</sub>, <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub>2</sub>, and <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub>3</sub>, will be:</span></p><dir> <dir> <p><span style="font-family:Arial;font-size:100%;"><i><span style="font-size:85%;"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> </span></i><span style="font-size:85%;"><sub>1</sub>=<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>1</sub>+<span style="font-family:Symbol;">e</span> <i><sub>B</sub></i>-<span style="font-family:Symbol;">e</span> <i><sub>C</sub></i></span></span></p> <p><span style="font-family:Arial;font-size:100%;"><span style="font-size:85%;"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub>2</sub>=<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>2</sub>-<span style="font-family:Symbol;">e</span> <i><sub>B </sub></i></span></span></p> <p><span style="font-family:Arial;font-size:100%;"><span style="font-size:85%;"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub>3</sub>=<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>3</sub>-<span style="font-family:Symbol;">e</span> <i><sub>C</sub></i></span></span></p></dir> </dir> <p><span style="font-family:Arial;font-size:100%;">Jennison defined the quantity <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub><i>C</i></sub> for the three antennas as:</span></p><dir> <dir> <span style="font-family:Arial;font-size:100%;"><span style="font-size:85%;"><p><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub><i>C</i></sub>=<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub>1</sub>+<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub>2</sub>-<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub>3</sub> =<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>1</sub>+<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>2</sub>-<img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>3</sub></p></span></span></dir></dir> <p><span style="font-family:Arial;font-size:100%;"><img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <sub><i>C</i></sub> is often called the <i>closure phase</i><sup><a href="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/refer.html#12">12</a></sup>.</span></p> <p><span style="font-family:Arial;font-size:100%;">The contributions to <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <i><sub>C</sub></i> from errors <span style="font-family:Symbol;">e</span> <i><sub>B</sub></i> and <span style="font-family:Symbol;">e</span> <i><sub>C</sub></i> in the signal phases cancel out allowing accurate measurement. Using measurements of <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/psi.gif" /> <i><sub>C</sub></i>, <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>3</sub> can be written in terms of <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>1</sub> and <img src="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/phi.gif" /> <sub>2</sub>, the unknown phases. If many closure phase measurements are made the complex visibility can be written as a function of several unknown phases. In order to produce an image of the sky the unknown phases must be estimated so that the complex visibility function can be calculated. This is usually done using iterative algorythms<sup><a href="http://web.archive.org/web/20040331004625/http://www.geocities.com/CapeCanaveral/2309/refer.html#13">13,14,15</a></sup> which attempt to minimise unphysical properties of the image, such as areas of negative brightness (black areas above and below the source in figures 8 and 9) and large fluctuations in the background radio noise well away from the known location of the source. In radio astronomy visibilities are typically measured on more than three baselines simultaneously, providing more information about the source than Jennison's closure phase technique. The mapping algorithms are designed to retreive the maximum amount of information from the measurements performed without adding artificial detail. <a href="http://web.archive.org/web/20040331004625/http://www.vsop.isas.ac.jp/general/Images.html">Images</a> have been produced with baselines of many thousands of kilometers and resolution higher than one milliarcsecond.</span></p>Funkhttp://www.blogger.com/profile/13086037268464989921noreply@blogger.com1tag:blogger.com,1999:blog-4267496207290097069.post-43536612437906811472009-11-01T09:18:00.002-08:002009-11-01T09:19:36.830-08:00Modern Optical Interferometry<span style="font-family:Arial;font-size:130%;"> </span><span style="font-family:Arial;font-size:100%;">The first two-telescope optical interferometer<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#16">16</a></sup> was constructed in 1974 by A. Labeyrie using light beams from small telescopes at the Nice observatory. This was followed by the construction of the Mark 1 prototype interferometer, which used modern optical detectors and mechanical control systems to measure and track phase variations produced by fluctuations in the Earth's atmosphere<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#17">17</a></sup>. These fluctuations are produced by turbulence and local density variations in the atmosphere. A light ray from a distant source refracted by an angle <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> and displaced laterally by a distance <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> <i>a</i> is shown in Figure 15. Experimental measurements<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#18">18</a></sup> have shown that both <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> and the optical path length fluctuate significantly in as little as five milliseconds. Fluctuations in <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> <i>a</i> are generally very much smaller than the interferometer baseline, and do not significantly affect its performance. For interferometric purposes the atmosphere can be successfully modelled as a phase screen, such as that shown in Figure 16. Variations in <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> <i><sub>A</sub></i> and <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/delta.gif" alt="delta" /> <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/theta.gif" /> <i><sub>B</sub></i> can be eliminated by tracking the light source with computer controlled tilt mirrors. Changes in the optical path lengths <i>l<sub>A</sub></i> and <i>l<sub>B</sub></i> can be reduced with computer controlled delay lines, but as with very long baseline radio interferometry, fluctuations in the phase of the incident light cannot be completely eliminated. There are two principle approaches to imaging under these conditions: </span><ol><span style="font-family:Arial;font-size:100%;"><li><i>Phase-referenced imaging</i> -- utilise measurements of a bright "reference" star very close to your target and accurately measure the phase difference between interference fringes formed with light from this reference star with those from the target </li><li><i>Closure phase imaging</i> -- use the fringe signal from the target or from a faint reference star nearby to get the optical paths approximately correct for three or more telescopes, and then measure Jennison's closure phase for the science target </li></span></ol> <p><span style="font-family:Arial;font-size:100%;">Method <b>1</b> has the benefit that it can be used on fainter science targets, but unfortunately the need for a very bright reference star extremely close to the target of interest is very limiting, so it is expected that only a small number of astronomical sources could be observed in this way. Method <b>2</b> is applicable to a wider range of sources as a fainter reference star can be used (even the science target itself), and the reference star can be a greater distance from the science target. The principle disadvantage is that much longer observations are required using this approach in order to get results of the same quality as method <b>1</b>. </span></p><p><span style="font-family:Arial;font-size:100%;">The limited applicability of method <b>1</b> due to the need for bright nearby reference stars meant that the first imaging interferometry experiments used the approach described in method <b>2</b>. In 1984 a team from Cambridge<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#19">19</a></sup> succeeded in making the first astronomical measurements of Jennison's closure phase at optical wavelengths by placing aperture masks over an 88 inch telescope on Mauna Kea and recording the fringe patterns electronically. This soon led to the production of crude images<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#20">20</a></sup> using the aperture synthesis techniques described on the <a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/page3.html">previous page</a>.</span></p> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/image55.gif" /></td></tr> <tr><td width="100%"><i><b>Figure 15 - Refraction of light in the atmosphere</b> </i></td></tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;"> </span></p> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/image56.gif" width="684" height="440" /></td></tr> <tr><td width="100%"><i><b>Figure 16 - For interferometric purposes the atmosphere can be satisfactorily modelled as an oscillating phase screen</b> </i></td></tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;">In 1988 construction of the <a href="http://web.archive.org/web/20050310101008/http://www.mrao.cam.ac.uk/telescopes/coast/">Cambridge Optical Aperture Synthesis Telescope (COAST)</a><sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#21">21,22</a></sup> began. The layout of this instrument is shown in Figure 17. Light beams from four 400mm tip-tilt corrected telescope mirrors are combined and fringe visibilities are measured in the visible and infrared. A fifth telescope has now been added, and the light from this telescope can be switched in place of the light from telescope 1 in Figure 17. The light from each telescope enters the main building at point H. In the figure, the beams from telescopes 1-4 are reflected from computer controlled mobile mirrors labelled A-D respectively. These mirrors compensate for the differences in optical path length from the source. The beams are then either reflected by dichroic<sup>*</sup> reflectors placed at point E into the infrared beam combining apparatus or by dichroics placed at point F into the visual (red light) beam combining apparatus. The dichroics allow blue light to pass into the autoguiding system, which controls tilt mirrors within each telescope.</span></p> <p><span style="font-family:Arial;font-size:100%;"><sup>*</sup>these dichroics reflect red and infrared light but are transparent to blue light</span></p> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody><tr><td width="100%"><img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/image57.gif" width="522" height="826" /></td></tr> <tr><td width="100%"><i><b>Figure 17 - Layout of the <a href="http://web.archive.org/web/20050310101008/http://www.mrao.cam.ac.uk/telescopes/coast/">COAST</a> array</b> </i></td></tr> </tbody></table> <p><span style="font-family:Arial;font-size:100%;">Mirrors A-D scan backwards and forwards about their ideal position so that interference between the beams produces oscillating intensities at the detectors. The position of these mirrors is tracked to an accuracy of a few nanometres using laser interferometers. This allows phase changes in the complex visibility to be measured, giving calculated closure phases to about 2% accuracy. In the beam combiners the beam from each telescope is split into four using an arrangement of beam splitters<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#22">22</a></sup> which enables accurate measurement of fringe visibility on all six baselines.</span></p> <p><span style="font-family:Arial;font-size:100%;">The first images<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#23">23</a></sup> produced from visibility and closure phase measurements made with <a href="http://web.archive.org/web/20050310101008/http://www.mrao.cam.ac.uk/telescopes/coast/">COAST</a> are shown in Figures 18 and 19. These have a resolution of approximately twenty milliarcseconds, and a dynamic range of about fifty. The sensitivity of the COAST array presently allows observation of objects as faint as 7th magnitude. The sensitivity and near-infrared capability of the COAST array makes it particularly suited to the observation of surface structure on red giant, supergiant and long period variable stars. Local atmospheric effects limit the transverse coherence of light to Fried lengths<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#24">24</a></sup> of fifty to two hundred millimetres. This limits the usable aperture of the telescopes to between one hundred and four hundred millimetres, depending on the wavelength used and the local weather conditions. Changes in the optical path length through the atmosphere due to atmospheric turbulence are significant on time scales of five to ten milliseconds, limiting integration times (analogous to camera shutter speed) to millisecond time scales.</span></p> <p><span style="font-family:Arial;font-size:100%;">A fringe tracking capability is currently being installed at COAST allowing optical path differences to be actively corrected during observations. The geometric arrangement of the COAST array allows the optical path differences on the longest baselines at COAST to be calculated from measurements made on short baselines (<i>baseline bootstrapping</i>). This is essential for observations of highly resolved sources.</span></p> <table width="522"> <tbody><tr> <td width="50%"> <span style="font-family:Arial Rounded MT Bold;">� Cavendish Astrophysics Group</span> </td><td width="50%" align="RIGHT"> <span style="font-family:Arial Rounded MT Bold;font-size:78%;">IMAGES KINDLY PROVIDED BY THE CAVENDISH ASTROPHYSICS GROUP, CAMBRIDGE </span></td> </tr> <tr> <td width="50%" align="CENTER"> <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/capella1.jpg" width="256" height="256" /> </td> <td width="50%" align="CENTER"> <img src="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/capella2.jpg" width="256" height="256" hspace="8" /></td> </tr> <tr> <td width="50%" align="CENTER"><i><b>Figure 18, September 13 1995</b></i></td> <td width="50%" align="CENTER"><i><b>Figure 19, September 28 1995</b></i></td></tr> </tbody></table> <table width="522"> <tbody><tr><td width="512"><i><b><span style="font-family:Arial;font-size:100%;">Capella in 1995</span></b></i> </td></tr> <tr><td width="512"> <i><b>Images of Capella from measurements taken on September 13 and September 28 1995. The images clearly show the rotation of this binary star over the fifteen-day period. J. A. Anderson determined the orbital motions of the Capella binary system in 1920 using a Michelson interferometer</b></i><sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#5">5</a></sup><i><b>, the first interferometric measurement of an object outside the solar system. In 1995 it became the first object to be imaged with a separate element interferometer.</b></i> </td></tr></tbody></table> <span style="font-family:Arial;font-size:100%;"> </span><p><span style="font-family:Arial;font-size:100%;"> </span></p> <p><span style="font-family:Arial;font-size:100%;"><a href="http://web.archive.org/web/20050310101008/http://ftp.nofs.navy.mil/projects/npoi/">The Navy Prototype Optical Interferometer (NPOI)</a> astrometric and imaging arrays on Anderson Mesa, Arizona have recently produced a number of images<sup><a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/refer.html#25">25</a></sup>. The NPOI array currently uses siderostats with small apertures, but these will soon be upgraded using a beam compression system. Both of the arrays are capable of very high precision visibility amplitude and closure phase measurements both at thirty-two wavelength bands in the visible spectrum and also at infra-red wavelengths. Observations are currently limited to sources brighter than 4th magnitude, largely due to the small apertures and because of the narrow spectral channels used. The high spectral resolution, long baselines and high measurement precision of the NPOI arrays make them particularly suitable for surface structure and stellar diameter measurements of bright stars and for accurate orbital determination of binary systems. The geometry of the NPOI array is ideal for baseline bootstrapping, and should allow very high resolution imaging of complex resolved sources.</span></p><p> <span style="font-family:Arial;font-size:100%;">Most of the imaging interferometer arrays under construction are aiming for 1-10 mas resolution with much higher sensitivity than COAST and NPOI. Two examples are the <a href="http://web.archive.org/web/20050310101008/http://planetquest.jpl.nasa.gov/Keck/keck_index.html">Keck</a> and <a href="http://web.archive.org/web/20050310101008/http://www.eso.org/projects/vlti/">VLTI</a> interferometers. Both of these are based around groups of very large conventional optical telescopes. Both arrays have relatively poor coverage of the u-v plane and limited capabilities for baseline bootstrapping. Initially they will only be suited to observations of compact sources, and will not be capable of generating accurate maps of complex sources. The introduction of a phase referencing system at the VLTI should eventually allow imaging of a limited number of complex sources - those which happen to reside close to a star bright enough for the phase-referenced imaging system. The introduction of a closure phase measurement instrument (AMBER) and the development of tracking of the fringe envelope (group delay tracking) will later allow the VLTI to undertake observations using both fainter reference stars and reference stars which are further away in the sky from the science target using the closure phase imaging approach. This will allow observations of a larger range of astronomical targets than can be observed using phase-referenced imaging. The main telescopes on both the VLTI and Keck sites will also be used for conventional astronomy, and it is expected that only a small fraction of the observing time will be available for interferometric imaging.</span></p><p> <span style="font-family:Arial;font-size:100%;">One of the most promising ground based very high resolution imaging arrays currently under construction is the <a href="http://web.archive.org/web/20050310101008/http://www.chara.gsu.edu/CHARA/">CHARA array</a>. This instrument consists of six fixed telescopes with separations of up to 350m, allowing imaging of simple sources with 0.2 mas resolution. An alternative UK designed array has also been proposed which would have ten movable telescopes with baselines up to 400m. The greater number of telescopes would allow more complex sources to be imaged. Two of the primary goals of this instrument would be near infra-red imaging of matter around extra-galactic black holes (AGN broad line regions) and imaging of dust disks around newly formed stars. This array is currently being studied by the <a href="http://web.archive.org/web/20050310101008/http://www.mro.nmt.edu/">Magdelena Ridge Observatory Consortium</a> in collaboration with <a href="http://web.archive.org/web/20050310101008/http://www.mrao.cam.ac.uk/telescopes/coast/">Cambridge University</a>. A list of other optical and infra-red interferometry projects can be found at the <a href="http://web.archive.org/web/20050310101008/http://olbin.jpl.nasa.gov/links/">OLBIN links webpage</a>. </span></p><p><span style="font-family:Arial;font-size:100%;">NASA's <a href="http://web.archive.org/web/20050310101008/http://planetquest.jpl.nasa.gov/SIM/sim_index.html">Space Interferometry Mission</a> may have some imaging capability at optical wavelengths. This mission will allow direct measurement of visibility phase for optical sources. </span></p><p><span style="font-family:Arial;font-size:100%;"> </span> <span style="font-family:Arial;font-size:100%;"> <p> </p> <a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/page5.html">Next Page</a> / <a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/page3.html">Previous Page</a> / <a href="http://web.archive.org/web/20050310101008/http://www.geocities.com/CapeCanaveral/2309/page1.html">Abstract and Contents Page</a>. <p> </p> <p>Return to <a href="http://web.archive.org/web/20050310101008/http://bigfoot.com/%7Ebob.tubbs">Bob's Home Page</a>.</p> </span> </p>Funkhttp://www.blogger.com/profile/13086037268464989921noreply@blogger.com0tag:blogger.com,1999:blog-4267496207290097069.post-56361992792896730052009-11-01T09:18:00.001-08:002009-11-01T09:18:42.507-08:00Conclusions<span style="font-family:Arial;font-size:130%;"> </span>Optical interferometry was suggested as an astronomical measurement technique over a century ago, but the technology required to exploit its full potential has only recently been developed. Present experimental arrays are already producing images with higher resolution than conventional terrestrial and space-based telescopes. Unfortunately atmospheric effects limit usable aperture sizes and integration times, making imaging impossible for objects fainter than about 10th magnitude<sup><a href="http://web.archive.org/web/20021201233616/http://www.geocities.com/CapeCanaveral/2309/refer.html#26">26</a></sup> with present arrays. The greatest scientific potential for optical interferometry seems to be the study of local stars and star systems. Long baseline optical interferometry represents the only feasible technique for imaging stellar surfaces, and planned observations of gas giants and long period variable stars<sup><a href="http://web.archive.org/web/20021201233616/http://www.geocities.com/CapeCanaveral/2309/refer.html#26">26</a></sup> are likely to improve our understanding of stellar astrophysics. The development of new arrays with higher sensitivity will open up new areas of research, such as the study of the environment around large black holes at the centre of galaxies.<span style="font-family:Arial;font-size:100%;"> <p>The optical arrays currently capable of aperture synthesis imaging were designed as prototype instruments and are incapable of mapping complex sources. Future arrays must incorporate many more telescopes on a site with good atmospheric conditions if the full potential of astronomical optical interferometry is to be realised. Experience from present optical arrays suggests that in the near future interferometers sensitive to red or infrared wavelengths, operating with the complex visibility and closure phase measurement approach described in this report, are likely to have the greatest success. Maximum sensitivity is achieved with aperture diameters a few times greater than the atmospheric transverse coherence length<sup><a href="http://web.archive.org/web/20021201233616/http://www.geocities.com/CapeCanaveral/2309/refer.html#24">24</a></sup>, wide spectral channels and the minimum number of optical elements. Higher measurement precision can be obtained if slightly smaller apertures and narrow spectral channels are used. In order to measure the complex visibility on every baseline in an array of many telescopes the light beams from each telescope must be split many times, reducing the sensitivity of the instrument. For faint astronomical objects it may be necessary to treat an array as a set of smaller arrays with fewer beam-splitting operations on the light from each telescope. In future arrays, separate beam combining apparatus could allow simultaneous operation of several groups of three or four telescopes.</p> <p>Optical interferometry and aperture synthesis have great potential as astronomical techniques. It is unlikely that terrestrial aperture synthesis arrays will ever have the sensitivity of large conventional telescopes, but long baseline interferometry represents a cost- effective technique for high resolution imaging of bright astronomical objects. The most ambitious proposals suggest arrays of terrestrial telescopes covering many kilometres, and arrays of orbital or lunar telescopes hundreds of kilometres in extent<sup><a href="http://web.archive.org/web/20021201233616/http://www.geocities.com/CapeCanaveral/2309/refer.html#27">27</a></sup>. These would permit a detailed search for planetary systems around other stars. These projects are still many decades in the future, but the experience gained with current arrays will be essential to their success.</p> <p> </p> <a href="http://web.archive.org/web/20021201233616/http://olbin.jpl.nasa.gov/">Latest Long Baseline Interferometry News</a>. <table width="960"> </table> <span style="font-family:Arial;font-size:100%;"> </span></span><p><span style="font-family:Arial;font-size:100%;"><span style="font-family:Arial;font-size:100%;"> </span></span></p> <span style="font-family:Arial;font-size:100%;"><span style="font-family:Arial;font-size:100%;"><a href="http://web.archive.org/web/20021201233616/http://www.geocities.com/CapeCanaveral/2309/page4.html">Previous Page</a> / <a href="http://web.archive.org/web/20021201233616/http://www.geocities.com/CapeCanaveral/2309/page1.html">Abstract and Contents Page</a>. </span></span><p><span style="font-family:Arial;font-size:100%;"><span style="font-family:Arial;font-size:100%;"> </span></span></p> <p><span style="font-family:Arial;font-size:100%;"><span style="font-family:Arial;font-size:100%;">Return to <a href="http://web.archive.org/web/20021201233616/http://www.cus.cam.ac.uk/%7Ernt20/">Bob's Home Page</a>.</span></span></p>Funkhttp://www.blogger.com/profile/13086037268464989921noreply@blogger.com0tag:blogger.com,1999:blog-4267496207290097069.post-52024154458750595372009-11-01T09:17:00.000-08:002009-11-01T09:18:17.003-08:00References<span style="font-family:Arial;font-size:130%;"> <b></b> </span> <span style="font-family:Arial;font-size:100%;"> </span><ol start="1"><span style="font-family:Arial;font-size:100%;"><li><a name="1">A. H. Fizeau C. R. Acad. Sci. (Paris) 66:934 (1867), french</a></li> <li><a name="2">A. A. Michelson, <i>On the application of interference methods to astronomical measurements</i>, Philosophical Magazine vol 30 pp 1 (July 1890)</a></li> <li><a name="3">A. A. Michelson, <i>Measurement of Jupiters satellites by interference</i>, Nature vol 45 pp 160 (December 1891)</a></li> <li><a name="4">A. A. Michelson, <i>Visibility of interference-fringes in the focus of a telescope</i>, Philisophical Magazine vol 31 pp 256 (March 1891)</a></li> <li><a name="5">J. A. Anderson, <i>On the application of Michelson's interferometer method to the measurement of close double stars</i>, Astrophysical Journal vol 51 pp 263 (June 1920)</a></li> <li><a name="6">A. A. Michelson, F. G. Pease<i>, Measurement of the diameter of Alpha Orionis with the interferometer</i>, Astrophysical Journal vol 53 pp 249 (1921)</a></li> <li><a name="7">F. G. Pease, <i>The 50ft stellar interferometer at Mount Wilson</i>, Scientific American vol 143 pp 290 (1930)</a></li> <li><a name="8">R. Hanbury Brown, <i>Measurement of stellar diameters</i>, Annual Review of Astronomy and Astrophysics vol 6 pp 13 (1968)</a></li> <li><a name="9">K. G. Jansky, Nature 132 pp 66 (1933)</a></li> <li><a name="10">M. Ryle, D. Vonberg, <i>Solar radiation on 175Mc/s</i>, Nature 158 pp 339 (September 1946)</a></li> <li><a name="11">R. C. Jennison, <i>A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distributions of small angular extent</i>, Monthly Notices of the Royal Astronomical Society vol 118 pp 276 (1958)</a></li> <li><a name="12">A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I. I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, <i>The structure of radio sources 3C 273B and 3C 84 deduced from the "closure" phases and visibility amplitudes observed with three-element interferometers</i>, Astrophysical Journal 193 pp 294 (October 1974)</a></li> <li><a name="13">T. J. Pearson and A. C. S. Readhead, <i>Image formation by self-calibration in radio astronomy</i>, Annual Review of Astronomy and Astrophysics vol 22 pp 97 (1984)</a></li> <li><a name="14">A. R. Thompson, J. M. Moran, G. W. Swenson JR, <i>Interferometry and synthesis in radio astronomy</i>, John Wiley and sons (1986)</a></li> <li><a name="15">T. J. Cornwell, <i>The applications of closure phase to astronomical imaging</i>, Science vol 245 pp 263 (July 1989)</a></li> <li><a name="16">A. Labyrie, <i>Interference fringes obtained on Vega with two optical telescopes</i>, Astrophysical Journal 196 pp L71 (March 1975)</a></li> <li><a name="17">M. Shao, D. H. Staelin, <i>First fringe measurements with a phase-tracking stellar interferometer</i>, Applied Optics 19 pp 1519 (1980)</a></li> <li><a name="18">D. F. Buscher, <i>A thousand and one nights of seeing on Mount Wilson</i>, SPIE Proceedings vol 2200 pp 260 (March 1994)</a></li> <li><a name="19">J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, <i>Closure phase in high-resolution optical imaging</i>, Nature vol 320 pp 595 (April 1986)</a></li> <li><a name="20">C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, <i>The first images from optical aperture synthesis</i>, Nature vol 328 pp 694 (August 1987)</a></li> <li><a name="21">C. A. Haniff, J. E. Baldwin, A. G. Basden, N. A. Bharmal, R. C. Boysen, D. F. Buscher, A. V. George, J. Keen, C. D. Mackay, B. O'Donovan, D. Pearson, H. Thorsteinsson, N. Thureau, R. N. Tubbs, P. J. Warner, D. M. Wilson, J. S. Young, <i>Progress at COAST 2000-2002</i>, SPIE Proceedings vol 4838 in press (August 2002)</a></li> <li><a name="22">J. E. Baldwin, R. C. Boysen, G. C. Cox, C. A. Haniff, J. Rogers, P. J. Warner, D. M. A. Wilson, C. D. Mackay, <i>Design and performance of COAST</i>, SPIE Proceedings vol 2200 pp 118 (March 1994)</a></li> <li><a name="23">J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, <i>The first images from an optical aperture synthesis array: mapping of Capella with COAST at two epochs</i>, Astronomy and Astrophysics vol 306 pp L13 (1996)</a></li> <li><a name="24">D. L. Fried, Journal of the Optical Society of America 55 pp 1427 (1965)</a></li> <li><a name="25">J. A. Benson, D. J. Hutter, N. M. Elias II, P. F. Bowers, K. J. Johnston, A. R. Hajian, J. T. Armstrong, D. Mozurkewich, T. A. Pauls, L. J. Rickard, C. A. Hummel, N. M. White, D. Black, C. S. Denison, <i>Multichannel optical aperture synthesis imaging of zeta(1) Ursae Majoris with the Navy Prototype Optical Interferometer</i>, Astronomical Journal vol 114 pp 1221 (1997)</a></li> <li><a name="26">MRAO Grant Review 1997</a></li> <li><a name="27">A. Labeyrie, <i>Current steps towards kilometric arrays of many telescopes: prospects for snapshot images with </i>10<sup>-4</sup><i> arcsec resolution</i>, SPIE Proceedings vol 2871 (June 1996)</a></li></span></ol> <p><span style="font-family:Arial;font-size:100%;"><a name="27">Introductory Textbooks:</a></span></p> <ol start="28"><span style="font-family:Arial;font-size:100%;"><li><a name="28">E. Hecht, <i>Optics</i>, Addison-Wesley (1987)</a></li> <li><a name="29">S. G. Lipson, H. Lipson, D. S. Tannhauser, <i>Optical Physics</i>, Cambridge University Press (1995)</a></li></span></ol> <p><span style="font-family:Arial;font-size:100%;"><a name="29">Other Reviews:</a></span></p> <ol start="30"><span style="font-family:Arial;font-size:100%;"><li><a name="30">J. D. Monnier, </a><i><a href="http://web.archive.org/web/20050324122733/http://www.iop.org/EJ/abstract/0034-4885/66/5/203">Optical interferometry in astronomy</a></i>, Reports on Progress in Physics, 66, 789-857 (2003 IoP) -- if you aren't an IOP member, you can download a lower quality version <a href="http://web.archive.org/web/20050324122733/http://www.astro.lsa.umich.edu/%7Emonnier/Publications/ROP2003_final.pdf">here</a>.</li> <li><a name="31">C. A. Haniff and D. F. Buscher, </a><i><a href="http://web.archive.org/web/20050324122733/http://physicsweb.org/article/world/16/5/9">Optical interferometry</a></i>, Physics World, 16 No. 5, 39-43, (May 2003).</li> <li><a name="32">P. R. Lawson, <i>Optical interferometry comes of age</i>, Sky and Telescope 105 (5), 30-39, (2003).</a></li> <li><a name="33">J. E. Baldwin and C. A. Haniff, <i>The application of interferometry to optical astronomical imaging</i>, Phil. Trans. A, 360, 969-986, (2002). Available as a </a><a href="http://web.archive.org/web/20050324122733/http://www.mrao.cam.ac.uk/telescopes/coast/papers/tyoung.ps">postscript</a> file for printing or to view using the <a href="http://web.archive.org/web/20050324122733/http://www.cs.wisc.edu/%7Eghost/">Ghostview</a> program.</li> </span></ol> <span style="font-family:Arial;font-size:100%;">Many of these papers can be found in <a href="http://web.archive.org/web/20050324122733/http://www.spie.org/web/abstracts/oepress/MS139.html">Selected Papers on Long Baseline Stellar Interferometry</a> </span>Funkhttp://www.blogger.com/profile/13086037268464989921noreply@blogger.com0tag:blogger.com,1999:blog-4267496207290097069.post-40055937906746401052009-11-01T09:07:00.000-08:002009-11-01T09:16:15.638-08:00Bob Tubbs's Interferometry PageBob Tubbs's Interferometry Page<br /><br />Some of my project work is listed below. If you want to read one of the projects simply click on the project title.<br />Astronomical Optical Interferometry.<br /><br />Literature Review, Bob Tubbs, April 1997.<br /><br />An historical account of the development of Optical Interferometry and discussion of the principles involved. Written by an undergraduate for an undergraduate audience.<br /><br />Areas covered explicitly<br /><br /> * Michelson's early work<br /> * The development of radio interferometry<br /> * Simple synthetic aperture imaging<br /> * Closure phase<br /> * Modern optical interferometers<br /><br />Areas not covered in depth<br /><br /> * Intensity interferometers<br /> * The use of closure relations in aperture synthesis imaging<br /><br />Accounts for approximately three weeks' work as part of the Natural Sciences Tripos undergraduate degree at <a href="http://www.cam.ac.uk/">Cambridge University</a>.<br /><br /><a href="http://www.strw.leidenuniv.nl/%7Etubbs/thesis/">My Ph.D. thesis</a> on <a href="http://www.strw.leidenuniv.nl/%7Etubbs/lucky/">Lucky Exposures / Video Astronomy.</a><br /><a href="http://web.archive.org/web/20050304042041/http://www.geocities.com/CapeCanaveral/2309/phase_track.html">Tracking and Characterising Atmospheric Phase Fluctuations at COAST.</a><br /><br />Part III Project, Bob Tubbs, May 1998.<br /><br />Techniques for studying the properties of interference fringes formed in Astronomical Optical Interferometers.<br /><br />Areas covered explicitly<br /><br /> * Tracking phase changes in interference fringes<br /> * Measuring the position of the fringe envelope<br /> * The effect of optical dispersion on closure phase measurements<br /> * Suggested new techniques for closure phase measurement<br /><br />Accounts for approximately six weeks' work as part of the Natural Sciences Tripos undergraduate degree at <a href="http://www.cam.ac.uk/">Cambridge University</a>.<br />Related links<br />This page hosted by Get your own Free Home Page<br /><br />Counter<br /><a href="http://web.archive.org/web/20050304042041/http://www.geocities.com/CapeCanaveral/Launchpad/5952/decon_notes.html">Deconvolution of Time-Domain Waveforms using the Nahman-Guillaume one-parameter filter.</a><br />A random collection of papers:<br /><br /> * Nature 320, 595, 1986, Closure phase in high-resolution optical imaging by J. E. Baldwin, C. A. Haniff, C. D. Mackay and P. J. Warner.<br /> * Astronomy and Astrophysics 306, L13, 1996, The first images from an optical aperture synthesis array: mapping of Capella with COAST at two epochs by J.E.Baldwin, M.G.Beckett, R.C.Boysen, D.Burns, D.F.Buscher, G.C.Cox, C.A.Haniff, C.D.Mackay, N.S.Nightingale, J.Rogers, P.A.G.Scheuer, T.R.Scott, P.G.Tuthill, P.J.Warner, D.M.A.Wilson and R.W.Wilson.<br /> * Nature 177, 27, 1956 Correlation between photons in two coherent beams of light by R.Hanbury Brown and R.Q.Twiss<br /> * Nature 178, 1046, 1956 A test of a new type of stellar interferometer on Sirius by R.Hanbury Brown and R.Q.Twiss<br /> * Nature 328, 694, 1987 The first images from optical aperture synthesis by C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin and P. J. Warner<br /> * Nature 429, 47, 2004 The central dusty torus in the active nucleus of NGC 1068W. Jaffe, K. Meisenheimer, H. J. A. Ro ttgering, Ch. Leinert, A. Richichi, O. Chesneau, D. Fraix-Burnet, A. Glazenborg-Kluttig, G.-L. Granato, U. Graser, B. Heijligers, R. Ko hler, F. Malbet, G. K. Miley, F. Paresce, J.-W. Pel, G. Perrin, F. Przygodda, M. Schoeller, H. Sol, L. B. F. M. Waters, G. Weigelt, J. Woillez and P. T. de Zeeuw<br /> * Astrophysical Journal 196, L71, 1975, Interference fringes obtained on Vega with two optical telescopes by A.Labeyrie<br /> * Proceedings of the Royal Society of London A 164, 476, 1938 Download "The spectrum of turbulence" by G.I.Taylor which defines Taylor's hypothesis (the basis for Taylor screens) -- bibliography code Proc. R. Soc. Lond. A, 164:476-490, 1938.<br /> * IAU Colloquium 50 -- Download "Optimum Exposure Time and Filter Bandwidth in Speckle Interferometry" by J. G. Walker<br /> * Monthly Notices of the Royal Astronomical Society Vol. XVII, Page 12 (1856) Download "Magnitudes of Thirty-six of the Minor Planets for the First Day of each Month of the Year 1857" by Norman PogsonFunkhttp://www.blogger.com/profile/13086037268464989921noreply@blogger.com0