ECEN 4606, UNDERGRADUATE OPTICS LAB

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1 ECEN 4606, UNDERGRADUATE OPTICS LAB Lab 5: Interferometry and Coherence SUMMARY: In this lab you will use interference of a temporally coherent (very narrow temporal frequency bandwidth) laser beam to reveal phase variations in the beam on the scale of a single wavelength. By passing one beam through a transparent object and detecting the phase accumulated, you will demonstrate thickness measurement with ~100 nm precision. By interfering the beam with a delayed copy of itself, you will determine the temporal frequency bandwidth of the laser. PRELAB: HOMEWORK PROBLEM 1: Pedrotti HOMEWORK PROBLEM 2: Pedrotti HOMEWORK PROBLEM 3: The phase of a wave at radio frequencies can be determined by direct measurement, e.g. on an oscilloscope. At optical frequencies, this is impossible. Instead, we detect the phase of a wave through interference with a known reference. Via constructive and destructive interference, this produces an intensity pattern which reveals the desired phase. To understand these intensity patterns, it is quite helpful to have a computer code which will plot an interference pattern for a given phase profile. Consider two plane waves of wavelength λ 0 = 633 nm incident on one of our CMOS camera chips. Let one be normally incident call this the reference. A) Write an expression proportional to the intensity (the absolute magnitude of the amplitude squared) on the CMOS camera when a second object beam of the same intensity is introduced with some angle of propagation θ relative to the camera normal. What is the maximum angle this system can detect reliably? Change the intensity of the beam under test relative to the reference and write an expression for the minima and maxima of the intensity pattern. If you wish to see this interference pattern by eye, what is the largest angle θ that there can be between the two beams? B) With the test beam at an angle less than the maximum detectable angle, introduce a physical deformation s(x,y) [meters] on the object beam wavefront of the form 2 2 ( x + y ) smax + ( ) 1 = x y R s x, y 2 R, x + y > R where R = 1 mm and s max = 5 λ 0. This could be created by a thin oil droplet on a glass slide, for example. Create a density plot vs. x and y of the resulting interference pattern. Modify your test beam angle and observe how the pattern changes can you use the pattern to determine s max? Change the sign of s max can you determine sign from the interference pattern? Hint: Below is a fragment of matlab code that will generate and plot s. Version 1.0, 9/17/09 Robert McLeod 1

2 % % Create and plot phase perturbation for lab 4 % lambda0 = 0.633; % Vacuum wavelength, microns R = 1000; % Object radius, microns smax = 5* lambda0; % Object peak phase, radians dx = 5.2; % Pixel size, microns Nx = 1280; % Number of pixels Ny = 1024; x = (-Nx/2+1:Nx/2)*dx; % x and y coordinate vectors y = (-Ny/2+1:Ny/2)*dx; [x_xy,y_xy] = meshgrid(x,y); % x and y 2D coordinate matrices % Generate s vs x and y s = smax*(1-(x_xy.^2+y_xy.^2)/r^2).*((x_xy.^2+y_xy.^2)<r^2); % Plot s pcolor(x,y,s); shading flat; colorbar; set(gca,'dataaspectratio',[1 11]); set(gcf,'color','w'); set(gca,'fontsize',12); set(gca,'fontname','times'); xlabel('x [{\mu}m]'); ylabel('y [{\mu}m]'); title( Wavefront deformation, s'); % 2D density plot % No grid % 1:1 aspect ratio % Background color = white % Use 12 pt Times font % Label the axes DESIGN PROBLEM: Using your computer tool and the understanding you get from it, design a procedure using the interferometer of Figure 1 that can: A) Measure the flatness (= variations in thickness) of a glass plate. How will you use the interference pattern to quantify the maximum deviation from perfectly flat? What accuracy do you expect to achieve? How will you distinguish hills (increased thicknesses) from valleys (decreased thickness)? B) Measure the temperature change in the air above a hot object. The index of refraction of air is, to first order n = P P where P o = 760 torr, T o = 293 o K, P is the pressure, and T is the temperature of the air. Assuming pressure is invariant, what temperature accuracy to you expect? 0 T T 0 Version 1.0, 9/17/09 Robert McLeod 2

3 Mirror Object Mirror Rail HeNe Obj. P.H. Collimate Image Camera Figure 1. Imaging Michelson (aka Twyman-Green) interferometer. The lower lens images the object plane to the camera plane with magnification = -1. The rail will allow one arm of the interferometer to be changed in length. TECHNICAL RESOURCES: TEXTBOOK: LECTURE NOTES: Chapter 8, particularly sections 8-1 through 8-3, although the remainder of the chapter should also be accessible. Lecture 5, Interferometers. EQUIPMENT AVAILABLE: A spatially-filtered, JDS Uniphase 1103P-3020 helium neon laser. The laser wavelength is 633 nm. Lenses: focal lengths of 50, 100 and 250 mm A mounted beam splitter and several optical-grade (flat) mirrors. 24 long optical rail and rail carriers for easy linear translation of one arm of the interferometer. Nearly flat transparent test object of index ~1.5. A monochrome, digitizing USB CMOS camera. Specifications and Matlab routines to read and plot lines from the image files are on the references page. A PC for viewing the images in real time and saving data files. Bring a USB stick to take the data away with you. LAB PROCEDURE: STEP 1: SET UP THE INTERFEROMETER Collimate the HeNe, considering the diameter of the resulting beam and your camera active area. Place the beam splitter cube so that the beam is roughly centered on the cube face and roughly retro-reflecting, then intentionally tilt the BS slightly so that the back reflection is NOT retro-reflecting (why?). Next, mount a mirror on a rail carrier. Version 1.0, 9/17/09 Robert McLeod 3

4 Place it so that, again, the beam is centered and roughly retro-reflecting. Use your knowledge on basic optical alignment from lab 1 to set up these conditions. Do the same for the object leg of the interferometer -- this mirror doesn't need to move. The distances of these two legs should be the roughly the same (± 1 cm). Place a screen in the camera location for now (no imaging lens, yet). Two waves are created by the BS, retro-reflect from the mirrors and are recombined by the BS. There is thus an interference pattern traveling down in Error! Reference source not found.. But unless you ve accidentally aligned the two beams to within the tolerance you found in the prelab, the fringe spacing will be too small to see. How can you align the two beams to be within this angle tolerance? Hint: you ve already practiced a technique which would work. Once you have done so and can see fringes, use the fringe pattern itself to continue the alignment. Try to get a completely uniform pattern. In your lab book: Record your setup (focal lengths, positions). If the pattern isn t totally uniform, why? How precisely have you aligned the angles of the two beams in this experiment? Compare to lab 1. Interferometer instruments can routinely measure the phase difference between the two beams to within 1/100 th wave. What angular tolerance would this achieve in your setup? If you propagated the beams a kilometer, how far would their centers diverge due to this angle? STEP 2: REFINE YOUR COLLIMATION Observe the fringe patterns as you Rotate the lens a small amount on the post holder Move the lens sideways or along the beam. Use the interferometer to optimize the lens position in order to get maximally flat fringes. You may find that a translation stage under the lens mount is handy for fine positioning. Now swap the lens front to back by rotating the lens mount on the post holder. Reoptimize. Is there a difference between the two orientations? Why? If necessary, return to the optimum setup. In your lab book: Record any observations or comments your set up is showing you the details of lens aberrations, so careful observation will be educational. Note the largest area of the unmagnified beam that you can force to be uniform in intensity this is your measurement area. STEP 3: MEASURE FLATNESS OF A TRANSPARENT OBJECT An appropriate part will be in the lab, but feel free to bring something if you want to have fun. Set the interferometer for maximally flat fringes, then tilt one mirror to make a linear fringe pattern. Block the reference arm, insert a lithographic mask in the object arm and use a doublet lens to image this target approximately 1:1 onto the CMOS Version 1.0, 9/17/09 Robert McLeod 4

5 camera. Remove the mask and reference block and insert your test object in one arm of the interferometer and observe the pattern. In your lab book: 1. Sketch or photograph the pattern, then use this to estimate the maximum thickness variation of the part (show your calculation!). This will be related to your calculation from prelab, but you must now consider how an extra thickness, t, deforms the wavefront s. Hint: Compare the optical phase for a slab of material, index n, and the same slab of air, index ~1. 2. Attempt to determine the sign of the thickness variation (is it thinner or thicker at the peak of the deviation?). Describe your procedure. 3. How would you apply this procedure to an opaque object? STEP 4: TEMPERATURE MEASUREMENT OF THE AIR First place your hand directly under one arm of the interferometer and see if you can observe the thermal currents in the air it will depend on the lab temperature. Then carefully light a tea candle under the object arm at the object location. The flame should not be in the field of view of your imaging system. Capture an image of the thermal plume, then extinguish the candle. In your lab book: Use the image to calculate the temperature change of the air. STEP 5: COHERENCE LENGTH Remove the object, tilt one beam to get linear fringes and optimize your interferometer for maximum contrast on the CMOS camera. This includes the optical set up and the camera settings. Try to get maximally dark nulls and maximally bring peaks without saturating the camera at either end. Record an image with the two arms at approximately equal length, then move the mirror on the rail carrier to change the relative delay between the two arms. You should observe that the contrast of your fringes decreases as you move the mirror back, although not necessarily monotonically. Record images on the camera. Note: With a slight alteration, this setup measures the Fourier transform of the optical spectrum which can then be recovered with an inverse transform. This forms the basis of an extremely common instrument in chemistry called the Fourier Transform Infrared Spectrometer (FTIR). In your lab book: Calculate and plot the fringe visibility V I I max max I + I min min as a function of the relative path length difference (remember the factor of two due to double-pass on the rail). This is the coherence function of the laser and reveals its spectral width. The coherence length is the path difference in which the fringe visibility falls off to a specified value (first null, 3dB etc). Using the coherence length you measured, calculate the laser temporal frequency bandwidth. Calculate the fractional bandwidth, the frequency bandwidth divided by the center frequency. Version 1.0, 9/17/09 Robert McLeod 5