Manual2023_Michelson

CHAPTER 1. RULES AND RESOURCES 2 for example, you bend down to the beam level or an optic is being moved or removed with the laser not being blocked. A small index card (provided) is a much better means of observing the path of the beam. It is also important to watch for stray reflections off of mirrors or other reflective surfaces. This means that any watches, rings, or bracelets should be removed before beginning this experiment. If you do get a laser beam or scatter in your eye please warn your partner and do not try to repeat the process to figure out what happened. If possible block the beam or turn off the laser and report the incident to lab staff. If you want to check for laser scatter one method is to put your BACK to the experiment. With the room lights off place a white sheet of paper in FRONT of you where you would like to check if the laser light is hitting the paper. Laser goggles are provided and should be used when you are aligning the optical components in your setup. If you feel uncomfortable with the light intensity or anything else laser related, please seek out assistance. Curtains and doors are provided to protect others from your laser equipment. Please be mindful to appropriately block out access to random bystanders so they do not have direct line of sight to your optical equipment. Do Not touch any optical surface It does not take many impurities on the optical surfaces (e.g. scratches, finger- prints, and even dust) to prevent lasing action from occurring in the HeNe lab, to ruin the cavity mode in the Cavity lab, to mess up the interference pattern in the Michelson lab, or to destroy the Fourier image in the Fourier lab. Either hold optical elements by their mounts or on their edges as applicable. Do Not eat or drink in the lab rooms If you must eat, do so in the hallway. Do Not move optics mounts or other hardware fixed to the bench with screws painted in red These are elements that are already in their proper place and moving them will make doing the lab impossible. If you don’t have a tool to loosen a screw, you probably shouldn’t be trying to move that component. If you do move an optical element held with red screws or suspect that such an element is not in its proper place, please immediately contact the lab TA or professor to help you reset this element - obtaining good data and completing the lab may depend on it. Do Not place foreign objects close to the optical setup It feels very convenient to place your laptop, or your paper notebook, right by your setup and type/write your report as you work. Do not do it! A computer monitor or a pen can easily scatter light into your eyes and cause serious injuries. Plastic covers, e.g. in the HeNe lab, were not designed or tested to hold extra weight and will easily brake if used as desk surfaces.

CHAPTER 1. RULES AND RESOURCES 5 an approval from your TA first. Close similarities between any two lab reports will be considered as plagiarism and will be treated with utmost seriousness following an official UBC policy on academic misconduct.

CHAPTER 2. THE MICHELSON INTERFEROMETER 8 Figure 2.1: An aligned interferometer with the interference fringes visible on the ground glass plate ( GG ). The laser path is shown with orange lines for reference. this does not have to be the case. In reality, the laser, the beamsplitter and mirror M 1 may be placed casually on an optical table, with no more care than is needed to assure that the laser beam strikes the mirror and its reflection returns to the beamsplitter. If mirror M 2 is angled so that the reflection from it goes to the same spot on the beam splitter, the instrument will be symmetric and will produce an interference pattern with high contrast fringes. You may start seeing the beam flickering as soon as the dots overlap – a result of the interference you are looking for. To enlarge the beams and observe the interference pattern, insert lens L in front of the beamsplitter cube, as shown in Figure 2.1. (A) When the interferometer is aligned properly, you should have a nice ex- panded interference pattern which looks similar to the one pictured in the upper corner of Figure 2.2. Record the image and explain the circular symmetry of the fringes. What happens to the interference pattern when you tilt one of the mirrors? Record the corresponding image and explain whether the circular symmetry disappears and why? Hint: If you are having difficulty recognizing circular symmetry, try moving the stage to another position. Do not get stuck on the search for the perfectly round fringes. In practice, the fringes may be distorted due to imperfections in the flatness of the mirrors. In fact, Michelson and other interferometers are used in industrial fabrication plants exactly for this purpose - to verify the flatness of mirrors. (i) What determines the spacing between the concentric fringes? To answer

CHAPTER 2. THE MICHELSON INTERFEROMETER 11 Vacuum Gauge M2 “leak valve” “pump valve” pump on/off switch Figure 2.4: Vacuum pump, valves, and vacuum gauge for the vacuum cell. NOTE: the gauge reads inches of mercury below atmospheric pressure, so the better the vacuum the higher the number! (B) Start by taking a single image and use it for defining the region of interest for the following scans. Now start the data acquisition and slowly open the “leak valve” to let in air while you record the fringes. You may need some practice introducing the air fast enough that the pressure rises to atmospheric pressure during the time you have set while not rising too quickly so that you miss the passage of fringes due to the finite sampling time of the video capture. At the end, process the video file and plot the measured fringe brightness in the region of interest as a function of time. Hint: the time should be extracted from the known frame rate and frame number. (C) Would you have seen the same number of fringes if you had used the Na lamp instead of the HeNe laser for this measurement? Why or why not? (D) What is your value for the index of refraction of air at λ = 632 . 8 nm? Does this agree with the “accepted” value? What are your sources of error? (i) Could you use this apparatus as a barometer or thermometer? If yes, how? If not, why not? 2.3.3 Calibrating the interferometer In many applications of the interferometer, one scans the path length difference between the two arms. In the next set of experiments, you will be doing this by moving the translation stage over a set distance. However, the stage may not move exactly the distance, or at the speed, you expected it to move (due to

CHAPTER 2. THE MICHELSON INTERFEROMETER 14 (i) Process your data using a complementary (and very powerful) method of “Fourier Transform (FT) spectroscopy”. To execute the latter, make sure to read Appendix C, including the sub-section on the numerical “FFT analysis with MATLAB”. Carry out your own Fourier transform of the col- lected data, and plot the calculated light spectrum ( G ( k ) in Appendix C). Find the average wavelength and compare with your findings in task (B) above. (ii) In comparison to the more “primitive” method of fringe counting, what extra information is provided by the method of FT spectroscopy? 2.3.5 Line splitting of the sodium lamp As mentioned in the previous section, the emission peak from sodium atoms is split into two lines with wavelengths λ 1 and λ 2 , known as D1 and D2 lines. This “fine structure” splitting occurs due to the electronic spin-orbit coupling. Each spectral line generates its own fringe pattern, and the two patterns overlap with one another. Consider a position of the translation stage, for which the bright fringes from the D1 line overlap with the bright D2 fringes. Due to the very small wavelength difference, the dark D1 and D2 fringes will also overlap with one another, and the resulting interference pattern will exhibit high bright- to-dark contrast (as if from a single-wavelength monochromatic source) – the fringes are said to have “high visibility”. Now imagine changing the arm length d of the interferometer in such a way as to satisfy: 2 d high → low = ( N + 1 / 2) λ 1 = Nλ 2 ( λ 1 < λ 2 ) , (2.3) where N is an integer. Since we moved a half-integer number of λ 1 , the D1 line is now producing a dark fringe (the two arms are interfering destructively), whereas the D2 line is still interfering constructively (moved by an integer num- ber of λ 2 ). Now the dark fringes from the D1 line will overlap with the bright D2 fringes, and vice versa, leading to a much lower contrast of the overall in- terference pattern – the fringes are said to attain “low visibility”. Hence, the end result of the line splitting is the modulation of the fringe visibility, which becomes apparent if the scan is extended to much longer length scales than what you used in the previous section, i.e. well beyond a few tens of microns. An example of such modulation is shown in Figure 2.6. One can extract the wavelength difference Δ λ from the modulated fringe pattern using Equation 2.3: Δ λ ≡ λ 2 - λ 1 = 2 d high → low 1 N - 1 N + 1 / 2 ≈ d high → low N 2 ≈ ¯ λ 2 4 d high → low , (2.4) where ¯ λ ≈ λ 1 ≈ λ 2 is the average wavelength found in the previous section, and we used the fact that N 1 (convince yourself that this assumption is correct). (A) Notice that in this task, you are seeking to measure the modulation period of the fringe contrast rather than the fringe intensity , which you recorded in your previous measurement. How would you calculate the fringe con- trast from the recorded CCD images? What speed and travel range of the translation stage would it be reasonable to use in this case? Hint,

CHAPTER 2. THE MICHELSON INTERFEROMETER 17 Figure 2.8: Example spectrum from a tungsten halogen lamp [Retzlaff, M.-G. et al. Journal of Sensors and Sensor Systems. 6. 171-184 (2017)], similar to the one used in this lab. the ZPL. Don’t get disappointed – there is one more trick to find them, based on your work in part (A) . Place a narrow band filter either in front of the camera or after the white light lamp. Putting the filter in front of the camera is better because the non filtered scattered white light will not pollute your images. If you do put the filter right after the white light source, please keep it at least a few cm away from the bulb. Similarly, the beam diffuser should not be placed too close to the white light source or it will melt! Slowly (at a speed of 10 μ m / s) scan (using the APT GUI) the region around what you believe is the ZPL position until you see the fringes appear on the camera web interface (again, you might want to check a few candidates for the ZPL). Be ready to quickly press the stop button on the APT user interface when you see the fringes appear. With the filter in place, the range over which fringes should be visible is about 50 microns. If you are spending more than 15 minutes without finding the fringes, ask a TA to check that you are doing everything correctly. Record the fringe intensity as a function of distance. Explain why does the filter help find the fringes. Hint: when you find the fringes, write down the stage position for future reference and note the direction you approached the zero-path length difference. The latter is important for mechanical backlash in the micrometer screw. Since you will be using a MATLAB script to take data, make sure that your code moves the stage forward (i.e. to higher values on the indexed stage) always approaching your position of zero-path length difference from below. When you repeat your scan, first move backwards beyond the initial starting point, then forward again, so as to reach the starting point also from below, thus avoiding the backlash. (C) Finally, remove the filter and find the interference fringes from the white light source by scanning the translation stage in the same narrow travel range, where you just saw the fringes with the filter installed. This time, the fringes will appear and disappear within 2 μ m, so you will probably have to adjust the max velocity and step distance of the stage: don’t go

APPENDIX A. RASPBERRY PI CAMERA OPERATION 20 Figure A.1: An example of the web interface for operating Raspberry Pi cameras in real time. help if the web interface is left in an odd state or if you want to return to the default settings. The web interface saves the last used settings even if the web browser is closed. Please note that more than one person can have a web interface open at the same time to see the camera but it is not advisable to be tweaking settings on different computers at the same time. Camera Settings Menu . There are many many options in the camera settings. You do not need to and should not play with most of them. The main settings of use are: 1. Exposure mode to ‘off’ or ‘auto’. ‘Off’ allows you to set the shutter speed yourself. 2. Shutter speed (in micro seconds) can be set when the exposure mode is set to off. Note that a shutter speed of 0 is auto exposure regardless of the exposure mode setting. 3. White Balance to ‘off’ of ‘auto’. This is useful when you want to correctly collect the red pixel values for data analysis. For the most part we are using HeNe lasers whose wavelength is red (633 nm) so it makes sense to set the white balance to ‘off’, for example, for taking cross sections of images where the red pixel value only is needed. 4. Other settings such as ISO may be useful. Sharpness, contrast, brightness, saturation can be played with but typically the shutter speed is the most valuable. 5. Image quality changes the amount of image compression and is mildly useful.

APPENDIX A. RASPBERRY PI CAMERA OPERATION 21 6. If you want truly uncompressed data for analysis, for example, for cross section analysis then the raw layer can be set to ‘On’. The raw Bayer information is appended to the end of the jpeg image file and must be extracted. This option makes the file size large so only use if needed for analysis. 7. Annotation and size can be changed if you want to have a descriptor written on your image. We do not recommend altering other settings unless you have a very good reason to. If you find that the camera view is strange, first try resetting the settings. If buttons are not working at all or the preview is updating only sporadically, the web interface will need to be reinstalled on the raspberry pi. Please ask for assistance if that is the case and do not do that on your own. Resources for more information: https://elinux.org/RPi-Cam-Web-Interface , which is based on the picamera module urlhttps://picamera.readthedocs.io/en/release- 1.13/. Raspberry Pi High Q camera image resolution and effective pixel size. It is important to realize that the effective pixel size of the RasPi camera changes depending on the image resolution specified. The specifications for the Raspberry Pi High Q camera are given as: Sony IMX477R stacked, back-illuminated sensor, 12.3 megapixels, 7.9 mm sensor diagonal, 1.55 μ m × 1.55 μ m pixel size. The maximum resolution of this camera is 4056 × 3040 pixels. Just to make sense of these numbers it means that there are more pixels along one direction than another which you can tell from looking at the rectangular shape of the CCD chip. If we multiply (4056 × 3040) we get 12330240 total pixels which agrees with the 12.3 megapixel specification. Each physical pixel is 1.55 μ m × 1.55 μ m in size which gives us an approximate 7.9 mm diagonal, as shown in Fig. A.2. Figure A.2: The CCD chip of the raspberry pi high Q camera has 4056 by 3040 pixels with a 7.9 mm diagonal dimension. Note that the image resolution can be specified by the user. For example, the web interface we use to collect raspberry pi images has a default image resolution of (2592 × 1944). This means that images collected have 2592 × 1944 elements

APPENDIX A. RASPBERRY PI CAMERA OPERATION 22 so that the effective pixel size has increased in comparison to the 1.55 μ m × 1.55 μ m size of the physical pixel elements. In that case the effective pixel size is instead 2.42 μ m × 2.42 μ m as calculated by: (4056 / 2592) × 1 . 55 μ m = 2 . 42 μ m , (3040 / 1944) × 1 . 55 μ m = 2 . 42 μ m . This is important for the HeNe lab where you are asked to measure the beam diameter with a camera. In that case it is important to know the separation distance between the elements in your image array. You need to take into account your image resolution used when saving your images to calculate the correct beam diameter.

Appendix B Thorlabs Advanced Positioning Technology (APT) Software In Cavity and Michelson labs, we have translation stages whose position can be controlled by software. The motorized actuators which change their length are either Z825B or Z625B from Thorlabs. The controllers that are used to interface from a computer to the actuators are either the TDC001 or the OptoDC Driver (ODC001) from Thorlabs. These controllers have a USB connection to the lab computer that controls it. There are two ways to control the motor length, and therefore translation stage position. First, there is a software GUI provided by the Thorlabs company called the APT (for Advanced Positioning Technology) User software. The sec- ond means of communications is by programming using, for example, MATLAB to send custom Active X Control commands to do things such as set the desired speed or position of the motor actuator. This Appendix will explain how to use the APT user software and will briefly discuss the programming through MATLAB. APT GUI To open the APT user software click on the APT User icon on the desktop or go to the Start Menu and to the Thorlabs option and click on APT user. You should see a GUI show up as shown in Figure B.1 below. On this GUI the position of the motor is shown with a value between 0 and 25 mm. Pressing the Home button will make the motor go to its home position which is 0 mm. If you click inside the box where the position is shown, in our case 11.0000 mm, then a box will appear where you can type in the position you would like the motor to go to. If, while you are moving, you want the motor to stop - press the Stop button. There are two ways the motor can move, either by being told as mentioned above to go to a certain position, or by performing a ‘jog’. A jog means the motor moves by a certain pre-set amount from its current position. The jog button up and down arrow will increase or decrease the motor position by an 23

APPENDIX B. THORLABS ADVANCED POSITIONING TECHNOLOGY (APT) SOFTWARE 24 Figure B.1: The APT User Software GUI. amount set by the ‘Jog Step Distance’. That Step Distance can be set by clicking on the Settings tab on the bottom right and a window will appear as shown in Figure B.2. The most useful setting in the Motor Driver Settings window is the ‘Max Vel’ input box at the top left under ‘Moves - Velocity Profile’. This is the speed at which the motor will move when you input a specific position to go to as described earlier by clicking in the position indicator box and inputting the desired position. The other settings in this Motor Driver Settings window are of less importance and should not necessarily be played with. In addition, the ‘Stage/Axis’ and ‘Advanced’ tabs are not recommended to be altered. To re-iterate, the main usage of this Motor Driver Settings window is to set the maximum velocity the motor moves at when a new position is asked of it. Troubleshooting of the APT User GUI: If the APT user software stops work- ing it may be needed to unplug the power supply of the controller from the wall, wait for about five seconds and then plug it back in. PLEASE DO NOT UN- PLUG THE POWER OF THE CONTROLLER BY DISCONNECTING THE POWER CONNECTOR PLUGGED INTO THE CONTROLLER FROM THE CONTROLLER. The controller can be damaged by doing that. Leave the power connector plugged into the controller and unplug the power supply from the wall instead. When the controller has been power cycled, it will read a zero position when the APT user software is reopened. This is an incorrect reading since the motor position is still at wherever it was when the power was turned off. To fix this press the Home button on the APT User GUI and watch the position just before it goes to home and resets itself to zero. It may say something like negative 11.5 mm which means that it had to travel back approximately 11.5 mm to get home. Then, to get back to the approximate position where you started, type in for example 11.5 mm into the position window and it will go back to the last position before the power was cycled. If all of this confuses you, please ask one

APPENDIX B. THORLABS ADVANCED POSITIONING TECHNOLOGY (APT) SOFTWARE 25 Figure B.2: The Motor Driver Settings window in which the velocity at which the motor moves to a new position can be set as well as other settings. of the instructors for help before you destroy something! Note: Please, properly close the APT User software when you are finished using the computer and properly sign out of the computer. This will help the next user to not have troubles due to existing unclosed connections. MATLAB control of the motor The second way to move the motor is by sending commands to the controller us- ing MATLAB. The possible commands are many and complex and are fully doc- umented in the thorlabsStage ProgrammingManual.pdf posted in the Hardware/Thorlabs Translation Stage folder. The commands that you will need are given in the thorlabsStage.m MATLAB script as an example. This script gives example of how to initiate a connection to the controller, how to set the speed and po- sition of the motor actuator, and finally how to cleanly close the connection to the controller and the motor. In the optical cavity lab, you are asked to com- bine commands from the thorlabsStage.m script with commands that record data from an oscilloscope in order to perform the “knife edge” scan. In the Michelson interferometer lab, you are asked to combine commands from the thorlabsStage.m script with commands found in raspiCamera.m script to col- lect data from the Raspberry Pi camera at the same time as the motor is being controlled. Note 1: If you run the thorlabsStage.m script (or any script that you create) that controls the motor, please first close any APT User GUI window

APPENDIX B. THORLABS ADVANCED POSITIONING TECHNOLOGY (APT) SOFTWARE 26 that you have open. If you desire to have a Figure B.1 window appear as When you control the motor by command in MATLAB, a window looking similar to Figure B.1 will appear as a MATLAB figure. Please be aware that it is NOT the APT User GUI window, so please do not click in it to control the motor. That window shows up just so you can see what is happening with the motor position as your script is running and that window should be closed when you are done running your script. Note 2: Please, be aware that the serial number found on the controller must be correctly input by hand into either the thorlabsStage.m script or any script you write yourself. In the thorlabsStage.m script, this is the line that looks like this: % Motor serial number (specific to each station) >> motorSN=83810101;

Appendix C Fourier Transform Spectroscopy So far you have investigated the behavior of the interference fringe pattern as a function of the optical path difference for different optical sources. In general the fringe pattern intensity versus the optical path difference (or equivalently versus the time delay between the two interfering beams) is related to the power spectrum of the light entering the amplitude division interferometer by a Fourier transform. Therefore one can easily measure the optical power spectrum using the measurement of the intensity variation as a function of stage position. Let’s see how this works 1 . The electric field incident on the camera is the sum of the electric fields E 1 and E 2 arriving from mirrors M1 and M2 respectively after having passed through the beamsplitter. Assuming for the moment that the input source is a purely monochromatic wave, we can characterize these electric fields with a vector amplitude (encoding the strength and polarization of the wave) and a spatial and time dependent phase: E 1 = A 1 e i ( k 1 · r - ωt + φ 1 ) (C.1) E 2 = A 2 e i ( k 2 · r - ωt + φ 2 ) . (C.2) The phase includes terms ( φ 1 = | k 1 | l 1 and φ 2 = | k 2 | l 2 ) which depend on the optical path lengths from the beamsplitter along path l 1 (encountering M1) and path l 2 (encountering M2). The intensity on the camera is simply the square of the total electric field: I = | E | 2 = E · E * = ( E 1 + E 2 ) · ( E * 1 + E * 2 ) (C.3) = | A 1 | 2 + | A 2 | 2 + 2 A 1 · A 2 cos θ (C.4) = I 1 + I 2 + 2 p I 1 I 2 cos θ (C.5) where we have assumed in the last step that the waves have the same polarization and that θ = ( k 1 - k 2 ) · r + φ 1 - φ 2 . You might be wondering about a white light source which is probably completely unpolarized. Is this assumption still valid in that case? This assumption is justified since the two fields E 1 and E 2 are copies of the incident field generated by the beam splitter. As long as the 1 This derivation follows that in “Introduction to Modern Optics” by Grant R. Fowles 27

APPENDIX C. FOURIER TRANSFORM SPECTROSCOPY 28 beam splitter and the optics which follow preserve the polarization of the light, these two fields will arrive at the detector with exactly the same polarization (whatever it was at the input of the interferometer). If the interferometer is well aligned so that the waves are co-linear ( k 1 = k 2 ) and the 50/50 beamsplitter generates two waves of the same intensity ( I 1 = I 2 = I/ 2), then the interference pattern is simply I ( x ) = I (1 + cos kx ) (C.6) where x = l 1 - l 2 is the path length difference and k = | k 1 | = 2 π/λ is the magnitude of the wavevector. Okay, now in general, the light illuminating the interferometer will be com- posed of many different frequencies. So, let’s model the power spectrum of the incident light as a continuous distribution, G ( ω ). Then the total optical power emitted into the interferometer in the frequency range from ω 0 to ω 0 + dω (where dω is an infinitesimally small frequency band) is simply G ( ω 0 ) dω . For convenience, we can instead use the distribution over wavelength G ( k ) (where ω = ck ). Then the optical power, P , hitting the detector from illuminating the interferometer with this polychromatic source will be the sum of the interfer- ence patterns from each monochromatic wave composing G ( k ). Assuming the interferometer is ideal and has no losses, we have then: P ( x ) = Z ∞ 0 G ( k ) (1 + cos kx ) dk (C.7) = Z ∞ 0 G ( k ) dk + Z ∞ 0 G ( k ) e ikx + e - ikx 2 dk (C.8) = 1 2 P (0) + 1 2 Z ∞ -∞ G ( k ) e ikx dk (C.9) where P (0) is simply the power measured at zero path length difference. We can rearrange this expression and define the power function W ( x ) W ( x ) = 2 P ( x ) - P (0) = Z ∞ -∞ G ( k ) e ikx dk. (C.10) We see then that the power function W ( x ) is the inverse Fourier transform of the power spectral density G ( k ). Inverting this expression we have that the power spectrum is the Fourier transform of the power function G ( k ) = 1 2 π Z ∞ -∞ W ( x ) e - ikx dx, (C.11) or equivalently G ( ν ) = 1 2 π Z ∞ -∞ W ( x ) e - i 2 πνx/c dx. (C.12) FFT analysis with MATLAB In order to compute the power spectra from your fringe pattern data, you will need to do some data processing. First, you will need to properly truncate your data because it may include points when the stage was moving backwards

APPENDIX C. FOURIER TRANSFORM SPECTROSCOPY 29 Monochromatic (laser) Two wavelengths close together (NaD lamp) Wide band-pass filter (white light) Narrow band- pass filter (orange filter) mirror position coherence length Figure C.1: Fringe patterns plotted as a function of the mirror position. Notice how the fringe visibility collapses and then revives as a function of the mirror position when the spectrum is discrete. Also, note that if the mirror is moved a distance d , the optical path length between the two arms in the interferometer change by 2 d . and then sitting stationary before and after the slow sweep through the ZPL position. Assuming that your calibrated relative position of the Thorlabs stage is stored in vector x , and the retrieved fringe brightness is stored in vector Yx ,

APPENDIX C. FOURIER TRANSFORM SPECTROSCOPY 30 start by plotting the raw data: >> plot(x,Yx); Find the indices i1 and i2 , between which the recorded brightness data look reasonable and truncate both vectors: >>x=x(i1:i2); >>Yx=Yx(i1:i2); >>plot(x,Yx); Condition your data by subtracting the mean value. Doing so will remove the (usually dominating, but not very useful) dc component from the Fourier Transform: >>Yx0=Yx-mean(Yx); Now take the Fourier Transform using the built-in fft function: >>Yk0=fft(Yx0); >>Yk=fftshift(Yk0); The second line above re-arranges the Fourier spectrum from the lowest neg- ative to the highest positive spatial frequency. To create the vector of spatial frequencies, use: >>dx=x(2)-x(1); >>k=linspace(-pi/dx,pi/dx,length(x)); Finally, plot the Fourier power spectrum, using the absolute values of the calculated vector Yk (remember that FT returns complex values): >>plot(k,abs(Yk)); Finding the main peak on the plotted Fourier spectrum will now allow you to determine the (spatial) period in the recorded data, and from it – the wavelength of light. In doing these final calculations, remember that dx in the example above is the distance that the stage moved between the two consecutive images, so the amount of the total optical path length varied between consecutive frames is twice that distance!