Optical Resonator Calculator - Physics
Contents
1. red 10 green and 20 blue Note that two modes can be resonant at the same frequency if their phase shifts are the same 12 a v v F we o 2 25 frequency offsel kHz Figure 2 7 Internal power of a cavity plotted for three modes 2 3 Cavity lengths and tuning The distance between optical components in a cavity is conventionally de scribed as a sum of macroscopic and microscopic distances Some properties of the cavity such as the free spectral range depend on the macroscopic distance between the optics For those properites changing the separation between the mirrors on the order of one laser wavelength will not affect their values However the resonance condition of the cavity depends on the mi croscopic displacements of the mirrors and thus both parameters must be considered The distance D between the mirrors is written as D L T 2 8 where L is the macroscopic length and T is the tuning or the microscopic displacement from the exact given macroscopic length T here is clearly a distance measurement but it can be rewritten as a phase shift of the 13 Gaussian beam We will call this phase shift governed by the microscopic tuning Az 3 Now Figure 2 5 plots the internal power of the cavity against frequency It shows for example that the cavity is resonant at the frequencies that correspond to the peaks in the graph Note that Figure 2 7 plots the internal power against A2. break case GTextField SET break case GTextField ENTERED line1l R2textfield getText temp_val parseFloat line1 R2slider setValue temp_val break if tfield RoCtextfield switch RoCtextfield eventType case GTextField CHANGED break case GTextField SET break case GTextField ENTERED line1 RoCtextfield getText temp_val parseFloat line1 RoCslider setValue temp_val break 37 Acknowledgements I would like to thank everyone at the University of Birmingham and the Uni versity of Florida s IREU for their help with this project and my scientific learning this summer In particular my thanks go to my project supervisor Dr Andreas Freise for all his patience and organization with the summer projects to all the people in Andreas s group Dr Stefan Hild Dr Simon Chelkowski Paul Fulda Antonio Perreca Jonathan Hallam Deepali Lod hia and Keiko Kokeyama for always being available to help to the other summer students Charlotte Bond Emil Schreiber and Matthew Arnold for helping so much to make this project successful particularly Charlotte who contributed code for the Optical Resonator Calculator and to Daniel Brown another summer student who contributed extensive amount of code and libraries to this project and was my go to person for most of my pro gramming questions I d also like to acknowledge Professor Bernard Whit ing Professor Guido Mueller and all others involved
3. psiRT 2 atan L zR Power10 log 1 R1 1 R1 R2 2 sqrt R1 sqrt R2 cos delta_x psiRT n m 1 log 10 return Power10 class InternalPower20 implements GraphCallback int n m float Power20 float zR psiRT public float computePoint float t_delta_x delta_x t_delta_x n 2 m 0 zR sqrt L RoC L 35 psiRT 2 atan L zR Power20 log 1 R1 1 R1 R2 2 sqrt R1 sqrt R2 cos delta_xt psiRT nt m 1 log 10 return Power20 This method tells the prorgam what to do if the user moves the sliders void handleSliderEvents gwSlider slider float precision 100 if slider Rislider Ritextfield setText String format 2f slider getValuef if slider R2slider R2textfield setText String format 2f slider getValuef J if slider RoCslider RoCtextfield setText String format 2f slider getValuef This method tells the program what to do if the user types a value into a textfield and hits Enter void handleTextFieldEvents GTextField tfield String linel float temp_val if tfield Ritextfield switch Ritextfield eventType case GTextField CHANGED break case GTextField SET break case GTextField ENTERED line1l Ritextfield getText temp_val parseFloat line1 36 Rislider setValue temp_val break if tfield R2textfield switch R2textfield eventType case GTextField CHANGED
4. reflectivities 15 of each mirror the radius of curvature of the second mirror and the value of Az the tuning With this setup this version of the calculator should help users to visualize the internal power function the Guoy phase the power output the FSR and the overall phase shift of a beam Also it should depict how changing the reflectivities or radii of curvature of a cavity s mirrors affects the beam inside the cavity 3 2 Project Overview The Processing IDE which we chose to use for our program already had code libraries for simple GUI components For example it provides code for check boxes text fields buttons and some other components However it had very little or no libraries pertaining to sliders meters or graphs Our first task then was to write extendable libraries for the components we needed Each student in our group had a different main project but each of us wrote some library functions that could be used by all of us and by others I will focus only on the small pieces that are directly used in the Optical Resonator Calculator One student in our group Daniel Brown wrote the library for among many other things the sliders Charlotte Bond wrote the code for the LEDMeter the meter which is used in this case to show the cavity s transmitted power I then wrote a two dimensional plotting library This plotting library allows a user to plot many traces and many separate graphs It can plot an
5. slider Therefore it runs many times to update the graphics In this sketch this and the functions doing the calculations seem to be consuming a lot of processing power Slight code variations should be implemented to reduce this problem For example telling the draw function to run only if some event has actually occurred Other modifications could be made to make the sketch more efficient Finally due to my relative lack of experience in programming there are some limitations in the two dimensional plotting class It is quite sensitive to the precise numbers being use for say the axis length and the tick mark spacing This issue is described in the comments of this code but it should be highlighted here as well In order for the scaling between the pixels and the graph units to be accurate when calculating the actual plot one must be very sure that the axis length is an exact multiple of the spacing between the tick marks If this is not quite possible the plots may be slightly inaccurate Now the spacing between the slider s tick marks is on the other hand calculated by dividing the axis length by the number of tick marks the user wants on the slider This can lead to small rounding issues because pixels must be represented as integers If the quotient done in this 24 calculation is not exactly an integer the quotient is rounded or truncated In the Optical Resonator Calculator the x axis of the graph is supposed to be
6. with this program at the University of Florida for all their work with the International REU program Last but not least my gratitude goes to the National Science Foundation NSF for funding and supporting the REU program 38 Bibliography 1 Ben Fry and Casey Reas Processing Website 2009 http www processing org 2 Antonio Perreca Lecture Notes Summer lecture notes University of Birmingham July 2009 3 Dr Andreas Freise Lasers and Quantum Optics Year 4 Lecture Notes University of Birmingham April 2009 39 40
7. 0 red 10 green and 20 blue The shder below the graph controls the tuning of the second mirror Below the meter shows the percentage of the input power that is outputted through the second mirror at the current value of delta x the tuning The cavity is drawn below the graph Each mirror has a reflectivity slider The 1 RoC slider controls the inverse of the Radius of Curvature of the second mirror 1 RoC 0 represents a flat mirror infinite RoC and 1 RoC 1 L represents a radius of curvature equal to the length of the cavity 1m Using the sliders you should be able to notice changes in the Guoy phase and the FSR for the different modes The sliders can be controlled with the mouse or left right arrow keys J Show white ine for deta x Percent of Input Power Transmitted 0 1 2 4 9 1 trans Reflectivity Reflectivity oss 096 0 00 EJ 0 00 059 gw optcs org Figure 3 1 A screenshot of the optical resonator calculator 3 3 1 Introducing the GUI components The image on the bottom left of the calculator in Figure 3 1 is a diagram of the cavity This very simple cavity has essentially the same basic setup and functionality as the cavities in each arm of a gravitational wave inter ferometer It consists of a laser two mirrors in this case one flat and one with a variable radius of curvature with variable reflectivites a beamsplit ter with two photodetectors and a photodetector for the light transmitted through th
8. 2 alltraces addTrace mode20 traceConfig3 Initialize and add all the sliders Rislider new gwSlider this 200 559 110 Rislider setTickColour 255 255 255 Rislider setFontColour 255 255 255 Rislider setValueType ValueType DECIMAL 1 00 reflectivity is unrealistic so we set the limit to 0 99 Rislider setLimits 0 96 0 00 0 99 Rislider setTickCount 5 Rislider setPrecision 2 This sets the textfield value for before you move the slider Ritextfield new GTextField this String value0f 0 96 160 557 36 15 R2slider new gwSlider this 370 559 110 R2slider setTickColour 255 255 255 30 R2slider setFontColour 255 255 255 R2slider setValueType ValueType DECIMAL R2slider setLimits 0 96 0 0 99 R2slider setTickCount 5 R2slider setPrecision 2 R2textfield new GTextField this String valueDf 0 96f 332 557 36 15 RoCslider new gwSlider this 370 452 110 RoCslider setTickColour 255 255 255 RoCslider setFontColour 255 255 255 RoCslider setValueType ValueType DECIMAL represents 1 RoC so the RoC can range from L to infinity RoCslider setLimits 0 5 0 1 RoCslider setTickCount 5 RoCslider setPrecision 2 RoCtextfield new GTextField this String valueDf 0 50f 332 450 36 15 doesn t exactly line up with axis but close delta_x_slider new gwSlider this 54 368 478 delta_x_slider setTickColour 255 255 255 default position for slide
9. 2 traceConfig3 tracecolor color 0 0 255 traceConfig3 traceweight 2 Initialize the Axes and PlotAreaBackground x pos y pos width height config object areal new PlotAreaBackground 5 15 540 389 plotconfig getConfig x1 y1 x2 y2 axis label config axis zero xaxis new Axis 60 350 526 350 delta x degrees axisconfig getConfig 293 yaxis new Axis 60 25 60 350 log of Intensity axisconfig getConfig 188 Tick label formatting xaxis setTickLblFormat 0f no decimal places for the ticks length of x axis in pixels its length in graph space 33 28571429px graph unit xaxis setMajorSpacing 33 yaxis setMajorSpacing 54 major tick labels start O and increment by 90 xaxis setMajorLabels 0 90 29 major tick labels start O and increment by 1 yaxis setMajorLabels 0 1 xaxis isZeroTickDrawn true yaxis isZeroTickDrawn true Axis label text formatting xaxis setTextPosition 300 400 yaxis setTextPosition 25 200 Initialize the trace objects need to pass the alltraces object the axes to be drawn on alltraces new Plotting xaxis yaxis these are the internal power equations being used one for each mode mode00 new InternalPower00 modei0 new InternalPower10 mode20 new InternalPower20 tell alltraces which traces and configs you want alltraces addTrace mode00 traceConfig1 alltraces addTrace mode10 traceConfig
10. Optical Resonator Calculator Gravitational Wave Detector Cavity Simulations with Processing School of Physics and Astronomy University of Birmingham University of Florida International REU 2009 Jamie Dougherty University of Rochester 7 August 2009 Project Supervisor Dr Andreas Freise Contents Table of Contents 1 Chapter 1 1 1 Introduction ooa a 2000000000004 2 Taski s adma a wt GR Oe BOE ee a a Ow e aE ss 2 Chapter 2 2 1 Optical cavities oaoa a 2 1 1 Types of optical resonators 2 1 2 The internal power of the cavity 2 2 Guoy phase shifts oaa a 2 3 Cavity lengths and tuning oaoa a 2 4 Behavior of a cavity aoaaa aa e 3 Chapter 3 3 1 Optical Resonator Calculator The Idea oaoa 3 2 Project Overview aoaaa a eee ee 3 3 The User s Manual 00 0000 eee ee 3 3 1 Introducing the GUI components 3 3 2 The mathematical details 3 3 3 How to use the GUI controls 3 3 4 Configurations to try 000 3 3 5 Intentional limitations 0 0 4 Chapter 4 4 1 Remaining Problems and Bugs 4 2 Possible Future Directions 004 11 13 14 15 15 16 17 17 18 20 20 22 A Appendix A Acknowledgements 27 38 Chapter 1 Introduction 1 1 Introduction In physics education and research simulations of physical systems play an important r
11. alculates the y value for each of these points in graph units and finally chooses to plot the maximum of these as the y value corresponding to the original x pixel 23 For various detailed reasons having to do with object passing and private variables this same loop is not easy to use for calculating the transmitted power Currently the calculation for the transmitted power does not directly read the y values off of the graph Instead it computes the y value given the value of the Az slider without the loop just described Therefore I believe the values being plotted are not quite the same values computed at the precise value of the Az slider This may cause the slight discrepancy for sharp peaks This may not be the actual reason though This bug was not resolved after several attempts and adjustments and therefore has been left for future revisions The remaining problems are not necessarily bugs but rather improvements that should probably be made to the code Currently the program or sketch as it is called in Processing invokes a lot of processing time from the computer running it In Processing sketches there are two main functions setup and draw The setup function is meant to contain all the object and variable declarations initializations and other fixed parameters It only runs once when the sketch runs The draw function however is meant to react to changes made while the sketch is running such as moving a
12. box declare variables objects and fonts gwolider Rislider GTextField Ritextfield gwolider R2slider GTextField R2textfield 27 gwSlider RoCslider GTextField RoCtextfield gwSlider delta_x_slider LEDMeter meter LEDMeterAxis meterAxis AxisConfig axisconfig PlotConfig plotconfig TraceConfig traceConfigl traceConfig2 traceConfig3 PlotAreaBackground areal Axis xaxis yaxis Plotting alltraces InternalPower00 mode00 InternalPower10 mode10 InternalPower20 mode20 PFont font float R1 R2 delta_x L RoC PImage cavityimage Logo _logo PFont titleFont PFont expFont PFont meterTitleFont String explanationtext GCheckbox chkWhiteLine void setup size 900 600 smooth TS length of the cavity is fixed to 1 meter initialize the axis plot and trace config objects axisconfig new AxisConfig plotconfig new PlotConfig traceConfig1 new TraceConfig traceConfig2 new TraceConfig traceConfig3 new TraceConfig set the config settings plotconfig bordercolor color 255 plotconfig fillcolor color 0 plotconfig borderweight 3 axisconfig axiscolor color 255 axisconfig textcolor color 255 axisconfig tickcolor color 255 traceConfig1 tracecolor color 255 0 0 each trace has its own traceConfig1 traceweight 2 color so its own config object traceConfig2 tracecolor color 0 255 0 traceConfig2 traceweight
13. d Space Research Group was to experiment with the programming IDE Processing in an attempt to develop some tools for simulating the optics used in gravitational wave detection It is our hope that the graphical simulations and code libaries we have developed can be used in gravitational wave detection and interferometry research and educa tion For example simple visual simulations of optics equations should allow students to gain a more intuitive understanding of the physical meaning of an equation by outputting graphs which change when the student changes system parameters 1 2 Tasks To achieve this goal we initially focused our efforts on developing some code libraries relevant to gravitational wave optics Since we are developing the code for visual simulations and for use by beginners in programming the Java based IDE Processing was chosen as our IDE Processing is an open source IDE developed for artists and designers with little experience in programming It is designed to simplify the programming of visual out put It also provides simple exporting tools for creating PDF files and Java Applets useful for posting the simulations on the Internet 1 Processing provides many libraries which are constantly in development but there are few provided libraries relevant to physics Therefore we first sought to create a library that we could use later for our simulations In particular I developed a set of code to be used in making tw
14. e second mirror Note that there are three sliders and accompa 17 nying text fields overlaying the cavity image The bottom left slider controls the reflectivity of the first mirror while the bottom right slider controls the reflectivity of the second rightmost mirror These reflectivities can range from 0 00 to 0 99 as one cannot have a mirror with a perfect reflectivity of 1 00 These will be referred to respectively as the R slider and the R slider Likewise the left mirror is mirror 1 and the right mirror is mirror 2 The remaining slider above these controls the inverse of the radius of curvature of the second mirror and will commonly be referred to as the RoC slider The graph above the cavity schematic is the graph of the logarithm of the cavity s internal power as a function of the microscopic tuning of the second mirror The logarithm of the internal power is plotted for three different transverse modes TEMoo in red TE Mj in green and TE Mop in blue The use of the logarithmic scaling will be explained later The tuning is measured in degrees where 360 is a microscopic displacement of the second mirror by one wavelength Below the graph there is a slider called the Az slider which can be placed anywhere along the x axis By default this slider has a white line attached to it that indicates precisely where the slider is lining up with the plots above Finally the meter on the bottom
15. fficient Optical resonators are made to amplify the light within the cavity so the mirrors used are highly reflective Essentially light enters the cavity through one mirror reflects off the opposite mirror and returns to the first mirror while some of it is transmitted exits the cavity through each mirror This light transmitted through the first mirror in each arm is the light that interferes at the beam splitter to form the signal Fabry Perot cavity il al al a2 r a4 a3 a3 rl tl r2 t2 Figure 2 2 A Fabry Perot cavity 2 2 1 1 Types of optical resonators When designing an optical resonator one has to ensure its stability That is certain frequencies of light will form a standing wave within the cavity The cavity is considered stable if the focal points of the mirrors and the distance between the mirrors are such that the internal beam does not continually grow in size after many reflections In other words the cavity is designed so that the beam will remain entirely within the cavity s mirrors There are five different types of stable two mirror optical cavities as shown in Figure 2 3 These types of resonators differ in their focal lengths of the mirrors governed by the mirror s radius of curvature and in their distance between the mirrors cavity length As you can see from Figure 2 3 some beams have different shapes within the cavity and are thus chosen for dif ferent purposes R plane paral
16. g are the power reflectivities of the first and second mirror R r L is the length of the cavity rt is the round trip Guoy phase explained later and n and m represent the transverse modes of the Guassian light hemispherical 0 1 plane parallel confocal concentric ee concave convex Figure 2 4 The stability diagram 2 beam mode TEMym This equation is plotted in Figure 2 5 The argu ment of the cosine term is the total phase shift of the light in the cavity The maximum internal power is achieved when the cosine function is equal to one This condition is called the cavity resonance while the minimum power is achieved at the cavity s anti resonance When one designs a cavity resonance is desired 3 Many factors can affect the resonance properties of the cavity as can easily be seen from Equation 2 3 Most notably the periodicity of the function is governed by the cosine term 2kL can be rewritten as E 2L nf 2kL Inf ER 2 4 10 resonance a g va o 2 Fl anti resonance frequency offsel kHz Figure 2 5 A graph of the internal power of a cavity where f is the light s frequency c is the speed of light and FSR is the free spectral range of the cavity the frequency separation between resonant frequencies see Figure 2 5 2 2 Guoy phase shifts As mentioned previously the beams used in these cavities are Gaussian beam
17. gain though numerical output could be easily included in future versions of the calculator if so desired 22 Chapter 4 Bugs and Future Directions 4 1 Remaining Problems and Bugs There are only a few remaining problems with the current version of the Optical Resonator Calculator These are described in detail below For the more code involved problems it would probably help to know how the actual code works in the libraries and the source code included in Appendix A However the explanations of the problems below should not require much prior knowledge of the code The most notable bug can be seen when the reflectivities of the mirrors are very high 0 97 In this case it is difficult to align the Az slider with the peaks precisely and the transmitted power meter does not quite display the correct value For instance if Ry Ro 0 99 and 1 RoC 0 5 if the cavity is on resonance the meter should read for each beam mode a transmitted power of iW The value it actually shows is closer to 0 1 W This bug is probably due to an inherent resolution issue with the graphical output To increase the resolution around sharp peaks the two dimensional plotting function implements a loop which is not also implemented exactly by the transmitted power meter calculations In this loop for each pixel along the x axis the plotting function converts the pixel number into graph units takes six points in graph units between each pixel c
18. in self consistent and simple portraying only a few concepts in each simulation With the foundational projects the students working with Dr Andreas Freise at the University of Birmingham have done this summer more sim ulations and educational and research tools should begin to be created for gravitational wave detector optics For additional information concerning the projects done this summer please refer to 25 www gwoptics org processing Also more details about the source code can be found in Appendix A complete with all comments 26 Appendix A Appendix A This is the exact Processing source code as of 7 August 2009 being used for the main file of the optical resonator calculator It uses quite a few libraries as listed at the top of the code and it also uses the classes Axis Axis Config GraphCallback LEDMeter LEDMeterAxis PlotAreaBackground PlotConfig Plotting and TraceConfig This source code may be slightly modified for the final version to be posted online but the changes will only be minor alterations The final version s source code and libraries will be posted online also For more detail about these classes one must refer to the code files for them which are not included in this report for the sake of brevity import org gwoptics gui slider import org gwoptics ValueType import guicomponents import org gwoptics Logo import org gwoptics LogoSize import guicomponents GCheck
19. ity can transmit on resonance one hundred percent of the input power 3 14 Chapter 3 The Optical Resonator Calculator 3 1 Optical Resonator Calculator The Idea It was our goal to express some of the equations and properties of optical cavities as described in Chapter 2 visually A graphical representation can often teach much more about an equation s properties than doing the calculations by hand It can give a student or researcher a more intu itive understanding of what the equation actually means in a physical sense Therefore with the background knowledge about the physics of optical sys tems we hope to create a useful visual tool and reusable code to simulate the properties discussed above as they pertain to gravitational wave detector interferometers For our first version of the calculator we chose to model the internal power of a two mirror optical cavity as a function of the tuning The cavity being modeled consists of one flat mirror and one spherical mirror It is important to model the internal power because it shows graphically the tuning values for which the cavity is resonant or anti resonant We wanted to plot the internal power for three different transverse modes TE Moo TEM10 and TEMa Also we thought it would be interesting to visually depict the output power of the cavity at a user defined value for tuning We gave the user the ability to control via simple sliders and text fields the
20. lel Ry 0 lt lt L eee R L2 concentric spherical Ry L 2 gt lt q R L confocal R L 4 R L hemispherical R RL concave convex Ry L R Figure 2 3 The five possible types of stable cavities 2 There are simple mathematical formulae that indicate whether or not a cavity is stable In its simplest form the rule can be stated as follows Given a cavity made of two spherical mirrors of radii of curvature Ry and Ro separated by a distance L the cavity is stable if 0 lt g92 lt 1 2 1 where L L 1 and g 1 2 2 g Ri and g2 Ro Graphically Figure 2 4 shows a plot of the stability region of cavities If g and ga are such that their intersection lies within the shaded region of this diagram then the cavity is stable Our theoretical cavity to be discussed later will consist of one flat mirror in finite radius of curvature and one mirror with adjustable radius of curvature from L the cavity length to infinity This is essentially a hemispherical type cavity 2 1 2 The internal power of the cavity The light standing wave inside the cavity circulates between the two mir rors a process known as power circulation The power circulation causes power amplification of the beam The power inside the cavity is given by T P j Ri Re 2rirecos 2kL Urt n m 1 2 3 where T is the power transmission of the first mirror given by T E Ry and R
21. o dimensional plots Other students developed GUI graphical user interface components and a three dimensional plotting program This library was then used to develop an optical resonator calculator This calculator is meant to simulate an optical cavity such as those used in grav itational wave detector interferometers It provides a simple GUI through which a user can change parameters of the cavity s mirrors and observe di rectly how those changes affect the points at which the cavity is resonant More importantly the code can fairly easily be altered and expanded to include much more functionality making it possible for future revisions Other students developed different simulations which will be described later All projects are posted on the Internet at www gwoptics org processing cavity_calculator Before describing our work in further detail some background information about the physics of the optics and gravitational wave detection is needed Chapter 2 The Physics of an Optical Cavity Gravitational wave detectors are in principle quite simple They consist of two perpendicular arms each of which contains an optical cavity see Figure 2 1 Generally these are Fabry Perot cavities At the intersection of the two arms is a beam splitter A laser pumps light a Gaussian beam to the beam splitter at which point half the light goes into each identical arm bounces off the mirrors test masses in the cavitie
22. oC 1 L represents a radius of curvature equal to the length of the cavity 1m Using the sliders you should be able to notice changes in the Guoy phase and the FSR for the different modes The sliders can be controlled with the mouse or left right arrow keys Adds a checkbox to select whether or not we want the white line on the delta_x slider to show Defaulted to yes true chkWhiteLine new GCheckbox this Show white line for delta x 550 380 200 chkWhiteLine setSelected true void draw background 200 200 200 32 Draw plotting area and the traces areal drawRect xaxis drawAxis yaxis drawAxis xaxis drawMajorTicks yaxis drawMajorTicks xaxis drawMajorTickLabels yaxis drawMajorTickLabels need to know the x and y axis ranges in graph space alltraces plotTraces 3 5 PI 3 5 PI 3 3 Draw the cavity image image cavityimage 10 412 Assign the slider values to variables R1 Rislider getValuef R2 R2slider getValuef delta_x delta_x_slider getValuef RoC 1 RoCslider getValuef Set and write the Title to the screen textFont titleFont 111 0 text Optical Resonator Calculator 720 30 Set and write a label for the meter textFont meterTitleFont 111 0 text Percent of Input Power Transmitted 720 480 Explanation box stroke 255 111 190 204 223 rect 575 45 300 328
23. ole They allow researchers and educators to approximate real systems simply and quickly conveying the important relations within the system Such simulations are common in physics but often they are numer ical tools rather than graphical While numerical simulations are powerful they are usually not very user friendly They sometimes require the user to understand how the program works and how to interpret the results before the user can successfully use the program This then can prevent students or non scientists from learning from these programs as well as often being cumbersome for experienced scientists who are seeking just a simple answer Also for newcomers to programming it is often difficult or frustrating to create new simulations from scratch Most people create programs by using bits and pieces of code made by others or by using libraries of code that take care of most of the difficult programming details However such code libraries or sample pieces of code are not always available for a particular problem or field These problems exists in the field of gravitational wave detection and ad vanced interferometry Numerical tools are used to simulate the behavior of optical systems but few graphical tools exist Also little to no code libraries relevant to these fields exist in simple programming languages Therefore the goal of the group of students brought together at the University of Birm ingham s Astrophysics an
24. play its current value The value is represented by the corresponding tick marks on the x axis of the graph By default this slider has a white line linked to it which helps to see where on the graph the slider is positioned This line can be switched off though by unselecting the check box to the right of the graph labelled Show white line for delta x 3 3 4 Configurations to try In most optical resonators highly reflective mirrors are used to enhance the power within the resonator This is true for gravitational wave interferom eters If you set the R and R sliders to 0 99 reflectivity you will see that 20 the peaks in the internal power plot become very sharp Equation 3 3 shows that the transmitted power is proportional to T When R is very close to 1 00 Tz becomes very small Therefore the internal power is only large for resonance conditions This makes the precise tuning of the cavity extremely important if one is to achieve resonance The calculator will also allow you to visualize the differences among un dercoupled overcoupled and impedance matched cavities These configu rations were described in detail in Chapter 2 The calculator s default is an impedance matched cavity with high reflectivities R R2 0 96 This is the only condition under which on resonance the cavity can transmit all of the input power in this case 1 W Altering the radius of curvature of the second mirror using the RoC
25. precisely linked with the Az slider Because of this difference in the coding of these two components one has to be very careful and very precise about the values chosen for each There does not seem to be a problem in this program but if modifications are made this conflict must be taken into account 4 2 Possible Future Directions One of the main purposes of this project was to establish a foundation for possible future simulations for gravitational wave detector optics A large amount of the time was spent writing library codes in the hope that they would be used in other projects The simulations created like the Optical Resonator Calculator were intended to be used on the Internet but also to show other people what can be done with the library code This calculator is simply the first version created of what could be a series of different simulation programs applets With this in mind some suggestions have been compiled for possible newer versions of the calculator e Output some numerical values given the slider or text field input e Allow adjustments to more parameters cavity length beam modes input power mirror 1 radius of curvature cavity configuration etc e Output an image of the beam shape e Calculate waist size of the beam at a chosen point These are by no means all the possible things that could be implemented or useful but some would probably be a good starting point We hope that the simulations will rema
26. r min value max value delta_x_slider setLimits 0 3 5 PI 3 5 PI delta_x_slider setTickCount 13 delta_x_slider setValueType ValueType DECIMAL delta_x_slider setPrecision 2 delta_x_slider setRenderMaxMinLabel false delta_x_slider setRenderValueLabel false x y length num height min max meter new LEDMeter 557 490 349 30 20 0 0 1 0 meter setGradient meter setGapSize 0 xpos ypos num labels xstep min max meterAxis new LEDMeterAxis 557 500 10 33 0 1 meterAxis setFontSize 12 31 Importing images cavityimage loadImage cavitypng png _logo new Logo this 770 570 true LogoSize Size25 Create fonts titleFont createFont verdana 20 meterTitleFont createFont verdana 13 expFont createFont verdana 11 Write the explanation here explanationtext This applet graphs the internal power of an optical cavity as a function of tuning for transverse modes 00 red 10 green and 20 blue The slider below the graph controls the tuning of the second mirror Below the meter shows the percentage of the input power that is outputted through the second mirror at the current value of delta x the tuning The cavity is drawn below the graph Each mirror has a reflectivity slider The 1 RoC slider controls the inverse of the Radius of Curvature of the second mirror 1 RoC 0 represents a flat mirror infinite RoC and 1 R
27. right of the calculator below the text box indicates the percentage of the input power that is outputted through the second mirror reaching the rightmost photodetector We will call this meter the transmitted power meter or simply the meter This meter is linked to the Az slider so it shows the total power output for the tuning value given by the Az slider 3 3 2 The mathematical details This calculator is essentially only using two equations It is graphing the base 10 logarithm of the internal power function given in Chapter 2 by Equa tion 2 3 The actual function being plotted then is Ti 1 Ri Re 2rirecos 2kL wre n m 1 log P1 log 3 1 18 where Vrt is the Guoy phase given by Equation 2 7 Then the calculator computes the transmitted output power of the cavity by adding at the chosen value of Az the internal power functions for each mode and multi plying the sum by the transmission coefficient of the second mirror Since R T 1 this coefficient is simply given by T 1 Ro 3 2 where R is controlled by the R slider Therefore for a given beam with 1 W of input power and mode TEMym the transmitted power Prans iS calculated as Pirans 101090 i To 3 3 In the cavity used in this calculator there are three beams three different transverse modes Each beam for the sake of overall simplicity has an input power of iW so that the total input power for the cavi
28. s Figure 2 6 shows the beam profile along the z axis of a Gaussian beam In contrast to plane waves when a Gaussian beam propogates it ac quires a phase shift This effect is most pronounced for Hermite Gaussmodes close to the beam waist where the phase velocity is slightly slower than com pared to plane waves 3 The Guoy phase y can be calculated along the z axis by Z z0 p z arctan 2 5 ZR 11 where zp is known as the Rayleigh range of the beam given by 2 6 beam waist Figure 2 6 The beam profile along z of a Gaussian beam The Guoy phase calculated with Equation 2 5 is the phase shift of the beam in one direction For our purposes we are interested in the round trip Guoy phase the phase it acquires by the time it reflects off the second mirror and returns to the first The round trip Guoy phase needed in Equation 2 3 is simply two times the Guoy phase given by Equation 2 5 Ori 24 p z 2 arctan 2 7 Since in Equation 2 3 the Guoy phase is multiplied by the term n m 1 it can be seen that the overall phase shift is larger for higher order modes higher values for n and m Therefore beams with different mode orders will have different phase shifts and the plots of their power enhancement functions will be offset from one another They will also then have different resonance frequencies for a given cavity Figure 2 7 shows a plot of Equation 2 3 for modes 00
29. s and returns to the beam splitter Once the light reaches the beam splitter again the two beams interfere and the signal is detected by a photodetector When a gravitational wave is incident on the interferometer it should appear as an oscillation of the distance between the freely falling test masses mirrors at the frequency of the graviational wave Free fall is approximated by suspending the mirrors by thin wires within a vibration isolation system The mirrors move due to the distortion in space and the interference pattern of the light at the beam splitter changes accordingly 2 1 Optical cavities The important components for our purposes are the optical cavities also known as optical resonators within the interferometer arms We will focus on these for the remainder of the text Please note that we are assuming PDtrans PDrefi Figure 2 1 Diagram of a basic interferometer a loss less cavity for our simulations A Fabry Perot cavity as shown in Figure 2 2 consists of two highly reflecting mirrors The reflectivity R of a mirror is the percentage of the light s incident power that is reflected by that mirror r also represents the reflectivity of the mirror but is the percentage of the incident amplitude that is reflected They are related by R r The light that is not reflected from the mirror passes through it a process called transmission governed by R T 1 where T t is the transmission coe
30. slider changes the Rayleigh range zp of the beam and therefore the round trip Guoy phase see Equation 2 7 In particular the larger the radius of curvature as 1 RoC approaches 0 the smaller the Guoy phase shift This is why as you decrease 1 RoC the peaks of the internal power for the different modes become very close and eventually overlap At an infinite radius of curvature the Guoy phase shift is 0 and multiplying the phase shift by n m 1 as in Equation 2 3 will not change the overall phase shift for different modes An undercoupled cavity is one in which T is less than T so Ry is greater than R This type of cavity has little circulating power If you set the R slider to be greater than the Ro slider you can notice that the internal power becomes small and even on resonance there is very little transmitted power compared to an impedance matched cavity An overcoupled cavity is one in which R is less than R It has a high circulating power compared to an impedence matched cavity which has re flectivities equal to the lower reflectivity in the overcoupled cavity However with R high T is low and there is little transmitted power These are some of the simple cavity properties that this calculator illustrates At the very least it should give users a more intuitive sense about how the cavity parameters affect its resonance properties It is probably most useful if one views the circulating internal po
31. slider to its desired position Also if you set the program s focus to a particular slider by clicking on it that slider can then be moved by one pixel at a time using the left and right arrow keys This helps mostly for the Ax slider when trying to choose a precise tuning value When the slider value is changed in this way the associated text field changes to the chosen slider value Additionally the slider value can be changed by directly typing the desired value into the text field and then hitting the Enter key Note that if Enter is not pressed the slider will not update to that value The R R2 and Az sliders all have a maximum precision of two decimal places If you type in a value of say 0 487 the slider should round this number to 0 49 However it is best if at most two decimal places are used so unexpected rounding does not occur Finally please note that the RoC slider actually controls the inverse of the radius of curvature of the second mirror That is the value represented by the RoC slider is actually To This is done so the radius of curvature can range from L the cavity length 1 m to infinity a flat mirror Infinity can not be represented by a slider but 1 in finity is essentially zero Therefore the RoC slider ranges from 0 0 for an infinite radius of curvature to 1 0 for a radius of curvature equal to the cavity length of 1m The Az slider does not have a text field and it does not dis
32. textFont expFont f i11 0 33 text explanationtext 579 49 295 340 Add logo _logo draw Power meter Apparently in Processing log is natural log Also now each beam has an input power of 1 3 W so the total input power is 1W float sum_delta_x pow 10 mode00 computePoint delta_x pow 10 mode10 computePoint delta_x pow 10 mode20 computePoint delta_x 3 float transmitted_power sum_delta_x 1 R2 meter setValue transmitted_power meter response meterAxis display Adjusting the line drawn off the slider for rounding errors if chkWhiteLine isSelected line delta_x_slider getValuef float 478 22 293 368 delta_x_slider getValuef float 478 22 293 15 These are the functions that are being plotted All the same except for different values of n class InternalPower00 implements GraphCallback int n m float Power00 float zR psiRT public float computePoint float t_delta_x delta_x t_delta_x n 0 m 0 zR sqrt L RoC L psiRT 2 atan L zR 34 Need to convert the natural log to log base 10 Power00 log 1 R1 1 R1 R2 2 sqrt R1 sqrt R2 cos delta_x psiRT n m 1 log 10 return Power00 class InternalPower10 implements GraphCallback int n m float Power10 float zR psiRT public float computePoint float t_delta_x delta_x t_delta_x n 1 m 0 zR sqrt L RoC L
33. ty is 1 W Therefore the maximum amount of power that can be transmitted through the second mirror is 1 W as shown on the transmitted power meter As explained in Chapter 2 this can only occur in an impedence matched cavity It is convenient for the transmitted power meter to range in value from 0 to 1 because it can then be thought of as representing the percentage of the input power that is being transmitted e g a value of 0 89 is 89 of the input power To improve the appearance of the plot of the cavity s internal power it is typically graphed on a logarithmic scale This is done because the peaks in the plot on a normal scale become extremely sharp especially for highly reflective mirrors Plotting on a logarithmic scale smooths out the graph a little bit When you look at the calculator s graph you may wonder why we are plotting the logarithm of the power rather than plotting the power on a logarithmic scale We chose to do this because having axes with evenly spaced tick marks is much more compatible with the two dimensional plotting function that I wrote than having axes on a logarithmic scale When viewing the graph the user still has the ability to see what value the power has and it does not make the calculations much more difficult 19 3 3 3 How to use the GUI controls The controls have been designed to be as simple as possible to use Each slider can be controlled by the mouse by either clicking or dragging the
34. wer function Equation 2 3 while running the program to see how the plots are being affected 21 3 3 5 Intentional limitations In the first and current version of this calculator the user only has limited control over the cavity s parameters Only four parameters can actually be set by the user the reflectivity of the first mirror the reflectivity of the second mirror the radius of curvature of the second mirror and the tuning of the cavity Giving the user many more values to change would make it more difficult for the user to really learn a few basic concepts about cavities and could be confusing We wanted to make this version simple so that it would illustrate only a few key points Therefore the user cannot change parameters such as the configuration of the cavity the cavity length the curvature of the first mirror the beam modes etc We felt that this would be too much information to include in an educational or reference tool though it probably would not be very difficult to include in future versions Also this applet is not quite a calculator in the usual sense of the word In a usual calculator one would enter some values and get precise numerical answers in return While this is certainly useful outputting exact numbers is not necessarily useful when simply portraying relations between parameters It may also become overwhelming or confusing for a student using the tool if many numbers are outputted to the screen A
35. y two dimensional function Other and probably better plotting libraries exist in Java or Processing but it was an integral part of my education in programming to write a plotting library from scratch Rather than simply using a pre made library I designed my own as an exercise in becoming more familiar with the language and object oriented programming All three of these libraries were designed to be very general so they could be incorporated into various programs I will not go into further detail here about how these libraries work because it is all explained in the actual code The code for the optical resonator calculator which uses these libraries is included in Appendix A This should give a more complete picture of how the various libraries and components are linked to create the Optical Resonator Calculator 16 3 3 The User s Manual The Optical Resonator Calculator program has been uploaded to the project s webpage as a Java applet with some explantory text This Java applet mod els some features of an optical resonator also known as an optical cavity using a graphical user interface GUI The GUI allows a user to change some of the cavity s parameters Figure 3 1 is a screenshot of the actual calculator Please refer to this image throughout the following discussion i la x Optical Resonator Calculator This applet graphs the internal power of an optical cavity as a function of tuning for transverse modes 0
36. z a value expressed in degrees These are the same graphs but plotted as functions of different equivalent variables The difference lies in rewriting the argument of the cosine term from Equation 2 3 with Ax instead of 2kL Az is expressed as an angle measurement where 27 radians or 360 corresponds to a displacement of the mirror by one wavelength We have chosen to plot over Ax instead of f because we are assuming a system with a fixed laser frequency and a tunable mirror displacement 2 4 Behavior of a cavity For the moment we are interested in the resonant properties of a cavity as it affects the internal power of the cavity and the transmitted power through the second mirror This behavior for a two mirror cavity depends on the length of the cavity and on the reflectivities of the mirrors There are then three different cases that can govern this behavior If Ti lt To the cavity is undercoupled If Ti To the cavity is impedance matched If T gt To the cavity is overcoupled Note that these can be rewritten as If R gt Ro the cavity is undercoupled if Ry lt R the cavity is overcoupled An undercoupled cavity will transmit little power through mirror 2 and will not have much circulating power An overcoupled cavity will have the greatest circulating power and other internal resonance conditions However an impedance matched cavity maximizes the power transmission and it is the only condition under which the cav
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