2015 Laboratory C: Mie Scattering and Analysis of Focused Beam Propagation

I. Properties of Rayleigh and Mie Scattering

Goal: This portion of the GUI Interaction provides insight into the characteristics of Rayleigh and Mie Scattering.
  1. Launch Mie Simulator (MieSimulator_v1_05R1.exe) GUI tool. This tool calculates Scattering Coefficient, Scattering Cross Section, Reduced Scattering Coefficient, Phase Function, Average Cosine of Phase Function (g), S1 and S2 for a given wavelength range. Number density is given in a volume of 1mm3.
  2. Select "Mono Disperse".
  3. Consider visible and NIR spectral region by entering Start, End, and Step Values of 500, 1200, and 5, respectively in the Wavelength (nm) panel. Unless otherwise stated, use this wavelength range for following simulations.
  4. Enter parameters representative of small cellular organelles: Diameter of 0.1 μm, Volume Fraction of 0.01, Sphere refractive index of 1.4 and Medium refractive index of 1.37.
  5. Click the Run Simulation button.
  6. Examine carefully the plots of Scattering Cross Section, Scattering Coefficient, Reduced Scattering Coefficient and Average Cosine of Phase Function (g) vs Wavelength. If need to zoom those plots, use the wheel in your mouse. Click Display Data button to visualize current data.
  7. We use A[fRay(λ/λ1000)-4+(1-fRay)(λ/λ1000)-bMie] fitting relationship specified in Steve L. Jacques's review paper ("Optical properties of biological tissues: a review" Phys. Med & Bio. 58(2013) R37-R61) to find fitting parameters fRay and bMie in μs' Power Law Fitting. λ1000 is 1000nm and A is μs' at 1000nm. Select μs' Power Law Fitting tab and click Best Fit button to find fitting parameters. You may also use fRay and bMie sliders for manual fitting. Take note of A, fRay and bMie values to understand the magnitude and wavelength dependence of μs'.
  8. Repeat the steps I.4-I.7 for parameters representative of a nuclei: Diameter of 4.0 μm, Volume Fraction of 0.01, Sphere refractive index of 1.39 and Mediumrefractive index of 1.37. Note key differences relative to scattering characteristics of small organelles. Change the Concentration to observe the changes in μs'.
  9. You need to prepare a scattering phantom with 804nm diameter polystyrene spheres in a aqueous medium. Expected reduced scattering coefficient is 0.1mm-1at 632.8nm. Refractive indexes of polystyrene sphere and water are 1.59 and 1.33 respectively. Find the concentration of polystyrene spheres in a 1ml volume.
  10. Enter Diameter of 0.1 μm, Concentration of 1e9 spheres/mm3, Sphere refractive index of 1.40 and Medium refractive index of 1.33 and Run simulation. Observe Phase Function profiles for Parallel(Para) and Perpendicular(Perp) polarization. Select Para or Perp and slide the wavelength slider from 500nm to 1200nm. Understand the relationship between the Phase Function and Average Cosine of Phase Function.
  11. Following table shows three poly disperse Log Normal scatterer distributions in a tissue model (J. Nguyen et al., Biomed. Exp. 4(10) 2013). The distribution 1, 2 and 3 represent small protein complexes, organelles such as lysosomes and mitochondria, and nuclei respectively. Calculate the scattering coefficient of each distribution at 620nm. Select Poly Disperse and change starting and ending wavelength to 620nm. Use 51 discrete sphere sizes and apply following parameters. What is the scattering coefficient of all distributions? (Hint: μs All = Σ μs)
 Distribution  Mean Diameter ± Std.Dev.(μm)  Number Density(mm-3)  nScatterer    nMedium
       1             0.06 ± 0.4                  4x1010             1.46       1.33       
       2             0.9 ± 0.3                   5x107              1.40       1.35
       3             9.6 ± 0.1                   5x104              1.39       1.37

II. Effect of numerical aperture (NA) on focus beam propagation

Goal: Get familiar with the "FocusedBeamSimulator" GUI tool and understand the effect of NA on the focal volume.

We consider flat "x-polarized" plane wave incident upon an apalantic lens and Cartesian coordinates. The lens is placed 1000μm below the origin to have its nominal focal point at the origin. The XY and XZ detectors are placed parallel to the x-y and x-z planes respectively. The plots in the left column shows the amplitude and phase obtained from the analytical solution (Richard and Wolf, Proc. Royal Soc. Lond. A 253 (1959)). The analytical solution results are used as a reference in this lab exercise. The plots in the right column shows the results obtained from Huygens-Fresnel approach. The Huygens-Fresnel method can provide results for both scattering and non scattering medium.

You may increase the detector resolution by sliding Detector Resolution slider to improve image quality. Note that each increment approximately doubles the simulation time. We recommend setting Detector Resolution to Medium for following exercises.

  1. Launch the Focused Beam Simulator (FocusedBeamSimulator.exe) GUI tool.
  2. Set Numerical Aperture to 0.4.
  3. In the Output Detector Plane Selection, select XY plane (Z = 0).
  4. Click Run Simulation.
  5. Wait until you see HF Inc. Done! in the progress slot in between Run Simulation and Close.
  6. Choose different electric field components in the Electric Field Component Selection. Change Plot Scale (Amplitude) to Log10 see details. Observe dominant Excomponent for x-polarized plane wave incident.
  7. Use the wheel in your mouse to zoom the plot. The theoretical Airy disk radius is given by 0.61 λ / NA. Estimate Airy disk radius from your results and compare it with the theoretical value.
  8. Repeat steps 3-7 for Numerical Aperture = 1.1. Compare your results for the different numerical apertures.
  9. Which numerical aperture provides the tightest focal spot?
  10. Set Numerical Aperture to 0.4.
  11. In the Output Detector Plane Selection, select XZ plane (Y = 0).
  12. Click Run Simulation.
  13. Observe amplitude and phase profiles in the focal volume.
  14. Repeat steps 11-13 for Numerical Aperture = 1.1. Compare the results for the different numerical apertures.
  1. This GUI tool is designed for 'flat' x-polarized incident. Laser sources are producing 'Gaussian' beams and you are asked to modify the model for Gaussian x-polarized incident. What additional information do you need? How do you implement it?

III. Focal field distortions

Goal: Understand the focal field distortion by spherical scatterers.

This GUI tool finds the maximum amplitude and displays it above the amplitude plot. The phase at the nominal focal point (Phase @ (0,0):) is shown above the phase plot.

  1. Set Numerical Aperture to 0.7
  2. Click Load Mie Data button to load three pre-computed Mie data tables to study the distortion by single and two scatterers.
  3. The first part of this lab is to understand the effect of single scatterer.
  4. In the Output Detector Plane Selection, select XZ plane (Y = 0) and in the Electric Field Component Selection panel, select Ex.
  5. Check Scatterer 1 and select the diameter as 5.0 μm.
  6. Set the location of the scatterer center X:, Y: and Z: to -2, 0, -4 μm, respectively.
  7. In the Huygens-Fresnel Approach panel, select Scattered.
  8. Click Run Simulation.
  9. Change Plot Scale to Log10 and observe amplitude and phase profiles. Note the Max. Amp.
  10. Toggle between Incident and Incident+Scattered radio buttons and observe the amplitude and phase changes.
  11. Change the diameter of the Scatterer 1 to 2.5 μm and repeat steps 6 - 10. If time permits, change the diameter of the Scatterer 1 to 1.0 μm. and repeat steps 6 - 10.
  12. Select the XY plane (Z = 0) and repeat steps 5 - 10.
  13. The second part of this lab is to understand the effect of two scatterers.
  14. Select the XY plane (Z = 0).
  15. Check Scatterer 1 and select the diameter as 5.0 μm. Set the location of the scatterer center X:, Y: and Z: to 0, 0, -10 μm, respectively.
  16. Click Run Simulation.
  17. Select Incident+Scattered radio button and observe amplitude and phase profiles. Note the Max. Amp.
  18. Uncheck Scatterer 1. Check Scatterer 2 and select the diameter as 2.5 μm. Set the location of the scatterer center X:, Y: and Z: to 0, 0, -3 μm, respectively.
  19. Click Run Simulation.
  20. Observe changes in amplitude and phase and note the Max. Amp.
  21. Check Scatterer 1 and Scatterer 2. Click Run Simulation.
  22. Compare changes in Max. Amp for different cases.

IV. Amplitude change and focal spot displacement

Goal: Understand the amplitude change by a single spherical scatterer.
  1. Set Numerical Aperture to 0.7.
  2. In the Output Detector Plane Selection, select XZ plane (Y = 0).
  3. in the Electric Field Component Selection panel, select Ex.
  4. Select Linear Scale.
  5. Click Run Simulation. Record Max. Amp. for non scattering case.
  6. Check Scatterer 1 and select the diameter as 5.0 μm.
  7. Set location of scatterer center X:, Y: and Z: to 0, 0, -2.51 μm, respectively.
  8. Click Run simulation. Record Max. Amp. and location of the maximum amplitude.
  9. Repeat the step 8 for Z = -5, -7.5, -10, -12.5, -15, -30 and -45 μm.
  10. Calculate the amplitude change and focal spot displacement relative to non scattering case at each scatterer location.
  11. Plot Z vs amplitude change and Z vs focal spot displacement.