Utilizing COMSOL Multiphysics, a program developed for finite element modeling of complex engineering systems, we developed a mathematically rigorous finite element model of light using Maxwell's Equations in the frequency domain. This model was used to determine the effects of frequency, power, permeability, permittivity, and multiple laser sources on the effective range of a LIDAR system used in self-driving cars.

1. LIDAR
ALEX BERTINO
STEPHANIE HALBERT
CO-ADVISOR: RYAN BURKS

2. WHAT IS LIDAR?
• LiDAR (Light Detection and
Ranging)
• 𝐷 =
𝑐∗𝑡
2
• Micropulse system
• Coherent
• Eye safe
• Pulsating laser
• Cloud Point

3. VELODYNE MODEL HDL-64E
• 64 channels
• 26.9° FOV
• Wavelength: 903 nm
• 60W w/ 12 V – 32 V @ 4
amps
• Single or Double return mode
• Single: 1,300,000 pts/s
• Double: 2,200,000 pts/s
• Rotates @ 5 Hz – 20 Hz
• 4 blocks with 16 lasers
• Transmitter is the side circles
and receiver is center
• 120 meter cars and foliage
(.80 reflective) and 50 meter
for pavement (.10 reflective)
• 5 ns pulse
• 8” Outer Diameter
• Temp range: -10°C - +60°C

4. DERIVATION OF WAVE EQUATION
• Maxwell’s Equations
• Take curl of 𝛻 × 𝐸 to get wave eqn
• Euler’s Equation
• Fourier Transform
• Assume uncharged medium
(𝜌 = 0)
𝛻 × 𝐵 = 𝜇𝜎𝐸 + 𝜇𝜖
𝜕𝐸
𝜕𝑡
𝛻 × 𝛻 × = 𝛻 𝛻 ∙ − 𝛻2
𝛻 𝛻 ∙ 𝐸 = 𝛻 𝜌/𝜖𝛻 ∙ 𝐸 = 𝜌/𝜖
𝛻 𝜌/𝜖 − 𝛻2 𝐸 = −
𝜕
𝜕𝑡
𝜇𝜎𝐸 + 𝜇𝜖
𝜕𝐸
𝜕𝑡
0 − 𝛻2 𝐸 = −𝜇𝜎
𝜕𝐸
𝜕𝑡
− 𝜇𝜖
𝜕2 𝐸
𝜕𝑡2
𝑒 𝑖𝜔𝑡
= cos(𝜔𝑡) + 𝑖sin(𝜔𝑡)
𝜕
𝜕𝑡
𝑒 𝑖𝜔𝑡
= 𝑖𝜔𝑒 𝑖𝜔𝑡
,
𝜕2
𝜕𝑡2 𝑒 𝑖𝜔𝑡
= −𝜔2
𝑒 𝑖𝜔𝑡
𝛻2
𝐸 + 𝜇𝜖𝜔2
− 𝑖𝜇𝜎𝜔 𝐸 = 0
𝛻 × 𝐸 = −
𝜕𝐵
𝜕𝑡
𝛻 × 𝛻 × 𝐸 = −
𝜕
𝜕𝑡
𝛻 × 𝐵

5. FREQUENCY DOMAIN:
WAVE
ELECTROMAGNETIC
EQUATION
• E: Electric Field
• μ: Permittivity of Medium
• 𝜖: Permeability of Medium
• ω: Angular Wave Frequency
• i: Imaginary Unit

6. DIFFICULTIES
WITH FINITE
ELEMENT
ANALYSIS
• Resolution of Small
Frequencies (10 elements
per wavelength)
• Computationally
Expensive (Especially 3D)
• Boundary Interactions
• Reflection
• Refraction
• Metallic Surfaces
• Scattering

7. SIMULATION COMPROMISES
• Lower Frequency Waves (30 MHZ)
• Larger Laser Width (10 m)
• 2D Simulation
• Continuous Medium

8. EFFECTS STUDIED
• Frequency ( 10 – 100 MHz )
• Power ( 1 – 1016 W/m )
• Relative Permeability ( 1 – 10 )
• Relative Permittivity ( 1 – 10 )
• 2 Sources spaced apart w/ one source being at angle towards
other

9. LIDAR SOURCE
BOUNDARY
CONDITION
• Sinusoidally Varying
Frequency
• Plane Wave

10. SCATTERING
BOUNDARY
CONDITION
• Approximates Effects of Far
Field
• Flux Boundary Condition
• 1st Order Approximation
• Valid for Contact Angles
Close to 90o

11. LASER INTENSITY -
GAUSSIAN
• c: Speed of Light through Medium
• 𝜖 o: Permittivity of Free Space
• n: Material Refractive Index
• Units of W/m2
• Power Flux in the Direction of
Laser Propagation
• Es = E: Electrical Field
• r: Distance from the center of the
beam
• ω0: Radius where intensity has
decreased to 1/e (0.135) of its

12. WAVE PROPAGATION (TIME DOMAIN)
30 MHZ 100 MHZ

13. DEFAULT
RESULTS
• Frequency: 30 MHz
• Electric Field: 3.1 * 104
V/m
• Distance: 100 m

14. EFFECT OF
FREQUENCY
• 10 – 100 MHz
• Laser Intensity Increases
at High Frequencies
• Slowly Decaying Relative
Intensity

15. EFFECT OF
POWER
• 1 – 1016 W/m
• Linear Increase with
Power
• Exponential Increase with
Electric Field
• Rapidly Decaying Laser
Intensity

16. EFFECT OF
RELATIVE
PERMEABILITY
• 1 – 10 (unitless)
• Similar Results to
Increased Frequency
• Shortened Lidar
Wavelength
• Neglects Conductivity,
Non-Continuous Medium

17. EFFECT OF
RELATIVE
PERMITTIVITY
• 1 – 10 (unitless)
• Identical Results to
Permeability

18. INTENSITY
INCREASED FREQUENCY VS. INCREASED
POWER
FREQUENCY POWER

19. INTERFERENCE
• Have 2 (or more) source
points
• Separation between sources
>> λ
• Constructive vs. Destructive
• Light waves
• Can’t see but intensity can be
measured

20. DERIVATION OF INTENSITY FOR MULTI-
SOURCES
𝐸 = 𝐸0 cos 𝛼 + 𝜔𝑡 = 𝐸0 𝑒 𝑖(𝛼+𝜔𝑡)
𝐸0
2
= (𝐸0 𝑒 𝑖𝛼)(𝐸0 𝑒 𝑖𝛼)
∗
𝛼 = − 𝑥𝑘 + 𝜀 = −(
2𝜋𝑥
λ
𝑥 + 𝜀)
𝐼 = 𝐼3 + 𝐼2 + 𝐼12
𝐸0
2
=
𝑖=1
𝑁
𝐸0𝑖
2
+
𝑗>𝑖
𝑁
𝑖=1
𝑁
𝐸0𝑖 𝐸0𝑗
𝐸0
2
= 𝑁2 𝐸01
2
𝐼 = 𝜖𝑣𝐸2
𝐼1 = 𝐸1
2
, 𝐼2 = 𝐸2
2
, 𝐼12 = (𝐸1 ∙ 𝐸2)2
𝐸0
2
=
𝑖=1
𝑁
𝐸0𝑖
2
+ 2
𝑖=1
𝑁
𝑗=1
𝑁
𝐸0𝑖 𝐸 𝑜𝑗 cos(𝛼𝑖 − 𝛼𝑗)
• Where E: electric field, I: intensity, α:
phase

21. d
x1
x2
L
𝑥2 = 𝑥1 + 𝐿
𝑑 sin 𝜃 = 𝑚𝐿 = 𝑚∆𝑥
∆𝛼 = 𝑥2 𝑘 + 𝜀 − 𝑥1 𝑘 + 𝜀 = ∆𝑥
2𝜋
λ
∆𝛼 =
2𝜋
λ
𝑑 sin 𝜃
𝑚
𝐿 = 𝑥2 − 𝑥1 = ∆𝑥
θ
𝐸0
2
=
𝑖=1
𝑁
𝐸0𝑖
2
+ 2
𝑖=1
𝑁
𝑗=1
𝑁
𝐸0𝑖 𝐸 𝑜𝑗 cos(𝛼𝑖 − 𝛼𝑗)
• Young’s Two-Split Experiment
• Finding the intensity @ any
point
O
LIGHT
INTERFERENCE
GEOMETRY
𝐼 = 𝜖𝑣𝐸2

22. EFFECT OF ANGLE
• Angle 18°- 48°
• Assumptions:
monochromatic coherent
wave
• Intensity decreases as
angle increases

23. CONCLUSION
• Increasing Frequency Leads to a More Favorable Intensity
Distribution
• Increased Permeability/Permittivity favorable in a Non-
Conducting, Continuous Medium
• Increase of Intensity with a lower intersection Angle

24. FUTURE GOALS
• Implement Beam Envelope/Ray Optics
• Reflection/Refraction Across a Non-Continuous/Conducting
Medium
• Larger Lidar Frequency
• Smaller Beam Width
• Beam Pulse and Response

25. REFERENCES
• https://www.quora.com/How-does-LIDAR-work
• https://autonomoustuff.com/wp-content/uploads/2018/04/63-
9194_Rev-G_HDL-64E_S3_Spec-Sheet_Web.pdf
• http://www.hizook.com/blog/2009/01/04/velodyne-hdl-64e-laser-
rangefinder-lidar-pseudo-disassembled
• http://felix.rohrba.ch/en/2015/lidar-footprint-diameter/
• http://velodynelidar.com/hdl-64e.html
• https://en.wikipedia.org/wiki/Lidar
• https://www.comsol.com/blogs/autonomous-vehicles-putting-
technology-in-the-drivers-seat/

26. • https://en.wikipedia.org/wiki/Inhomogeneous_electromagnetic_wav
e_equation
• https://www.comsol.com/blogs/modeling-metallic-objects-in-
wave-electromagnetics-problems/
• https://em.geosci.xyz/content/maxwell1_fundamentals/frequency_
domain_equations.html
• https://www.rp-photonics.com/optical_intensity.html
• https://www.newport.com/n/gaussian-beam-optics
• https://www.comsol.com/blogs/modeling-of-materials-in-wave-
electromagnetics-problems/
• https://www.comsol.com/video/equation-based-modeling-custom-
simulations-comsol-multiphysics

27. • https://www.comsol.com/blogs/using-perfectly-matched-layers-
and-scattering-boundary-conditions-for-wave-electromagnetics-
problems/
• https://www.comsol.com/blogs/taking-care-of-fast-oscillations-
wave-optics-module/
• https://www.comsol.com/blogs/guide-to-frequency-domain-wave-
electromagnetics-modeling/
• https://www.comsol.com/blogs/introducing-ray-optics-module/
• https://www.comsol.com/blogs/computational-electromagnetics-
modeling-which-module-to-use/
• https://www.comsol.com/blogs/how-to-use-the-beam-envelopes-
method-for-wave-optics-simulations/

28. • https://en.wikipedia.org/wiki/Leonhard_Euler#Mathematical_n
otation
• https://en.wikipedia.org/wiki/Fourier_transform
• https://www.rp-photonics.com/gaussian_beams.html
• https://en.wikipedia.org/wiki/Wave_interference
• http://www.physicsclassroom.com/class/light/Lesson-1/Two-
Point-Source-Interference
• https://www.microscopyu.com/techniques/polarized-
light/principles-of-interference
• Hecht, Eugene. Optics. 4th ed., Addison Wesley, 2002.
• http://1.bp.blogspot.com/-
OLSZv2U7Mvs/T9ONFCHhD1I/AAAAAAAACSM