Problem 9 - Optical Compass

Team Brazil – University of Campinas

Maria Carolina Volpato and Denise Christovam

1 Team Brazil – University of Campinas Problem Statement

Bees locate themselves in space using their eyes’ sensitivity to light polarization. Design an inexpensive optical compass using polarization effects to obtain the best accuracy. How would the presence of clouds in the sky change this accuracy?

2 Team Brazil – University of Campinas Visualization of the phenomenon

Problem 9- Optical Compass 3 Team Brazil – University of Campinas Light Scattering Incident

Scattered Backward Forward scattering scattering

Scattered Light

Incoming Light 4 Problem 9- Optical Compass Particle Team Brazil – University of Campinas Light Scattering

Rayleigh Scattering Mie Scattering

Conditions to Rayleigh scattering are satisfied!

Problem 9- Optical Compass 5 Team Brazil – University of Campinas Linear Polarization

➔ For the observer at PMP polarization plane collapses into a line ➔ 90o from the source of light

Problem 9- Optical Compass 6 Team Brazil – University of Campinas Mapping the sky

➔ Rayleigh sky model describes how the maximally polarized light stripe varies with rotation

Degree of polarization at sunset or sunrise. https://en.wikipedia.org/wiki/Rayleigh_sky_model Problem 9- Optical Compass 7 Team Brazil – University of Campinas Materials and design ➔ Polarizing sheets ➔ Guillotine Paper Cutting Machine ➔ Adhesive tape ~ R$ 30.00 ($6.00) ➔ Cardboard A4 (120 g/m²)

o 12 petals – 30 72 petals – 5o 18 petals – 20o Problem 9- Optical Compass 8 Team Brazil – University of Campinas Angular Resolution - Malus Law

Angular Resolution ~ 20º

Polarization Direction Problem 9- Optical Compass 9 Team Brazil – University of Campinas Setup- Vertical and Horizontal Mapping

Vertical

Horizontal

Problem 9- Optical Compass 10 Team Brazil – University of Campinas Vertical Angle χ Φ= 335º Ө

0º 40º 80º 120º

Problem 9- Optical Compass 11 Team Brazil – University of Campinas Horizontal Angle

Φ Ө ~ 10º 0º 60º 120º

180º 240º 300º Problem 9- Optical Compass 12 Team Brazil – University of Campinas Temporal evolution of the shadow

➔ Angle in relation to the ground Ө= 25±1º ➔ Approximate geographical position: ◆ Latitude: 23º 00’ 21” S ◆ Longitude: 46º 50’ 20” W ➔ Window facing SSW (276º form north) ➔ Total time of acquisition 266 min ( from 12 PM to sunset) ➔ Performed at 09/24/2019

https://www.timeanddate.com/sun/@3460598?month=9&year Problem 9- Optical Compass =2019 13 Team Brazil – University of Campinas Temporal evolution of the shadow

Problem 9- Optical Compass 14 Team Brazil – University of Campinas Presence of clouds

Clear sky Partly Cloudy Cloudy

Typical radius of the droplets ~ 10² nm ~ λ

Mie Scattering dominates

Problem 9- Optical Compass 15 Team Brazil – University of Campinas Conclusions ➔ It is possible to build a cheap device R$ 30.00 - $ 6.00

➔ It indicates N-S direction Rayleigh Scattering

➔ Clouds reduces the accuracy Mie Scattering

Problem 9- Optical Compass 16 Team Brazil – University of Campinas

Thank You!

17 Team Brazil – University of Campinas How do bees locate themselves?

https://www.beeculture.com/bees-see-matters/ Problem 9- Optical Compass 18 Team Brazil – University of Campinas A model for radiation

The electric fields propagate radially

Time-averaged Poynting vector:

“p” wave

Total time-averaged power radiated:

David J. Griffiths, Introduction to Electrodynamics, Prentice Hall, 1999

Problem 9- Optical Compass 19 Team Brazil – University of Campinas Rayleigh x Mie If α≪λ : Rayleigh

Angular dependence Distance between Same dependence particle and with frequency observer

We recover dipole radiation!

If α is close to λ: Mie Distortion of the radiation in one direction

Problem 9- Optical Compass 20 Team Brazil – University of Campinas Rayleigh x Mie Recovering the toroidal profile of dipole radiation:

dipole z

Light scatters at xy plane – polarization depends on our position in relation to the Sun 21 Team Brazil – University of Campinas Atmospheric Conditions

% Kinetic Diameter (nm)

N2 78 0.364

O2 21 0.346

Conditions to Rayleigh scattering are satisfied! CO2 0.03 0.330

H2O 3 0.265

Ar 0.9 0.340

Problem 9- Optical Compass 22 Team Brazil – University of Campinas Linear polarization

At PMP, the polarization Plane: L

L L

Plane the observer sees Observer’s Plane of view polarization

At B, the observer sees E⊥ :

Sunbeam

L Observer Plane the observer sees

23 Problem 9- Optical Compass Team Brazil – University of Campinas Degree of polarization

At B, the observer sees E⊥ : Sunbeam

L Observer Plane the observer sees

As sunlight is unpolarized at first

We can define the degree of polarization P(γ):

24 Problem 9- Optical Compass Team Brazil – University of Campinas Navigation

Scattering Angle Sun Observation point

We always need some other info, like the time of the day (or measure for extended periods of time)

y (W)

z (N)

Problem 9- Optical Compass 25 Team Brazil – University of Campinas Analyse the data

➔ ImageJ

Problem 9- Optical Compass 26 Team Brazil – University of Campinas Deﬁning angular resolution

Problem 9- Optical Compass 27 Team Brazil – University of Campinas Angular resolution - Malus Law

Deviation from Malus Law Intrinsic width of shadow: Limited due to small ~20o in 180o (“efficiency” of misalignments/anisotropies compass: ~89%) resolution Problem 9- Optical Compass 28 Team Brazil – University of Campinas Temporal evolution of the shadow

Setup

Problem 9- Optical Compass 29 Gaussian – Calibration

Deviation from Malus Law due to small misalignments/ anisotropies Calibration curve for degree of polarization and sun 72 petals – 5o Intrinsic width of shadow: position ~20o in 180o (“efficiency” of compass: ~89%)

Limited

resolution 30 Timelapse – T0 = 1 PM

31 Timelapse – T0 = 5:26 PM

32 Temporal evolution of the shadow

• Shadow sharpens • Strip rotates indicating solar position

33 Team Brazil – University of Campinas Accuracy

Problem 9- Optical Compass 34 Team Brazil – University of Campinas Error Analysis - Intensity

Problem 9- Optical Compass 35 Team Brazil – University of Campinas Error Analysis - Position

土5º

Problem 9- Optical Compass 土8º 36 Team Brazil – University of Campinas Error Analysis - Position

Problem 9- Optical Compass 37 Team Brazil – University of Campinas Atmospheric conditions - Cloudy weather

Conditions to Rayleigh scattering are not satisfied

Mie Scattering dominates

Typical radius of the ~ λ droplets ~ 10² nm No pattern formed Problem 9- Optical Compass 38 Team Brazil – University of Campinas References

Wehner, R.D., 1976. Polarized-light navigation by insects. Scientific American, 235(1), pp.106-115.

Rossel, S., 1993. Navigation by bees using polarized skylight. Comparative Biochemistry and Physiology Part A: Physiology, 104(4), pp.695-708.