15

Thick lens

Martin Murin

12

15

2

12. Thick Lens:

“A bottle filled with a liquid can work as a lens. Arguably, such a

bottle is dangerous if left on a table on a sunny day.

Can one use such a ‘lens’ to scorch a surface?“

15

3

Bottle as a thick lens

𝑅1

𝑅2

𝑓

15

4

How it worksincoming light

𝐼 0 Concentrated light= power

Material heats up

Ignition

Section 1 Section 2

Light passing through the bottle Energy of lightReflection, absorption, dissipationIntensity of incident light

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5

Different shapes of bottles

SphereTop view:CylinderUndefined

shape

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6

Intensification of light - experiment

Ratio of light intensitiesin these points

= intensification

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7

Patterns and intensifications observed

25×5×5×

1500×undefinedfocus

focusis a line

focusis a line

focusis a point

Spherical bottle is the most dangerous

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8

Section 1:Theoretical estimate of intensification

Two theoretical models

Geometrical model Numerical ray tracing

𝑅1

𝑅2𝑓

15

GEOMETRICAL MODEL

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10

Focus of a sphere

1𝑓 =(𝑛−1 )[ 1𝑅1− 1

𝑅2+

(𝑛−1 )𝑑𝑛𝑅1𝑅2 ]

“Lensmaker’s equation”

For a sphere: 𝑅2=−R1=−R 𝑑=2𝑅

𝑓 = 𝑛𝑅2𝑛−2 𝑛=1.33≈ 43 𝑓 =2𝑅

For water

EXPERIMENTALLY CONFIRMED

( )

15

𝐼=𝑘 𝐼 0𝑆𝑙𝑒𝑛𝑠

𝑆𝑠𝑜𝑙𝑀2

𝑖𝑚𝑎𝑔𝑒𝑎𝑟𝑒𝑎=𝑆𝑜𝑙𝑎𝑟 𝑎𝑟𝑒𝑎×𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 ²

𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛=𝑓

𝑓 −𝑑𝑠𝑜𝑙

𝐼=𝑘 𝐼 0𝑆 𝑙𝑒𝑛𝑠

𝑆𝑠𝑜𝑙

𝑑𝑠𝑜𝑙2

𝑓 2

Maximum achievable intensity?

𝑑𝑠𝑜𝑙

𝑓𝑆𝑠𝑜𝑙×𝑀 2

𝑆𝑠𝑜𝑙

𝑆 𝑙𝑒𝑛𝑠𝐼 0

(power is conserved)

Spherical aberration

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15

12

ABERRATION EXPERIMENT

Displacement of the ray(x-axis)

Horizontal offset of concentrated lightfrom the focus

(y-axis)

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13

Effective area – ABERRATION EXPERIMENT

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

Displacement of the ray on the bottle [radius]

Displacement of the ray on

the paper [radius]

0,5

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14

Theoretical prediction of maximal intensity

𝐼=𝑘 𝐼 0

𝜋 𝑅2

4𝑆𝑠𝑜𝑙

𝑑𝑠𝑜𝑙2

4𝑅2

𝐼=𝑘 14000 𝜋16 𝐼 0≅ 2750𝑘 𝐼0

𝐼=𝑘 𝐼 0𝑆 𝑙𝑒𝑛𝑠

𝑆𝑠𝑜𝑙

𝑑𝑠𝑜𝑙2

𝑓 2

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15

Comparison with the experiment

1500×

𝐼=𝑘 14000 𝜋16 𝐼 0≅ 2750𝑘 𝐼0Permeability constant includes• Reflection on plastics-air interface• Reflection on plastics-liquid interface (2x)• Scattering in the liquid• Absorption in the liquid (depends

on the frequency of the light)

+ other undeterminable losses

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16

0 0.02 0.04 0.06 0.08 0.1 0.120

200

400

600

800

1000

1200

1400

1600

Intensification as a function of distance from the focal point

Distance from the focal point [radius]

Inte

nsifi

catio

n [1

]

0,008 (measured)

1500(measured)

Unrealistic profile

Motivation for better model

Expected profile

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NUMERICAL RAY TRACING

Advantages:accuracy – spherical aberration, reflection, dispersion were assumedresults – full image of intensity profile curve

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18

(width)

𝜃01

𝑤

𝑛1𝑛2𝑛3

𝜃04h𝑑02𝑑01

h01 Assumed negligible

𝑤→0

Reflections are conserved

Reflection + refraction at the interfaces

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19

Ray tracing – ins and outs

𝜃1𝜃2

h 𝑟

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3

𝜃3 𝜃4𝜃1

𝜃5𝛾 2𝛾 3 𝛾 4

𝜃6

𝑛1 𝑛2

𝑑1

𝑛3

𝜆

Input parameters

Output parameters

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• Aberration and comaincluded by geometry

• Reflection – Fresnel equations

Method: included effects

𝑓

h3 h 𝑖𝜃6

*unpolarized light assumed

𝑅=|𝑛1𝑐𝑜𝑠 𝜃𝑖−𝑛2𝑐𝑜𝑠𝜃 𝑡

𝑛1𝑐𝑜𝑠𝜃𝑖+𝑛2𝑐𝑜𝑠𝜃𝑡 |2

+|𝑛1𝑐𝑜𝑠𝜃𝑡−𝑛2𝑐𝑜𝑠𝜃𝑖𝑛1𝑐𝑜𝑠𝜃 𝑡+𝑛2𝑐𝑜𝑠 𝜃𝑖|2

2

𝑛1

𝑛2 𝑛3 𝑛2𝑛1

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• Dispersion – Cauchy’s equation

Method: included effects

[1]

[1] Water Refractive Index in Dependence on Temperature and Wavelength, by Alexey N. Bashkatov, Elina A. Genina, Optics Department Saratov State University, Saratov, Russia

*for water:

• Sun’s spectrum- black body radiation

𝐼(𝑣 ,𝑇 )  =2h𝑣3

𝑐21

𝑒h𝑣𝑘𝑇 −1

[ 𝑊𝑚2𝑠𝑟 𝐻𝑧 ]

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Ray tracing – processed output

-0.00999999999999999 0.01 0.03 0.050

200

400

600

800

1000

1200

1400

1600

1800

2000

Intensification

without tracing tracingtracing + reflection tracking + reflection + dispersion

Distance from Focal Point [radius]

Inte

nsifi

catio

n [1

]

EVALUATION> Exact intensification profile

with maximum 1270×> Reflection is significant> Dispersion has little enhancing effect

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Perform more accurate measurement

Aperture: f/2.8ISO: 400Shutter speed: 1/30 sec

Aperture: f/22ISO: 100Shutter speed: 1/400 sec

1280×Maximum intensification:

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Ray tracing vs. accurate measurement

99% correlation!

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SECTION 2: DISSIPATION FROM MATERIAL

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26

Heating equation

Change in internal energy

Net power input

Power dissipatedby radiation

Heat dissipated to surroundings by…

conduction convection

Typical times of ignition (without considering losses)

𝑡=𝑚𝑘 (𝑇 h𝑠𝑐𝑜𝑟𝑐 𝑖𝑛𝑔−𝑇 0 )

𝜖 𝐼 𝑖𝑛𝑆=1.76 s 𝑚=7×10−7𝑘𝑔𝑇 0

𝑇𝑆𝛻𝑇

𝑚𝑘𝕕𝑇𝕕𝑡 =𝜖 𝐼𝑖𝑛𝑺−(𝑐 𝑺𝒍 +h𝑺)Δ𝑇 −𝜎𝜖𝑺𝑇 4

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27

Significance of albedo

didn’t ignite

61,5%

8,4%

59,2% 47,3 %

5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.00

2

4

6

8

10

12

14

16

albedo (%)

Tim

e of

scor

chin

g [s

]

Albedo has little influence Albedo hassignificant influence

Albedo of the material is VERY important.

Dark paper

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Basic condition for ignition

𝑃 𝑖𝑛𝑐𝑜𝑚𝑖𝑛𝑔≥ 𝑃 𝑙𝑜𝑠𝑠𝑒𝑠(𝑇 𝑖𝑔𝑛𝑖𝑡𝑖𝑜𝑛)To heat the material further, this condition must be satisfied:

Conduction RadiationReflection Convection

200 300 400 500 600 700 800 9000

0.0050.01

0.0150.02

0.025

Comparison of dissipated heat by conduction, convection and radiation

conductionconvectionradiation

Temperature [K]

Diss

ipat

ed p

ower

[W]All of them are comparably significant.

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Condition for ignition𝑃 𝑖𝑛 (𝑇 )≥𝜖 𝑃 𝑖𝑛+(𝑐 𝑆𝑙 +h𝑆) 𝛥𝑇+𝜎𝜖 𝑆𝑇 4

If incoming intensity is sufficiently large,

temperature rises until ignitionWhat is sufficient?

Critical intensity (peak) intensity of incoming radiation that ignites the surface

Measurement…

Calculation

𝑃𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑒𝑑=𝑐𝑆𝑙 Δ𝑇 +h𝑆 𝛥𝑇+𝜎𝜖 𝑆𝑇4 At ignition temperature for wood:

𝐼 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙≅ 33𝑘𝑊𝑚2

0.015W +0.0074W +0.0114W=0.033W

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Critical intensities (the measurement)• We change distance between the

lens and the target

• What is the range of distancesin which the material scorch?

• Conditions of the experiment– Sunlight intensity – Typical shifts:

Scorched in this region

Critical intensityvalue calculatedfrom geometry

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ComparisonCalculated light intensity from the condition of ignition

Light intensity achieved by lens in the experimentto scorch the wooden sample

Heat flux used to ignite a wooden sample in the article*

𝐼 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙≅ 33𝑘𝑊𝑚2

𝐼 h𝑐 𝑎𝑟𝑟𝑖𝑛𝑔=14 ±1.4𝑘𝑊𝑚2

𝜙h𝑒𝑎𝑡=15−30𝑘𝑊𝑚2

Li, Yudong and Drysdale, D.D., 1992. Measurement Of The Ignition Temperature Of Wood. AOFST 1

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Measurements

Material Critical Intensity* () [W/m²]

Bond paper (white) 500 000Dot matrix printing paper 175 000Cardboard 10 000Wood 14 000Black scarf (100% polyacryl)

4 000

Thin blue plastic bag (polyethylen)

3 000

*to damage the surface

scorchedor burned

melted

Theoretical maximumfor Brusnianka is

(depends on Sunlight intensity)

CONFIRMED BY AN EXPERIMENT

?

Albedo > 60%¼ energy absorbed by water in IR region

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33

Secti

on 1

Secti

on 2

ConclusionThank you for your attention!

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THANK YOU FOR YOUR ATTENTION!

Martin Murin

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APPENDICES

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36

Appendix A: Derivation of intensification

𝑆 𝑖𝑚𝑎𝑔𝑒=𝑆𝑠𝑜𝑙×𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 ²

𝐼=𝑘 𝐼 0𝑆 𝑙𝑒𝑛𝑠

𝑆𝑠𝑜𝑙

𝑑𝑠𝑜𝑙2

𝑓 2

Power is conserved

𝐼×𝑆𝑖𝑚𝑎𝑔𝑒=𝑘𝐼 0×𝑆 𝑙𝑒𝑛𝑠

𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛=( h′h =𝑎′𝑎=

𝑓𝑎𝑓 −𝑎𝑎 =

𝑓𝑓 −𝑎 )= 𝑓

𝑓 −𝑑𝑠𝑜𝑙𝑆𝑠𝑜𝑙=1.52×10

18𝑚2

𝐼=𝑘 𝐼 0𝑆𝑙𝑒𝑛𝑠

𝑆𝑖𝑚𝑎𝑔𝑒

𝐼=𝑘 𝐼 0𝑆𝑙𝑒𝑛𝑠

𝑆𝑠𝑜𝑙𝑀2

𝐼=𝑘 𝐼 0𝑆 𝑙𝑒𝑛𝑠

𝑆𝑠𝑜𝑙( 𝑓𝑓 −𝑑𝑠𝑜𝑙 )

2

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37

• The only relevant optical property of liquid is the index of refraction

Appendix B: Different liquids in bottle

1𝑓 =(𝑛−1 )[ 1𝑅1− 1

𝑅2+

(𝑛−1 )𝑑𝑛𝑅1𝑅2 ]

“Lensmaker’s equation”

𝑓 =𝑛𝑅2𝑛−2(for sphere)

• The smaller the focal length, the smaller the magnification

• The smaller the magnification, the higher the intensity

𝐼=𝑘 𝐼 0𝑆 𝑙𝑒𝑛𝑠

𝑆𝑠𝑜𝑙

𝑑𝑠𝑜𝑙2

𝑓 2

𝑖𝑚𝑎𝑔𝑒𝑎𝑟𝑒𝑎=𝑆𝑜𝑙𝑎𝑟 𝑎𝑟𝑒𝑎×𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 ²𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛=

𝑓𝑓 −𝑑𝑠𝑜𝑙

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250 300 350 400 450 500 550 600 650 700 750870

880

890

900

910

920

f(x) = − 0.000062205107143 x² − 0.0029416157143 x + 920.21121479714R² = 0.99981565166762

Plotted relation

Temperature [K]

Rate

of c

hang

e of

tem

pera

ture

[K

/s]

Appendix C: Time evolution of temperature

𝕕𝑇𝕕 𝑡 =−6×10−5𝑇 2−2.9×10− 3𝑇 +920.21

𝑇=−3379.95+3940.47𝑒0.47 𝑡

0.868404+𝑒0.47 𝑡

Using constants for ignition of wood

𝑚=7×10−7 kg 𝑘=2000 Jkg K

𝑆=10−6m ²𝑐=0.05 Wm K

h=20 Wm2 K

𝜖=0.5𝜎=5.67×10− 8 W

m2K 4

𝑇 0=300K

𝑙=5×10−4m

𝐼 𝑖𝑛=1000Wm2

𝑚𝑘𝕕𝑇𝕕𝑡 = (1−𝜖 ) 𝐼𝑖𝑛𝑆+(𝑐 𝑆𝑙 +h𝑆) Δ𝑇 −𝜎𝜖𝑆𝑇 4

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𝑚𝑘𝕕𝑇𝕕𝑡 = (1−𝜖 ) 𝐼𝑖𝑛𝑆+𝑇 0(𝑐 𝐴𝑙 +h𝑆)−(𝑐 𝐴𝑙 +h𝑆)𝑇 −𝜎𝜖 𝑆𝑇 4

Appendix C: Time evolution of temperature

250 300 350 400 450 500 550 600 650 700 750870

880

890

900

910

920

f(x) = − 0.000062205107143 x² − 0.0029416157143 x + 920.21121479714R² = 0.99981565166762

Plotted relation

Temperature [K]

Rate

of c

hang

e of

te

mpe

ratu

re [K

/s]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100200300400500600700800900

1000

Temperature of wood

Time [s]Te

mpe

ratu

re [K

]

𝕕𝑇𝕕 𝑡 =−6×10−5𝑇 2−2.9×10− 3𝑇 +920.21 𝑇=

−3379.95+3940.47𝑒0.47 𝑡

0.868404+𝑒0.47 𝑡

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Appendix D: Used materials – dot matrix paper

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Appendix E: Experiments – conditions during critical intensity measurement

Lens parameters:focal lengthradius of curvaturelens radiusindex of refraction* * crown glass, 589 nm, encyclopedia Britannica

Measurement data:date and time 6th January 2015 at 12:00 - 13:00weather sunny, no cloudssunlight intensity [W/m²] 380 (Bratislava airport), 330 (Koliba)

Position of the Sunaltitude 12:10 19° 16´ 37´´

12:30 18° 59´ 57´´13:00 17° 59´ 47´´

apparent diameter 00° 32' 32''Solar diameterdistance from Earth

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Appendix E: Experiments – time vs albedo

Paper with different color (albedo) measured by albedometer

Round bottom boiling flask

𝑟=4.2cm

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43

Appendix F: Light intensity ratio from a photo

1. Capture in RAW format

2. Convert to TIFF using dcraw* program- No color interpolations, compression, white balance…

3. Set linear colorspace in Photoshop

4. Open linear TIFF in PS canceling any suggestions for “improvements”

5. Read RGB values and compare them within one image

6. Try different exposition times and verify that2x longer exposition gives 2x grater RGB values

http://www.cybercom.net/~dcoffin/dcraw/

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Appendix G: Visual data – caustic curve

Image courtesy of Jakub Trávník at http://jtra.cz

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45

Appendix G: Visual data – focused light on CMOS

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Appendix G: Visual data – focused light on CMOS(comma aberration)

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Appendix H: Additional data – electromagnetic absorption by water

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48

Appendix H: Additional data – intensity of solar irradiance during the day

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

Time of the day [hours]

Dire

ct R

adia

tion

[kW

/m^2

]

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Appendix H: Additional data – ignition data

Li, Yudong and Drysdale, D.D., 1992. Measurement Of The Ignition Temperature Of Wood. AOFST 1

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50

Appendix H: Additional data – thermal conductivities

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Appendix H: Additional data – Fresnel equations

𝑟 𝑠=𝑛1cos (𝜃𝑖 )−𝑛2 cos (𝜃𝑡 )𝑛1 cos (𝜃𝑖 )+𝑛2cos (𝜃𝑡)

𝑟𝑝=𝑛2 cos (𝜃𝑖 )−𝑛1 cos (𝜃𝑡 )𝑛1 cos (𝜃𝑡 )+𝑛2cos (𝜃𝑖)

𝑡 𝑠=2𝑛1 cos (𝜃𝑖 )

𝑛1 cos (𝜃 𝑖 )+𝑛2cos (𝜃𝑡 )

𝑡𝑝=2𝑛1 cos (𝜃𝑖 )

𝑛1 cos (𝜃 𝑡 )+𝑛2 cos (𝜃 𝑖)