15 thick lens martin murin 12. 15 12. thick lens: “a bottle filled with a liquid can work as a...
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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?“
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3
Bottle as a thick lens
𝑅1
𝑅2
𝑓
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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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Intensification of light - experiment
Ratio of light intensitiesin these points
= intensification
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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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Section 1:Theoretical estimate of intensification
Two theoretical models
Geometrical model Numerical ray tracing
𝑅1
𝑅2𝑓
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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
( )
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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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ABERRATION EXPERIMENT
Displacement of the ray(x-axis)
Horizontal offset of concentrated lightfrom the focus
(y-axis)
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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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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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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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(width)
𝜃01
𝑤
𝑛1𝑛2𝑛3
𝜃04h𝑑02𝑑01
h01 Assumed negligible
𝑤→0
Reflections are conserved
Reflection + refraction at the interfaces
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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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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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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
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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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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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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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• 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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39
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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41
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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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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Appendix G: Visual data – focused light on CMOS
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46
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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49
Appendix H: Additional data – ignition data
Li, Yudong and Drysdale, D.D., 1992. Measurement Of The Ignition Temperature Of Wood. AOFST 1
15
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)
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