Post on 14-Dec-2015
Problem # 8
• Liquid light guide • A transparent vessel is filled with a
liquid (e.g. water). A jet flows out of the vessel. A light source is placed so that a horizontal beam enters the liquid jet. Under what conditions does the jet operate like a light guide?
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 2
Overview• Introduction
– Observation– Problem Analysis
• Theoretical Model• Experiment• Results and Discussion
– Direct Transmission– Other Factors
• Conclusions & Summary
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 3
Introduction• Observation• A laser beam can perform
internal reflections in a jet. This makes the jet capable of serving as a light guide. However, when the jet strength decreases to a point, the jet can no longer act as a light guide.
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 4
Introduction• Problem Analysis• Water jet: near parabolic path• Laser beam first comes into
contact with the upper surface of jet
• Incident angle > critical angle: total internal reflection
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 5
Introduction
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 6
Fig.1 The liquid light guide apparatus
h=46cm
Theoretical Model
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 7
According to Bernoulli’s equation,
2 21 1 1 2 2 2
1 1
2 2P v gy P v gy
2v
1v
1P
2P
2 1y y
Theoretical Model
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 8
1 2 12 ( )v g y y
2 0v Assuming that and1 2 atmP P P
2v
1v
1P
2P
2 1y y
We can write 21 1 2
1
2v gy gy
And get
In our case, 2 1H y y
We get 2v gH
Theoretical Model
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 9
2
4
xy h
H
The water jet should follow a path similar to a horizontal projectile motion.
Therefore, we will use the formula to simulate the path of our light guide.
An experiment is done to verify our hypothesis.
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 10
Horizontal D
istance (cm)
Height of Water (cm)
Relationship Between Horizontal Distance and Water Height
Theoretical Prediction Experimental Results
Theoretical Model
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 11
•Assumption that surfaces of jet has the same shape as the center of the jet.•Ray tracing technique-> Coordinate geometry
2
4 2
x dy h
H
Theoretical Model
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 12
Let Then the reflecting point is
Substituting P into we can write
1 (1,0)d
( , )P t h2
4 2
x dy h
H
2
4 2
t dh h
H 2t dH
( 2 , )P dH h
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 13
To get the tangential line at a given point, it’s slope is:
Tangential line is:
Normal vector can be written:
2
dy x
dx H
1
2:
2
dHT y x h d
H
1 ( 2 ,2 )n dH H
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 14
Reflecting light beam is
Because the incident angle and reflecting angle are the same, we can write
2 (1, )d a
1 1 2 2
1 1 2 2
cosn d n d
n d n d
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 15
According to Snell’s law,
The critical angle for water is:
When the stream acts as a light guide, the incident angle must be greater than the critical angle,
Thus,
48.8
2
20 cos 48.8 0.66
2 4
dH
dH H
2
2
(1 0.66 ) 2
4 0.66
dH
0.648H d
1 1 2 2sin sinn n
1 2
1
sin ( ) 48.8n
n
• Continuous ray tracing – Incident angle grows larger with each reflection.
• Theoretically, the jet will operate as a light guide if the first reflection is a total internal reflection.
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 16
Experiment• Experimental methods• Water is filled into the reservoir and
the jet ejects into a basin.• The laser beam is turned on and
aimed through the hole. The height of the beam is then adjusted to pass through the center of the jet.
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 17
Experiment• Experimental methods• Pictures are taken from the side of the
basin to record the path of light inside the jet.
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 18
Results and Discussion
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 19
Chart 2. The actual critical water height for direct transmission and its theoretical prediction.
Direct Transmission
Hole Diameter
Critical H
eight (cm
)
Results and Discussion
• In the theory, the jet should act as a light guide until the water reaches the critical height.
• However, various other effects that causes the light guide to fail are observed.
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 20
Results and Discussion
• Other Influential Factors:– Partially Functional
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Diameter: 1cmWater Height: 5cm
Results and Discussion• Other Influential Factors
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Diameter 1cm 1.5cm 2cm 2.5cmDeformed End <6cm <4cm <3.5cm <3cmDrops <3.5cm N/A N/A N/A
Results and Discussion
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 25
DiameterEffect
1cm 1.5cm 2cm 2.5cm
Deformed End <6cm <4cm <3.5cm <3cmDrops <3.5cm N/A N/A N/ADeformed Flow <2.5cm <3cm <2.5cm <2.5cmDirect Pass <0.61cm <0.87cm <1.2cm <1.6cmOverall Range >6cm >4cm >3.5cm >3cm
Table 1. The critical height for all effects to happen and the overall range for the beam to work as a light guide.
Conclusions & Summary
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 26
• In our experiment so far, the theory shows the trend of the critical height of the water.
•Possible causes of bias are:1. Path of jet is not completely parabolic2. Turbulent jet surface
Conclusions & Summary
112/04/18 Reporter: 儲君宇 Joshua Jun-Yu Chu 27
•The jet will act as a light guide except in the following cases:
1. Incident angle < Critical angle- Speed of jet is too slow- Surface becomes too turbulent- Jet becomes deformed2. The flow breaks into drops