Atmospheric RefractionA Study of Mirage

Supervisor:

Mr H.Y. CHENG

Team Members:

*Wai Fu, CHAN

Tsz Hin, KONG

Hon Lam, LEE

To Chin, YU

August 13, 2012

SKH Lam Woo Memorial Secondary School

Team No.: 100030

This report is aimed at studying the phenomenon of mirage, using a methodthat is much simpler than those appeared in previous work ([12, 19]). It isdivided in to three parts. In the �rst part we look into the trajectory oflight in an inhomogeneous medium and develop the mathematical methodand equations that will be needed later. In the next part, we introduced acouple of models to simulate real-life atmospheric conditions in which miragecan take place. We also point out the need for a numerical approach dueto the complexity of the models. We have developed a program to simulatethe process of refraction. In the third part we use the program to simulatevarious cases of mirage.The method used can be also applied on all kinds of other phenomena such

as Novaya-Zemlyz e�ect and green �ash despite they are not introduced inthis paper.

Introduction

In this report we hope to obtain a complete mathematical picture of mirage. We beginby deriving the basic equation for trajectory of light in inhomogeneous medium fromFermat's principle, which states that light travels along the path with minimum (or

1

more precisely, stationary) travelling time.[3, 4] Mathematically, Fermat's principle canbe stated as

δT [f ] = 0 (0.1)

,where T [f ] is the time needed for light to travel along the path y = f(x) from a �xedinitial point x = a to a �xed �nal point x = b. The delta sign symbolises functionalderivative w.r.t. f(x).From this the equation of trajectory can be found to be

[1 + (y′)2]∂n

∂y− [1 + (y′)2]y′

∂n

∂x− ny′′ = 0 (0.2)

We then study the behavior of the equation by considering some toy models for therefractivity, namely n = a

y and n = ay + b. These models are not realistic but areanalytically solvable.For more realistic modelling of the atmosphere, we begin by �rst �nding the refractivity

pro�le based on a standard atmospheric model.[10, 11, 14]{P (y) = P0exp(−gMy

RT0)

T (y) = T0 − Ly(0.3)

Using Ideal Gas Law[15] and Gladstone-Dale relation[8], we obtain the expression forrefractivity in a standard atmosphere:

n(y) =KairP0M

R(T0 − Ly)exp(−gMy

RT0) + 1 (0.4)

However, the e�ect of refraction is very weak in the standard atmosphere, which makesit almost impossible for mirage to occur. Thus we need to modify (de-standardize) thestandard model in order to obtain models that are closer to the actual atmosphericconditions under which mirage could occur. We developed two di�erent temperaturepro�les by adding extra terms to the standard temperature pro�le:T (y) = T0 − Ly + ∆− ∆

1+exp(− y−yba

), F ermi (inferiormirage)

T (y) = T0 − Ly + c exp(−12(y−yba )2) , Gauss (superiormirage)

(0.5)

and hence the refractivity pro�les:n(y) = KairP0M

R(T0−Ly+∆− ∆

1+exp(− y−yba )

)exp(−gMy

RT0) + 1 (Fermi)

n(y) = KairP0M

R(T0−Ly+c exp(− 12

(y−yb

a)2))

exp(−gMyRT0

) + 1 (Gauss)(0.6)

The meanings of the various symbols that appear here will become evident later.These expressions are too complicated for analytical treatment. Therefore, we develop

a computer program to numerically solve the equation of trajectory. Using the program,we can �nd the position and size of the image(s), as well as the position of the vanishingline, thus concluding our project.

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Contents

I. Mathematics of Light Trajectory 4

1. Introduction to Mirage 4

2. Deriving the equation of trajectory using Fermat's princple 5

3. Solving the equation for n = n(y) 7

4. Summary 10

II. Mathematics of the Atmosphere 10

5. The Standard Atmosphere 10

6. The Fermi Atmosphere (Inferior Mirage) 11

7. The Gauss Atmosphere (Superior Mirage) 12

8. Summary 12

III. Application 12

9. Inferior Mirage 139.1. Typical Inferior Image (with vanishing line) . . . . . . . . . . . . . . . . . 149.2. Re�ection-Only Inferior Mirage . . . . . . . . . . . . . . . . . . . . . . . . 159.3. Vanishing zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

10.Superior Mirage 1610.1. Typical Superior Mirage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1810.2. Oscillating Light Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

11.Summary 19

IV. Further Discussion and Application 20

12.Lens 2012.1. Atmospheric Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2012.2. Liquid Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

13.More complicated atmospheres 2113.1. Multi-layered atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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13.2. Di�erent types of atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . 2113.3. Non-ideal atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

14.The reverse problem 22

15.Summary 22

16.Introduction 22

17.Principle of method 23

18.Accuracy 24

19.Source Code 25

Part I.

Mathematics of Light Trajectory

1. Introduction to Mirage

It is well-known that the Earth's atmosphere is not homogeneous. As a result, lighttravelling in the atmosphere will undergo refraction and the travelling path will deviatefrom a straight line. This is the cause of various optical phenomena such as mirage andthe green �ash during sunset.We will focus on mirage in this report but the equations and tools that we develop here

can be applied to other phenomena involving refraction in an inhomogeneous medium aswell.There are two basic types of mirage: inferior and superior. They correspond to di�erent

temperature pro�les. Inferior image can form when the temperature decreases withaltitude, while superior image requires that the temperature increases with altitude.For both types of mirage, usually (but not necessarily) two images will be formed as

two physically feasible light trajectories can be drawn from the object to the observer'seyes, thus forming one erect image and one inverted image.[2, 5] When the temperaturedecreases with altitude, the light will bend upwards so the inverted image is locatedbelow the erect image, as if the erect image is re�ected by (fake) water. (Figure 1.1left) As for superior image, the temperature increases with altitude. The light will benddownwards so the inverted image will be formed above the erect one. (Figure 1.1 right)Another interesting feature about mirage is that part of the object may become invis-

ible to the observer because the plane of re�ection may lie across the object and causethe part of object below the �vanishing line� to disappear. [1] The phenomenon will bediscussed in further detail in the last part of our report.

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Figure 1.1: (left-top) Example of inferior mirage on a highway. (right-top) Example ofsuperior mirage of a ship.(left-bottom) Ray diagram for inferior image. (right-bottom) Ray diagramfor superior image.[Source: Google Image Search(http:// images.google.com), Science PhotoLibrary(http://www.sciencephoto.com/)]

We now begin by �nding the trajectory of light in an inhomogeneous medium withgiven refractivity.

2. Deriving the equation of trajectory using Fermat'sprincple

Any trajectory under the rectangular coordinate system can be represented by a functiony = f(x). According to Fermat's Principle, which states that the trajectory of lightcorresponds to the path along which the time of travel is stationary[3], we wish to �nd apath y = f(x) that satis�es

δT [f ] = 0 (2.1)

,where T [f ] is the time needed for light to travel along the path y = f(x) from a �xedinitial point x = a to a �xed �nal point x = b. The delta sign symbolises variation of Twhen f(x) is varied. We can use calculus of variation to �nd the equation of trajectory.Since n = c

v , where c is the speed of light in vaccum and v is the speed of light in medium,

5

we can further expand T [f ] as1

T [f ] =

ˆ x=b

x=an(x, y)dl (2.2)

=

ˆ x=b

x=an(x, y)

√(dx)2 + (dy)2 (2.3)

=

ˆ x=b

x=an(x, y)

√1 + (y′)2dx (2.4)

=

ˆ x=b

x=an(x, f(x))

√1 + (f ′(x))2dx (2.5)

,where n(x, y) represents the refractivity at the point (x, y). For clarity, the domainof integration will be omitted and the integral sign is understood to represent a de�niteintegral.The functional derivative can be calculated as follows. First, consider an arbitrary

�rst-order variation in f(x), the time of travel will become

T [f + δf ] =

ˆn(x, f(x) + δf(x))

√1 + (f ′(x) + δf ′(x))2dx (2.6)

Now we use Taylor's formula to expand the terms inside the integral.

n(x, f(x) + δf(x)) = n(x, f(x)) +∂n

∂yδf(x) + ... (2.7)

√1 + (f ′(x) + δf ′(x))2 =

√1 + (f ′(x))2 +

f ′(x)√1 + (f ′(x))2

δf ′(x) + ... (2.8)

As the variation δf is a �rst-order small term, we can neglect second or higher orderterms. Combining the above equations, we have

δT [f ] = T [f + δf ]− T [f ] (2.9)

=

ˆn(x, f(x) + δf(x))

√1 + (f ′(x) + δf ′(x))2dx (2.10)

−ˆn(x, f(x))

√1 + (f ′(x))2dx

=

ˆ{[n(x, f(x)) +

∂n

∂yδf(x)][

√1 + (f ′(x))2 +

f ′(x)√1 + (f ′(x))2

δf ′(x)](2.11)

−n(x, f(x))√

1 + (f ′(x))2}dx

=

ˆ[n(x, f(x))

f ′(x)√1 + (f ′(x))2

δf ′(x) +∂n

∂y

√1 + (f ′(x))2δf(x)]dx (2.12)

1The time has a extra factor of c. However as we only want to �nd the curve corresponding to the

stationary time, the extra factor does not a�ect our result, as if T is stationary for the curve f , thencT will also be stationary for the same curve.

6

Next we integrate the �rst term by part.

δT [f ] =

ˆ[n(x, f(x))

f ′(x)√1 + (f ′(x))2

δf ′(x) +∂n

∂y

√1 + (f ′(x))2δf(x)]dx (2.13)

= { n(x, f(x))f ′(x)√

1 + (f ′(x))2δf(x)}x=b

x=a −ˆδf(x)

d

dx(n(x, f(x))

f ′(x)√1 + (f ′(x))2

)dx

+

ˆ∂n

∂y

√1 + (f ′(x))2δf(x)dx

Since the end-points (x = a, b) are �xed (δf(a) = δf(b) = 0) , the �rst term vanishes.Now we calculate the derivative in the second term.

d

dx(n(x, f(x))

f ′(x)√1 + (f ′(x))2

) =dn

dx

f ′(x)√1 + (f ′(x))2

+n(x, f(x))f ′′(x)√

1 + (f ′(x))2− n(x, f(x))[f ′(x)]2

[1 + (f ′(x))2]32

= (∂n

∂x+∂n

∂yf ′(x))

f ′(x)√1 + (f ′(x))2

+n(x, f(x))f ′′(x)√

1 + (f ′(x))2− n(x, f(x))[f ′(x)]2

[1 + (f ′(x))2]32

Combining the above equations, we have,

δT [f ] =

ˆ{∂n∂y

√1 + (f ′(x))2 − d

dx(n(x, f(x))

f ′(x)√1 + (f ′(x))2

)}δf(x)dx (2.14)

As δf(x) is arbitrary, the only way for δT [f ] = 0 for any �rst-order variation is for

∂n

∂y

√1 + (f ′(x))2 − d

dx(n(x, f(x))

f ′(x)√1 + (f ′(x))2

) = 0

∂n

∂y

√1 + (f ′(x))2 − (

∂n

∂x+∂n

∂yf ′(x))

f ′(x)√1 + (f ′(x))2

− n(x, f(x))f ′′(x)√1 + (f ′(x))2

+n(x, f(x))[f ′(x)]2

[1 + (f ′(x))2]32

= 0

After simplifying, we get our �nal result:

[1 + (f ′(x))2]∂n

∂y− [1 + (f ′(x))2]f ′(x)

∂n

∂x− nf ′′(x) = 0 (2.15)

We can replace f(x) by y and write:

[1 + (y′)2]∂n

∂y− [1 + (y′)2]y′

∂n

∂x− ny′′ = 0 (2.16)

This equation governs the trajectory of light on a plane.

3. Solving the equation for n = n(y)

Under many circumstances, including those that will be considered later in this paper,the refractivity only depends on the altitude (i.e. y), thus the case when n is independentof x is of particular interest.

7

To solve equation 2.16 when the refractivity is independent of x (i.e. n = n(y)), letv(y) = dy

dx . Then we have

y′′ =dv

dx(3.1)

=dv

dy

dy

dx(3.2)

= v′v (3.3)

Hence, equation 2.16 will become

ny(1 + v2)− nvv′ = 0 (3.4)

,where ny = dndy . Now we can solve for v(y).

nvv′ = ny(1 + v2) (3.5)

vv′

(1 + v2)=

nyn

(3.6)

ˆv dv

1 + v2=

ˆnyndy (3.7)

1

2ln|v2 + 1|+ const. = ln(n) (3.8)

v2 + 1 = C1n2 (3.9)

v = ±√C1n2 − 1 (3.10)

,where C1 is an undetermined constant. From here we can then �nd out the solution.

dy

dx= ±

√C1n2 − 1 (3.11)

±ˆ

dy√C1n2 − 1

= x+ C2 (3.12)

±η(y) = x+ C2 (3.13)

,where η(y) =´ dy√

C1n2−1and C2 is another undetermined constant. We can determine

both C1 and C2 if we are given either the initial or the boundary conditions.Before we move on, it is bene�cial to study the behavior of light in a couple of toy

models in order to illustrate the method more clearly.

Example 1: n = ay

Let us consider the case in which n = ay . We calculate η(y) as

η(y) =

ˆdy√

C1n2 − 1(3.14)

=

ˆdy√

C1(ay )2 − 1(3.15)

= −√a2C1 − y2 (3.16)

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Hence the equation of path is

x+ C2 = ±√a2C1 − y2 (3.17)

x2 + 2C2x+ (C2)2 = a2C1 − y2 (3.18)

x2 + y2 + 2C1x+ (C2)2 − a2C1 = 0 (3.19)

which corresponds to a circular arc.

Example 2: n = ay + b

Now, let us consider the case in which the refractivity is a linear function of y.

η(y) =

ˆdy√

C1n2 − 1(3.20)

=

ˆdy√

C1(ay + b)2 − 1(3.21)

To calculate the integral in line 3.21, we let u(y) =√C1(ay + b),

η(y) =1√C1a

ˆdu√u2 − 1

(3.22)

=1√C1a

ln|u+√u2 − 1| (3.23)

=1√C1a

ln|√C1(ay + b) +

√C1(ay + b)2 − 1| (3.24)

Hence, by equation 3.13, the equation for the path of light will be

± 1√C1a

ln|√C1(ay + b) +

√C1(ay + b)2 − 1| = x+ C2 (3.25)√

C1(ay + b) +√C1(ay + b)2 − 1 = e±

√C1a(x+C2) (3.26)

y =1

a[

1

2√C1

(ea√C1(x+C2) + e−a

√C1(x+C2))− b]

The absolute sign has been removed in the second line since the refractivity (ay + b)

must be larger than zero. We then note that ex+e−x

2 = cosh(x) and relabel the constantsto give the �nal form of the solution:

y =1

Acosh(Ax+B)− b

a(3.27)

,where a√C1 is replaced by A and a

√C1C2 by B. Hereby we have obtained the solution

for the case when n = ay + b.

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4. Summary

In this part we have developed the equation governing the trajectory of light (eq. 2.16),as well as studying some mathematically simple toy models. These models may have littlepractical value but are good exercises for us to gain further intuition into atmosphericrefraction. In the following parts we will turn to some more realistic models which arenot analytically solvable.

Part II.

Mathematics of the Atmosphere

In this part we introduce two more realistic models of the atmosphere, namely the FermiAtmosphere and the Gauss Atmosphere. The Fermi Atmosphere allows inferior mirageto take place while in the Gauss Atmosphere, superior mirage can be found. We startfrom the Standard Atmosphere and derive from it the two atmospheres that we will useto study mirage.

5. The Standard Atmosphere

Within the troposphere, the altitude y and the pressure P has the following relationship:[11]

P (y) = P0exp(−gMy

RT0) (5.1)

,where P0 is the atmospheric pressure at sea level (101325Pa), g is the gravitationalacceleration (9.81m/s2), M is the molar mass of air (0.02896kg/mol), R is the universalgas constant (8.31 J/(molK)), and T0 is the temperature at sea level.Meanwhile the temperature can be expressed as a linear function.[14]

T (y) = T0 − Ly (5.2)

,where L is called the lapse rate. (Standard value for L is 0.0065K/m[14])We can treat air as an ideal gas[15] and apply the ideal gas law to obtain the expression

for density.[16]

ρ(y) =P (y)M

RT (y)(5.3)

=P0M

R(T0 − Ly)exp(−gMy

RT0) (5.4)

To �nd the refractivity, we can substitute the expression for density into the Gladstone-Dale Relation.[7, 8, 13]

n− 1 = Kairρ (5.5)

10

,where Kair is the Gladstone-Dale constant for air. (Kair for visible light is approxi-mately 0.00023m3/kg [6]) Hence

n(y) =KairP0M

R(T0 − Ly)exp(−gMy

RT0) + 1 (5.6)

Using standard values (L = 0.0065K/m, T0 = 288.15), we can plot n versus y as

Figure 5.1: Refractivity in a standard atmosphere

From the graph it is clear that refraction is not very signi�cant in such atmospheresince the graph becomes �at very quickly. We need to develop other atmospheric modelsin which refraction is a more signi�cant e�ect.

6. The Fermi Atmosphere (Inferior Mirage)

Most inferior images are due to a hot surface, such as a heated asphalt road or desertsand. If so we have to modify the temperature distribution to include the thermal e�ectof the hot surface. We chose to add an additional Fermi-distribution-like term to ourexpression for temperature, as is suggested in [12] and [19]. The temperature pro�le canbe written as

T (y) = T0 − Ly + ∆− ∆

1 + exp(−y−yba )

(6.1)

,where ∆ is the temperature di�erence between the surface and T0, yb represents theposition of the boundary layer, i.e. the point where the e�ect of the hot surface begins tofade, and a represents the �decay rate�, i.e. how fast the e�ect of the hot surface fades.After such modi�cation, the refractivity becomes

n(y) =KairP0M

R(T0 − Ly + ∆− ∆

1+exp(− y−yba

))exp(−gMy

RT0) + 1 (6.2)

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We will call this model the Fermi Atmosphere.

7. The Gauss Atmosphere (Superior Mirage)

For superior mirage, the temperature has to increase with attitude at some point. There-fore we have to insert an �inversion layer� into the atmosphere to simulate this.We have chosen to add a Gaussian term to the temperature in order to model the

inversion layer, so the temperature pro�le becomes

T (y) = T0 − Ly + c exp(−1

2(y − yba

)2) (7.1)

,where yb is the position of the inversion layer, c represents the temperature rise of theinversion layer, i.e. larger c corresponds to greater temperature di�erence, and a is thewidth of the inversion layer.The refractivity becomes

n(y) =KairP0M

R(T0 − Ly + c exp(−12(y−yba )2))

exp(−gMy

RT0) + 1 (7.2)

We will call this model the Gauss Atmosphere.

8. Summary

In this part we have introduced two models for the atmosphere to replace the standardmodel in order to model mirage. However, due to the complexity of the model, it is notpossible to obtain analytical results from direct application of our previous equations.Hence we have to rely on numerical method to study refraction in such atmosphere usinga computer simulation.In the next part we will discuss the working principle and results of our computer

simulation. An exerpt of the source code (written in C#) can be found in the Appendix.

Part III.

Application

With the simulation, we are now in a position capable of dealing with realistic refrac-tivity distributions. In this �nal part, we will showcase several ray diagrams of miragesproduced with the simulation.

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9. Inferior Mirage

Recall the Fermi Atmosphere (eq. 6.2)

n(y) =KairP0M

R(T0 − Ly + ∆− ∆

1+exp(− y−yba

))exp(−gMy

RT0) + 1 (9.1)

So the temperature pro�le is

T (y) = T0 − Ly + ∆− ∆

1 + exp(−y−yba )

(9.2)

We have chosen the following set of parameters:

Symbol Value

T0 300.15 K

L 0.0065 K/m

∆ 86 K

yb 0 m

a 1 m

Figure 14.1 shows a plot of the temperature pro�le. The surface (y = 0) is at around343 K (70 oC), which is quite typical for hot roads or the surface of desert. [20]

Figure 9.1: Plot of temperature pro�le for a Fermi atmosphere

13

Using this temperature pro�le, we can use our program to generate various light tra-jectories. Here are some interesting cases that we have found using our program. Alllight trajectories are generated by our program and plotted using the software Graph.[24]

9.1. Typical Inferior Image (with vanishing line)

Figure 9.2: Ray diagram of inferior mirage

Figure 14.2 shows a very typical mirage where part of the object vanishes and one invertedimage of the object is formed below the erect image. The vanishing line is indicated bythe thick black line. Any object below the vanishing line cannot be seen by the observeras rays from those points cannot reach the observer regardless of the initial angle. Figure14.3 shows a similar situation in real life. The dunes undergo similar mirage that isshown in �gure 14.2.

Figure 9.3: Real-life inferior mirage [Source: Google Image Search]

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9.2. Re�ection-Only Inferior Mirage

Figure 9.4: Ray diagram of re�ection-only mirage

Figure 14.4 shows a very interesting case of mirage where ONLY an inverted image isformed. This happens in real situations, as shown in �gure 14.5.

Figure 9.5: Real-life re�ection-only mirage [Source: Google Image Search]

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9.3. Vanishing zone

For every object point there is an area which the light cannot reach. This area is calledthe vanishing zone, shaded in grey in Figure 14.6. When the observer is located insidethe vanishing zone, the object will be invisible to the observer as no light ray can reachthe observer. We can �nd the vanishing zone using our program as well.

Figure 9.6: Multiple light trajectories and vanishing zone

10. Superior Mirage

Superior mirage is simulated using the Gauss Atmosphere:

n(y) =KairP0M

R(T0 − Ly + c exp(−12(y−yba )2))

exp(−gMy

RT0) + 1 (10.1)

The temperature pro�le is

T (y) = T0 − Ly + c exp(−1

2(y − yba

)2) (10.2)

We have chosen the following set of parameters:

Symbol Value

T0 273.15 K

L 0.0065 K/m

c 10 K

yb 20 m

a 4 m

16

Figure 15.1 shows a plot of the temperature pro�le.

Figure 10.1: Plot of temperature pro�le for a Gauss atmosphere

Using this temperatuer pro�le, we can �nd various light trajectories.

17

10.1. Typical Superior Mirage

Figure 10.2: Ray diagram of superior mirage

Figure 15.2 shows a typical superior mirage where an inverted image is formed above theerect image. We may compare our plot with real superior mirage shown in �gure 15.3.

Figure 10.3: Real-life superior mirage [Source: Google Image Search]

If the reader is careful enough, he may notice that there is another erect image abovethe inverted image, which is not found in the ray diagram. The reason for this is that a

18

real-life superior mirage usually happens in an atmosphere with more than one inversionlayer. Therefore superior mirages usually gives more than two images, as opposed to thatsuggested by the ray diagram.

10.2. Oscillating Light Rays

While studying the Gauss Atmosphere, we found something quite special. When theinitial angle of the light ray is small enough, the light ray will follow a wave-like trajectoryshown in �gure 15.4. Oscillatory light rays are the key to understand more complexphenomena such as the Novaya Zemlya e�ect[19] , which is beyond the scope of thispaper.

Figure 10.4: Plot of an oscillating trajectory

11. Summary

In this part various cases of mirage is demonstrated through ray diagrams generated byour simulation program. The result of our simulation resembles closely to what happensin actual situations. The formation of mirage can be well-understood through studyingthe ray diagrams. Therefore, we have achieved our goals in this report.

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Part IV.

Further Discussion and

Application

The method and tools that we have developed in earlier parts is not yet utilised at fullpotential in this paper. In this part we point out some possible directions for furtherstudy and research based on what had been developed in earlier parts.

12. Lens

12.1. Atmospheric Lensing

Equation 2.16 incorporated variation of refractivity in the x-direction. However, through-out this report we only deal with refractivities that are independent to x. If x-coordinatedependence are added, the equation can be used the study the phenomenon of atmo-spheric lensing. We will give a quick qualitative description here.Consider a hot object, e.g. a hot iron cylinder, placed in the atmosphere, as shown in

�gure 17.1, the hot object will heat up the air nearby and hence changing the refractivityof nearby air. The density of air will be lower near the object so light will bend awayfrom it. Hence the image of the background seen by the observer will be distorted.

Figure 12.1: Atmospheric lensing e�ect

With our equation and computer program, we can solve for the light trajectories andsimulate the distortion.

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12.2. Liquid Lens

Liquid lens are becoming popular due to their easy recon�gurability and �exibility. Equa-tion 2.16 can be used to study or design liquid lens with cirular symmetry.

Figure 12.2: Ray diagrams of light passing through liquid lens at di�erent states [Source:Optilux (http://optilux.com/technology/)

13. More complicated atmospheres

13.1. Multi-layered atmospheres

The atmospheric models that we used are the simplest of all models used to simulatemirages. For better results (esp. when simulating superior mirage), we may introducemulti-layered pro�les. For example, we can add an additional inversion layer to ouroriginal Gauss atmosphere to create a dual-inversion atmosphere with the following tem-perature pro�le:

T (y) = T0 − Ly + c1 exp(−1

2(y − yb1a1

)2) + c2 exp(−1

2(y − yb2a2

)2) (13.1)

We can continue to add further inversion layers to model even more complicated phe-nomena.

13.2. Di�erent types of atmospheres

Apart from adding more terms to the temperature pro�le, we can add other functionsbesides those we used to the expression. For example, we can try adding an exponentialterm used in [12]:

T (y) = T0 − Ly + ∆ exp(− y

y0) (13.2)

13.3. Non-ideal atmosphere

We can also consider an atmosphere composed of non-ideal medium such that the idealgas law is invalid. For example, we can consider refractivity in an van der Waals gas

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obeying the van der Waals equation:[25]

(P +a

V 2)(V − b) = kT (13.3)

The calculation is more di�cult than when using ideal gas law, as the van der Waalsequation does not explicitly contain the density of the gas.

14. The reverse problem

Another thing that can be done is to study the reverse of the problem dealt in thispaper, i.e. to �nd the refractivity distribution from given light trajectories. This is amuch harder problem but may have useful applications such as measuring gas densitydistribution without the need to disturb the gas.In order to do this, we can use equation 2.16, which reads

[1 + (y′)2]∂n

∂y− [1 + (y′)2]y′

∂n

∂x− ny′′ = 0 (14.1)

di�erent from the way we treat it in this paper. This time we view n(x, y) as the unknown.Given a light trajectory y(x), with know derivatives y′(x) and y′′(x), we may solve forn(x, y) along the trajectory by using equation 2.16 as a �rst-order PARTIAL di�erentialequation in n(x, y). This is more di�cult than what we have done in this paper.

15. Summary

As can be seen, there are still many interesting topics concerning atmospheric refractionthat are not treated in this paper. These topics can be tackled using the methods andequations that are developed in this paper. Perhaps they will be treated by the authorsor other people in the future.

Appendix I � Algorithm and

Source Code for our simulation

16. Introduction

The program has two functions. It can trace the trajectory of light starting from a giveninitial point and a given initial slope (i.e. solve the Initial Condition Problem of eq.2.16). It can also �nd the trajectory between two given points (i.e. solve the BoundaryValue Problem of eq. 2.16). The algorithm is based on the Taylor method for solvingODE. The output of the program is a text �le containing the point series which can beused to plot the path of the light ray, using Excel or other graphing software.

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17. Principle of method

This program solves eq. 2.16 using 2nd-Order Taylor method.[17] We assume the solutiony is smooth enough such that y can be approximated by its Taylor series within a smallrange. We also assume that the object point is lying on the y-axis.The program begins by initializing the position of object (y0), distance between object

and observer (x), height of observer (y(x)) and iteration step (h).The iteration begins at (0, y0) , where y0 is the initial height (position of the object). At

each iteration, the x-coordinated will be increased by h. Thus, during the i-th iteration,the hortzontal displacement is xi = i ∗ h.At the beginning of each iteration, we �rst �nd y′′(xi) at the current point using eq.

2.16:

[1 + (y′)2]∂n

∂y− [1 + (y′)2]y′

∂n

∂x− ny′′ = 0 (17.1)

y′′(xi) =[1 + (y′(xi))

2] (ny(xi, yi)− y′(xi)nx(xi, yi))

n(xi, yi)

,and y(xi+1) and y′(xi+1) can be found by the iteration formulae:

y(xi+1) = y(xi) + y′(xi)h+ 0.5 y′′(xi)h2 (17.2)

y′(xi+1) = y′(xi) + y′′(xi)h (17.3)

By MVT, the error term for y(xi+1) is given by 16y′′′(c)h3 [18], where c is some value

between xi and xi+1. In our program, we estimate the error by

Err = 16y′′(xi+1)−y′′(xi)

h h3 (17.4)

= 16 [y′′(xi+1)− y′′(xi)]h2 (17.5)

Below are the pseudo-codes representing our algorithm: 2

2Calculation of estimated error is not shown, an exerpt of the source code written in C# can be found

in the Appendix.

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program TAYLOR_IVP ( y0 , yp0 ){i=0h=0.1y[0] = y0y'[0] = yp0UPPER_BOUND = 1000

while( y[i] < UPPER_BOUND ){//n := refractivity , nx := dn/dx , ny := dn/dyy�[i] = ( ny(i*h, y[i]) - y'[i] * nx(i*h, y[i]) ) * (1 + y'[i] ^ 2) / n(i*h, y[i])y[i+1] = y[i] + 0.5 * y'[i] * h + 1/6 * y�[i] * h^2y'[i+1] = y'[i] + 0.5 * y�[i] * hi++}return y[]}

To �nd a path between two given points, the program is con�gured to test a range ofstarting incidence angle. For each angle a unique curve of path of light y(x) is obtained,the di�erence between the position of the observer (d, h1) and y(d) is calculated.

18. Accuracy

The output of our program for n = 1+0.1y is compared with the analytical solution (eq.3.27) we have obtained in Part 1, for the case y0 = 0, y′0 = 1:

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Figure 18.1: Comparison of numerical and analytical results

We can see from the plot that the simulation produces remarkably accurate results.In fact, for h = 0.1, the percentage error is 0.8%. For h = 0.01, the percentage error isonly 0.5%.

19. Source Code

const double UBound = 50;

public double n(x,y){return 0.1y+1;}

public double ny(x,y){return 0.1;}

public double nx(x,y){return 0;}

public List<double> BVP(double _y0, double _x, double _yx, double _h, out double_angle, out double _err){

25

double TanAngle = 0;double SmallY = 100000000000;double StepAngle = 1;double StartAngle = 0;double EndAngle = 90;double StartAngle_ = 0;double EndAngle_ = 90;_angle = 666;_err = 666;

for (int i = 1; i < 6; i++){

for (double angle = StartAngle; angle < EndAngle; angle=angle+StepAngle){TanAngle = Math.Tan(angle * Math.PI / 180);outputs = IVPn(_y0, TanAngle, _h, Convert.ToInt32(_x/_h));if (Math.Abs(_yx - outputs[outputs.Count() - 1]) < SmallY){

SmallY = Math.Abs(_yx - outputs[outputs.Count() - 1]);StartAngle_ = angle - StepAngle;EndAngle_ = angle + StepAngle;outputsf = new List<double>(outputs);err�nal = new List<double>(err);_angle = angle;_err = SmallY;

}}

StepAngle = StepAngle / 10;StartAngle = StartAngle_;EndAngle = EndAngle_;}err = new List<double>(err�nal);return outputsf;}

public List<double> IVP(double _y0, double _yp0, double _h){i=0;y.Clear();yp.Clear();ypp.Clear();err.Clear();yp.Add(_yp0);y.Add(_y0);

26

ypp.Add((ny(0, _y0)-nx(0,_y0)*_yp0) * (1 + _yp0*_yp0) / n(0, _y0));err.Add(0); h = _h;while (y[i] >= 0 && y[i] < UBound){

y.Add(y[i] + yp[i] * h+0.5*ypp[i]*h*h);yp.Add(yp[i]+ypp[i]*h);i++;ypp.Add((ny(i*h, y[i]) - nx(i*h,y[i]) * yp[i]) * (1 + yp[i]*yp[i]) / n(i*h,y[i]));err.Add((ypp[i] - ypp[i-1]) * h *h/6);

}return y;}

public List<double> IVPn(double _y0, double _yp0, double _h, int _n){i = 0;y.Clear();yp.Clear();ypp.Clear();err.Clear();yp.Add(_yp0);y.Add(_y0);ypp.Add((ny(0, _y0) - nx(0, _y0) * _yp0) * (1 + _yp0 * _yp0) / n(0, _y0));err.Add(0);h = _h;while (y[i] >= 0 && y[i] < UBound && i <_n){

y.Add(y[i] + yp[i] * h + 0.5 * ypp[i] * h * h);yp.Add(yp[i] + ypp[i] * h);i++;ypp.Add((ny(i * h, y[i]) - nx(i * h, y[i]) * yp[i]) * (1 + yp[i] * yp[i]) / n(i * h,

y[i]));err.Add((ypp[i] - ypp[i - 1]) * h * h / 6);

}return y;}

References

[1] Optics Picture of the Day - Baltic Mirage (http://www.atoptics.co.uk/fz545.htm)

[2] An Introduction to Mirages (http://mintaka.sdsu.edu/GF/mirages/mirintro.html)

[3] Wikipedia - Fermat's Principle (http://en.wikipedia.org/wiki/Fermat's_principle)

27

[4] Feynman R.P. , The Feynman Lectures on Physics (Vol. II)

[5] Greenler R., Rainbows, Halos, and Glories

[6] Merzkirch W., Flow Visualization (2nd Ed.), Academic Press

[7] Wikipedia - Gladstone-Dale Relation(http://en.wikipedia.org/wiki/Gladstone%E2%80%93Dale_relation)

[8] Mineralogy Database - Gladstone-Dale Relationships(http://webmineral.com/help/Gladstone-Dale.shtml)

[9] Wikipedia - Visible Spectrum (http://en.wikipedia.org/wiki/Visible_spectrum)

[10] Hydrostatic Equilibrium and the Homogeneous Atmosphere(http://mintaka.sdsu.edu/GF/explain/thermal/hydrostatic.html)

[11] Wikipedia - Atmospheric Pressure (http://en.wikipedia.org/wiki/Atmospheric_pressure)

[12] Gutierrez D. et al., Simulation of atmosphere phenomena, Elsevier Computer &Graphics (2006)

[13] Dale T.P., Gladstone J. H., On the In�uence of Temperature on the Refraction ofLight (1858)

[14] Wikipedia - US Standard Atmosphere (http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere)

[15] Wikipedia - Ideal Gas (http://en.wikipedia.org/wiki/Ideal_gas)

[16] Wikipedia - Ideal gas law (http://en.wikipedia.org/wiki/Ideal_gas_law)

[17] Taylor's Method for ODE (http://math.fullerton.edu/mathews/n2003/TaylorDEMod.html)

[18] Error Estimation(http://www.millersville.edu/∼bikenaga/calculus/tayerr/tayerr.html)

[19] van der Werf S. Y. et al., Novaya Zemlya e�ect and sunsets, Applied Optics, Vol.42, No. 3 (Jan., 2003), pp. 367-378

[20] Buxton P. A., The Temperature of the Surface of Deserts, Journal of Ecology, Vol.12, No. 1 (Jan., 1924), pp. 127-134

[21] Atmospheric Optics - Multiple Road Mirage (http://www.atoptics.co.uk/fz311.htm)

[22] M. Vollmer, Mirrors in the air: mirages in nature and in the laboratory, PhysicsEducation, Vol. 44, No. 1 (Mar., 2009), pp. 165-174, IOP Publishing

[23] The Omega Sunset (and Sunrise) (http://mintaka.sdsu.edu/GF/explain/simulations/inf-mir/Omega.html#pro�le)

[24] Graph O�cial Site (http://www.padowan.dk/)

[25] Wikipedia - van der Waals equation (http://en.wikipedia.org/wiki/Van_der_Waals_equation)

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