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Lab Experiments 295 KamalJeeth Instrumentation and Service Unit
Experiment-351
A
EXPERIMENTAL VERIFICATION OF FRESNEL’S EQUATIONS FOR REFLECTION OF ELECTROMAGNETIC WAVES K Chandrasekhara Reddy and D Sailaja Department of Physics, S.S.B.N.Degree & P.G. College (Autonomous), Anantapur-515001.A.P. INDIA.
E-mail:[email protected] Abstract The Fresnel’s equations, deduced by Augustin-Jean Fresnel, describe the behaviour of light when moving through media of different refractive indices. The equations relating the amplitudes of the reflected and transmitted waves with the amplitude of the incident waves are known as Fresnel’s equations or Fresnel’s formulae. The reflection of light predicted by these equations is known as the Fresnel reflection. Using a polarizer analyzer pair, Fresnel’s equations for reflection of electromagnetic waves are experimentally verified.
Introduction Light waves are electromagnetic in nature. Light waves coming directly from a source usually vibrate randomly in all directions at right angles to the wave normal. When light is plane polarized, the components of its vibrations in some plane are transmitted or reflected to the exclusion of the other components. The eye is unable to detect such a selection of components. Although the intensity is reduced to 50 % in the most efficient polarizer, its rotation causes no change in intensity. However, if a second polarizing device operates on the polarized light, it will transmit or reflect only the cosine components in its plane of vibration. By considering the behaviour of electric and magnetic fields at a dielectric boundary (between air and glass), Fresnel derived equations which relate the amplitude and intensity of reflected and transmitted waves with the corresponding value of the incident wave [1-3].
Theory Fresnel’s equations for reflection of electromagnetic waves at the interface of two nonmagnetic dielectric media (air and glass) are
Ars sin(θ1 − θ 2 ) = Ais sin(θ1 + θ 2 ) Arp Aip
=
…1
tan(θ1 − θ 2 ) tan(θ1 + θ 2 )
…2
Vol-11, No-4, Dec.-2011
Lab Experiments 296 KamalJeeth Instrumentation and Service Unit Here Ai and Ar are the amplitude of the electric field vectors of the incident and the reflected wave respectively. The subscript ‘p’ indicates that the electric vector is parallel to the plane of incidence (p-polarization), and the subscript ‘s’ indicates that the electric vector is perpendicular to the plane of incidence (s-polarization). θ1 and θ2 are the angles of incidence and refraction respectively. Dividing Equation-1 by Equation-2 gives
tanα r cos(θ1 − θ 2 ) =tanα i cos(θ1 + θ 2 )
…3
Here α r is the angle between the plane of polarization of the reflected light and the plane of incidence. α i is the angle between the plane of polarization of the incident light and the plane of incidence. The validity of Equation-3 would, therefore, imply the verification of Fresnel’s equations. The angles θ1 and θ2 are related by the well known Snell’s Law:
n=
sin θ1 sin θ 2
…4
where n is the refractive index of the refracting medium. When the angle of incidence θ1 equals Brewster’s angle θ B , where tan θ B = n, the reflected light does not contain p-polarization. Here the reflected and the refracted rays are at right angles to each other such that θ1 + θ 2 =900. The experiment is performed with the help of a prism, the refractive index n of its material is determined by n = Sin( A + D m ) / 2 SinA / 2
…5
where is the angle of the prism and Dm is the angle of the minimum deviation of a ray of monochromatic light refracted through its principle section.
Apparatus used Spectrometer, prism, a pair of polaroids, sodium light, spirit level etc.
Experimental procedure Preliminary adjustments of the spectrometer are made. The refractive index ‘n’ of the material of the prism is determined for sodium light by using the spectrometer (n=1.6). Now the polarizing angle or Brewster’s angle θ B is determined for the material of the prism with the help of Brewster’s law −1
Tan θ B = n (or) θ B = tan n (Table-I)
Vol-11, No-4, Dec.-2011
Lab Experiments 297 KamalJeeth Instrumentation and Service Unit Table-1 Refractive Index n
θ B = tan −1 n
1.6
580
Brewster’s angle θ B The prism is removed from the prism table, and the telescope is set to receive the direct image of the slit. The junction of the cross wires is made to coincide with the centre of the image by moving the telescope slowly with the help of its tangent screw. This initial position of the telescope is recorded. The telescope is then rotated through an angle (180-2 θ B ) and is clamped at this position. The reflecting faces of the prism are cleaned and the prism is placed on the prism table which is turned such that the parallel beam of light from the collimator is reflected from the face of the prism and enters the telescope (Figure- 1).The image of the slit can now be observed through the telescope. The centre of the image is made to coincide at the junction of the cross wires. At this position, the angle of incidence on the reflecting surface of the prism is θ B and the reflected light does not contain the p-polarization.
Figure-1: Determination θB
Figure-2: Experimental Set-up
Polaroid with a graduated circular scale and capable of rotation is mounted in front of the telescope. The polaroid is rotated till the light is extinguished or the intensity of the image produced by the slit becomes minimum. This initial position of the Polaroid which is now set parallel to the plane of incidence is recorded.
Vol-11, No-4, Dec.-2011
Lab Experiments 298 KamalJeeth Instrumentation and Service Unit The prism is removed and the telescope with the Polaroid is brought to its initial position. Another Polaroid is mounted on the collimator which is rotated till the light gets cut off, as seen through the telescope. For this setting of the collimator Polaroid, the plane of polarization of the incident light is vertical, that is perpendicular to the plane of incidence. This position of the collimator Polaroid is recorded in Table- 2. Table-2 Initial Positions Telescope 1800
00
Telescope Polaroid
Collimator Polaroid
357030’
72030’
Initial positions for the experiment The collimator polaroid is rotated through an angle, θ C =300, so that the plane of polarization of the incident light is inclined to the plane of incidence (which is horizontal) at an angle αi =900- θ C . The collimator polaroid is rotated through an angle, θ C =300, so that the plane of polarization of the incident light is inclined to the plane of incidence (which is horizontal) at an angle
θ1 = (180o − β ) / 2 .For θ1 =400, β =1000. The telescope Polaroid is rotated from its initial setting through an angle θ t , so that the light is extinguished. It may be noted that θ t can be both positive and negative, depending on the direction of rotation with respect to the initial position. Thus the plane of polarization of the reflected light is inclined at the plane of incidence by an angle
α r =900- θ t . The values of α r for several different values of the angle of incidence, θ t , are recorded in table-3 and experiment is repeated for θC = 450 and 600 (Table-3 , 4, and 5) . No. of Observations
1 2 3
αi (=90- θC) degrees 600 600 600
Table-3 Telescope Angle of Rotation incidence β θ1= degrees (1800-β)/2 degrees
Telescope Polaroid Rotation θt degrees
1000 400 15012’ 0 0 90 45 110 0 0 80 50 07006’ Collimator Polaroid rotation θC=300
Vol-11, No-4, Dec.-2011
αr (=90- θt) degrees 74052’ 790 82054’
tan α r tan α i
2.134 2.970 4.635
Lab Experiments 299 KamalJeeth Instrumentation and Service Unit Table-4 Angle of Telescope incidence Rotation β θ1= degrees (1800-β)/2 degrees
αi No.of Observations
(=90- θC) degrees 450 450 450
1 2 3
1000 900 800
Telescope Polaroid Rotation θt degrees
400 450 500
250 19018’ 12012’
αr
tan α r tan α i
(=90- θt) degrees 650 70042’ 77048’
2.144 2.855 4.625
Collimator polaroid rotation, θC=450
No.of Observation s
αi (=90- θC) degrees
Table-5 Angle of Telescope incidence Rotation θ1= β (1800-β)/2 degrees degrees
300 300 300
1 2 3
1000 900 800
400 450 500
Telescope Polaroid Rotation θt degrees
(=90- θt) degrees
38006’ 31006’ 200
51054’ 58054’ 700
αr
tan α r tan α i 2.208 2.871 4.758
Collimator Polaroid rotation, θC=600 Table-6
Set
I (θC=300)
II (θC=450)
III (θC=600)
No. of Obs erv atio ns
θ1 Degr ees
1
θ2 = (sin-1
sin θ 1 n )
cos(θ1 − θ tan α r
Degrees
(θ1- θ2) Degree s
(θ1+ θ2) Degrees
40
23040’
16019’
63040’
2.254
2.134
2
45
26012’
18048’
71012’
2.937
2.970
3
50
28.6
21024’
78036’
4.710
4.635
1
40
0
23 40
0
16 19
0
’
63 40
2.254
2.144
2
45
26012’
18048’
71012’
2.937
2.855
3
50
28.6
21024’
78036’
4.710
4.625
1
40
23040’
16019’
63040’
2.254
2.208
2
45
26012’
18048’
71012’
2.937
2.871
3
50
28.6
21024’
78036’
4.710
4.758
’
’
cos(θ1 + θ tan α i Remarks
The values of
tan α r tan α i and
-
cos(θ1 − θ 2 ) cos(θ1 + θ 2 ) are nearly equal.
Verification of Fresnel’s equations
Vol-11, No-4, Dec.-2011
Lab Experiments 300 KamalJeeth Instrumentation and Service Unit For different values of angle of incidence θ1, the corresponding values of the angle of refraction θ2 are determined using the Snell’s law (Equation-4).
cos(θ1 − θ 2 ) cos(θ1 + θ 2 )
The values of -
cos(θ1 − θ 2 ) cos(θ1 + θ 2 )
are calculated and presented in Table-6.
tan α r cos(θ1 − θ 2 ) and which is a straight line with slope tan α i cos(θ1 + θ 2 ) of 450, thus ing the Fresnel’s equations.
A graph is plotted between
Results tan α r tan α i
cos(θ1 − θ 2 ) are found to be nearly equal, thus ing cos(θ1 + θ 2 ) tan α r cos(θ1 − θ 2 ) and Fresnel’s equations experimentally. The graph plotted between tan α i cos(θ1 + θ 2 ) 0 is a straight line, with slope 45 (Figure-2), ing Fresnel’s equations graphically. The values of
and -
5.0
B
enter text here
5.0
5.0
4.5
4.5 4.5
4.0
4.0 4.0
3.5
3.5 3.5
3.0
3.0 3.0
2.5
2.5 2.5
2.0
2.0 2.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0 2.0
2.5
3.0
3.5
4.0
4.5
5.0
2.0
2.5
3.0
3.5
4.0
4.5
X Axis Title
tan α r tanαi /
tan α r tanαi /
tan α r tanαi /
Figure-2: variation of Cos (θ1-θ2)/ Cos (θ1+θ2) versus tanαr tanαi
References [1]
Joseph Valasec, Introduction to theoretical and experimental optics , John Wiley & sons Inc, New York.
[2]
F.A.Jenkins and H.E White, Fundamentals of Optics (4th edition) McGraw Hill, NewYork, 1976.
[3]
D.Chatopadhyay and P.C.Rakshit, An advanced course in Practical Physics-, New Central Book Agency (P) Ltd, Kolkata.
Vol-11, No-4, Dec.-2011
5.0
Experiment-351
A
EXPERIMENTAL VERIFICATION OF FRESNEL’S EQUATIONS FOR REFLECTION OF ELECTROMAGNETIC WAVES K Chandrasekhara Reddy and D Sailaja Department of Physics, S.S.B.N.Degree & P.G. College (Autonomous), Anantapur-515001.A.P. INDIA.
E-mail:[email protected] Abstract The Fresnel’s equations, deduced by Augustin-Jean Fresnel, describe the behaviour of light when moving through media of different refractive indices. The equations relating the amplitudes of the reflected and transmitted waves with the amplitude of the incident waves are known as Fresnel’s equations or Fresnel’s formulae. The reflection of light predicted by these equations is known as the Fresnel reflection. Using a polarizer analyzer pair, Fresnel’s equations for reflection of electromagnetic waves are experimentally verified.
Introduction Light waves are electromagnetic in nature. Light waves coming directly from a source usually vibrate randomly in all directions at right angles to the wave normal. When light is plane polarized, the components of its vibrations in some plane are transmitted or reflected to the exclusion of the other components. The eye is unable to detect such a selection of components. Although the intensity is reduced to 50 % in the most efficient polarizer, its rotation causes no change in intensity. However, if a second polarizing device operates on the polarized light, it will transmit or reflect only the cosine components in its plane of vibration. By considering the behaviour of electric and magnetic fields at a dielectric boundary (between air and glass), Fresnel derived equations which relate the amplitude and intensity of reflected and transmitted waves with the corresponding value of the incident wave [1-3].
Theory Fresnel’s equations for reflection of electromagnetic waves at the interface of two nonmagnetic dielectric media (air and glass) are
Ars sin(θ1 − θ 2 ) = Ais sin(θ1 + θ 2 ) Arp Aip
=
…1
tan(θ1 − θ 2 ) tan(θ1 + θ 2 )
…2
Vol-11, No-4, Dec.-2011
Lab Experiments 296 KamalJeeth Instrumentation and Service Unit Here Ai and Ar are the amplitude of the electric field vectors of the incident and the reflected wave respectively. The subscript ‘p’ indicates that the electric vector is parallel to the plane of incidence (p-polarization), and the subscript ‘s’ indicates that the electric vector is perpendicular to the plane of incidence (s-polarization). θ1 and θ2 are the angles of incidence and refraction respectively. Dividing Equation-1 by Equation-2 gives
tanα r cos(θ1 − θ 2 ) =tanα i cos(θ1 + θ 2 )
…3
Here α r is the angle between the plane of polarization of the reflected light and the plane of incidence. α i is the angle between the plane of polarization of the incident light and the plane of incidence. The validity of Equation-3 would, therefore, imply the verification of Fresnel’s equations. The angles θ1 and θ2 are related by the well known Snell’s Law:
n=
sin θ1 sin θ 2
…4
where n is the refractive index of the refracting medium. When the angle of incidence θ1 equals Brewster’s angle θ B , where tan θ B = n, the reflected light does not contain p-polarization. Here the reflected and the refracted rays are at right angles to each other such that θ1 + θ 2 =900. The experiment is performed with the help of a prism, the refractive index n of its material is determined by n = Sin( A + D m ) / 2 SinA / 2
…5
where is the angle of the prism and Dm is the angle of the minimum deviation of a ray of monochromatic light refracted through its principle section.
Apparatus used Spectrometer, prism, a pair of polaroids, sodium light, spirit level etc.
Experimental procedure Preliminary adjustments of the spectrometer are made. The refractive index ‘n’ of the material of the prism is determined for sodium light by using the spectrometer (n=1.6). Now the polarizing angle or Brewster’s angle θ B is determined for the material of the prism with the help of Brewster’s law −1
Tan θ B = n (or) θ B = tan n (Table-I)
Vol-11, No-4, Dec.-2011
Lab Experiments 297 KamalJeeth Instrumentation and Service Unit Table-1 Refractive Index n
θ B = tan −1 n
1.6
580
Brewster’s angle θ B The prism is removed from the prism table, and the telescope is set to receive the direct image of the slit. The junction of the cross wires is made to coincide with the centre of the image by moving the telescope slowly with the help of its tangent screw. This initial position of the telescope is recorded. The telescope is then rotated through an angle (180-2 θ B ) and is clamped at this position. The reflecting faces of the prism are cleaned and the prism is placed on the prism table which is turned such that the parallel beam of light from the collimator is reflected from the face of the prism and enters the telescope (Figure- 1).The image of the slit can now be observed through the telescope. The centre of the image is made to coincide at the junction of the cross wires. At this position, the angle of incidence on the reflecting surface of the prism is θ B and the reflected light does not contain the p-polarization.
Figure-1: Determination θB
Figure-2: Experimental Set-up
Polaroid with a graduated circular scale and capable of rotation is mounted in front of the telescope. The polaroid is rotated till the light is extinguished or the intensity of the image produced by the slit becomes minimum. This initial position of the Polaroid which is now set parallel to the plane of incidence is recorded.
Vol-11, No-4, Dec.-2011
Lab Experiments 298 KamalJeeth Instrumentation and Service Unit The prism is removed and the telescope with the Polaroid is brought to its initial position. Another Polaroid is mounted on the collimator which is rotated till the light gets cut off, as seen through the telescope. For this setting of the collimator Polaroid, the plane of polarization of the incident light is vertical, that is perpendicular to the plane of incidence. This position of the collimator Polaroid is recorded in Table- 2. Table-2 Initial Positions Telescope 1800
00
Telescope Polaroid
Collimator Polaroid
357030’
72030’
Initial positions for the experiment The collimator polaroid is rotated through an angle, θ C =300, so that the plane of polarization of the incident light is inclined to the plane of incidence (which is horizontal) at an angle αi =900- θ C . The collimator polaroid is rotated through an angle, θ C =300, so that the plane of polarization of the incident light is inclined to the plane of incidence (which is horizontal) at an angle
θ1 = (180o − β ) / 2 .For θ1 =400, β =1000. The telescope Polaroid is rotated from its initial setting through an angle θ t , so that the light is extinguished. It may be noted that θ t can be both positive and negative, depending on the direction of rotation with respect to the initial position. Thus the plane of polarization of the reflected light is inclined at the plane of incidence by an angle
α r =900- θ t . The values of α r for several different values of the angle of incidence, θ t , are recorded in table-3 and experiment is repeated for θC = 450 and 600 (Table-3 , 4, and 5) . No. of Observations
1 2 3
αi (=90- θC) degrees 600 600 600
Table-3 Telescope Angle of Rotation incidence β θ1= degrees (1800-β)/2 degrees
Telescope Polaroid Rotation θt degrees
1000 400 15012’ 0 0 90 45 110 0 0 80 50 07006’ Collimator Polaroid rotation θC=300
Vol-11, No-4, Dec.-2011
αr (=90- θt) degrees 74052’ 790 82054’
tan α r tan α i
2.134 2.970 4.635
Lab Experiments 299 KamalJeeth Instrumentation and Service Unit Table-4 Angle of Telescope incidence Rotation β θ1= degrees (1800-β)/2 degrees
αi No.of Observations
(=90- θC) degrees 450 450 450
1 2 3
1000 900 800
Telescope Polaroid Rotation θt degrees
400 450 500
250 19018’ 12012’
αr
tan α r tan α i
(=90- θt) degrees 650 70042’ 77048’
2.144 2.855 4.625
Collimator polaroid rotation, θC=450
No.of Observation s
αi (=90- θC) degrees
Table-5 Angle of Telescope incidence Rotation θ1= β (1800-β)/2 degrees degrees
300 300 300
1 2 3
1000 900 800
400 450 500
Telescope Polaroid Rotation θt degrees
(=90- θt) degrees
38006’ 31006’ 200
51054’ 58054’ 700
αr
tan α r tan α i 2.208 2.871 4.758
Collimator Polaroid rotation, θC=600 Table-6
Set
I (θC=300)
II (θC=450)
III (θC=600)
No. of Obs erv atio ns
θ1 Degr ees
1
θ2 = (sin-1
sin θ 1 n )
cos(θ1 − θ tan α r
Degrees
(θ1- θ2) Degree s
(θ1+ θ2) Degrees
40
23040’
16019’
63040’
2.254
2.134
2
45
26012’
18048’
71012’
2.937
2.970
3
50
28.6
21024’
78036’
4.710
4.635
1
40
0
23 40
0
16 19
0
’
63 40
2.254
2.144
2
45
26012’
18048’
71012’
2.937
2.855
3
50
28.6
21024’
78036’
4.710
4.625
1
40
23040’
16019’
63040’
2.254
2.208
2
45
26012’
18048’
71012’
2.937
2.871
3
50
28.6
21024’
78036’
4.710
4.758
’
’
cos(θ1 + θ tan α i Remarks
The values of
tan α r tan α i and
-
cos(θ1 − θ 2 ) cos(θ1 + θ 2 ) are nearly equal.
Verification of Fresnel’s equations
Vol-11, No-4, Dec.-2011
Lab Experiments 300 KamalJeeth Instrumentation and Service Unit For different values of angle of incidence θ1, the corresponding values of the angle of refraction θ2 are determined using the Snell’s law (Equation-4).
cos(θ1 − θ 2 ) cos(θ1 + θ 2 )
The values of -
cos(θ1 − θ 2 ) cos(θ1 + θ 2 )
are calculated and presented in Table-6.
tan α r cos(θ1 − θ 2 ) and which is a straight line with slope tan α i cos(θ1 + θ 2 ) of 450, thus ing the Fresnel’s equations.
A graph is plotted between
Results tan α r tan α i
cos(θ1 − θ 2 ) are found to be nearly equal, thus ing cos(θ1 + θ 2 ) tan α r cos(θ1 − θ 2 ) and Fresnel’s equations experimentally. The graph plotted between tan α i cos(θ1 + θ 2 ) 0 is a straight line, with slope 45 (Figure-2), ing Fresnel’s equations graphically. The values of
and -
5.0
B
enter text here
5.0
5.0
4.5
4.5 4.5
4.0
4.0 4.0
3.5
3.5 3.5
3.0
3.0 3.0
2.5
2.5 2.5
2.0
2.0 2.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0 2.0
2.5
3.0
3.5
4.0
4.5
5.0
2.0
2.5
3.0
3.5
4.0
4.5
X Axis Title
tan α r tanαi /
tan α r tanαi /
tan α r tanαi /
Figure-2: variation of Cos (θ1-θ2)/ Cos (θ1+θ2) versus tanαr tanαi
References [1]
Joseph Valasec, Introduction to theoretical and experimental optics , John Wiley & sons Inc, New York.
[2]
F.A.Jenkins and H.E White, Fundamentals of Optics (4th edition) McGraw Hill, NewYork, 1976.
[3]
D.Chatopadhyay and P.C.Rakshit, An advanced course in Practical Physics-, New Central Book Agency (P) Ltd, Kolkata.
Vol-11, No-4, Dec.-2011
5.0
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