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Chapter: 2 General physics
    Section: 2.5 Radiation and optics
        SubSection: 2.5.9 Light reflection

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2.5.9 Light reflection

The reflectance R at normal incidence at the interface between two non-absorbing media of refractive index n0 and n1 is given by

      R = [(n0n1)/(n0 + n1)]2

R(%) for various values of n1 and n0 = 1 (air) are tabulated below

   

n1

1.46

1.52

1.60

1.80

2.0

  3.0   4.0
   

R(%)

3.5

4.3

5.3

8.2

11.1

 25.0

 36.0

For light at an angle of incidence θ0 and angle or refraction θ1 in the two media, the reflectance R depends on the state of polarization of the light and is given by

         R = [(N0N1)/(N0 + N1)]2

where for light polarized with the electric vector parallel to the plane of incidence (p vibration, TM wave)

         N0 = n0/cosθ0   N1 = n1/cosθ1

while for light with the electric vector perpendicular to the plane of incidence (s vibration, TE wave)

         N0 = n0cosθ0   N1 = n1cosθ1

For incident unpolarized light, the reflectance is the mean of Rp and Rs. Values tabulated below are for an air/glass interface (n0 = 1, n1 = 1.52)

Angle of incidence

Rp (%)

Rs (%)

Angle of incidence

Rp (%)

Rs (%)

 

 

 

 

 

 

  0°

4.3

4.3

   56.7°

  0.0

15.7

15°

3.9

4.7

60°

  0.2

18.3

30°

2.7

6.1

75°

 10.6

40.8

45°

0.9

9.7

85°

 49.2

73.8

 

 

 

 

 

 

The angle at which Rp = 0 is given by tan θ0 = n1/n0 and is known as the Brewster angle.

At grazing incidence (θ0 = 90°), Rp = Rs = 100%.

The reflection from a dielectric surface may readily be modified by the application of thin film coatings. For a single dielectric film located between the two bulk media of refractive indices n0 and n1, the reflectance varies with the film thickness df and refractive index nf. When the film optical thickness (nfdf) equals one quarter of the wavelength of the incident light, the reflectance is a minimum if nf is between n0 and n1, or a maximum if it lies outside this range. The reflectance R at normal incidence is then given by


     

R =

nf2n0n1

2

nf2 + n0n1

Typical refractive indices of common thin film materials (wavelength 550 nm) and the reflectance of quarter-wave films on glass (n0 = l, n1 = 1.52) are shown below. The reflectances of quarter-wave multi-layer stacks of alternately high and low refractive index films on a glass substrate are also tabulated.



Material

Refractive Index (nf)

Reflectance (%)

 

 

 

Cryolite (Na3AlF6)     .    .    .    .    .    .    .    .    .    .    .    .

1.35

  0.8

Magnesium fluoride    .    .    .    .    .    .    .    .    .    .    .    .

1.38

  1.3

Silicon dioxide            .    .    .    .    .    .    .    .    .    .    .    .

1.46

  2.8

Aluminium oxide         .    .    .    .    .    .    .    .    .    .    .    .

1.63

  7.4

Lead fluoride              .    .    .    .    .    .    .    .    .    .    .    .

1.75

 11.3

Zirconium dioxide       .    .    .    .    .    .    .    .    .    .    .    .

2.05

 22.0

Tantalum pentoxide     .    .    .    .    .    .    .    .    .    .    .    .

2.15

 25.5

Zinc sulphide              .    .    .    .    .    .    .    .    .    .    .    .

2.35

 32.5

Titanium dioxide         .    .    .    .    .    .    .    .    .    .    .    .

2.2–2.7

27.3–42.9

 

 

 

Multilayer stack (HL ... LH)

 

 

(ZnS/MgF2)    
3 layer   68.3
5 layer   87.7
7 layer   95.6
9 layer   98.5
11 layer     99.5

 

 

 




For complete elimination of the reflected beam the refractive index of a single film should equal , when the reflectance will be zero for the wavelength at which the film is quarter-wave in optical thickness. With the range of thin film materials available, complete suppression of the reflected beam cannot be achieved using a single film on common optical glasses. However, with a multilayer design, high-efficiency antireflection coatings on glass can be produced with a reflectance of less than 0.5% through the visible spectrum.

High refractive index films can usefully be employed as beam splitters. At oblique angles of incidence, appropriate design of a multilayer system allows the intensity and state of polarization of the transmitted and reflected components to be varied over wide limits.

For metals and other opaque media the normal incidence reflectance R is given by


    R = (n1n0)2 + k12
(n1 + n0)2 + k12

where n0 is the refractive index of the entrance medium and n1 and k1 are the real and imaginary parts of the complex refractive index n1 = n1ik1of the absorbing medium. n1 and k1 are usually known as the refractive index and absorption index respectively.


Typical optical constants and reflectance (normal incidence) of metals and semi-conductors in air

Material

Wavelength

μm

n

k

R(%)

 

 

 

 

 

Aluminium        .    .    .    .    .    .    .    .    .    .    .    .    .    .

0.22

0.14 

2.35

91.8 

 

  0.546

0.82 

5.99

91.6 

 

 1.0  

 1.4   

 9.5  

94.2 

 

 10.0   

 25.4     

  67.3    

98.1 

Silver     .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .

 0.55

 0.055

 3.32

98.2 

 

  10.0    

 10.7     

  69.0   

99.1 

Gold .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .

 0.55

0.33 

 2.32

81.5 

 

 1.0  

0.27 

 7.07

97.9 

 

 10.0   

7.4   

 53.4   

99.0 

Rhodium     .    .    .    .    .    .    .    .    .    .    .    .    .    .    .

   0.546

1.62 

 4.63

77.1 

Chromium   .    .    .    .    .    .    .    .    .    .    .    .    .    .    .

   0.546

2.51 

 2.66

48.2 

Nickel    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .

 0.54

1.85 

 3.27

60.7 

Copper  .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .

 0.55

0.76 

 2.46

66.8 

Steel      .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .

 0.55

 2.4   

 3.4  

58.5 

Silicon   .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .

   0.546

4.05 

 0.03

36.5 

 

  1.50

3.48 

  0     

30.6 

 

 5.0

 3.422

  0     

30.0 

Germanium      .    .    .    .    .    .    .    .    .    .    .    .    .    .

    0.545

5.15 

  2.15 

51.5 

 

   1.50

4.28 

  0     

38.6 

 

  10.0  

 4.003

  0     

36.0 

Gallium arsenide   .    .    .    .    .    .    .    .    .    .    .    .    .

     0.55  

4.04 

  0.3  

36.6 

 

  1.5

3.38 

  0     

29.5 

 

  5.0

3.30 

  0     

28.6 

Indium phosphide      .    .    .    .    .    .    .    .    .    .    .    .

    0.55

3.65 

 0.4  

33.0 

 

  1.5

3.17 

  0     

27.1 

 

  5.0

3.09 

  0     

26.1 

 

 

 

 

 




Values of the optical constants exhibit significant variations depending on whether the material is in bulk or thin film form and its method of preparation. The reflectance of a metal surface is altered by the build up of oxide or other surface layers, while very dramatic changes can be produced by the deposition of thicker dielectric films.

Opaque metal films, e.g. aluminium, silver and gold (in the infrared), are commonly used as high reflectance mirrors, while very thin semi-transparent films (i.e. a few nm thick) can be employed as beam splitters, although there is significant light loss due to absorption.




R.J.King / K.W.Raine

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