Abstract

Polarizers with transmittances of 35 to 45% are common (Nicol prisms, dichroic polarizers); with non-reflecting coatings, such polarizers can be made with transmittances approaching 50%. Beam-splitting polarizers can be made with transmittances approaching 100%, but these polarizers are not “spathic”: The geometry of the emergent beam is quite different from the geometry of the entering beam.

In this paper it is shown that a spathic polarizer with a transmittance of 100% is impossible; such a polarizer would violate the laws of thermodynamics. It is shown how one can construct in principle a spathic polarizer with a transmittance of about 89%, the exact figure depending on the brightness and wavelength of the light. It is further shown that as far as thermodynamic limitations alone are concerned, a spathic polarizer having a transmittance of about 99% is possible.

The quantitative results are based on Planck’s thermodynamic theory of heat radiation.

© 1962 Optical Society of America

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References

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  1. M. Planck, Vorlesungen über die Theorie der Wärmestrahlung (Johann Ambrosius Barth, Leipzig, 1913), 2nd ed.
  2. M. Planck, The Theory of Heat Radiation (P. Blackiston’s Son and Company, Philadelphia, 1914; Dover Publications, Inc., New York, 1959). Translation of reference 1 by M. Masius.
  3. J. K. Roberts, Heat and Thermodynamics (Blackie and Son Limited, London, 1933), 2nd ed., p. 396.
  4. Reference 3, p. 393.
  5. M. W. Zemansky, Heat and Thermodynamics (McGraw-Hill Book Company, Inc., New York, 1951), 3rd ed. p. 192.

Planck, M.

M. Planck, Vorlesungen über die Theorie der Wärmestrahlung (Johann Ambrosius Barth, Leipzig, 1913), 2nd ed.

M. Planck, The Theory of Heat Radiation (P. Blackiston’s Son and Company, Philadelphia, 1914; Dover Publications, Inc., New York, 1959). Translation of reference 1 by M. Masius.

Roberts, J. K.

J. K. Roberts, Heat and Thermodynamics (Blackie and Son Limited, London, 1933), 2nd ed., p. 396.

Zemansky, M. W.

M. W. Zemansky, Heat and Thermodynamics (McGraw-Hill Book Company, Inc., New York, 1951), 3rd ed. p. 192.

Other (5)

M. Planck, Vorlesungen über die Theorie der Wärmestrahlung (Johann Ambrosius Barth, Leipzig, 1913), 2nd ed.

M. Planck, The Theory of Heat Radiation (P. Blackiston’s Son and Company, Philadelphia, 1914; Dover Publications, Inc., New York, 1959). Translation of reference 1 by M. Masius.

J. K. Roberts, Heat and Thermodynamics (Blackie and Son Limited, London, 1933), 2nd ed., p. 396.

Reference 3, p. 393.

M. W. Zemansky, Heat and Thermodynamics (McGraw-Hill Book Company, Inc., New York, 1951), 3rd ed. p. 192.

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Figures (4)

Fig. 1
Fig. 1

A nonspathic polarizer with a transmittance of 100% (except for surface reflections). The initial beam of unpolarized light is split into two parallel beams of polarized light by the block of calcite. The upper of the two beams then passes through a suitably oriented half-wave plate, which changes the plane of polarization of the upper beam so that it is the same as that of the lower beam. All of the energy of the initial unpolarized beam is thus converted to polarized light of the same state of polarization.

Fig. 2
Fig. 2

The two parallel screens that define the geometry of a pencil of radiation.

Fig. 3
Fig. 3

The relation between the initial entropy 2 L ( K ) of the pencil of light before it passes through the spathic polarizer of transmittance α and the final entropy L ( 2 α K ) + L p of the light-plus-polarizer system after the pencil has been transmitted by the polarizer. The plot is calculated for incident unpolarized light with the wavelength 0.55 μ and with the temperature 3000°K. The two lines show that the final entropy would be less than the initial entropy if the transmittance α were greater than 99.1%. Since a decrease of entropy in a natural process is impossible according to the second law of thermodynamics, the transmittance α cannot be greater than 99.1%.

Fig. 4
Fig. 4

Showing the maximum possible transmittance αm of a room temperature spathic polarizer as a function of the temperature of incident unpolarized light of λ = 0.55 μ. It may be noted that the maximum possible transmittance is fairly close to 99% for all light of practical interest.

Equations (38)

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A Ω = A 1 A 2 / r 2
T 1 = h ν / k log e [ 1 + ( h ν 3 / c 2 N 1 ) ]
L 1 = k ν 2 c 2 [ ( 1 + c 2 N 1 h ν 3 ) log e ( 1 + c 2 N 1 h ν 3 ) c 2 N 1 h ν 3 log e c 2 N 1 h ν 3 ]
K 1 c 2 N 1 h ν 3 , L 1 c 2 L 1 k ν 2 , T 1 k T 1 h ν ,
T 1 = 1 / log e ( 1 + 1 / K 1 )
K 1 = 1 / ( exp ( 1 / T 1 ) 1 ) .
L 1 = ( 1 + K 1 ) log e ( 1 + K 1 ) K 1 log K 1 ,
L ( 2 K 1 ) < 2 L ( K 1 ) .
T < 0.201 ( λ < λ m ) ,
exp ( 1 / T ) > 144 ( λ < λ m ) .
K 1 = exp ( 1 / T 1 ) ( λ < λ m ) ,
L 1 = K 1 ( 1 log e K 1 ) ( λ < λ m )
L 1 = ( 1 + 1 / T 1 ) exp ( 1 / T 1 ) ( λ < λ m ) ,
G A Ω B D h ν 3 / c 2 , H A Ω B D k ν 2 / c 2 ,
U 1 = G K 1 ,
S 1 = H L 1
U a = U i U f T 0 ( S i S f ) .
U i = 2 G K ,
S i = 2 H L ( K ) .
T 0 = ( h ν / k ) T 0 .
U f = S f = 0.
U a = G [ 2 K 2 T 0 L ( K ) ] .
E = U a / U i = 1 T 0 L ( K ) / K .
E 1 T 0 ( T + 1 ) / T ( λ < λ m )
T = 0.115 , K = 1.67 × 10 4 , L ( K ) = 16.2 × 10 4 ,
T 0 = T / 10 = 0.0115.
E = 0.89.
U i = 2 G K , S i = 2 H L ( K ) .
U f = 2 G α K , S f = H L ( 2 α K )
U p = ( 1 α ) U 1 = 2 G ( 1 α ) K , S p = U p / T p = 2 H ( 1 α ) K / T p ,
U i = U f + U p .
S i S f + S p ,
2 L ( K ) L ( 2 α K ) + L p ,
L p 2 ( 1 α ) K / T p
2 L ( K ) L ( 2 K ) .
2 L ( K ) = L ( 2 α m K ) + 2 ( 1 α m ) K / T p .
T = 0.115 , K = 1.67 × 10 4 , L ( K ) = 16.2 × 10 4 , L ( 2 K ) = 30.0 × 10 4 , T p = 0.0115.
1 α m = 1 / [ 1 + ( log 2 e ) ( T p 1 T 1 ) ] ( λ < λ m ) ,