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Analysis of polarized light with imperfect components
Volume
18, number
3
OPTICS
ANALYSIS OF POLARIZED
COMMUNICATIONS
LIGHT WITH IMPERFECT
August
1976
COMPONENTS
M.P. KOTHIYAL Fine Technics Laboratory, Mechanical Engineering Department, Indian Institute of Technology, Madras - 600 036, India Received
26 November
1975, revised version
received
26 April 1976
The effect of analyzer imperfections on the formulas for the ellipticity and the azimuth of polarized light, when a nonideal compensator and an imperfect analyzer are used for its analysis, has been investigated. It is shown that the imperfections of the analyzer do not influence the measured values of the ellipticity and the azimuth to the first order if an ideal or a nearly ideal compensator is used.
1. Introduction
(1)
Elliptically polarized light is analyzed using a combination of a compensator and an analyzer which are independently adjustable. Ideally the compensator should be a quarter-wave plate with slow to fast axis amplitude transmittance ratio, p, equal to unity and the analyzer free from any defects like birefringence, etc. The compensator which receives the elliptically polarized light is so aligned that the light leaving it is plane polarized. The analyzer is then aligned in cross position such that zero is indicated by the detector. The azimuth and the ellipticity of the polarized light can be calculated from the azimuths of the compensator and the analyzer at the zero condition. For the case of a nonideal compensator having any retardation, A, the nulling azimuths of the compensator and the analyzer are different than those in the ideal case. Formulas for the ellipticity and the azimuth of the polarized light in terms of the new azimuths have been given by various authors [l-4]. The purpose of the present communication is to investigate how these formulas are modified when an imperfect analyzer is used along with a nonideal compensator for the analysis of the polarized light.
The polarized light falls on a compensator with its fast axis at an azimuth C followed by analyzer at azimuth A. The light vector incident on the analyzer described in its principle frame is given by E %A
cosA
sinA
cost
-sin A
cosA
sin C
=
( E ~4 )(
- sin C cos c
cos c -sin
[cosCcosA*
=
(
[-cosCsinA*
C -p, -pc
t [sinCcosA*
sinCsinA*!E, sinCcosA*]E, tp,
cosCsinA*]E:,
+ (2) where pc = p exp (-i A) is the complex transmittance ratio of the compensator and A* = A - C. The Jones matrix of an imperfect analyzer is given by [VI
2. Theory Let the polarized light be represented 310
by a vector
where aA, eA, and rA are the complex first-order correction parameters and represent the most general
deviation of the analyzer from its ideal counter-part. flux will pass through the analyzer when E x/j +$&,A
August 1976
OPTICS COMMUNICATIONS
Volume 18, number 3
It has been shown by Azzam and Bashara [5] that a minimum
=o,
(3)
where.second order terms have been neglected. It is to be noted that the correction contribute to the minimum setting to a first order. From eqs. (2) and (3), we get
parameters c~,, and TA do not
E y _ pc tan A* tan C- 1 + 9 (tan A* + pc tan C) -tan Ct PC tan A* + $t @, - tanA* tan C) J%
(4)
Eq. (4) represents the state of polarization of the polarized light which is being analyzed in terms of the component azimuths,A and C, the compensator transmittance ratio pc and the analyzer imperfection parameter CA. Since eA is a complex quantity we may write eA = elA + i E2A. elA has an effect similar to a first order rotation of the azimuth angle A of the analyzer [5,8]. Its effect can therefore be removed at any fixed wavelength by redefining the azimuth A. It has been suggested, however, that EIA is negligible for good quality polarization prisms [8], Therefore, we have eA = i “2A. The incident polarized light is also represented by a vector [6,7]
Ex -=
cos/3 cos8-i
Er
sin/3cos0+icosflsin0
where tan 0 is the ellipticity light in terms of 8 and fl is
sin/3 sin0 (5)
’
and /3 the azimuth of the elliptically
polarized light. The polarization
state of the
(6) Equating the rhs of eqs. (4) and (6) and seperating the real and the imaginary parts we obtain the following equations: (1 t tan fl tan C) t f2A p sin A(tan p- tan C) - e2A p cos A tan
e(i + tan/3 tan C)
=tanA*[-pcosA(tanp-tanC)-psinAtane(1ttanBtanC)te2A
- tan
tane(tanp-tanC),
e (tan p - tan C) t f2A p cos A(tan p - tan C) + e&t p sin A tan e (1
=tanA*[psinA(tanp-tanC)-pcosAtane(1
2. I. Influence
(7)
+ tan 0 tan C)
ttanptanC)te&t(l+tanfltanC).
(8)
of analyzer imperfections on azimuth
In order to study the influence of the analyzer imperfections on the azimuth of the elliptically polarized light, use has been made of the eqs. (7) and (8). After eliminating tanA* from eqs. (7) and (8), we obtain atan2C+btanC+c=0,
(9) 311
Volume
18, number
3
OPTICS COMMUNICATIONS
August
1976
where 2A [sin20 cos2~ + sin28 cos2p t p2(cos2/3 cos20 + sin20 sin2P)],
a=psinAsin2/3cos20+psinAsin28-2e
b = 2p sin A cos 2p cos 28 - 2 ezA sin 2/3 cos 28 (1 - p2), c=pcosAsin2f?-psinAsin2~cos28-2e2A[ Eq. (9) is a second-order
equation
(10)
cos2p cos28 t sin20 sin20 tp2(sin28
cos2p t sin2p
in tan Cgiving two values of tan C If Cl and C2 are the two azimuths then
tan(C1 t C,) = - b/(a - c). Substituting
c0Ge)l.
(11)
the values from eq. (10) into eq. (11) we have
tan(C1 + C2) = -
2/3 1 -
cot
‘2.4
(I - P2)
(12)
p sin A sin 20 cos 20
The second term on the rhs of eq. (12) is zero when e2A = 0. This reduces eq. (12) to tan(C1 + C2) = - cot 20, which gives fl=f
(C, tc,-
n/2).
(13)
This gives the azimuth of the polarized light in terms of the compensator azimuths. This equation is similar to that given by Hall [l]. The second term in eq. (12) will also be zero when p = 1. For a non-ideal compensator p = 1+6p where 6p Q 1, and A = 90” + 6A where 6A is small, the second term on the rhs of eq. (12) will produce only a seond-order error term (a eZA 6p) and can be neglected. Eq. (13) is, therefore, valid even in the presence of small analyzer imperfections. Eqs. (7) and (8) can be rearranged to give tan(/.LC)[pcosAtanA*te2ApsinA-e2A
tanA*tanB]t1tpsinAtan0tanA*-e2ApcosAtan0=0, (14)
and tan(&C)[-p
sin A tanA* -tan
e + EzA p cos A] +p cm A tan e tan
A* - eZA tanA* + eZA p sin A tan e = 0. (15)
Eliminating tan@- C) from eqs. (14) and (15) and simplyfing, we obtain the following equation for the ellipticity parameter 0 : sin2e=_2ptanA*sinA p2 tan2A* + 1
+~E~/,@cosA~~~~~;~~~. (16)
IfQ
= 0, we haye
sin 20 =
- 2p tan A* sin A p2 tan2A* t 1
’
(17)
which is the same as the relation given by Holmes [3] 312
From eq. (16) we see that for A = 90’) the analyzer imperfection eZA does not influence the measured value of the ellipticity. For a non-ideal compensator the 2nd term in eq. (16) will again be a second-order error term (a eZA &A). This term can be neglected reducing eq. (16) to eq. (17). ,The above analysis shows that if we use a compensator with only first-order deviations from the ideal values, the analyzer imperfections do not contribute to the first-order error in the measured values of the ellipticity and the azimuth of the elliptically polarized light.
Volume 18, number 3
OPTICS COMMUNICATIONS
Acknowledgement The author is grateful to Dr.R.S. Sirohi for his help in preparation of this paper and to the referee for his valuable comments.
References
August 1976
[2] R.C. Plumb, J. Opt. Sot. Am. 50 (1960) 892. (31 D.A. Holmes, J. Opt. Sot. Am. 54 (1964) 1115. [4] M.P. Kothiyal, J. Opt. Sot. Am. 65 (1975) 352. [5] R.M.A. Azzam and N.M. Bashara, J. Opt. Sot. Am. 61 (1971) 1236. [6] R.M.A. Azzam and N.M. Bhashara, J. Opt. Sot. Am. 62 (1972) 222. 171 K. Leonhardt, Optik 34 (1971) 4. i8i D.E. Aspnes, J. Opt. Sot. Am. 61 (1971) 1077.
[l] AC. Hall, J. Opt. Sot. Am. 53 (1964) 801.
313
18, number
3
OPTICS
ANALYSIS OF POLARIZED
COMMUNICATIONS
LIGHT WITH IMPERFECT
August
1976
COMPONENTS
M.P. KOTHIYAL Fine Technics Laboratory, Mechanical Engineering Department, Indian Institute of Technology, Madras - 600 036, India Received
26 November
1975, revised version
received
26 April 1976
The effect of analyzer imperfections on the formulas for the ellipticity and the azimuth of polarized light, when a nonideal compensator and an imperfect analyzer are used for its analysis, has been investigated. It is shown that the imperfections of the analyzer do not influence the measured values of the ellipticity and the azimuth to the first order if an ideal or a nearly ideal compensator is used.
1. Introduction
(1)
Elliptically polarized light is analyzed using a combination of a compensator and an analyzer which are independently adjustable. Ideally the compensator should be a quarter-wave plate with slow to fast axis amplitude transmittance ratio, p, equal to unity and the analyzer free from any defects like birefringence, etc. The compensator which receives the elliptically polarized light is so aligned that the light leaving it is plane polarized. The analyzer is then aligned in cross position such that zero is indicated by the detector. The azimuth and the ellipticity of the polarized light can be calculated from the azimuths of the compensator and the analyzer at the zero condition. For the case of a nonideal compensator having any retardation, A, the nulling azimuths of the compensator and the analyzer are different than those in the ideal case. Formulas for the ellipticity and the azimuth of the polarized light in terms of the new azimuths have been given by various authors [l-4]. The purpose of the present communication is to investigate how these formulas are modified when an imperfect analyzer is used along with a nonideal compensator for the analysis of the polarized light.
The polarized light falls on a compensator with its fast axis at an azimuth C followed by analyzer at azimuth A. The light vector incident on the analyzer described in its principle frame is given by E %A
cosA
sinA
cost
-sin A
cosA
sin C
=
( E ~4 )(
- sin C cos c
cos c -sin
[cosCcosA*
=
(
[-cosCsinA*
C -p, -pc
t [sinCcosA*
sinCsinA*!E, sinCcosA*]E, tp,
cosCsinA*]E:,
+ (2) where pc = p exp (-i A) is the complex transmittance ratio of the compensator and A* = A - C. The Jones matrix of an imperfect analyzer is given by [VI
2. Theory Let the polarized light be represented 310
by a vector
where aA, eA, and rA are the complex first-order correction parameters and represent the most general
deviation of the analyzer from its ideal counter-part. flux will pass through the analyzer when E x/j +$&,A
August 1976
OPTICS COMMUNICATIONS
Volume 18, number 3
It has been shown by Azzam and Bashara [5] that a minimum
=o,
(3)
where.second order terms have been neglected. It is to be noted that the correction contribute to the minimum setting to a first order. From eqs. (2) and (3), we get
parameters c~,, and TA do not
E y _ pc tan A* tan C- 1 + 9 (tan A* + pc tan C) -tan Ct PC tan A* + $t @, - tanA* tan C) J%
(4)
Eq. (4) represents the state of polarization of the polarized light which is being analyzed in terms of the component azimuths,A and C, the compensator transmittance ratio pc and the analyzer imperfection parameter CA. Since eA is a complex quantity we may write eA = elA + i E2A. elA has an effect similar to a first order rotation of the azimuth angle A of the analyzer [5,8]. Its effect can therefore be removed at any fixed wavelength by redefining the azimuth A. It has been suggested, however, that EIA is negligible for good quality polarization prisms [8], Therefore, we have eA = i “2A. The incident polarized light is also represented by a vector [6,7]
Ex -=
cos/3 cos8-i
Er
sin/3cos0+icosflsin0
where tan 0 is the ellipticity light in terms of 8 and fl is
sin/3 sin0 (5)
’
and /3 the azimuth of the elliptically
polarized light. The polarization
state of the
(6) Equating the rhs of eqs. (4) and (6) and seperating the real and the imaginary parts we obtain the following equations: (1 t tan fl tan C) t f2A p sin A(tan p- tan C) - e2A p cos A tan
e(i + tan/3 tan C)
=tanA*[-pcosA(tanp-tanC)-psinAtane(1ttanBtanC)te2A
- tan
tane(tanp-tanC),
e (tan p - tan C) t f2A p cos A(tan p - tan C) + e&t p sin A tan e (1
=tanA*[psinA(tanp-tanC)-pcosAtane(1
2. I. Influence
(7)
+ tan 0 tan C)
ttanptanC)te&t(l+tanfltanC).
(8)
of analyzer imperfections on azimuth
In order to study the influence of the analyzer imperfections on the azimuth of the elliptically polarized light, use has been made of the eqs. (7) and (8). After eliminating tanA* from eqs. (7) and (8), we obtain atan2C+btanC+c=0,
(9) 311
Volume
18, number
3
OPTICS COMMUNICATIONS
August
1976
where 2A [sin20 cos2~ + sin28 cos2p t p2(cos2/3 cos20 + sin20 sin2P)],
a=psinAsin2/3cos20+psinAsin28-2e
b = 2p sin A cos 2p cos 28 - 2 ezA sin 2/3 cos 28 (1 - p2), c=pcosAsin2f?-psinAsin2~cos28-2e2A[ Eq. (9) is a second-order
equation
(10)
cos2p cos28 t sin20 sin20 tp2(sin28
cos2p t sin2p
in tan Cgiving two values of tan C If Cl and C2 are the two azimuths then
tan(C1 t C,) = - b/(a - c). Substituting
c0Ge)l.
(11)
the values from eq. (10) into eq. (11) we have
tan(C1 + C2) = -
2/3 1 -
cot
‘2.4
(I - P2)
(12)
p sin A sin 20 cos 20
The second term on the rhs of eq. (12) is zero when e2A = 0. This reduces eq. (12) to tan(C1 + C2) = - cot 20, which gives fl=f
(C, tc,-
n/2).
(13)
This gives the azimuth of the polarized light in terms of the compensator azimuths. This equation is similar to that given by Hall [l]. The second term in eq. (12) will also be zero when p = 1. For a non-ideal compensator p = 1+6p where 6p Q 1, and A = 90” + 6A where 6A is small, the second term on the rhs of eq. (12) will produce only a seond-order error term (a eZA 6p) and can be neglected. Eq. (13) is, therefore, valid even in the presence of small analyzer imperfections. Eqs. (7) and (8) can be rearranged to give tan(/.LC)[pcosAtanA*te2ApsinA-e2A
tanA*tanB]t1tpsinAtan0tanA*-e2ApcosAtan0=0, (14)
and tan(&C)[-p
sin A tanA* -tan
e + EzA p cos A] +p cm A tan e tan
A* - eZA tanA* + eZA p sin A tan e = 0. (15)
Eliminating tan@- C) from eqs. (14) and (15) and simplyfing, we obtain the following equation for the ellipticity parameter 0 : sin2e=_2ptanA*sinA p2 tan2A* + 1
+~E~/,@cosA~~~~~;~~~. (16)
IfQ
= 0, we haye
sin 20 =
- 2p tan A* sin A p2 tan2A* t 1
’
(17)
which is the same as the relation given by Holmes [3] 312
From eq. (16) we see that for A = 90’) the analyzer imperfection eZA does not influence the measured value of the ellipticity. For a non-ideal compensator the 2nd term in eq. (16) will again be a second-order error term (a eZA &A). This term can be neglected reducing eq. (16) to eq. (17). ,The above analysis shows that if we use a compensator with only first-order deviations from the ideal values, the analyzer imperfections do not contribute to the first-order error in the measured values of the ellipticity and the azimuth of the elliptically polarized light.
Volume 18, number 3
OPTICS COMMUNICATIONS
Acknowledgement The author is grateful to Dr.R.S. Sirohi for his help in preparation of this paper and to the referee for his valuable comments.
References
August 1976
[2] R.C. Plumb, J. Opt. Sot. Am. 50 (1960) 892. (31 D.A. Holmes, J. Opt. Sot. Am. 54 (1964) 1115. [4] M.P. Kothiyal, J. Opt. Sot. Am. 65 (1975) 352. [5] R.M.A. Azzam and N.M. Bashara, J. Opt. Sot. Am. 61 (1971) 1236. [6] R.M.A. Azzam and N.M. Bhashara, J. Opt. Sot. Am. 62 (1972) 222. 171 K. Leonhardt, Optik 34 (1971) 4. i8i D.E. Aspnes, J. Opt. Sot. Am. 61 (1971) 1077.
[l] AC. Hall, J. Opt. Sot. Am. 53 (1964) 801.
313