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Correct applying of the Senarmont method with incorrect quarterwave plate
Optik 113, No. 1 (2002) 55–56 ª 2002 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
Autor, Titel
55
International Journal for Light and Electron Optics
Short Note
Correct applying of the Senarmont method with incorrect quarterwave plate F. Ratajczyk, P. Kurzynowski Institute of Physics, Wrocław University of Technology, Wybrzez˙e Wyspian`skiego 27, 50-370 Wrocław, Poland
Abstract: The purpose of the article is to discuss the Senarmont method of the measurement of the phase difference, which is effective even if in the measurement setup the quarterwave plate with the phase difference different from 90 is used. Key words: Birefringence – phase shift – compensator
possible to measure this change by clockwise rotation of the analyzer A till extinguishing of the light after the analyzer (the change of the azimuth from point A to point D on the sphere). The arc AD is equal the PC arc so g ¼ 2a :
ð1Þ
1. The principle of the Senarmont method of the measurement In this article the current method of the measurement of the phase difference g between eigenwaves which arises when the light passes through the birefringent medium is treated. The principle of the Senarmont method [1–3] of the phase difference measurement is shown in fig. 1. The measurement setup consists of the polarizer P, the object Ob with the azimuth 45 less than the polarizer’s azimuth, the quarterwave plate l/4 placed on the same azimuth like the polarizer and the analyzer A, which azimuth is 45 less than the azimuth of the object Ob as well. Values of azimuths, which are shown in fig. 1, are exemplary. If it is necessary the light is made monochromatic using the filter F. The principle of the method is shown on the Poincare´ sphere. The light polarized linearly by the polarizer P passes through the object Ob introducing the phase difference g between eigenwaves and changes the polarization state given from the point P to the point B on the Poincare´ sphere. This, elliptically in general, polarization state after passing the light through the quarterwave plate l/4 changes into the linear one (point C) which the azimuth is a ¼ g/2 less than the azimuth of the light incident on the object Ob. It is
Received 11 June 2001; accepted 1 September 2001. Correspondence to: P. Kurzynowski Fax: ++48-71-3283696 E-mail: [email protected]
Fig. 1. The principle of the Senarmont method measurement.
0030-4026/02/113/01-055 $ 15.00/0
56
F. Ratajczyk, P. Kurzynowski, Correct applying of the Senarmont method with incorrect quarterwave plate
2. The principle of the Senarmont method of the measurement with incorrect quarterwave plate The quarterwave plate can be a weak point of the Senarmont measurement. The incorrectness of the quarterwave plate’s realization is Dgq 9. An additional deviation becomes from using the plate in the measurement setup with a source of the wave with the wavelength different from the nominal one, it means this for which the phase difference gq is equal 90 . If the phase plate is of zero order and is a quarterwave plate for the He-Ne laser light then using the sodium lamp light causes additional incorrectness of the phase difference Dgq 7, and for mercury lamp light l ¼ 546 nm Dgq 15:5 . If the quarterwave plate is of N-th order ðgq ¼ N 360 þ l=4Þ then mistakes Dgq resulting from mismatching of the wavelength are 5N times greater. In the laboratory practice using quarterwave plates for lights of different wavelengths is common. The way of avoiding this problem is presented below. The measurement setup is the same like in fig. 1. The light after passing through the polarizer P and the object Ob is elliptically polarized, which is represented by the point B on the sphere. The light after passing through considered not well-fitted quarterwave plate introducing the phase difference gq ¼ 90 þ Dgq is not linearly polarized like previously (point C) but elliptically (point C0 in fig. 2). In the result of rotating the analyzer A we do not receive total extinguishing of the light after the analyzer but we can identify the position (point D on the sphere) of the analyzer for which the intensity of the light after the analyzer is the smallest. Additionally the doubled azimuth change 2a of the analyzer is not equal to the phase difference g (arc AD 6¼ PC). But it is easy to find the correlation between these quantities. The meridian going through the point C0, the equator and the arc PC0 creates a right-angled spherical triangle. The arc PC0 being the hypotenuse of the triangle is equal to the arc PB, so its central angle is equal g. The cathetus PD0 is equal to the arc AD, so its value is 2a. The angle with the apex in the point P is equal to the mistake of the quarterwave plate Dgq. Therefore, using the identity formulae for the spherical triangle PD0 C0 one can obtain the equation for the phase difference g in the form: tan g ¼
tan 2a : cos Dgq
ð2Þ
Fig. 2. The principle of the Senarmont method measurement with the incorrect quarterwave plate.
If the mistake of the quarterwave plate Dgq will be replaced in eq. 2 by the phase difference gq ðDgq ¼ 90 gq Þ then eq. 2 takes a form: tan 2a tan g ¼ : ð3Þ sin gq If the mistake of the quarterwave plate is not greater than 90 , the angle 2a is smaller than the phase difference g in the odd quarters and greater in the even quarters.
3. Example If gq of the quarterwave plate is 70 or 110 and the azimuth change of the analyzer is 22.5 , then accordingly to eq. 1 the phase difference is 45 . Calculating this phase difference using improved method (eq. 3) gives in result value 46.8 . For gq equal 60 or 120 these phase differences are 45 and 49 , accordingly.
References [1] de Senarmont H: Ann. Chim. Phys. 73 (1840) 337 [2] Jerrard HG: J. Opt. Soc. Am. 38 (1948) 35–59 [3] Jerrard HG: J. Opt. Soc. Am. 44 (1954) 634
Autor, Titel
55
International Journal for Light and Electron Optics
Short Note
Correct applying of the Senarmont method with incorrect quarterwave plate F. Ratajczyk, P. Kurzynowski Institute of Physics, Wrocław University of Technology, Wybrzez˙e Wyspian`skiego 27, 50-370 Wrocław, Poland
Abstract: The purpose of the article is to discuss the Senarmont method of the measurement of the phase difference, which is effective even if in the measurement setup the quarterwave plate with the phase difference different from 90 is used. Key words: Birefringence – phase shift – compensator
possible to measure this change by clockwise rotation of the analyzer A till extinguishing of the light after the analyzer (the change of the azimuth from point A to point D on the sphere). The arc AD is equal the PC arc so g ¼ 2a :
ð1Þ
1. The principle of the Senarmont method of the measurement In this article the current method of the measurement of the phase difference g between eigenwaves which arises when the light passes through the birefringent medium is treated. The principle of the Senarmont method [1–3] of the phase difference measurement is shown in fig. 1. The measurement setup consists of the polarizer P, the object Ob with the azimuth 45 less than the polarizer’s azimuth, the quarterwave plate l/4 placed on the same azimuth like the polarizer and the analyzer A, which azimuth is 45 less than the azimuth of the object Ob as well. Values of azimuths, which are shown in fig. 1, are exemplary. If it is necessary the light is made monochromatic using the filter F. The principle of the method is shown on the Poincare´ sphere. The light polarized linearly by the polarizer P passes through the object Ob introducing the phase difference g between eigenwaves and changes the polarization state given from the point P to the point B on the Poincare´ sphere. This, elliptically in general, polarization state after passing the light through the quarterwave plate l/4 changes into the linear one (point C) which the azimuth is a ¼ g/2 less than the azimuth of the light incident on the object Ob. It is
Received 11 June 2001; accepted 1 September 2001. Correspondence to: P. Kurzynowski Fax: ++48-71-3283696 E-mail: [email protected]
Fig. 1. The principle of the Senarmont method measurement.
0030-4026/02/113/01-055 $ 15.00/0
56
F. Ratajczyk, P. Kurzynowski, Correct applying of the Senarmont method with incorrect quarterwave plate
2. The principle of the Senarmont method of the measurement with incorrect quarterwave plate The quarterwave plate can be a weak point of the Senarmont measurement. The incorrectness of the quarterwave plate’s realization is Dgq 9. An additional deviation becomes from using the plate in the measurement setup with a source of the wave with the wavelength different from the nominal one, it means this for which the phase difference gq is equal 90 . If the phase plate is of zero order and is a quarterwave plate for the He-Ne laser light then using the sodium lamp light causes additional incorrectness of the phase difference Dgq 7, and for mercury lamp light l ¼ 546 nm Dgq 15:5 . If the quarterwave plate is of N-th order ðgq ¼ N 360 þ l=4Þ then mistakes Dgq resulting from mismatching of the wavelength are 5N times greater. In the laboratory practice using quarterwave plates for lights of different wavelengths is common. The way of avoiding this problem is presented below. The measurement setup is the same like in fig. 1. The light after passing through the polarizer P and the object Ob is elliptically polarized, which is represented by the point B on the sphere. The light after passing through considered not well-fitted quarterwave plate introducing the phase difference gq ¼ 90 þ Dgq is not linearly polarized like previously (point C) but elliptically (point C0 in fig. 2). In the result of rotating the analyzer A we do not receive total extinguishing of the light after the analyzer but we can identify the position (point D on the sphere) of the analyzer for which the intensity of the light after the analyzer is the smallest. Additionally the doubled azimuth change 2a of the analyzer is not equal to the phase difference g (arc AD 6¼ PC). But it is easy to find the correlation between these quantities. The meridian going through the point C0, the equator and the arc PC0 creates a right-angled spherical triangle. The arc PC0 being the hypotenuse of the triangle is equal to the arc PB, so its central angle is equal g. The cathetus PD0 is equal to the arc AD, so its value is 2a. The angle with the apex in the point P is equal to the mistake of the quarterwave plate Dgq. Therefore, using the identity formulae for the spherical triangle PD0 C0 one can obtain the equation for the phase difference g in the form: tan g ¼
tan 2a : cos Dgq
ð2Þ
Fig. 2. The principle of the Senarmont method measurement with the incorrect quarterwave plate.
If the mistake of the quarterwave plate Dgq will be replaced in eq. 2 by the phase difference gq ðDgq ¼ 90 gq Þ then eq. 2 takes a form: tan 2a tan g ¼ : ð3Þ sin gq If the mistake of the quarterwave plate is not greater than 90 , the angle 2a is smaller than the phase difference g in the odd quarters and greater in the even quarters.
3. Example If gq of the quarterwave plate is 70 or 110 and the azimuth change of the analyzer is 22.5 , then accordingly to eq. 1 the phase difference is 45 . Calculating this phase difference using improved method (eq. 3) gives in result value 46.8 . For gq equal 60 or 120 these phase differences are 45 and 49 , accordingly.
References [1] de Senarmont H: Ann. Chim. Phys. 73 (1840) 337 [2] Jerrard HG: J. Opt. Soc. Am. 38 (1948) 35–59 [3] Jerrard HG: J. Opt. Soc. Am. 44 (1954) 634