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Simultaneous spectral calibration of two phase plates
Optics & Laser Technology 31 (1999) 517±519
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Simultaneous spectral calibration of two phase plates N.N. Nagib*, S.A. Khodier, M.S. Khalil, H.M. Sidki National Institute for Standards, PO Box 136 Giza, 12211, Giza, Egypt Received 13 September 1999; received in revised form 11 November 1999; accepted 15 November 1999
Abstract Based on a new model for the spectral calibration of phase plates in pairs, experimental results are presented for three mica plates between 430 and 684 nm. One of the plates is 36.4 mm thick and the remaining two are identical, each being 28.6 mm thick. In the studied spectral range, the retardances varied from 123.58 to 86.88 for the ®rst plate and from 96.98 to 688 for each of the two other plates. Dierent procedures for performing the calibration of the plates and the special case of two identical plates are discussed. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Phase plates; Phase retardance; Mica
1. Introduction A monochromatic light beam of wavelength l crossing the optical system P±C1±C2±A (where P and A are the polarizer and the analyzer and, C1 and C2 are two phase plates) could be extinguished at two pairs ( p1, a1) and ( p2, a2) of the polarizer and the analyzer settings where p1 ?p2 and a1 ?a2 [1]. If C1 and C2 are oriented at 458 and 08 (angles are measured CCW from the horizontal direction looking into the beam with respect to the fast axes of the plates and with respect to the transmission axes of the polarizing elements), then [2] cos d1 ÿcos 2ai = cos 2pi
1
and cos d2 ÿsin 2pi = sin 2ai
2
where d1 and d2 are the retardances of C1 and C2 and i = 1, 2. Eqs. (1) and (2) uniquely de®ne d1 and d2 provided that d1 < p and d2 < p. A single pair of P and A readings is then sucient to calibrate both of the two * Corresponding author. Fax: +20-2-3867-451. E-mail address: [email protected] (N.N. Nagib).
plates simultaneously at l. No previous method has been reported that allows for calibrating two plates at the same time and settings of the optical elements. Obviously, the procedure can be repeated at other wavelengths to ®nd the spectral dependencies of the retardances of the two plates. This paper presents the experimental results for the calibration of two phase plates simultaneously in the spectral range 430±684 nm. Dierent experimental procedures for calibrating the plates are mentioned and results for the special case of two identical plates are also presented.
2. Experimental Three muscovite mica plates of complete lamination structure and free from defects were studied in this work. The ®rst plate C1 is of thickness 36.4 mm and the other two plates C2 and C20 were cut from a single plate of thickness 28.6 mm. Each plate was embedded in optical cement between two glass covers and its fast axis was identi®ed by Tutton test, see Ref. [3]. During measurements, the fast axis of each plate was precisely oriented at the required setting (08 or 458). Thus, to adjust the fast axis at 08, the plate is inserted between
0030-3992/99/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 9 ) 0 0 1 0 9 - 7
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N.N. Nagib et al. / Optics & Laser Technology 31 (1999) 517±519
the two polarizing elements P and A with p = 08 and a = 908. With the fast axis at 008, the plate is smoothly rotated about this position until the light leaving the analyzer is extinguished. The fast axis is then oriented at 08 with an estimated accuracy of 20.028. For the plate at 458, the same procedure is repeated with a = 458, p=ÿ458 and the fast axis at 0458. Orienting the plates can be performed with an incident white light. To check that C2 and C20 were identical, the two plates were introduced with their fast axes at 458 and ÿ458 next to the polarizer ( p = 0). The analyzer is rotated for extinction. The plates are of identical retardances if a= 2 908 which was the case with our plates within the limits of experimental error. The same optical and detecting systems used in [3] are also employed here, but some aspects concerning the measurements need to be discussed. The two plates C1 and C2 are introduced in the optical system with C1 following the polarizer and is oriented at 458 and C2 oriented at 08, see Fig. 1. For a speci®c wavelength value l, two pairs of P and A settings exist at which light could be extinguished. However, we found it more appropriate to ®x the polarizer at a de®nite setting p and vary l and a simultaneously for extinction. First, we start by setting p = 0. It follows from Eqs. (1) and (2) that at extinction, d1 cosÿ1 ÿcos 2a
3
and d2 p=2
polarizer and the analyzer settings would provide extinction corresponding to each value of l. Since we are ®xing the value of p instead of l, another pair (a ', l ) could be obtained for the polarizer setting p ' where p '_p. There is no confusion in determining whether p '=pÿ(p/2) or p '=p+(p/2) since the polarizer settings are restricted to ÿp/2 R p R p/2. The rule is simple; if p is positive then p ' is negative and equals pÿ(p/2) while if p is negative then p ' is positive and equals p+(p/2). The same rule applies for a '. On the other hand, another set of p, a and l readings could be obtained if the two plates are interchanged in positions and orientations. This second set provides additional data for the spectral calibration of the retardances d1 and d2 of the two plates. For the spectral calibration of two identical plates (d1=d2=d ), it follows from Eqs. (1) and (2), see also Ref. [2] that a ÿ p 458
The two plates C2 and C20 are oriented as in the previous case (C2 at 458 and C20 at 08) while P and A are ®xed at the settings p and ÿ( p + 458) for a de®nite p value. The wavelength l is then scanned for extinction. The process is then repeated for dierent p values to ®nd the spectral dependence of d. For a wavelength value l, it follows from Eqs. (1), (2) and (5) that at extinction, cos d tan 2p
4
The previous step may not provide extinction if the ellipticity introduced by the ®rst plate cannot be compensated by the second plate in the studied spectral range. Note that extinction of light must exist at p = 0 for particular values of l and a, but this may take place at some wavelength value outside the working range of the detecting system. In this case, other values of p (positive or negative) might be tried until extinction is attained. Next, the polarizer setting p is changed in steps (selecting appropriate values) and the corresponding l and a values at extinction are registered each time. As mentioned before, two pairs of the
Fig. 1. The optical system P±C1 ±C2±A. P.M is the photomultiplier and H is the horizontal direction.
5
6
or cos d cot 2a
Fig. 2. The calibration curves of the two plates C1 and C2.
7
N.N. Nagib et al. / Optics & Laser Technology 31 (1999) 517±519
3. Results and discussion As mentioned above, there are dierent procedures for calibrating two phase plates simultaneously. One of the three parameters l, p and a is kept ®xed while the other two are varied for extinction. We found it more suitable to ®x p and vary both l and a. Another set of readings could be obtained by interchanging the positions and the orientations of the two plates. The results presented in Fig. 2 are for two sets of readings (one set with C1 preceding C2 and the other with C1 following C2). Each plate is then calibrated twice. Between 430 and 684 nm, d1 varied from 123.58 to 86.88, while in the same spectral range d2 varied from 96.98 to 688. d1 and d2 are of exact l/4 retardances at 656 nm and 475 nm, respectively, in conformity with previous results for the same samples [3]. Now, the retardance d of a mica plate is given as d ÿ2pd ne ÿ no =l
8
where d is the thickness of the plate and (neÿno) is the birefringence at the wavelength l. Since the two plates were cleaved from the same crystal, it is reasonable to consider that they have the same spectral dependence of birefringence [3]. Therefore, if d1 and d2 denote the thicknesses of the plates C1 and C2 then l1 l2 l2 d1 =d2 dl1 1 =d2 d1 =d2 l2 l1 l2 dl1 1 ÿ d1 = d2 ÿ d2
9
where l1 and l2 refer to measurements at 430 and 684 nm, respectively. Our results give: d1/d2=1.273; l1 l2 l2 l1 l2 l1 dl1 1 =d2 1:275; d1 =d2 1:276; and d1 ÿ d1 = d2 ÿ
519
dl2 2 1:270, which ensures the accuracy of our measurements. Fig. 3 represents the calibration curve of the two identical plates C2 and C20 : Two sets of measurements were carried out where the second set was obtained with the two plates interchanged. Note that when the two plates are identical, all the optical elements are kept ®xed in orientations and only l is scanned for extinction. Obviously, this procedure is only allowed if the plates were checked to be identical as described above. If the plates were not exactly identical, a broadening in the null point will occur as described in [3]. In this latter case, the plates should be calibrated as if they were of dierent retardances (the previous case of C1 and C2). The results for the two sets of measurements of C2 and C20 are almost identical and only one of them is presented in Fig. 3, which also includes the data of d2 as reproduced from Fig. 2. 4. Conclusion We investigated the use of new formulas for phase retardance measurements. The optical system is similar to that of the Senarmont compensator, but without requiring an exact l/4 plate and measurements could be performed at dierent wavelengths in the simplest possible way. Three plates were calibrated in the spectral range from 430 to 684 nm. In comparison with other known methods for phase plate calibrations, the presented method is advantageous in that it is the only reported method for calibrating two plates simultaneously. In addition, the optical system and the mathematical expressions are very simple which save much time and eort. The accuracy of the retardance measurements depends on the quality of the optical elements, the spectral bandwidth of the incident radiation, the sensitivity of the detecting system and the precision of measuring the angles and orienting the plates. Our results are highly consistent as was discussed in Section 3 and in view of previous results for the same samples [3]. Detailed analysis of the uncertainty in null polarimetric methods for calibrating phase plates was discussed in [3,4] and it was estimated that the accuracy of our retardance measurements is better than 0.3%. References
Fig. 3. The retardance d of the identical plates C2 and C20 (q). The data of d2 (P) is reproduced from Fig. 2.
[1] Aben KhK. Phase plates in the measurement of phase dierence. Opt Spectrosc 1963;14:124±7. [2] Nagib NN. New formulas for phase retardance measurements of birefringent plates. Optics & Laser Technology 1999;31(4):309± 13. [3] El-Bahrawi MS, Nagib NN, Khodier SA, Sidki HM. Birefringence of muscovite mica. Optics & Laser Technology 1998;30(6±7):411±5. [4] Rochford KB, Wang CM. Uncertainty in null polarimetric measurements. Natl Inst Stand Technol IR 5055, 1996.
www.elsevier.com/locate/optlastec
Simultaneous spectral calibration of two phase plates N.N. Nagib*, S.A. Khodier, M.S. Khalil, H.M. Sidki National Institute for Standards, PO Box 136 Giza, 12211, Giza, Egypt Received 13 September 1999; received in revised form 11 November 1999; accepted 15 November 1999
Abstract Based on a new model for the spectral calibration of phase plates in pairs, experimental results are presented for three mica plates between 430 and 684 nm. One of the plates is 36.4 mm thick and the remaining two are identical, each being 28.6 mm thick. In the studied spectral range, the retardances varied from 123.58 to 86.88 for the ®rst plate and from 96.98 to 688 for each of the two other plates. Dierent procedures for performing the calibration of the plates and the special case of two identical plates are discussed. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Phase plates; Phase retardance; Mica
1. Introduction A monochromatic light beam of wavelength l crossing the optical system P±C1±C2±A (where P and A are the polarizer and the analyzer and, C1 and C2 are two phase plates) could be extinguished at two pairs ( p1, a1) and ( p2, a2) of the polarizer and the analyzer settings where p1 ?p2 and a1 ?a2 [1]. If C1 and C2 are oriented at 458 and 08 (angles are measured CCW from the horizontal direction looking into the beam with respect to the fast axes of the plates and with respect to the transmission axes of the polarizing elements), then [2] cos d1 ÿcos 2ai = cos 2pi
1
and cos d2 ÿsin 2pi = sin 2ai
2
where d1 and d2 are the retardances of C1 and C2 and i = 1, 2. Eqs. (1) and (2) uniquely de®ne d1 and d2 provided that d1 < p and d2 < p. A single pair of P and A readings is then sucient to calibrate both of the two * Corresponding author. Fax: +20-2-3867-451. E-mail address: [email protected] (N.N. Nagib).
plates simultaneously at l. No previous method has been reported that allows for calibrating two plates at the same time and settings of the optical elements. Obviously, the procedure can be repeated at other wavelengths to ®nd the spectral dependencies of the retardances of the two plates. This paper presents the experimental results for the calibration of two phase plates simultaneously in the spectral range 430±684 nm. Dierent experimental procedures for calibrating the plates are mentioned and results for the special case of two identical plates are also presented.
2. Experimental Three muscovite mica plates of complete lamination structure and free from defects were studied in this work. The ®rst plate C1 is of thickness 36.4 mm and the other two plates C2 and C20 were cut from a single plate of thickness 28.6 mm. Each plate was embedded in optical cement between two glass covers and its fast axis was identi®ed by Tutton test, see Ref. [3]. During measurements, the fast axis of each plate was precisely oriented at the required setting (08 or 458). Thus, to adjust the fast axis at 08, the plate is inserted between
0030-3992/99/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 9 ) 0 0 1 0 9 - 7
518
N.N. Nagib et al. / Optics & Laser Technology 31 (1999) 517±519
the two polarizing elements P and A with p = 08 and a = 908. With the fast axis at 008, the plate is smoothly rotated about this position until the light leaving the analyzer is extinguished. The fast axis is then oriented at 08 with an estimated accuracy of 20.028. For the plate at 458, the same procedure is repeated with a = 458, p=ÿ458 and the fast axis at 0458. Orienting the plates can be performed with an incident white light. To check that C2 and C20 were identical, the two plates were introduced with their fast axes at 458 and ÿ458 next to the polarizer ( p = 0). The analyzer is rotated for extinction. The plates are of identical retardances if a= 2 908 which was the case with our plates within the limits of experimental error. The same optical and detecting systems used in [3] are also employed here, but some aspects concerning the measurements need to be discussed. The two plates C1 and C2 are introduced in the optical system with C1 following the polarizer and is oriented at 458 and C2 oriented at 08, see Fig. 1. For a speci®c wavelength value l, two pairs of P and A settings exist at which light could be extinguished. However, we found it more appropriate to ®x the polarizer at a de®nite setting p and vary l and a simultaneously for extinction. First, we start by setting p = 0. It follows from Eqs. (1) and (2) that at extinction, d1 cosÿ1 ÿcos 2a
3
and d2 p=2
polarizer and the analyzer settings would provide extinction corresponding to each value of l. Since we are ®xing the value of p instead of l, another pair (a ', l ) could be obtained for the polarizer setting p ' where p '_p. There is no confusion in determining whether p '=pÿ(p/2) or p '=p+(p/2) since the polarizer settings are restricted to ÿp/2 R p R p/2. The rule is simple; if p is positive then p ' is negative and equals pÿ(p/2) while if p is negative then p ' is positive and equals p+(p/2). The same rule applies for a '. On the other hand, another set of p, a and l readings could be obtained if the two plates are interchanged in positions and orientations. This second set provides additional data for the spectral calibration of the retardances d1 and d2 of the two plates. For the spectral calibration of two identical plates (d1=d2=d ), it follows from Eqs. (1) and (2), see also Ref. [2] that a ÿ p 458
The two plates C2 and C20 are oriented as in the previous case (C2 at 458 and C20 at 08) while P and A are ®xed at the settings p and ÿ( p + 458) for a de®nite p value. The wavelength l is then scanned for extinction. The process is then repeated for dierent p values to ®nd the spectral dependence of d. For a wavelength value l, it follows from Eqs. (1), (2) and (5) that at extinction, cos d tan 2p
4
The previous step may not provide extinction if the ellipticity introduced by the ®rst plate cannot be compensated by the second plate in the studied spectral range. Note that extinction of light must exist at p = 0 for particular values of l and a, but this may take place at some wavelength value outside the working range of the detecting system. In this case, other values of p (positive or negative) might be tried until extinction is attained. Next, the polarizer setting p is changed in steps (selecting appropriate values) and the corresponding l and a values at extinction are registered each time. As mentioned before, two pairs of the
Fig. 1. The optical system P±C1 ±C2±A. P.M is the photomultiplier and H is the horizontal direction.
5
6
or cos d cot 2a
Fig. 2. The calibration curves of the two plates C1 and C2.
7
N.N. Nagib et al. / Optics & Laser Technology 31 (1999) 517±519
3. Results and discussion As mentioned above, there are dierent procedures for calibrating two phase plates simultaneously. One of the three parameters l, p and a is kept ®xed while the other two are varied for extinction. We found it more suitable to ®x p and vary both l and a. Another set of readings could be obtained by interchanging the positions and the orientations of the two plates. The results presented in Fig. 2 are for two sets of readings (one set with C1 preceding C2 and the other with C1 following C2). Each plate is then calibrated twice. Between 430 and 684 nm, d1 varied from 123.58 to 86.88, while in the same spectral range d2 varied from 96.98 to 688. d1 and d2 are of exact l/4 retardances at 656 nm and 475 nm, respectively, in conformity with previous results for the same samples [3]. Now, the retardance d of a mica plate is given as d ÿ2pd ne ÿ no =l
8
where d is the thickness of the plate and (neÿno) is the birefringence at the wavelength l. Since the two plates were cleaved from the same crystal, it is reasonable to consider that they have the same spectral dependence of birefringence [3]. Therefore, if d1 and d2 denote the thicknesses of the plates C1 and C2 then l1 l2 l2 d1 =d2 dl1 1 =d2 d1 =d2 l2 l1 l2 dl1 1 ÿ d1 = d2 ÿ d2
9
where l1 and l2 refer to measurements at 430 and 684 nm, respectively. Our results give: d1/d2=1.273; l1 l2 l2 l1 l2 l1 dl1 1 =d2 1:275; d1 =d2 1:276; and d1 ÿ d1 = d2 ÿ
519
dl2 2 1:270, which ensures the accuracy of our measurements. Fig. 3 represents the calibration curve of the two identical plates C2 and C20 : Two sets of measurements were carried out where the second set was obtained with the two plates interchanged. Note that when the two plates are identical, all the optical elements are kept ®xed in orientations and only l is scanned for extinction. Obviously, this procedure is only allowed if the plates were checked to be identical as described above. If the plates were not exactly identical, a broadening in the null point will occur as described in [3]. In this latter case, the plates should be calibrated as if they were of dierent retardances (the previous case of C1 and C2). The results for the two sets of measurements of C2 and C20 are almost identical and only one of them is presented in Fig. 3, which also includes the data of d2 as reproduced from Fig. 2. 4. Conclusion We investigated the use of new formulas for phase retardance measurements. The optical system is similar to that of the Senarmont compensator, but without requiring an exact l/4 plate and measurements could be performed at dierent wavelengths in the simplest possible way. Three plates were calibrated in the spectral range from 430 to 684 nm. In comparison with other known methods for phase plate calibrations, the presented method is advantageous in that it is the only reported method for calibrating two plates simultaneously. In addition, the optical system and the mathematical expressions are very simple which save much time and eort. The accuracy of the retardance measurements depends on the quality of the optical elements, the spectral bandwidth of the incident radiation, the sensitivity of the detecting system and the precision of measuring the angles and orienting the plates. Our results are highly consistent as was discussed in Section 3 and in view of previous results for the same samples [3]. Detailed analysis of the uncertainty in null polarimetric methods for calibrating phase plates was discussed in [3,4] and it was estimated that the accuracy of our retardance measurements is better than 0.3%. References
Fig. 3. The retardance d of the identical plates C2 and C20 (q). The data of d2 (P) is reproduced from Fig. 2.
[1] Aben KhK. Phase plates in the measurement of phase dierence. Opt Spectrosc 1963;14:124±7. [2] Nagib NN. New formulas for phase retardance measurements of birefringent plates. Optics & Laser Technology 1999;31(4):309± 13. [3] El-Bahrawi MS, Nagib NN, Khodier SA, Sidki HM. Birefringence of muscovite mica. Optics & Laser Technology 1998;30(6±7):411±5. [4] Rochford KB, Wang CM. Uncertainty in null polarimetric measurements. Natl Inst Stand Technol IR 5055, 1996.