- Email: [email protected]
Calibration of absolute spectral radiance in UV and VUV regions by using synchrotron radiation
Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1087–1091
Calibration of absolute spectral radiance in UV and VUV regions by using synchrotron radiation Tatsuya Zama ∗ , Ichiro Saito National Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology (NMIJ/AIST), AIST Tsukuba Central 3, 1-1 Umezono 1-Chome, Tsukuba-shi, Ibaraki-ken 305-8563, Japan Available online 24 February 2005
Abstract A synchrotron radiation was used as an absolute spectral radiance source in ultraviolet and vacuum ultraviolet regions and that scale was transferred to a deuterium lamp. A calibration system was used for comparing each spectral radiant flux. For the absolute calibration, it is important to determine the polarization dependence of the calibration system, because polarization of radiant flux is different between synchrotron radiation and the deuterium lamp. A polarimeter was introduced to determine the polarization dependence of our calibration system. The polarization dependence and that uncertainty of our system are calculated by using a new calculation method. © 2005 Elsevier B.V. All rights reserved. Keywords: Polarization; Synchrotron radiation; Absolute calibration; Vacuum ultraviolet
1. Introduction Quantitative evaluation is important for the elucidation of natural phenomena, and forms the foundation of experimental studies. Such quantitative evaluation requires a scale. In the visible region and a part of the ultraviolet and infrared region, blackbody (and Plank’s law) comes into common use for establishing the absolute scale. But in the shorter wavelength region [1–7], no common method has yet been established. The radiant flux of blackbody is not sufficiently strong for a primary source in this region. We have been attempting to establish an absolute scale of spectral radiance in ultraviolet (UV) and vacuum ultraviolet (VUV) regions by using synchrotron radiation (SR) as the primary standard radiant source [8–11]. The spectral radiance scale is transferred to an undertest light source (a deuterium lamp) by comparing the light source with SR from an electron storage ring TERAS in the National Institute of Advanced In∗
Corresponding author. E-mail address: [email protected] (T. Zama).
0368-2048/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2005.01.165
dustrial Science and Technology (AIST). A calibration system that consists of a monochromator and a focusing mirror is used for the comparison. For the absolute calibration, it is important to consider the polarization dependence of our calibration system; because SR is strongly polarized, most light sources we can use conveniently are unpolarised, and the efficiency of the monochromator and mirror depends on polarization of incident flux. This aspect is different from the calibration in the visible region (blackbody and most convenient light sources are unpolarised). In UV and VUV regions, there is not good material for a polarizer or depolarizer. One of the methods used to determine polarization dependence involves rotating the calibration system around the incident flux axis [5,6]. However, we cannot introduce this method because our laboratory does not have sufficient space to rotate our calibration system. We introduced a polarimeter in order to determine the polarization dependence of our calibration system [12,13]. The polarization dependence and the calibration uncertainty were estimated by a simple method [10]. In this paper, we present a new uncertainty evaluation method and make a closer evaluation.
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T. Zama, I. Saito / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1087–1091
Fig. 1. Schematic diagram of the calibration by SR.
2. Principle of calibration
tor of the undertest light source and that of the source’s radiation, which passes through our calibration system (SSR,out = Mc SSR , Sun,out = Mc Sun ). PSR and Pun show the radiant flux ratio of the perpendicular component to the parallel component of SR and that of the undertest light 2 /E2 2 2 source (PSR = ESR,s SR,p , Pun = Eun,s /Eun,p , the subscripts p and s show the parallel and perpendicular electric vectors). Expressing the components of Stokes vector 2 of SR as SSR = (SSR,0 , SSR,1 , SSR,2 , SSR,3 ) (SSR,0 = ESR,p + 2 2 2 ESR,s , SSR,1 = Eun,p − Eun,s ) and other Stokes vector components in the same manner, the following equation is derived. 2 T SSR,0 s S SR,out,0 = T 2 1 + PSR 2 , (1 + PSR ) p Tp 2 SSR,0 T S SR,out,1 = T 2 1 − PSR s2 , (1 + PSR ) p Tp T2 2 S un,out,0 = Eun,p Tp2 1 + Pun s2 Tp
Fig. 1 shows the schematic diagram of our calibration by SR. The undertest light source (a deuterium lamp) is calibrated by comparing spectral radiant flux between SR and the deuterium lamp. A polarimeter is introduced for determining the polarization dependence. The polarimeter is comprised of two sets of four metal mirrors. Each mirror set keeps the The following equation is derived from the above equation. axis of incident radiation and that of outgoing radiation on the same line, and can rotate around the optical axis in a Sun,out,0 2
∴ Sun,0 = (1 + Pun )Eun,p = (1 + Pun ) vacuum. The rotations of these mirror sets are independent 2 2 1 + P Ts T un T 2 p of each other. A set of mirrors functions as an incomplete p polarizer and a phase shifter.
SSR,0 Sun,out,0 The polarization dependence of the calibration system is = (1 + Pun ) SSR,out,0 (1 + PSR ) determined by comparing the polarization between the inci
dent radiant flux and the outgoing radiant flux of our caliTs2 1 + PSR T 2 bration system. The radiant flux and the polarization of SR p
are calculable from Schwinger’s theory [14]. The polariza× (2) Ts2 tion of outgoing radiant flux of SR is determined by using the 1 + Pun T 2 p polarimeter. Stokes vector and Mueller matrix are used for the polarThe following equation is derived from SSR,out,0 and SSR,out,1 , ization dependence calculation. We assume our calibration above. system as an incomplete polarizer and a phase shifter. The Mueller matrix Mc (Tp , Ts , ∆) of our calibration system can be shown as follows. 2 Tp + Ts2 Tp2 − Ts2 0 0 2 2 2 2 0 0 1 Tp − Ts Tp + Ts (1) M c (Tp , Ts , ∆) = 2 0 0 2Tp Ts cos(∆) 2Tp Ts sin(∆) 0 0 −2Tp Ts sin(∆) 2Tp Ts cos(∆) Tp and Ts are the reflectance of our calibration system related to the radiation whose electric vector is parallel and that whose electric vector is perpendicular to the plane of incidence of our calibration system, respectively. ∆ is the phase shift between incident and outgoing radiant flux that results from transmitting the radiant flux over our calibration system. In the following, SSR and SSR,out show the Stokes vector of SR and that of SR, which passes through our calibration system. Sun and Sun,out show the Stokes vec-
SSR,out,1 SSR,out,0
Ts2 1 − PSR 2 1− Tp T2 1 ∴ s2 = = Tp PSR T2 1+ 1 + PSR s2 Tp
SSR,out,1 SSR,out,0
SSR,out,1 SSR,out,0 (3)
T. Zama, I. Saito / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1087–1091
The Eq. (2) shows that the total radiant flux of the undertest light source is determined from (a) the total radiant flux of SR (SSR,out,0 ), (b) the radiant flux ratio of the undertest light source to SR (Sun,out,0 /SSR,out,0 ), (c) the radiant flux ratio of the perpendicular component to the parallel component of SR (PSR ), (d) that of the undertest light source (Pun ), and (e) the reflectance ratio of the perpendicular component to the parallel component of our calibration system (Ts /Tp ). The Eq. (3) shows that the reflectance ratio (Ts /Tp ) is derived from the Stokes vectors of SR (SSR,out,0 and SSR,out,1 ). Both (a) and (c) are calculable from Schwinger’s theory; (d) is almost equal to 1 (most light sources we can use conveniently are unpolarized); (b) is determined from the comparison between SR and the undertest light source using our calibration system and (e) is evaluated by using our polarimeter. This evaluation method of (e) is shown as follows.
3. Evaluation of polarization dependence The polarization dependence (Ts /Tp ) of our calibration system is determined using the Eq. (3). The polarization (Stokes vector) of SR which passes through our calibration system (SSR,out,0 and SSR,out,1 ), is determined by our polarimeter, and the polarization of the incident SR (PSR ) is calculated from Schwinger’s theory. The incident polarization component of our calibration system is equalized, by letting the SR component that has the same angular distance from the electron orbital plane, into our calibration system. The Mueller matrix of each mirror set of the polarimeter is expressed in the same manner as that shown in Eq. (1). Each mirror set can rotate round the optical axis. Because the ψ rotation of the mirror set corresponds to −ψ rotation of the incident radiant flux and ψ rotation of the outgoing radiant flux, the Mueller matrix of the polarimeter P(T1,x , T1,y , T2,x , T2,y , δ1 , ψ1 , δ2 , ψ2 ) is derived as follows (the suffixes 1 and 2 indicate the parameters related to the mirror sets 1 and 2, respectively). P(T1,x , T1,y , T2,x , T2,y , δ1 , ψ1 , δ2 , ψ2 ) = R(ψ2 )Mp,2 (T2,x , T2,y , δ2 )R(−ψ2 ) ×R(ψ1 )Mp,1 (T1,x , T1,y , δ1 )R(−ψ1 ) 1 0 0 0 0 cos(2ψ ) − sin(2ψ ) 0 i i R(ψi ) = , 0 sin(2ψi ) cos(2ψi ) 0 0 0 0 1
(i = 1, 2)
Mp (Ti,x , Ti,y , δi ) shows the Mueller matrix of a mirror set and R(ψi ) shows that of the rotation. Ti,x and Ti,y are the reflectance related to the radiation whose electric vector is parallel and perpendicular. δi is the phase shift.
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The Stokes vector of incident radiant flux S and that of outgoing radiant flux S is expressed as follows. S = P(T1,x , T1,y , T2,x , T2,y , δ1 , ψ1 , δ2 , ψ2 )S 2 T2 T1,x 2,x IM = RS0 , F= RS0 , 4 ρ1 =
T1,y , T1,x
ρ2 =
T2,y T2,x
IM = IM,DC (ψ1 ) + IM,cos (ψ1 ) cos(2ψ2 ) + IM,sin (ψ1 ) sin(2ψ2 ) S1 2 IM,DC (ψ1 ) = F (1 + ρ2 ) (1 + ρ12 ) + (1 − ρ12 ) cos(2ψ1 ) S0
S2 , +(1 − ρ12 ) sin(2ψ1 ) S0 IM,cos (ψ1 ) = F (1 − ρ2 2 ) sin(2ψ1 ) cos(2ψ1 ) 1 − ρ12 1 + ρ12 × + + 2ρ1 cos(δ1 ) tan(2ψ1 ) sin(2ψ1 ) tan(2ψ1 ) S2 S1 + (1 + ρ12 − 2ρ1 cos(δ1 )) S0 S0
sin(δ1 ) S3 , −2ρ1 cos(2ψ1 ) S0
×
IM,sin (ψ1 ) = F (1 − ρ22 ) sin(2ψ1 ) cos(2ψ1 ) 1 − ρ12 S1 × + (1 + ρ1 2 − 2ρ1 cos(δ1 )) cos(2ψ1 ) S0
cos(δ1 ) + tan(2ψ1 ) + 2ρ1 tan(2ψ1 )
sin(δ1 ) S3 +2ρ1 sin(2ψ1 ) S0 (1 + ρ12 )
S2 S0
(4)
A detector is set at the normal incidence position at the back of the polarimeter. R and IM are the responsivity of the detector and the detector signal, respectively, and a new constant F is introduced. The detector signal consists of three components: a constant IM,DC (ψ1 ), cos(2ψ2 ) (whose coefficient is IM,cos (ψ1 )), and sin(2ψ2 ) (whose coefficient is IM,sin (ψ1 )). Therefore, from the measurement of IM with rotation ψ2 and by fitting the result by sine and cosine function, the three parameters IM,DC (ψ1 ), IM,cos (ψ1 ), and IM,sin (ψ1 ) can be evaluated. ρ and δ depend on the optical constants of mirror material, and these are therefore constant for the rotation.
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Setting the angle ψ1 = 0, π/4, π/2, and 3π/4, and determining IM,DC , IM,cos and IM,sin , the following results are derived.
π π 3π IM,cos (0) − IM,cos = IM,sin − IM,sin 2 4 4 2 2 = 2F (1 − ρ2 )(1 − ρ1 ) = C1 π S1 IM,cos (0) + IM,cos = 2F (1 − ρ22 )(1 + ρ12 ) = C2 2 S0
π π 3π IM,DC (0) + IM,DC = IMl,DC + IM,DC 2 4 4 2 2 = 2F (1 + ρ2 )(1 + ρ1 ) = C3 π S1 IM,DC (0) − IM,DC = 2F (1 + ρ22 )(1 − ρ12 ) = C4 2 S0 (5) In the above equation, the new variations C1 , C2 , C3 , and C4 are introduced, and the ratio of the Stokes vector on the incident radiant flux S1 /S0 is derived as follows.
S1 C3 C4 = (6) S0 C1 C2 In the above equation, we assume the horizontal component of the outgoing radiant flux is stronger than the vertical component, because the incident radiant flux (SR) is strongly polarized horizontally and all optics of our calibration system is set at an almost normal incidence position for the incident flux. Because the polarimeter is set at the back of our calibration system, the S1 /S0 in the above equation correspond to SSR,out,0 /SSR,out,1 . From (2) to (6), the absolute radiant flux of the undertest light source is determined.
4. Evaluation of uncertainty The C1 , C2 , C3 , and C4 determined by Eq. (5) have uncertainties. Expressing C1 , C2 , C3 , and C4 as the uncertainties, disregarding the terms of higher orders and rewriting Eq. (6), the following equation is derived.
S1 (C3 + C3 )(C4 + C4 ) = S0 (C1 + C1 )(C2 + C2 )
1 C3 C4 C1 C2 ∼ , + − − =D 1+ 2 C3 C4 C1 C2
C3 C4 D= C1 C2 The Eqs. (3) and (2) are rewritten as follows. Tp2 Ts2
=
1 1−D PSR 1 + D
D C3 C4 C1 C2 × 1− + − − 1 − D2 C3 C4 C1 C2
Sun,0 = (1 + Pun )
Sun,out,0 SSR,out,0
SSR 1 + PSR
1−D 1+D Pun 1 − D 1+ PSR 1 + D 1+
× 1 −
1 1 − PSR 1−D 1−D 1+ + 1+D Pun 1+D
D × (1 + D)2
C3 C4 C1 C2 × + − − C3 C4 C1 C2
In this evaluation, the mirror set 2 is rotated from 0 to 2 (this equals one revolution, and made it easy to check the motion error of mirror set 2), and the mirror set 1 is set to only ψ1 = 0, π/4, π/2, and 3π/4, which means the angle setting error of mirror set 1 is not as easy to check. We introduce a checking criterion by using (4) and (5). The first and third equations of Eq. (5) express that the two measurement results are equal. But if there are angle setting errors, the two measurement results are not equal. Expressing the angle setting error as ψ1 , the following equation is derived. IM,DC (0 + ψ1 ) = F (1 + ρ22 )(1 − ρ12 ) 1 + ρ12 S1 S2 × + cos(2ψ1 ) + sin(2ψ1 ) S0 S0 1 − ρ12 π IMl,DC + ψ1 4 = F (1 + ρ22 )(1 − ρ12 ) 1 + ρ12 S1 S2 × − sin(2ψ1 ) + cos(2ψ1 ) S0 S0 1 − ρ12 π IM,DC + ψ1 2 = F (1 + ρ22 )(1 − ρ12 ) 1 + ρ12 S1 S2 × − cos(2ψ1 ) − sin(2ψ1 ) S0 S0 1 − ρ12
3π IM,DC + ψ1 4 = F (1 + ρ22 )(1 − ρ12 ) 1 + ρ12 S1 S2 × + sin(2ψ1 ) − cos(2ψ1 ) S0 S0 1 − ρ12
T. Zama, I. Saito / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1087–1091
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noise. In order to minimize the angle setting error of mirror set 1, our checking criterion excludes the measurement result whose value in the above equation exceeds the detector noise level. Fig. 2 shows the excluded and the adopted results by the above checking criterion. By using this method, the uncertainty of our calibration system is improved from ±10% to ±7% and this improved uncertainty almost due to the noise of the detector. To improve the dark current and noise of the detector, we are introducing a new polarimeter with a large aperture and mirror.
5. Conclusions Fig. 2. Excluded and adopted results.
A similar equation is derived from IM,cos (ψ1 ) and IM,sin (ψ1 ). Assuming the angle setting errors of ψ1 = 0, π/4, π/2, and 3π/4 are ψ11 , ψ12 , ψ13 , and ψ14 , those are independent and the average angle setting uncertainty is ψ1 , the following equation is derived. π + ψ12 IM,DC (0 + ψ11 ) + IM,DC 2
π 3π − IM,DC + ψ13 + IM,DC + ψ14 4 4
√ S2 2 ≤ 2 2F (1 + ρ22 )(1 − ρ12 )(2 ψ1 ) S0 S2 = 2F (1 + ρ22 )(1 − ρ12 )(2 ψ1 ) S0 π + ψ12 IM,cos (0 + ψ11 ) − IM,cos 2
π 3π − IM,sin + ψ13 − IM,sin + ψ14 4 4
S2 2 ≤ 2F (1 − ρ22 )(1 + ρ1 2 − 2ρ1 cos(␦1 )) S0
S3 2 (2 ψ1 ) + 2(1 − ρ22 )ρ1 sin(␦1 ) S0 In the above equation, only angle setting errors are considered though there are other causes of error, such as detector
We established a new calculation method for evaluating the polarization effect of our calibration system and examined a new checking criterion. The uncertainty of the spectral radiance of the undertest light source is improved, and was evaluated to be ±7%.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
D. Einfeld, D. Stuck, B. Wende, Metrologia 14 (1978) 111. F. Riehle, B. Wende, Metrologia 22 (1986) 75. E. Tegeler, Nucl. Instr. Meth. A282 (1989) 706. J. Hollandt, U. Becker, W. Paustian, M. Richter, G. Ulm, Metrologia 37 (2000) 563. J. Hollandt, W. Jans, M. K¨uhne, F. Lindenlauf, B. Wende, Rev. Sci. Instrum. 63 (1) (1992) 1278. F. Lei, W. Paustian, E. Tegeler, Metrologia 32 (1996) 589. M. Richter, J. Hollandt, U. Kroth, W. Paustian, H. Rabus, R. Thornagel, G. Ulm, Metrologia 40 (2003) 107. T. Zama, K. Awazu, H. Onuki, J. Electron Spectrosc. Relat. Phenom. 80 (1996) 493. T. Zama, T. Saito, H. Onuki, J. Synchrotron Radiat. 5 (1998) 759. T. Zama, T. Saito, H. Onuki, J. Electron Spectrosc. Relat. Phenom. 101–103 (1999) 991. T. Zama, I. Saito, Metrologia 40 (2003) 115. T. Koide, T. Shidara, T. Miyahara, M. Yuri, Rev. Sci. Instrum. 66 (1995) 1923. L. Nahon, C. Alcaraz, Appl. Opt. 43 (2004) 1024. J. Schwinger, Phys. Rev. 75 (1949) 1912.
Calibration of absolute spectral radiance in UV and VUV regions by using synchrotron radiation Tatsuya Zama ∗ , Ichiro Saito National Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology (NMIJ/AIST), AIST Tsukuba Central 3, 1-1 Umezono 1-Chome, Tsukuba-shi, Ibaraki-ken 305-8563, Japan Available online 24 February 2005
Abstract A synchrotron radiation was used as an absolute spectral radiance source in ultraviolet and vacuum ultraviolet regions and that scale was transferred to a deuterium lamp. A calibration system was used for comparing each spectral radiant flux. For the absolute calibration, it is important to determine the polarization dependence of the calibration system, because polarization of radiant flux is different between synchrotron radiation and the deuterium lamp. A polarimeter was introduced to determine the polarization dependence of our calibration system. The polarization dependence and that uncertainty of our system are calculated by using a new calculation method. © 2005 Elsevier B.V. All rights reserved. Keywords: Polarization; Synchrotron radiation; Absolute calibration; Vacuum ultraviolet
1. Introduction Quantitative evaluation is important for the elucidation of natural phenomena, and forms the foundation of experimental studies. Such quantitative evaluation requires a scale. In the visible region and a part of the ultraviolet and infrared region, blackbody (and Plank’s law) comes into common use for establishing the absolute scale. But in the shorter wavelength region [1–7], no common method has yet been established. The radiant flux of blackbody is not sufficiently strong for a primary source in this region. We have been attempting to establish an absolute scale of spectral radiance in ultraviolet (UV) and vacuum ultraviolet (VUV) regions by using synchrotron radiation (SR) as the primary standard radiant source [8–11]. The spectral radiance scale is transferred to an undertest light source (a deuterium lamp) by comparing the light source with SR from an electron storage ring TERAS in the National Institute of Advanced In∗
Corresponding author. E-mail address: [email protected] (T. Zama).
0368-2048/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2005.01.165
dustrial Science and Technology (AIST). A calibration system that consists of a monochromator and a focusing mirror is used for the comparison. For the absolute calibration, it is important to consider the polarization dependence of our calibration system; because SR is strongly polarized, most light sources we can use conveniently are unpolarised, and the efficiency of the monochromator and mirror depends on polarization of incident flux. This aspect is different from the calibration in the visible region (blackbody and most convenient light sources are unpolarised). In UV and VUV regions, there is not good material for a polarizer or depolarizer. One of the methods used to determine polarization dependence involves rotating the calibration system around the incident flux axis [5,6]. However, we cannot introduce this method because our laboratory does not have sufficient space to rotate our calibration system. We introduced a polarimeter in order to determine the polarization dependence of our calibration system [12,13]. The polarization dependence and the calibration uncertainty were estimated by a simple method [10]. In this paper, we present a new uncertainty evaluation method and make a closer evaluation.
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T. Zama, I. Saito / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1087–1091
Fig. 1. Schematic diagram of the calibration by SR.
2. Principle of calibration
tor of the undertest light source and that of the source’s radiation, which passes through our calibration system (SSR,out = Mc SSR , Sun,out = Mc Sun ). PSR and Pun show the radiant flux ratio of the perpendicular component to the parallel component of SR and that of the undertest light 2 /E2 2 2 source (PSR = ESR,s SR,p , Pun = Eun,s /Eun,p , the subscripts p and s show the parallel and perpendicular electric vectors). Expressing the components of Stokes vector 2 of SR as SSR = (SSR,0 , SSR,1 , SSR,2 , SSR,3 ) (SSR,0 = ESR,p + 2 2 2 ESR,s , SSR,1 = Eun,p − Eun,s ) and other Stokes vector components in the same manner, the following equation is derived. 2 T SSR,0 s S SR,out,0 = T 2 1 + PSR 2 , (1 + PSR ) p Tp 2 SSR,0 T S SR,out,1 = T 2 1 − PSR s2 , (1 + PSR ) p Tp T2 2 S un,out,0 = Eun,p Tp2 1 + Pun s2 Tp
Fig. 1 shows the schematic diagram of our calibration by SR. The undertest light source (a deuterium lamp) is calibrated by comparing spectral radiant flux between SR and the deuterium lamp. A polarimeter is introduced for determining the polarization dependence. The polarimeter is comprised of two sets of four metal mirrors. Each mirror set keeps the The following equation is derived from the above equation. axis of incident radiation and that of outgoing radiation on the same line, and can rotate around the optical axis in a Sun,out,0 2
∴ Sun,0 = (1 + Pun )Eun,p = (1 + Pun ) vacuum. The rotations of these mirror sets are independent 2 2 1 + P Ts T un T 2 p of each other. A set of mirrors functions as an incomplete p polarizer and a phase shifter.
SSR,0 Sun,out,0 The polarization dependence of the calibration system is = (1 + Pun ) SSR,out,0 (1 + PSR ) determined by comparing the polarization between the inci
dent radiant flux and the outgoing radiant flux of our caliTs2 1 + PSR T 2 bration system. The radiant flux and the polarization of SR p
are calculable from Schwinger’s theory [14]. The polariza× (2) Ts2 tion of outgoing radiant flux of SR is determined by using the 1 + Pun T 2 p polarimeter. Stokes vector and Mueller matrix are used for the polarThe following equation is derived from SSR,out,0 and SSR,out,1 , ization dependence calculation. We assume our calibration above. system as an incomplete polarizer and a phase shifter. The Mueller matrix Mc (Tp , Ts , ∆) of our calibration system can be shown as follows. 2 Tp + Ts2 Tp2 − Ts2 0 0 2 2 2 2 0 0 1 Tp − Ts Tp + Ts (1) M c (Tp , Ts , ∆) = 2 0 0 2Tp Ts cos(∆) 2Tp Ts sin(∆) 0 0 −2Tp Ts sin(∆) 2Tp Ts cos(∆) Tp and Ts are the reflectance of our calibration system related to the radiation whose electric vector is parallel and that whose electric vector is perpendicular to the plane of incidence of our calibration system, respectively. ∆ is the phase shift between incident and outgoing radiant flux that results from transmitting the radiant flux over our calibration system. In the following, SSR and SSR,out show the Stokes vector of SR and that of SR, which passes through our calibration system. Sun and Sun,out show the Stokes vec-
SSR,out,1 SSR,out,0
Ts2 1 − PSR 2 1− Tp T2 1 ∴ s2 = = Tp PSR T2 1+ 1 + PSR s2 Tp
SSR,out,1 SSR,out,0
SSR,out,1 SSR,out,0 (3)
T. Zama, I. Saito / Journal of Electron Spectroscopy and Related Phenomena 144–147 (2005) 1087–1091
The Eq. (2) shows that the total radiant flux of the undertest light source is determined from (a) the total radiant flux of SR (SSR,out,0 ), (b) the radiant flux ratio of the undertest light source to SR (Sun,out,0 /SSR,out,0 ), (c) the radiant flux ratio of the perpendicular component to the parallel component of SR (PSR ), (d) that of the undertest light source (Pun ), and (e) the reflectance ratio of the perpendicular component to the parallel component of our calibration system (Ts /Tp ). The Eq. (3) shows that the reflectance ratio (Ts /Tp ) is derived from the Stokes vectors of SR (SSR,out,0 and SSR,out,1 ). Both (a) and (c) are calculable from Schwinger’s theory; (d) is almost equal to 1 (most light sources we can use conveniently are unpolarized); (b) is determined from the comparison between SR and the undertest light source using our calibration system and (e) is evaluated by using our polarimeter. This evaluation method of (e) is shown as follows.
3. Evaluation of polarization dependence The polarization dependence (Ts /Tp ) of our calibration system is determined using the Eq. (3). The polarization (Stokes vector) of SR which passes through our calibration system (SSR,out,0 and SSR,out,1 ), is determined by our polarimeter, and the polarization of the incident SR (PSR ) is calculated from Schwinger’s theory. The incident polarization component of our calibration system is equalized, by letting the SR component that has the same angular distance from the electron orbital plane, into our calibration system. The Mueller matrix of each mirror set of the polarimeter is expressed in the same manner as that shown in Eq. (1). Each mirror set can rotate round the optical axis. Because the ψ rotation of the mirror set corresponds to −ψ rotation of the incident radiant flux and ψ rotation of the outgoing radiant flux, the Mueller matrix of the polarimeter P(T1,x , T1,y , T2,x , T2,y , δ1 , ψ1 , δ2 , ψ2 ) is derived as follows (the suffixes 1 and 2 indicate the parameters related to the mirror sets 1 and 2, respectively). P(T1,x , T1,y , T2,x , T2,y , δ1 , ψ1 , δ2 , ψ2 ) = R(ψ2 )Mp,2 (T2,x , T2,y , δ2 )R(−ψ2 ) ×R(ψ1 )Mp,1 (T1,x , T1,y , δ1 )R(−ψ1 ) 1 0 0 0 0 cos(2ψ ) − sin(2ψ ) 0 i i R(ψi ) = , 0 sin(2ψi ) cos(2ψi ) 0 0 0 0 1
(i = 1, 2)
Mp (Ti,x , Ti,y , δi ) shows the Mueller matrix of a mirror set and R(ψi ) shows that of the rotation. Ti,x and Ti,y are the reflectance related to the radiation whose electric vector is parallel and perpendicular. δi is the phase shift.
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The Stokes vector of incident radiant flux S and that of outgoing radiant flux S is expressed as follows. S = P(T1,x , T1,y , T2,x , T2,y , δ1 , ψ1 , δ2 , ψ2 )S 2 T2 T1,x 2,x IM = RS0 , F= RS0 , 4 ρ1 =
T1,y , T1,x
ρ2 =
T2,y T2,x
IM = IM,DC (ψ1 ) + IM,cos (ψ1 ) cos(2ψ2 ) + IM,sin (ψ1 ) sin(2ψ2 ) S1 2 IM,DC (ψ1 ) = F (1 + ρ2 ) (1 + ρ12 ) + (1 − ρ12 ) cos(2ψ1 ) S0
S2 , +(1 − ρ12 ) sin(2ψ1 ) S0 IM,cos (ψ1 ) = F (1 − ρ2 2 ) sin(2ψ1 ) cos(2ψ1 ) 1 − ρ12 1 + ρ12 × + + 2ρ1 cos(δ1 ) tan(2ψ1 ) sin(2ψ1 ) tan(2ψ1 ) S2 S1 + (1 + ρ12 − 2ρ1 cos(δ1 )) S0 S0
sin(δ1 ) S3 , −2ρ1 cos(2ψ1 ) S0
×
IM,sin (ψ1 ) = F (1 − ρ22 ) sin(2ψ1 ) cos(2ψ1 ) 1 − ρ12 S1 × + (1 + ρ1 2 − 2ρ1 cos(δ1 )) cos(2ψ1 ) S0
cos(δ1 ) + tan(2ψ1 ) + 2ρ1 tan(2ψ1 )
sin(δ1 ) S3 +2ρ1 sin(2ψ1 ) S0 (1 + ρ12 )
S2 S0
(4)
A detector is set at the normal incidence position at the back of the polarimeter. R and IM are the responsivity of the detector and the detector signal, respectively, and a new constant F is introduced. The detector signal consists of three components: a constant IM,DC (ψ1 ), cos(2ψ2 ) (whose coefficient is IM,cos (ψ1 )), and sin(2ψ2 ) (whose coefficient is IM,sin (ψ1 )). Therefore, from the measurement of IM with rotation ψ2 and by fitting the result by sine and cosine function, the three parameters IM,DC (ψ1 ), IM,cos (ψ1 ), and IM,sin (ψ1 ) can be evaluated. ρ and δ depend on the optical constants of mirror material, and these are therefore constant for the rotation.
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Setting the angle ψ1 = 0, π/4, π/2, and 3π/4, and determining IM,DC , IM,cos and IM,sin , the following results are derived.
π π 3π IM,cos (0) − IM,cos = IM,sin − IM,sin 2 4 4 2 2 = 2F (1 − ρ2 )(1 − ρ1 ) = C1 π S1 IM,cos (0) + IM,cos = 2F (1 − ρ22 )(1 + ρ12 ) = C2 2 S0
π π 3π IM,DC (0) + IM,DC = IMl,DC + IM,DC 2 4 4 2 2 = 2F (1 + ρ2 )(1 + ρ1 ) = C3 π S1 IM,DC (0) − IM,DC = 2F (1 + ρ22 )(1 − ρ12 ) = C4 2 S0 (5) In the above equation, the new variations C1 , C2 , C3 , and C4 are introduced, and the ratio of the Stokes vector on the incident radiant flux S1 /S0 is derived as follows.
S1 C3 C4 = (6) S0 C1 C2 In the above equation, we assume the horizontal component of the outgoing radiant flux is stronger than the vertical component, because the incident radiant flux (SR) is strongly polarized horizontally and all optics of our calibration system is set at an almost normal incidence position for the incident flux. Because the polarimeter is set at the back of our calibration system, the S1 /S0 in the above equation correspond to SSR,out,0 /SSR,out,1 . From (2) to (6), the absolute radiant flux of the undertest light source is determined.
4. Evaluation of uncertainty The C1 , C2 , C3 , and C4 determined by Eq. (5) have uncertainties. Expressing C1 , C2 , C3 , and C4 as the uncertainties, disregarding the terms of higher orders and rewriting Eq. (6), the following equation is derived.
S1 (C3 + C3 )(C4 + C4 ) = S0 (C1 + C1 )(C2 + C2 )
1 C3 C4 C1 C2 ∼ , + − − =D 1+ 2 C3 C4 C1 C2
C3 C4 D= C1 C2 The Eqs. (3) and (2) are rewritten as follows. Tp2 Ts2
=
1 1−D PSR 1 + D
D C3 C4 C1 C2 × 1− + − − 1 − D2 C3 C4 C1 C2
Sun,0 = (1 + Pun )
Sun,out,0 SSR,out,0
SSR 1 + PSR
1−D 1+D Pun 1 − D 1+ PSR 1 + D 1+
× 1 −
1 1 − PSR 1−D 1−D 1+ + 1+D Pun 1+D
D × (1 + D)2
C3 C4 C1 C2 × + − − C3 C4 C1 C2
In this evaluation, the mirror set 2 is rotated from 0 to 2 (this equals one revolution, and made it easy to check the motion error of mirror set 2), and the mirror set 1 is set to only ψ1 = 0, π/4, π/2, and 3π/4, which means the angle setting error of mirror set 1 is not as easy to check. We introduce a checking criterion by using (4) and (5). The first and third equations of Eq. (5) express that the two measurement results are equal. But if there are angle setting errors, the two measurement results are not equal. Expressing the angle setting error as ψ1 , the following equation is derived. IM,DC (0 + ψ1 ) = F (1 + ρ22 )(1 − ρ12 ) 1 + ρ12 S1 S2 × + cos(2ψ1 ) + sin(2ψ1 ) S0 S0 1 − ρ12 π IMl,DC + ψ1 4 = F (1 + ρ22 )(1 − ρ12 ) 1 + ρ12 S1 S2 × − sin(2ψ1 ) + cos(2ψ1 ) S0 S0 1 − ρ12 π IM,DC + ψ1 2 = F (1 + ρ22 )(1 − ρ12 ) 1 + ρ12 S1 S2 × − cos(2ψ1 ) − sin(2ψ1 ) S0 S0 1 − ρ12
3π IM,DC + ψ1 4 = F (1 + ρ22 )(1 − ρ12 ) 1 + ρ12 S1 S2 × + sin(2ψ1 ) − cos(2ψ1 ) S0 S0 1 − ρ12
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noise. In order to minimize the angle setting error of mirror set 1, our checking criterion excludes the measurement result whose value in the above equation exceeds the detector noise level. Fig. 2 shows the excluded and the adopted results by the above checking criterion. By using this method, the uncertainty of our calibration system is improved from ±10% to ±7% and this improved uncertainty almost due to the noise of the detector. To improve the dark current and noise of the detector, we are introducing a new polarimeter with a large aperture and mirror.
5. Conclusions Fig. 2. Excluded and adopted results.
A similar equation is derived from IM,cos (ψ1 ) and IM,sin (ψ1 ). Assuming the angle setting errors of ψ1 = 0, π/4, π/2, and 3π/4 are ψ11 , ψ12 , ψ13 , and ψ14 , those are independent and the average angle setting uncertainty is ψ1 , the following equation is derived. π + ψ12 IM,DC (0 + ψ11 ) + IM,DC 2
π 3π − IM,DC + ψ13 + IM,DC + ψ14 4 4
√ S2 2 ≤ 2 2F (1 + ρ22 )(1 − ρ12 )(2 ψ1 ) S0 S2 = 2F (1 + ρ22 )(1 − ρ12 )(2 ψ1 ) S0 π + ψ12 IM,cos (0 + ψ11 ) − IM,cos 2
π 3π − IM,sin + ψ13 − IM,sin + ψ14 4 4
S2 2 ≤ 2F (1 − ρ22 )(1 + ρ1 2 − 2ρ1 cos(␦1 )) S0
S3 2 (2 ψ1 ) + 2(1 − ρ22 )ρ1 sin(␦1 ) S0 In the above equation, only angle setting errors are considered though there are other causes of error, such as detector
We established a new calculation method for evaluating the polarization effect of our calibration system and examined a new checking criterion. The uncertainty of the spectral radiance of the undertest light source is improved, and was evaluated to be ±7%.
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