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A high-purity circular polarization modulator: Application to birefringence and circular dichroism measurements on multidielectric mirrors
Volume 37, number 4
OPTICS COMMUNICATIONS
15 May 1981
A HIGH-PURITY CIRCULAR POLARIZATION MODULATOR: APPLICATION TO BIREFRINGENCE AND CIRCULAR DICHROISM MEASUREMENTS ON MULTIDIELECTRIC MIRRORS M.A. BOUCHIAT and L. POTTIER Laboratoire de Spectroscopie Hertzienne de L 'E.N.S., 75231 Paris Cedex 05, France Received 27 January 1981
We present a method for producing a modulated circular polarization of high purity, i.e. whose modulation frequency allows specific discrimination against finear polarization and unpolarized intensity. We apply it to measure the circular dichroism and birefringence of high reflectivity multidielectric mirrors: no circular dichroism is observed at a level of 10-6, but birefringence values from 3 × 10-4 to 2 × 10-3 radian per reflexion under normal incidence are obtained.
1. Motivation This work originates in an experiment designed to search the parity violation predicted to exist in atoms by the recent theories of weak neutral currents [ 1]. In this experiment, the effect under search can be simulated by a stray birefringence or circular dichroism of 10-4: it is therefore necessary to control these quantities to a level of about 10 - 5 . Since the optical parts we need to control are mainly light reflecting parts (i.e. mirrors), the methods used in usual dichrometers for transparent samples do not apply without modifications and we had to imagine new procedures.
2. Definition of birefringence and circular dichroism; principle of their measurement The intensity and polarization of a light beam can be represented by a two-dimensional complex polarization vector ¢, or equivalently by the four real quantities le21 + t41, l e 2 1 - tey2l, 2 Re e*ey and 2 Im e*ey, where x and y are coordinate axes orthogonal to the beam. For a better understanding of what will follow, let us first point out the physical interpretation of these four quantities: i) lex21 + le21 is the unpolarized intensity (UI); it can be measured by simply receiving the beam onto a detector.
ii) lex21 - le21 is the x y linear polarized intensity (xy LI); it can be measured by inserting an analyzer (e.g. a prism) before the detector, and taking the difference of the values measured with the analyzer oriented along Ox and Oy. iii) 2 Re exey * is the bb ' linearly polarized intensity (bb' LI); here b and b' denote the first and second bisectrices of the x y axes. The bb' LI is measured by taking the difference of the values obtained with the analyzer oriented along Ob and Ob'. iv) 2 Im e*ey is the circularly polarized intensity (CI). It can be measured by inserting a quarterwave plate before the prism and taking the difference of the two values obtained with the prism oriented along the two bisectrices of the principal axes of the plate. If the beam passes through some linear optical system, the four intensities corresponding to the transmitted beam will be linear combinations o f those o f the incident beam, with coefficients which of course reflect the properties of the optical system. If we measure one of the four intensities of the transmitted beam as explained above, and if by some technical trick (which will be actually the central topic of the next section) we modulate one and only one - of the four intensities of the incident beam, then by synchronous detection we can obtain directly this particular coefficient which couples the modulated quantity for the incident beam to the measured quantity for the transmitted beam. In particular, if we modulate the incident CI and detect the transmitted UI, then we
0 0 3 0 - 4 0 1 8 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
229
Volume 37, number 4
OPTICS COMMUNICATIONS
15 May 1981
measure the circular dichroisrn of the optical system; and if we modulate the incident CI and detect the transmitted xy (or bb') LI we measure the xy (or bb')
g
birefringence. The limitations of this method with respect to measuring very small birefringences or circular dichroisms are twofold: i) First, real modulators designed for modulating one of the four incident intensities actually produce spurious modulations of the three other ones as well, more or less at the level o f 10 2. In the next section of this paper, we will show how to improve the purity of a CI modulator so as to make it about two orders of magnitude better. ii) Second, real modulators generally produce a small spurious modulation of the direction of the transmitted beam. Therefore the positions of the points where the beam meets subsequent optical parts and the detector, are also modulated. Because the transparency of the optical parts and the sensitivity of the detector surface are always inhomogeneous to some extent, this generates a spurious modulation in the detector's signal. Moreover, the amplitude and phase of this spurious modulation critically depend on the positions o f the optical elements on the beam: therefore it becomes in practice impossible to master and highly non reproducible as soon as the optical parts are moved. iii) Then, the detected modulation contains not only the contribution of the optical system of interest, but also that of the other optical components of the experimental arrangement: windows of the modulator; surface of the detector; e t c . . These extra contributions are usually eliminated by taking the difference between two measurements performed with and without tire optical system of interest, which implicitly assumes that the path of the beam remains unchanged when the system is removed; this approximation is reasonable for a transparent optical element at normal incidence; however, in the case of mirrors (which reflect the beam backwards), specific methods have to be imagined; we shall present two in the last section.
Fig. 1. Usual circular polarization modulation: a) a Pockels cell; b) a rotating quarterwave plate. of an electrooptical or photoelastic element driven by a sine voltage of frequency co, with axes at 45 ° of the incident linear polarization (fig. la). An older system consists of a simple quarterwave plate mechanically rotated at frequency c o / 2 , 1 (fig. lb). We shall choose this latter modulator as our example because it leads to simpler modulations; however the following discussion is general and its extension to other modulators is straightforward. We consider a modulator that is intended to modulate the CI and only the CI at frequency co. For instance, in the case of our example modulator (fig. 1b), the output intensities are:
Idl + 141-- l, le21
To date's polarization modulators usually consist 230
= 1 (1 + cos 2cot),
2 Re exey* = ~- sin 2cot, 2 I r a exey * = sin cot, so that frequency co is in principle a specific label of the CI. In practice, however, if tbe beam transmitted by this modulator is received onto a detector, either directly or through a prism, by lock-in detection with sin cot as a reference, one will observe spurious sin cot modulations of the UI as well as LI's, with a typical order of magnitude of 10 2 ~2. Thus at the percent level the CI is no longer labeled specifically. +]
+2
3. Generation of high-purity modulated circular polarization
~2
Since a half-turn rotation of the plate restores the directions of the principal axes, the fundamental frequency of the optical modulation is co when the rotation frequency is 00/2. Actually, we have tried four modulators: a rotating 5M4 quartz plate (Fichou, France); a rotating X/4 Polaroid sheet; a transverse field Pockels modulator (Lasermetrics 3030 HFW); and a longitudinal field Pockels modulator (Lasermetrics 1080 P). The spurious modulations were different, but their order of magnitude was 10 -2 in the four cases.
Volume 37, number 4
OPTICS COMMUNICATIONS
The improvements which we propose are based on this simple property: a half-wave plate changes the
polarization vector into its symmetric with respect to the plate axes. We shall take advantage of this property in two steps: in the first step, we shall remove the spurious LI modulations; in the second step, the spurious UI ones: a) First, after the modulator we insert a half-wave plate (hereafter called output plate) which we mechanically rotate in its plane +3 at some frequency coO v~ co (fig. 2a). In principle, this plate will have no effect on the UI of the beam; the CI will undergo a simple sign reversal; on the other hand, the two LI's will be blurred out. More precisely, the LI modulations initially present at frequency co will be transferred to different frequencies co -+ 46o 0 +4 and can then be easily distinguished. b) So far the sin cot modulation still contains a spurious contribution from the UI. To discriminate it we now insert before the initial modulator one more rotating half-wave plate, which we shall call the input plate (fig. 2b). Its rotation frequency coi must of course be different from co and coO" Because o f this rotating input plate, the polarization vector E at the input of the initial modulator now rotates at frequency 2co i, which induces an extra modulation at frequency 4oo i on the CI and LI, while the UI remains unaffected. *3 To that end the plate is mounted in a plastic ring (rotor) injection of compressed air between rotor and stator provides frictionless suspension; air jets blown obliquely on the rotor drive the rotation. Typical speeds are 2000 rpm for a useful diameter ~ 10 cm and 4000 rpm for 65 cm. *4 The rotation of the plate axes at frequency co0 causes a rotation of the polarization vector £ at frequency 2ooo, which results in modulations at frequency 4coo in the LI's since these are quadratic in the components oflL
i) f/"~ ~ J~-~outputplate ~"
b)
~ i nplat*p f/-D',. u t
I ~ . ~ V modulator ~)°utputG),plate
Fig. 2. The two improvements: a) addition of the output plate; b) addition of the input plate.
15 May 1981
Finally, at the output of our improved modulator (i.e. at the output of the output plate), the various intensities are in principle modulated at the following frequency combinations: UI: 0;
CI: 4oo i + co;
LI's: 4oo i + 200 -+ 4co 0
Between the CI and each of its partners, there is now
a twofoM distinction: the CI is distinguished from the UI by both the input plate and the initial modulator; and it is distinguished from the LI's by both the initial modulator and the output plate. As we already stated, real modulators and plates are imperfect at a typical level of 10 - 2 , and that was the reason why the CI was affected by spurious modulations at that level with the initial modulator. The same argument shows that the spurious modulations of the CI in the case of the improved modulator will be at a typical level of ( I 0 - 2 ) 2 = 10 - 4 , i.e. 100 times smaller. This is actually what we observe experimentally. However, a detailed analysis shows that higher order combinations of imperfections o f the input and output plate themselves fix a limit to the quality of the final triple modulator independently o f the quality o f the initial modulator: this means that adding real (imperfect) input and output plates to an ideal (perfect) initial modulator would spoil the results. In other terms, for the triple modulator to be noticeably better than the initial one, the quality o f the input and output plate should noticeably exceed some minimum level determined by the quality of the initial modulator. Our experimental results indicate that this condition is well fulfilled with usual commercial modulators and plates.
4. Application to the measurement of the birefringence and circular dichroism of multidielectric mirrors Applied to mirrors under quasi-normal incidence, the concepts of birefringence and circular dichroism may sound strange, since the polarizations of the incident and reflected beams are expected to be identical: indeed, for ideally isotropic coatings, these quantities vanish. The results presented below show that real mirrors, that were yet excellent in other respects, possess an easily measurable birefringence; on the other hand, no circular dichroism was observable at our level of sensitivity. 231
Volume 37, number 4
OPTICS COMMUNICATIONS
Fig. 3. Simplified scheme of the multipass. The number of forward-backward passes (2 on this figure) can reach about 70 in our real case. The system that we studied was the multipass system used in our parity violation experiment [1]. Under multipass system, we understand a pair of spherical mirrors facing each other, one of which is provided with a small hole at its center (fig. 3). A laser beam comes in through the hole, slightly off axis (roughly 1 degree), then performs between the mirrors a number of forward-backward passes that can easily be as many as about 70, and finally escapes back through the hole. The output direction is symmetrical of the input one, just like in the reflection on a single mirror. With a c.w. laser, the interest of the multipass is that the high number of passes increases the effective available power; with a pulsed laser, the long folded path in the multipass is useful as an optical delay line }s By means of our improved modulator, we label the C! of the entering beam with a very specific modulation (cf. sect. 3). By lock-in detection, we measure the fraction of this modulation that is transferred onto the UI (resp. LI's) of the returned beam, which yields the circular dichroism (resp. birefringence) of the multipass plus contributions of the other optical components (see sect. 2). The distinction between the contribution of the multipass and that of the other components cannot be obtained by removing the multipass: for in that case the beam would proceed forwards instead of returning backwards; we would then have to move the prism and detector and place them on the new path of the beam, which would sufficiently change the value of their contribution to make the result o f the difference meaningless. Instead of this, we use the following two methods: a) Leaving the drilled mirror fixed and the output direction constant but changing the distance of the second mirror, we can vary the number of passes, i.e. vary proportionally the effect of the multipass without changing that of the other components. The measured modulation amplitude can then be plotted versus the *s A theory of the multipass system will be found in ref. [2]. 232
15 May 1981
I
Jq-
/
Fig. 4. Measured birefringence as a function of the nunther of forward-backward passes: experimental points and least square straight line. number N of forward-backward passes (which is known by counting the laser spots on the mirrors). As an example, fig. 4 shows experimental data for tile optical phase shift (birefringence angle) ¢ obtained when the principal axes of the mirrors are parallel, so that their effects add. The observed variation is a linear function ¢ = aN + b. The slope a is interpreted as the effect of one forward-backward pass: in the case of circular dichroism measurements, it is the difference between the circular dichroisms of the two mirrors; in the case of birefringence measurements, it is the sum of the two phase shifts corresponding to one reflection on each mirror. The quantity b is the contribution of the other optical components *6 b) In the case of birefringence, a second method consists in changing the angular position 0 of one of the mirrors about its normal: this changes the directions of the principal axes of this mirror. The two modulation amplitudes corresponding to the xy LI and bb' LI are then plotted versus 0. In the simplest case where the phase shift and principal directions are the same at every point of each mirror (uniform birefringence), the expected variations correspond to a pair of sine functions of the variable 20, with equal amplitudes and a phase difference of ~r/2. Their cornmon amplitude yields the birefringence angle of the rotated mirror; their phase yields the principal directions of this same mirror; their dc values yield the birefringence angle and principal directions of the fixed mirror. Fig. 5 shows an example of experimental data, obtained with N = 23 forward-backward passes.
~6 Since tile considered polarization changes are all very small, they can be treated to first order: then the effects of successive reflections simply add, whatever the principal direc.. tions on ttle two mirrors be.
Volume 37, number 4
OPTICS COMMUNICATIONS
'°'b\'%/''° I 'ZY \ 1 °
'.'ofY M '°-'1-/
\
/
\
.
Fig. 5. xy and bb' modulation depths versus polar angle of the mirror around its normal: experimental points and least square fit. (N = 23 forward-backward passes).
5. Results and discussion We measured several pairs of mirrors, purchased from two different sources (Matra, France; Spectra Physics, USA). All of them were excellent high reflectivity mirrors (R ~ 0.997 to 0.998 at the operating wavelength X = 540 nm). a) Concerning the birefringence, the behaviour of all these mirrors appears to be very well represented under the simple assumption of uniform birefringence, as is shown e.g. by the good agreement between fitted theoretical curves and experimental points of figs. 4 and 5. Typical measured birefringence values are 3 × 10 4 to 6 X 10 - 4 radian for the Spectra Physics
15 May 1981
mirrors, and a few times 10 3 radian for the Matra ones. This birefringence is not caused by mechanical stresses, for these have been carefully avoided; in addition, the results are unchanged when stresses are deliberately applied to the mirrors. b) In the case of circular dichroism (CD), our method does not yield the separate CD's of the two mirrors, but only the resulting CD of the pair, i.e. the difference between the two individual CD's. Thus if we study p mirrors, we can only measure p - 1 independent differences, but in principle we can never know the p values themselves. However, our experimental results, obtained from these two different types of mirrors, show that all those differences are zero to an accuracy of 10 6. It seems very unlikely that all these mirrors may have exactly the same large CD; the only reasonable conclusion is that for all of them the circular dichroism is smaller than 10 --6. Finally, to come back to the initial motivation of this work, the parity violation experiment, we arrive to the conclusion that at the present level, the birefringence of a multidielectric mirror is a non-negligible source of systematic error. We are presently testing methods to solve this difficulty.
References [1] M.A. Bouchiat and L. Pottier: Physics Lett. 62B (1976) 327, and ref. quoted there; Proc. Intern. Workshop on Neutral current interactions in atoms, ed. W.L. Williams, University of Michigan, Ann Arbor, Mich., USA (1980). [2] D. Herriott, H. Kogelnik and R. Kompfner, Appl. Optics 3 (1964) 523.
233
OPTICS COMMUNICATIONS
15 May 1981
A HIGH-PURITY CIRCULAR POLARIZATION MODULATOR: APPLICATION TO BIREFRINGENCE AND CIRCULAR DICHROISM MEASUREMENTS ON MULTIDIELECTRIC MIRRORS M.A. BOUCHIAT and L. POTTIER Laboratoire de Spectroscopie Hertzienne de L 'E.N.S., 75231 Paris Cedex 05, France Received 27 January 1981
We present a method for producing a modulated circular polarization of high purity, i.e. whose modulation frequency allows specific discrimination against finear polarization and unpolarized intensity. We apply it to measure the circular dichroism and birefringence of high reflectivity multidielectric mirrors: no circular dichroism is observed at a level of 10-6, but birefringence values from 3 × 10-4 to 2 × 10-3 radian per reflexion under normal incidence are obtained.
1. Motivation This work originates in an experiment designed to search the parity violation predicted to exist in atoms by the recent theories of weak neutral currents [ 1]. In this experiment, the effect under search can be simulated by a stray birefringence or circular dichroism of 10-4: it is therefore necessary to control these quantities to a level of about 10 - 5 . Since the optical parts we need to control are mainly light reflecting parts (i.e. mirrors), the methods used in usual dichrometers for transparent samples do not apply without modifications and we had to imagine new procedures.
2. Definition of birefringence and circular dichroism; principle of their measurement The intensity and polarization of a light beam can be represented by a two-dimensional complex polarization vector ¢, or equivalently by the four real quantities le21 + t41, l e 2 1 - tey2l, 2 Re e*ey and 2 Im e*ey, where x and y are coordinate axes orthogonal to the beam. For a better understanding of what will follow, let us first point out the physical interpretation of these four quantities: i) lex21 + le21 is the unpolarized intensity (UI); it can be measured by simply receiving the beam onto a detector.
ii) lex21 - le21 is the x y linear polarized intensity (xy LI); it can be measured by inserting an analyzer (e.g. a prism) before the detector, and taking the difference of the values measured with the analyzer oriented along Ox and Oy. iii) 2 Re exey * is the bb ' linearly polarized intensity (bb' LI); here b and b' denote the first and second bisectrices of the x y axes. The bb' LI is measured by taking the difference of the values obtained with the analyzer oriented along Ob and Ob'. iv) 2 Im e*ey is the circularly polarized intensity (CI). It can be measured by inserting a quarterwave plate before the prism and taking the difference of the two values obtained with the prism oriented along the two bisectrices of the principal axes of the plate. If the beam passes through some linear optical system, the four intensities corresponding to the transmitted beam will be linear combinations o f those o f the incident beam, with coefficients which of course reflect the properties of the optical system. If we measure one of the four intensities of the transmitted beam as explained above, and if by some technical trick (which will be actually the central topic of the next section) we modulate one and only one - of the four intensities of the incident beam, then by synchronous detection we can obtain directly this particular coefficient which couples the modulated quantity for the incident beam to the measured quantity for the transmitted beam. In particular, if we modulate the incident CI and detect the transmitted UI, then we
0 0 3 0 - 4 0 1 8 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
229
Volume 37, number 4
OPTICS COMMUNICATIONS
15 May 1981
measure the circular dichroisrn of the optical system; and if we modulate the incident CI and detect the transmitted xy (or bb') LI we measure the xy (or bb')
g
birefringence. The limitations of this method with respect to measuring very small birefringences or circular dichroisms are twofold: i) First, real modulators designed for modulating one of the four incident intensities actually produce spurious modulations of the three other ones as well, more or less at the level o f 10 2. In the next section of this paper, we will show how to improve the purity of a CI modulator so as to make it about two orders of magnitude better. ii) Second, real modulators generally produce a small spurious modulation of the direction of the transmitted beam. Therefore the positions of the points where the beam meets subsequent optical parts and the detector, are also modulated. Because the transparency of the optical parts and the sensitivity of the detector surface are always inhomogeneous to some extent, this generates a spurious modulation in the detector's signal. Moreover, the amplitude and phase of this spurious modulation critically depend on the positions o f the optical elements on the beam: therefore it becomes in practice impossible to master and highly non reproducible as soon as the optical parts are moved. iii) Then, the detected modulation contains not only the contribution of the optical system of interest, but also that of the other optical components of the experimental arrangement: windows of the modulator; surface of the detector; e t c . . These extra contributions are usually eliminated by taking the difference between two measurements performed with and without tire optical system of interest, which implicitly assumes that the path of the beam remains unchanged when the system is removed; this approximation is reasonable for a transparent optical element at normal incidence; however, in the case of mirrors (which reflect the beam backwards), specific methods have to be imagined; we shall present two in the last section.
Fig. 1. Usual circular polarization modulation: a) a Pockels cell; b) a rotating quarterwave plate. of an electrooptical or photoelastic element driven by a sine voltage of frequency co, with axes at 45 ° of the incident linear polarization (fig. la). An older system consists of a simple quarterwave plate mechanically rotated at frequency c o / 2 , 1 (fig. lb). We shall choose this latter modulator as our example because it leads to simpler modulations; however the following discussion is general and its extension to other modulators is straightforward. We consider a modulator that is intended to modulate the CI and only the CI at frequency co. For instance, in the case of our example modulator (fig. 1b), the output intensities are:
Idl + 141-- l, le21
To date's polarization modulators usually consist 230
= 1 (1 + cos 2cot),
2 Re exey* = ~- sin 2cot, 2 I r a exey * = sin cot, so that frequency co is in principle a specific label of the CI. In practice, however, if tbe beam transmitted by this modulator is received onto a detector, either directly or through a prism, by lock-in detection with sin cot as a reference, one will observe spurious sin cot modulations of the UI as well as LI's, with a typical order of magnitude of 10 2 ~2. Thus at the percent level the CI is no longer labeled specifically. +]
+2
3. Generation of high-purity modulated circular polarization
~2
Since a half-turn rotation of the plate restores the directions of the principal axes, the fundamental frequency of the optical modulation is co when the rotation frequency is 00/2. Actually, we have tried four modulators: a rotating 5M4 quartz plate (Fichou, France); a rotating X/4 Polaroid sheet; a transverse field Pockels modulator (Lasermetrics 3030 HFW); and a longitudinal field Pockels modulator (Lasermetrics 1080 P). The spurious modulations were different, but their order of magnitude was 10 -2 in the four cases.
Volume 37, number 4
OPTICS COMMUNICATIONS
The improvements which we propose are based on this simple property: a half-wave plate changes the
polarization vector into its symmetric with respect to the plate axes. We shall take advantage of this property in two steps: in the first step, we shall remove the spurious LI modulations; in the second step, the spurious UI ones: a) First, after the modulator we insert a half-wave plate (hereafter called output plate) which we mechanically rotate in its plane +3 at some frequency coO v~ co (fig. 2a). In principle, this plate will have no effect on the UI of the beam; the CI will undergo a simple sign reversal; on the other hand, the two LI's will be blurred out. More precisely, the LI modulations initially present at frequency co will be transferred to different frequencies co -+ 46o 0 +4 and can then be easily distinguished. b) So far the sin cot modulation still contains a spurious contribution from the UI. To discriminate it we now insert before the initial modulator one more rotating half-wave plate, which we shall call the input plate (fig. 2b). Its rotation frequency coi must of course be different from co and coO" Because o f this rotating input plate, the polarization vector E at the input of the initial modulator now rotates at frequency 2co i, which induces an extra modulation at frequency 4oo i on the CI and LI, while the UI remains unaffected. *3 To that end the plate is mounted in a plastic ring (rotor) injection of compressed air between rotor and stator provides frictionless suspension; air jets blown obliquely on the rotor drive the rotation. Typical speeds are 2000 rpm for a useful diameter ~ 10 cm and 4000 rpm for 65 cm. *4 The rotation of the plate axes at frequency co0 causes a rotation of the polarization vector £ at frequency 2ooo, which results in modulations at frequency 4coo in the LI's since these are quadratic in the components oflL
i) f/"~ ~ J~-~outputplate ~"
b)
~ i nplat*p f/-D',. u t
I ~ . ~ V modulator ~)°utputG),plate
Fig. 2. The two improvements: a) addition of the output plate; b) addition of the input plate.
15 May 1981
Finally, at the output of our improved modulator (i.e. at the output of the output plate), the various intensities are in principle modulated at the following frequency combinations: UI: 0;
CI: 4oo i + co;
LI's: 4oo i + 200 -+ 4co 0
Between the CI and each of its partners, there is now
a twofoM distinction: the CI is distinguished from the UI by both the input plate and the initial modulator; and it is distinguished from the LI's by both the initial modulator and the output plate. As we already stated, real modulators and plates are imperfect at a typical level of 10 - 2 , and that was the reason why the CI was affected by spurious modulations at that level with the initial modulator. The same argument shows that the spurious modulations of the CI in the case of the improved modulator will be at a typical level of ( I 0 - 2 ) 2 = 10 - 4 , i.e. 100 times smaller. This is actually what we observe experimentally. However, a detailed analysis shows that higher order combinations of imperfections o f the input and output plate themselves fix a limit to the quality of the final triple modulator independently o f the quality o f the initial modulator: this means that adding real (imperfect) input and output plates to an ideal (perfect) initial modulator would spoil the results. In other terms, for the triple modulator to be noticeably better than the initial one, the quality o f the input and output plate should noticeably exceed some minimum level determined by the quality of the initial modulator. Our experimental results indicate that this condition is well fulfilled with usual commercial modulators and plates.
4. Application to the measurement of the birefringence and circular dichroism of multidielectric mirrors Applied to mirrors under quasi-normal incidence, the concepts of birefringence and circular dichroism may sound strange, since the polarizations of the incident and reflected beams are expected to be identical: indeed, for ideally isotropic coatings, these quantities vanish. The results presented below show that real mirrors, that were yet excellent in other respects, possess an easily measurable birefringence; on the other hand, no circular dichroism was observable at our level of sensitivity. 231
Volume 37, number 4
OPTICS COMMUNICATIONS
Fig. 3. Simplified scheme of the multipass. The number of forward-backward passes (2 on this figure) can reach about 70 in our real case. The system that we studied was the multipass system used in our parity violation experiment [1]. Under multipass system, we understand a pair of spherical mirrors facing each other, one of which is provided with a small hole at its center (fig. 3). A laser beam comes in through the hole, slightly off axis (roughly 1 degree), then performs between the mirrors a number of forward-backward passes that can easily be as many as about 70, and finally escapes back through the hole. The output direction is symmetrical of the input one, just like in the reflection on a single mirror. With a c.w. laser, the interest of the multipass is that the high number of passes increases the effective available power; with a pulsed laser, the long folded path in the multipass is useful as an optical delay line }s By means of our improved modulator, we label the C! of the entering beam with a very specific modulation (cf. sect. 3). By lock-in detection, we measure the fraction of this modulation that is transferred onto the UI (resp. LI's) of the returned beam, which yields the circular dichroism (resp. birefringence) of the multipass plus contributions of the other optical components (see sect. 2). The distinction between the contribution of the multipass and that of the other components cannot be obtained by removing the multipass: for in that case the beam would proceed forwards instead of returning backwards; we would then have to move the prism and detector and place them on the new path of the beam, which would sufficiently change the value of their contribution to make the result o f the difference meaningless. Instead of this, we use the following two methods: a) Leaving the drilled mirror fixed and the output direction constant but changing the distance of the second mirror, we can vary the number of passes, i.e. vary proportionally the effect of the multipass without changing that of the other components. The measured modulation amplitude can then be plotted versus the *s A theory of the multipass system will be found in ref. [2]. 232
15 May 1981
I
Jq-
/
Fig. 4. Measured birefringence as a function of the nunther of forward-backward passes: experimental points and least square straight line. number N of forward-backward passes (which is known by counting the laser spots on the mirrors). As an example, fig. 4 shows experimental data for tile optical phase shift (birefringence angle) ¢ obtained when the principal axes of the mirrors are parallel, so that their effects add. The observed variation is a linear function ¢ = aN + b. The slope a is interpreted as the effect of one forward-backward pass: in the case of circular dichroism measurements, it is the difference between the circular dichroisms of the two mirrors; in the case of birefringence measurements, it is the sum of the two phase shifts corresponding to one reflection on each mirror. The quantity b is the contribution of the other optical components *6 b) In the case of birefringence, a second method consists in changing the angular position 0 of one of the mirrors about its normal: this changes the directions of the principal axes of this mirror. The two modulation amplitudes corresponding to the xy LI and bb' LI are then plotted versus 0. In the simplest case where the phase shift and principal directions are the same at every point of each mirror (uniform birefringence), the expected variations correspond to a pair of sine functions of the variable 20, with equal amplitudes and a phase difference of ~r/2. Their cornmon amplitude yields the birefringence angle of the rotated mirror; their phase yields the principal directions of this same mirror; their dc values yield the birefringence angle and principal directions of the fixed mirror. Fig. 5 shows an example of experimental data, obtained with N = 23 forward-backward passes.
~6 Since tile considered polarization changes are all very small, they can be treated to first order: then the effects of successive reflections simply add, whatever the principal direc.. tions on ttle two mirrors be.
Volume 37, number 4
OPTICS COMMUNICATIONS
'°'b\'%/''° I 'ZY \ 1 °
'.'ofY M '°-'1-/
\
/
\
.
Fig. 5. xy and bb' modulation depths versus polar angle of the mirror around its normal: experimental points and least square fit. (N = 23 forward-backward passes).
5. Results and discussion We measured several pairs of mirrors, purchased from two different sources (Matra, France; Spectra Physics, USA). All of them were excellent high reflectivity mirrors (R ~ 0.997 to 0.998 at the operating wavelength X = 540 nm). a) Concerning the birefringence, the behaviour of all these mirrors appears to be very well represented under the simple assumption of uniform birefringence, as is shown e.g. by the good agreement between fitted theoretical curves and experimental points of figs. 4 and 5. Typical measured birefringence values are 3 × 10 4 to 6 X 10 - 4 radian for the Spectra Physics
15 May 1981
mirrors, and a few times 10 3 radian for the Matra ones. This birefringence is not caused by mechanical stresses, for these have been carefully avoided; in addition, the results are unchanged when stresses are deliberately applied to the mirrors. b) In the case of circular dichroism (CD), our method does not yield the separate CD's of the two mirrors, but only the resulting CD of the pair, i.e. the difference between the two individual CD's. Thus if we study p mirrors, we can only measure p - 1 independent differences, but in principle we can never know the p values themselves. However, our experimental results, obtained from these two different types of mirrors, show that all those differences are zero to an accuracy of 10 6. It seems very unlikely that all these mirrors may have exactly the same large CD; the only reasonable conclusion is that for all of them the circular dichroism is smaller than 10 --6. Finally, to come back to the initial motivation of this work, the parity violation experiment, we arrive to the conclusion that at the present level, the birefringence of a multidielectric mirror is a non-negligible source of systematic error. We are presently testing methods to solve this difficulty.
References [1] M.A. Bouchiat and L. Pottier: Physics Lett. 62B (1976) 327, and ref. quoted there; Proc. Intern. Workshop on Neutral current interactions in atoms, ed. W.L. Williams, University of Michigan, Ann Arbor, Mich., USA (1980). [2] D. Herriott, H. Kogelnik and R. Kompfner, Appl. Optics 3 (1964) 523.
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