- Email: [email protected]
Phase closure bias versus dispersion in a stellar interferometer
Optics Communications 216 (2003) 329–334 www.elsevier.com/locate/optcom
Phase closure bias versus dispersion in a stellar interferometer G. Huss *, E. Longueteau, L. Delage, F. Reynaud Equipe Optique, IRCOM (UMR 6615), 123 Avenue Albert Thomas, 87060 Limoges Cedex, France Received 2 July 2002; received in revised form 17 December 2002; accepted 31 December 2002
Abstract Phase closure is a well-known technique able to eliminate phase biases in a three-arm interferometer. This property is very useful to avoid the phase effect of the atmosphere in stellar interferometry. In this paper we theoretically investigate the effect of differential dispersion in a three-arm interferometer. We demonstrate that phase closure can be corrupted by differential dispersion between the interferometric arms. We begin by a general demonstration of such effect and propose a few examples where symmetry properties avoid phase closure errors. An analytical model for gaussian shaped spectrum and second order dispersion development is proposed. This analytical solution is illustrated by experimental results on a fibre version of a stellar interferometer. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Phase closure; Dispersion; All fibre interferometer
1. Introduction The very high-resolution imaging devices implemented or to be implemented in the next future are based on the concept of synthesised apertures [1]. For diluted arrays the available data consist of interference fringes contrasts and phases corresponding to the object spatial intensity spectrum at the sampled spatial frequencies. If the modulus of the object complex visibility is quite easy to measure, the phase acquisition suffers a large number of disturbances such as interferometer instabilities or atmospheric turbulence. A solution to solve the
*
Corresponding author. Tel.: +33-5-55-45-74-15; fax: +33-555-45-72-53. E-mail address: [email protected] (G. Huss).
correlative problems is to combine the phase information in order to compute a composite data insensitive to these effects. The phase closure is commonly used to remove the atmospheric phase piston effect by means of a linear combination of the measured fringes phases [2,3]. This method is efficient for a quasi-monochromatic application or in the frame of non-dispersive instruments. Dispersion effects in an air-path stellar interferometer have been thoroughly described by Tango [4]. Dyer and Christensen [5] analysed the dispersion effects on the fringe visibility in fibre optic interferometry. In the frame of an all guided interferometer developed in our laboratory [6,7], we have experimentally investigated the effect of differential dispersion on the phase closure acquisition in a three-beam interferometer. We begin this paper by a theoretical approach. The disruptive influence of
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01072-1
330
G. Huss et al. / Optics Communications 216 (2003) 329–334
this effect is proposed in a general case. A few examples with particular experimental situation are given to make the reader more confident on this topic. This general frame is followed by a more restrictive one but giving rise to an analytical solution. Using a gaussian spectral model and a second order description of the dispersion effect, we derive the phase closure bias equation. This effect is computed in order to quantify the resulting phase shift according to the amount of differential parabolic phase shifts in the interferometer. The input data have been obtained from the experimental configuration that we reported in [6]. The numerical simulations fit accurately the experimental results demonstrating the precision of our model. These general results can be extrapolated to any configuration where a stellar interferometer suffers differential dispersion and are not only restricted to guided optics interferometers.
2. General presentation of phase closure data corruption resulting from differential dispersion In a three-beam interferometer dedicated to high-resolution imaging, the emerging information consists of interferometric mixing of the three collected optical fields. These signals are characterised by their complex degrees of coherence giving quantitative information on the correlation between the mixed optical fields. The Zernike–Van Cittert theorem indicates that the object intensity distribution is the Fourier transform of the interference fringeÕs complex visibility. In the following we will assume the visibility function to be the product of a spatial and spectral independent terms. In a three-beam configuration, three complex numbers are accessible. Assuming the spectrum to be identical in the two arms i and j, the corresponding complex degree of coherence cij , at the output of the interferometer is given by: Z 1 cij ¼ IðmÞ exp½jUij ðmÞ dm; ð1Þ 0
where (Im) is the power spectral density of the source, m the frequency, and Uij ðmÞ is the total phase difference in the interferometer between arms i and j.
This phase shift can be split in to three main contributions: Uij ðmÞ ¼ bi Li bj Lj þ /objij þ atmsi atmsj ;
ð2Þ
where bi Li bj Lj is the phase dispersion difference between arms i and j with a bi propagation constant and a Li propagation length, atmsi atmsj is the differential phase shift resulting from the atmosphere or instrument instability, and /objij is the phase of intensity object spectrum at the spatial frequency observed by the telescopes pair i/j. In the following the two last terms will be considered as achromatic. Conversely the first term is frequency dependent and can be developed by means of a Taylor series around the spectral central frequency m0 : bi Li bj Lj ¼ ½bi ðm0 ÞLi bj ðm0 ÞLj obj obi Li Lj ðm m0 Þ þ om om " # o2 b j o2 bi ðm m0 Þ2 L L þ i j om2 om2 2 " # 3 o3 b j o3 bi ðm m0 Þ þ þ L L i j om3 om3 3! ð3Þ In such a context the complex degree of coherence cij can be expressed as Z 1 cij ¼ IðmÞ exp½ jðbi Li bj Lj Þ dt 0
expðj½/objij þ atmsi atmsj Þ:
ð4Þ
If the source is quasi-monochromatic or in the case of a non-dispersive interferometer, the phase closure is able to provide a phase information without atmospheric or instrument bias w ¼ Argðc12 Þ þ Argðc23 Þ þ Argðc31 Þ
ð5Þ
Arg is the argument of the complex number w ¼ b1 ðm0 ÞL1 b2 ðm0 ÞL2 þ /obj12 þ atms1 atms2 þ b2 ðm0 ÞL2 b3 ðm0 ÞL3 þ /obj23 þ atms2 atms3 þ b3 ðm0 ÞL3 b1 ðm0 ÞL1 þ /obj31 þ atms3 atms1 w ¼ /obj12 þ /obj23 þ /obj31 :
ð6Þ
G. Huss et al. / Optics Communications 216 (2003) 329–334
331
Conversely, if differential dispersion becomes to be R1 significant the 0 IðmÞ exp½jðbi Li bj Lj Þ terms have moduli that decrease and a phase error resulting from the wrapping of expbjðbi Li bj Lj Þc phasors. As long as this operation is non-linear, it results in a phase closure error Dw Z 1 Dw ¼ Arg IðmÞ exp½jðb1 L1 b2 L2 Þ dm 0 Z 1 þ Arg IðmÞ exp½jðb2 L2 b3 L3 Þ dm Z0 1 þ Arg IðmÞ exp½jðb3 L3 b1 L1 Þ dm: 0
ð7Þ We can notice that this problem is independent of the atmosphere and object phase terms. In the following we will only consider the dispersion terms for numerical simulations, analytical solutions or experimental results. This situation corresponds to a point-like source illuminating the stellar interferometer. Fig. 1 illustrates a general case with an asymmetric spectrum. This computation uses normalised quantities and the only relevant parameter is the phase variation of the phase difference over the spectral span for the three interferometric mixing. As soon as the modulus of the complex degrees of
Fig. 2. Example of an asymmetric spectrum and interferometric spectral phase shift with second and third order dispersion terms. In this case the interferometric arms 1 and 2 are dispersion compensated. In spite of a coherence function modulus decay of jc12 j ¼ 1, jc23 j ¼ 0:66, and jc31 j ¼ 0:66, the resulting phase closure bias is Dw ¼ 0.
coherence begins to decrease, the phasor wrapping begins to induce phase closure errors. It is important to notice that if the differential dispersion coefficients and the spectral distribution are stable, this error is steady and can be corrected. In such a way the interferometer can be calibrated. However few examples lead to a simplification or error vanishing. First, if two interferometric arms are dispersion balanced, one interferometric mixing does not suffer wrapping effect and the two other ones are perfectly symmetrical. This situation is illustrated in Fig. 2 and leads to a phase closure bias equal to 0. Second, if the spectrum has a symmetric shape, the symmetry properties of the complex degree of coherence cij lead to make the phase closure error disappear for all odd dispersion terms. Fig. 3 gives an example with a third order term.
3. Experimental illustration and numerical fitting of the phase closure error Fig. 1. Example of an asymmetric spectrum and interferometric spectral phase shift with second and third order dispersion terms. In this case the phase closure bias is Dw ¼ 0:224 rad and the moduli jc12 j ¼ 0:88; jc23 j ¼ 0:88, and jc31 j ¼ 0:67.
The experiment is based on a three-arm fibre interferometer operating at 670 nm mean wavelength previously published in [8]. A schematic drawing of the experimental set-up is shown in
332
G. Huss et al. / Optics Communications 216 (2003) 329–334
Fig. 3. Example of a symmetric spectrum and interferometric spectral phase shift with third order dispersion terms. In spite of a coherence function modulus decay of jc12 j ¼ 0:81, jc23 j ¼ 0:81, and jc31 j ¼ 0:66, the resulting phase closure bias is Dw ¼ 0.
Fig. 4. The interferometer is implemented with polarisation maintaining fibre. A laser diode is launched in a polarisation maintaining fibre. This
point-like source illuminates the interferometer input pupils. In this case the object spectrum has a 0 phase over the spectral frequencies. The source is driven under the laser threshold in order to obtain a 10 nm spectral bandwidth. Three microscope objectives T1 , T2 , and T3 act as telescopes. T1 and T3 are placed on translation stages in order to be moved. The recombination device is made of polarisation maintaining couplers connected by splicing fusion or FC/PC connectors. Moving the telescopes T1 and T3 generates an optical path variation, so called piston. For each telescope configuration, the group delay equalisation is achieved by actuating fibre delay lines [9], and the contrasts and phase closure terms are acquired. The fibre delay lines being based on a stretching process, their actuations lead to a variation of the differential dispersions in the interferometer. Therefore for various positions of the telescopes the differential dispersion varies [10]. The higher the longitudinal spacing between the telescopes,
Fig. 4. Experimental set-up.
G. Huss et al. / Optics Communications 216 (2003) 329–334
333
The phase shift Uij ðmÞ due to the differential dispersion effects can be expressed as: 2
ðm m0 Þ Uij ðmÞ ¼ Tij þ Uij ðm0 Þ; 2 " # o2 b j o2 bi Tij ¼ Li 2 Lj : om2 om
Fig. 5. Experimental (points) and simulated (shaded area) phase closure term versus the air path difference (cm) between telescopes T1 =T2 and T2 =T3 .
the larger the differential effects of dispersion. The acquisition device achieves an averaging over 20 measurements for the fringe contrasts and phase closure. The experimental phase closure term is represented in Fig. 5 as a function of the relative position of the three telescopes, i.e., versus the induced variable dispersion. We restrict our study to the second order term of dispersion, considering the fact that for a narrow spectral bandwidth around a 680 nm mean wavelength, the effects of the higher terms are negligible. The linear term versus m frequency represents the group delay between the two interferometric arms and can be cancelled by adjusting the air delay line position. The second order term is satisfactory to accurately characterise the differential effects of dispersion leading to contrast decay in the interferometer. This term can be calibrated as previously demonstrated in the frame of a fibre interferometer [6]. In such context the phase closure bias Dw can be derived assuming the spectrum IðmÞ to be gaussian shaped with a m0 ¼ 441 THz central frequency corresponding to a 670 nm mean wavelength: "
2 # 2ðm m0 Þ IðmÞ ¼ A exp ln 2 ; ð8Þ Dm where Dm denotes the spectral bandwidth at halfmaximum of spectral power density.
ð9Þ
At the 0 group delay corresponding to the cancellation of the first order term, Tij represents the variable term of differential dispersion. The computation of the phase shift introduced by the differential effects of dispersion is achieved by making the Tij term vary. The detection of the interference fringes by a monopixel detector leads to a signal resulting from the superposition of all the frequency contributions. For each telescope pair configuration Ti =Tj we get Z cij ¼ IðmÞ exp½jUij ðmÞom m "
2 # Z þ1 2ðm m0 Þ exp ln 2 ¼ Dm 1
exp½jUij ðmÞ dm ¼
exp½jUij ðm0 Þ 1=2 : 2 2 T þ j 2pij lnp2 Dm
ð10Þ This is the complex number cij that can be decomposed as follows: cij ¼
1 4
T 2 1=4 þ 4pij2
Tij Dm2 2
exp j arctg þ Uij ðm0 Þ ; 8 ln 2 ðln 2Þ2 p2
Tij ¼
Di
2 Dm
2pC 2pC Li þ Dj 2 Lj 2 m m
ð11Þ
ð12Þ
with D the dispersion coefficient of each fibre arm. Using the three complex degrees of coherence, the phase closure bias Dw can be expressed as
T12 Dm2 T23 Dm2 Dw ¼ arctg þ arctg 8 ln 2 8 ln 2
2 T31 Dm þ arctg : ð13Þ 8 ln 2
334
G. Huss et al. / Optics Communications 216 (2003) 329–334
Fig. 5 shows the experimental and theoretical phase closure for a variable phase dispersion. Two fibre arms are stretched, generating a variable dispersion as a function of the delay line position. This variation is devoted to simulate the correction of air-path variation in a spatial interferometer. The shaded area plots this theoretical phase closure evolution versus dispersion effects assuming our analytical model. We can notice that the experimental points match the theoretical predictions very accurately. The standard deviation between experiment and theory is 0.03 rad averaging over 20 acquisitions. Assuming an initial gaussian spectrum, the modelling of the phase closure term evolution versus the air path compensated by the fibre delay line gives a result as a function of only three parameters: the dispersion value of the fibre, the differential geometrical length of the fibre arms 1 and 2 and the spectral bandwidth Dm. Consequently the evolution of the phase closure is directly associated to the differential dispersion variation that can be calibrated in a fibre interferometer [8]. Therefore, the phase closure term evolution can be calibrated in an all fibre interferometer or in any dispersive interferometer. The final relationship (13) allows us to predict the phase closure default versus the air path differences for a given spectral bandwidth. In the frame of our experiment, we have operated with a 10 nm spectral bandwidth so that phase closure computation is consistent with the experimental results, even when the chromatic dispersion description is restricted to a second order. In order to extrapolate these computations to a larger spectral bandwidth, it may be necessary to take into account the higher order terms of the dispersion. Notice that in such a context, no analytical solution is available and only the numerical simulation can predict the phase closure error. 4. Conclusion This theoretical and experimental work demonstrates that dispersion effects have to be taken into account for an accurate evaluation of acqui-
sition biases. Differential dispersion leads to a corruption of the phase closure measurement as soon as fringe contrasts begin to decrease. The experimental demonstration has been achieved with an all guided interferometers using fibre delay lines developed in our laboratory [10]. In this experimental framework, the variable dispersion results from the fibre stretching. The theoretical model being validated, these effects could be extrapolated to larger delay line strokes. In the case of narrow spectral bandwidth corresponding, for instance, to a spectroscopic analysis of the interferometric fringes, the error modelling can be achieved by only using a second order dispersion term. If each spectral channel is assumed to be gaussian shaped, the phase closure error evolution versus the dispersion has a simple analytical expression depending on three parameters: the dispersion value of the fibre, the position of the telescopes, and the spectral bandwidth. This work could be extended to other spectral distributions (new wavelength, new spectral bandwidth) but it would make necessary an accurate study of the higher terms of dispersion. Consequently the phase closure error is not incompatible with image reconstructions as long as they can be calibrated. Of course if the differential effects of dispersion become too large, interference fringe contrasts could be reduced leading to a high degradation of the signal to noise ratio for the image restoration. References [1] A. Glindemann et al., in: P.J. Lena, A. Quirrenbach (Eds.), Interferometry in Optical Astronomy, Proc. SPIE, 4006, 2000, p. 2. [2] D.H. Rogstag, Appl. Opt. 7 (1968) 585. [3] A.H. Greenaway, Opt. Commun. 42 (1982) 157. [4] W.J. Tango, Appl. Opt. 29 (1990) 516. [5] S.D. Dyer, D.A. Christensen, Opt. Eng. 36 (9) (1997) 2440. [6] G. Huss, L.M. Simohamed, F. Reynaud, Opt. Commun. 182 (2001) 71. [7] L. Delage, F. Reynaud, A. Lannes, Appl. Opt. 39 (34) (2000) 6406. [8] G. Huss, F. Reynaud, L. Delage, Opt. Commun. (2001). [9] L.M. Simohamed, F. Reynaud, Pure Appl. Opt. 6 (1997) 37. [10] L.M. Simohamed, F. Reynaud, Opt. Commun. 159 (1999) 118.
Phase closure bias versus dispersion in a stellar interferometer G. Huss *, E. Longueteau, L. Delage, F. Reynaud Equipe Optique, IRCOM (UMR 6615), 123 Avenue Albert Thomas, 87060 Limoges Cedex, France Received 2 July 2002; received in revised form 17 December 2002; accepted 31 December 2002
Abstract Phase closure is a well-known technique able to eliminate phase biases in a three-arm interferometer. This property is very useful to avoid the phase effect of the atmosphere in stellar interferometry. In this paper we theoretically investigate the effect of differential dispersion in a three-arm interferometer. We demonstrate that phase closure can be corrupted by differential dispersion between the interferometric arms. We begin by a general demonstration of such effect and propose a few examples where symmetry properties avoid phase closure errors. An analytical model for gaussian shaped spectrum and second order dispersion development is proposed. This analytical solution is illustrated by experimental results on a fibre version of a stellar interferometer. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Phase closure; Dispersion; All fibre interferometer
1. Introduction The very high-resolution imaging devices implemented or to be implemented in the next future are based on the concept of synthesised apertures [1]. For diluted arrays the available data consist of interference fringes contrasts and phases corresponding to the object spatial intensity spectrum at the sampled spatial frequencies. If the modulus of the object complex visibility is quite easy to measure, the phase acquisition suffers a large number of disturbances such as interferometer instabilities or atmospheric turbulence. A solution to solve the
*
Corresponding author. Tel.: +33-5-55-45-74-15; fax: +33-555-45-72-53. E-mail address: [email protected] (G. Huss).
correlative problems is to combine the phase information in order to compute a composite data insensitive to these effects. The phase closure is commonly used to remove the atmospheric phase piston effect by means of a linear combination of the measured fringes phases [2,3]. This method is efficient for a quasi-monochromatic application or in the frame of non-dispersive instruments. Dispersion effects in an air-path stellar interferometer have been thoroughly described by Tango [4]. Dyer and Christensen [5] analysed the dispersion effects on the fringe visibility in fibre optic interferometry. In the frame of an all guided interferometer developed in our laboratory [6,7], we have experimentally investigated the effect of differential dispersion on the phase closure acquisition in a three-beam interferometer. We begin this paper by a theoretical approach. The disruptive influence of
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01072-1
330
G. Huss et al. / Optics Communications 216 (2003) 329–334
this effect is proposed in a general case. A few examples with particular experimental situation are given to make the reader more confident on this topic. This general frame is followed by a more restrictive one but giving rise to an analytical solution. Using a gaussian spectral model and a second order description of the dispersion effect, we derive the phase closure bias equation. This effect is computed in order to quantify the resulting phase shift according to the amount of differential parabolic phase shifts in the interferometer. The input data have been obtained from the experimental configuration that we reported in [6]. The numerical simulations fit accurately the experimental results demonstrating the precision of our model. These general results can be extrapolated to any configuration where a stellar interferometer suffers differential dispersion and are not only restricted to guided optics interferometers.
2. General presentation of phase closure data corruption resulting from differential dispersion In a three-beam interferometer dedicated to high-resolution imaging, the emerging information consists of interferometric mixing of the three collected optical fields. These signals are characterised by their complex degrees of coherence giving quantitative information on the correlation between the mixed optical fields. The Zernike–Van Cittert theorem indicates that the object intensity distribution is the Fourier transform of the interference fringeÕs complex visibility. In the following we will assume the visibility function to be the product of a spatial and spectral independent terms. In a three-beam configuration, three complex numbers are accessible. Assuming the spectrum to be identical in the two arms i and j, the corresponding complex degree of coherence cij , at the output of the interferometer is given by: Z 1 cij ¼ IðmÞ exp½jUij ðmÞ dm; ð1Þ 0
where (Im) is the power spectral density of the source, m the frequency, and Uij ðmÞ is the total phase difference in the interferometer between arms i and j.
This phase shift can be split in to three main contributions: Uij ðmÞ ¼ bi Li bj Lj þ /objij þ atmsi atmsj ;
ð2Þ
where bi Li bj Lj is the phase dispersion difference between arms i and j with a bi propagation constant and a Li propagation length, atmsi atmsj is the differential phase shift resulting from the atmosphere or instrument instability, and /objij is the phase of intensity object spectrum at the spatial frequency observed by the telescopes pair i/j. In the following the two last terms will be considered as achromatic. Conversely the first term is frequency dependent and can be developed by means of a Taylor series around the spectral central frequency m0 : bi Li bj Lj ¼ ½bi ðm0 ÞLi bj ðm0 ÞLj obj obi Li Lj ðm m0 Þ þ om om " # o2 b j o2 bi ðm m0 Þ2 L L þ i j om2 om2 2 " # 3 o3 b j o3 bi ðm m0 Þ þ þ L L i j om3 om3 3! ð3Þ In such a context the complex degree of coherence cij can be expressed as Z 1 cij ¼ IðmÞ exp½ jðbi Li bj Lj Þ dt 0
expðj½/objij þ atmsi atmsj Þ:
ð4Þ
If the source is quasi-monochromatic or in the case of a non-dispersive interferometer, the phase closure is able to provide a phase information without atmospheric or instrument bias w ¼ Argðc12 Þ þ Argðc23 Þ þ Argðc31 Þ
ð5Þ
Arg is the argument of the complex number w ¼ b1 ðm0 ÞL1 b2 ðm0 ÞL2 þ /obj12 þ atms1 atms2 þ b2 ðm0 ÞL2 b3 ðm0 ÞL3 þ /obj23 þ atms2 atms3 þ b3 ðm0 ÞL3 b1 ðm0 ÞL1 þ /obj31 þ atms3 atms1 w ¼ /obj12 þ /obj23 þ /obj31 :
ð6Þ
G. Huss et al. / Optics Communications 216 (2003) 329–334
331
Conversely, if differential dispersion becomes to be R1 significant the 0 IðmÞ exp½jðbi Li bj Lj Þ terms have moduli that decrease and a phase error resulting from the wrapping of expbjðbi Li bj Lj Þc phasors. As long as this operation is non-linear, it results in a phase closure error Dw Z 1 Dw ¼ Arg IðmÞ exp½jðb1 L1 b2 L2 Þ dm 0 Z 1 þ Arg IðmÞ exp½jðb2 L2 b3 L3 Þ dm Z0 1 þ Arg IðmÞ exp½jðb3 L3 b1 L1 Þ dm: 0
ð7Þ We can notice that this problem is independent of the atmosphere and object phase terms. In the following we will only consider the dispersion terms for numerical simulations, analytical solutions or experimental results. This situation corresponds to a point-like source illuminating the stellar interferometer. Fig. 1 illustrates a general case with an asymmetric spectrum. This computation uses normalised quantities and the only relevant parameter is the phase variation of the phase difference over the spectral span for the three interferometric mixing. As soon as the modulus of the complex degrees of
Fig. 2. Example of an asymmetric spectrum and interferometric spectral phase shift with second and third order dispersion terms. In this case the interferometric arms 1 and 2 are dispersion compensated. In spite of a coherence function modulus decay of jc12 j ¼ 1, jc23 j ¼ 0:66, and jc31 j ¼ 0:66, the resulting phase closure bias is Dw ¼ 0.
coherence begins to decrease, the phasor wrapping begins to induce phase closure errors. It is important to notice that if the differential dispersion coefficients and the spectral distribution are stable, this error is steady and can be corrected. In such a way the interferometer can be calibrated. However few examples lead to a simplification or error vanishing. First, if two interferometric arms are dispersion balanced, one interferometric mixing does not suffer wrapping effect and the two other ones are perfectly symmetrical. This situation is illustrated in Fig. 2 and leads to a phase closure bias equal to 0. Second, if the spectrum has a symmetric shape, the symmetry properties of the complex degree of coherence cij lead to make the phase closure error disappear for all odd dispersion terms. Fig. 3 gives an example with a third order term.
3. Experimental illustration and numerical fitting of the phase closure error Fig. 1. Example of an asymmetric spectrum and interferometric spectral phase shift with second and third order dispersion terms. In this case the phase closure bias is Dw ¼ 0:224 rad and the moduli jc12 j ¼ 0:88; jc23 j ¼ 0:88, and jc31 j ¼ 0:67.
The experiment is based on a three-arm fibre interferometer operating at 670 nm mean wavelength previously published in [8]. A schematic drawing of the experimental set-up is shown in
332
G. Huss et al. / Optics Communications 216 (2003) 329–334
Fig. 3. Example of a symmetric spectrum and interferometric spectral phase shift with third order dispersion terms. In spite of a coherence function modulus decay of jc12 j ¼ 0:81, jc23 j ¼ 0:81, and jc31 j ¼ 0:66, the resulting phase closure bias is Dw ¼ 0.
Fig. 4. The interferometer is implemented with polarisation maintaining fibre. A laser diode is launched in a polarisation maintaining fibre. This
point-like source illuminates the interferometer input pupils. In this case the object spectrum has a 0 phase over the spectral frequencies. The source is driven under the laser threshold in order to obtain a 10 nm spectral bandwidth. Three microscope objectives T1 , T2 , and T3 act as telescopes. T1 and T3 are placed on translation stages in order to be moved. The recombination device is made of polarisation maintaining couplers connected by splicing fusion or FC/PC connectors. Moving the telescopes T1 and T3 generates an optical path variation, so called piston. For each telescope configuration, the group delay equalisation is achieved by actuating fibre delay lines [9], and the contrasts and phase closure terms are acquired. The fibre delay lines being based on a stretching process, their actuations lead to a variation of the differential dispersions in the interferometer. Therefore for various positions of the telescopes the differential dispersion varies [10]. The higher the longitudinal spacing between the telescopes,
Fig. 4. Experimental set-up.
G. Huss et al. / Optics Communications 216 (2003) 329–334
333
The phase shift Uij ðmÞ due to the differential dispersion effects can be expressed as: 2
ðm m0 Þ Uij ðmÞ ¼ Tij þ Uij ðm0 Þ; 2 " # o2 b j o2 bi Tij ¼ Li 2 Lj : om2 om
Fig. 5. Experimental (points) and simulated (shaded area) phase closure term versus the air path difference (cm) between telescopes T1 =T2 and T2 =T3 .
the larger the differential effects of dispersion. The acquisition device achieves an averaging over 20 measurements for the fringe contrasts and phase closure. The experimental phase closure term is represented in Fig. 5 as a function of the relative position of the three telescopes, i.e., versus the induced variable dispersion. We restrict our study to the second order term of dispersion, considering the fact that for a narrow spectral bandwidth around a 680 nm mean wavelength, the effects of the higher terms are negligible. The linear term versus m frequency represents the group delay between the two interferometric arms and can be cancelled by adjusting the air delay line position. The second order term is satisfactory to accurately characterise the differential effects of dispersion leading to contrast decay in the interferometer. This term can be calibrated as previously demonstrated in the frame of a fibre interferometer [6]. In such context the phase closure bias Dw can be derived assuming the spectrum IðmÞ to be gaussian shaped with a m0 ¼ 441 THz central frequency corresponding to a 670 nm mean wavelength: "
2 # 2ðm m0 Þ IðmÞ ¼ A exp ln 2 ; ð8Þ Dm where Dm denotes the spectral bandwidth at halfmaximum of spectral power density.
ð9Þ
At the 0 group delay corresponding to the cancellation of the first order term, Tij represents the variable term of differential dispersion. The computation of the phase shift introduced by the differential effects of dispersion is achieved by making the Tij term vary. The detection of the interference fringes by a monopixel detector leads to a signal resulting from the superposition of all the frequency contributions. For each telescope pair configuration Ti =Tj we get Z cij ¼ IðmÞ exp½jUij ðmÞom m "
2 # Z þ1 2ðm m0 Þ exp ln 2 ¼ Dm 1
exp½jUij ðmÞ dm ¼
exp½jUij ðm0 Þ 1=2 : 2 2 T þ j 2pij lnp2 Dm
ð10Þ This is the complex number cij that can be decomposed as follows: cij ¼
1 4
T 2 1=4 þ 4pij2
Tij Dm2 2
exp j arctg þ Uij ðm0 Þ ; 8 ln 2 ðln 2Þ2 p2
Tij ¼
Di
2 Dm
2pC 2pC Li þ Dj 2 Lj 2 m m
ð11Þ
ð12Þ
with D the dispersion coefficient of each fibre arm. Using the three complex degrees of coherence, the phase closure bias Dw can be expressed as
T12 Dm2 T23 Dm2 Dw ¼ arctg þ arctg 8 ln 2 8 ln 2
2 T31 Dm þ arctg : ð13Þ 8 ln 2
334
G. Huss et al. / Optics Communications 216 (2003) 329–334
Fig. 5 shows the experimental and theoretical phase closure for a variable phase dispersion. Two fibre arms are stretched, generating a variable dispersion as a function of the delay line position. This variation is devoted to simulate the correction of air-path variation in a spatial interferometer. The shaded area plots this theoretical phase closure evolution versus dispersion effects assuming our analytical model. We can notice that the experimental points match the theoretical predictions very accurately. The standard deviation between experiment and theory is 0.03 rad averaging over 20 acquisitions. Assuming an initial gaussian spectrum, the modelling of the phase closure term evolution versus the air path compensated by the fibre delay line gives a result as a function of only three parameters: the dispersion value of the fibre, the differential geometrical length of the fibre arms 1 and 2 and the spectral bandwidth Dm. Consequently the evolution of the phase closure is directly associated to the differential dispersion variation that can be calibrated in a fibre interferometer [8]. Therefore, the phase closure term evolution can be calibrated in an all fibre interferometer or in any dispersive interferometer. The final relationship (13) allows us to predict the phase closure default versus the air path differences for a given spectral bandwidth. In the frame of our experiment, we have operated with a 10 nm spectral bandwidth so that phase closure computation is consistent with the experimental results, even when the chromatic dispersion description is restricted to a second order. In order to extrapolate these computations to a larger spectral bandwidth, it may be necessary to take into account the higher order terms of the dispersion. Notice that in such a context, no analytical solution is available and only the numerical simulation can predict the phase closure error. 4. Conclusion This theoretical and experimental work demonstrates that dispersion effects have to be taken into account for an accurate evaluation of acqui-
sition biases. Differential dispersion leads to a corruption of the phase closure measurement as soon as fringe contrasts begin to decrease. The experimental demonstration has been achieved with an all guided interferometers using fibre delay lines developed in our laboratory [10]. In this experimental framework, the variable dispersion results from the fibre stretching. The theoretical model being validated, these effects could be extrapolated to larger delay line strokes. In the case of narrow spectral bandwidth corresponding, for instance, to a spectroscopic analysis of the interferometric fringes, the error modelling can be achieved by only using a second order dispersion term. If each spectral channel is assumed to be gaussian shaped, the phase closure error evolution versus the dispersion has a simple analytical expression depending on three parameters: the dispersion value of the fibre, the position of the telescopes, and the spectral bandwidth. This work could be extended to other spectral distributions (new wavelength, new spectral bandwidth) but it would make necessary an accurate study of the higher terms of dispersion. Consequently the phase closure error is not incompatible with image reconstructions as long as they can be calibrated. Of course if the differential effects of dispersion become too large, interference fringe contrasts could be reduced leading to a high degradation of the signal to noise ratio for the image restoration. References [1] A. Glindemann et al., in: P.J. Lena, A. Quirrenbach (Eds.), Interferometry in Optical Astronomy, Proc. SPIE, 4006, 2000, p. 2. [2] D.H. Rogstag, Appl. Opt. 7 (1968) 585. [3] A.H. Greenaway, Opt. Commun. 42 (1982) 157. [4] W.J. Tango, Appl. Opt. 29 (1990) 516. [5] S.D. Dyer, D.A. Christensen, Opt. Eng. 36 (9) (1997) 2440. [6] G. Huss, L.M. Simohamed, F. Reynaud, Opt. Commun. 182 (2001) 71. [7] L. Delage, F. Reynaud, A. Lannes, Appl. Opt. 39 (34) (2000) 6406. [8] G. Huss, F. Reynaud, L. Delage, Opt. Commun. (2001). [9] L.M. Simohamed, F. Reynaud, Pure Appl. Opt. 6 (1997) 37. [10] L.M. Simohamed, F. Reynaud, Opt. Commun. 159 (1999) 118.