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Slow-Light Fourier Transform Interferometer

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Slow-Light Fourier Transform Interferometer PHYSICAL REVIEW LETTERS week ending PRL 99, 240801 (2007) 14 DECEMBER 2007 Slow-Light Fourier Transform Interferometer Zhimin Shi* and Robert W. Boyd The Institute of Optics, University of Rochester, Rochester, New York 14627 USA Ryan M. Camacho, Praveen K. Vudyasetu, and John C. Howell Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627 USA (Received 11 July 2007; published 11 December 2007) We describe a new type of Fourier transform (FT) interferometer in which the tunable optical delay between the two arms is realized by using a continuously variable slow-light medium instead of a moving arm as in a conventional setup. The spectral resolution of such a FT interferometer exceeds that of a conventional setup of comparable size by a factor equal to the maximum group index of the slow-light medium. The scheme is experimentally demonstrated by using a rubidium atomic vapor cell as the tunable slow-light medium, and the spectral resolution is enhanced by a factor of approximately 100. DOI: 10.1103/PhysRevLett.99.240801 PACS numbers: 07.60.Ly, 07.60.Rd, 42.62.Fi Recently, there has been considerable interest in the interest. The frequency dependence of the refractive index development of slow- and fast-light techniques [1] to con- of such an ideal tunable slow-light medium in the vicinity trol the propagation velocity of light pulses through a of a reference frequency 0 is given by material system. While early work concentrated on tech- n0 0 niques to realize very large [2,3], very small [4], or even g n n 0 ; (1) negative [5] group indices, recent work has been aimed at 0 developing practical applications of slow-light methods where 0 ÿ is the frequency detuning and n0 [6–11]. In addition to applications in optical communica- 0 g n ÿ n is the relative group index. We assume that for such tion systems [1], it was recently shown that slow light can g a medium n0 can be varied continuously, for example, by also be used to enhance the spectral sensitivity of certain g changing the number density of an atomic vapor, from zero types of interferometers [12–14]. 0 Fourier transform (FT) interferometry [15] is a powerful to a maximum value ng;max. Note that 0 is a reference 0 technique that has an intrinsically high signal-to-noise ratio frequency chosen such that n 0 remains constant as ng is (SNR) and can have high resolving power. These proper- tuned. We consider a Mach-Zehnder (MZ) interferometer ties have led to its many applications in biomedical engi- with such a tunable slow-light medium of length L in one neering, metrology [16], astronomy, etc. A conventional arm and a nondispersive reference medium of length L2 FT interferometer [see Fig. 1(a)] is typically comprised of a and refractive index n2 in the other arm. For simplicity, we 2 fixed arm and a moving arm, both of which contain non- let I jE j . When the input field has multiple fre- dispersive media (typically air) with refractive index n. quency components, the output intensity at each of the two The length of the moving arm can be changed to achieve a ports of such a MZ interferometer (see Fig. 2) is given by variable optical delay time (ODT) nL=c, where L is Z 1 0 0 ikn 0 ng =0L ikn2L2 2 the length difference between the two arms, and c is the Iout; Iin je e j d; (2) 4 speed of light in vacuum. To resolve the spectrum of an input optical field with center frequency , needs to be where k 2=c is the wave number at frequency in tuned from zero to a maximum value max with a step size comparable to 1=. The spectral resolution is given by moving arm tunable slow- min 1= 2max [15]. To achieve a high spectral reso- detector detector 0.5Lmax light medium lution, one needs a large device size [typically with the beam beam order of c= 2nmin] and a large number of data acquis- splitter splitter ition steps [determined by = 2min] for each measurement. 0.5δL input field In this Letter, we propose and demonstrate a new type of input field FT interferometer that uses a continuously tunable slow- fixed arm fixed arms (a) (b) light medium to realize a tunable group delay between the two arms [see Fig. 1(b)]. We first develop the theory for the FIG. 1 (color online). Schematic diagrams of (a) a conven- ideal case in which the slow-light medium has a uniform tional FT interferometer with one moving arm and one fixed group index ng (defined by ng n dn=d) and thus no arm; (b) a FT interferometer with a tunable slow-light medium in group velocity dispersion over the frequency range of one of the two fixed arms. 0031-9007=07=99(24)=240801(4) 240801-1 © 2007 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 99, 240801 (2007) 14 DECEMBER 2007 beam splitter λ heater + controller ter [e.g., Eq. (11.4) in Ref. [15] ], except that in the present (BS) /2 polarization beam case the Fourier conjugate pair is the detuning 0 and the splitter (PBS) group delay g instead of the absolute frequency and the ODT . PBS Rb cell In the ideal case in which the slow-light medium is lossless, the spectral resolution of such a slow-light AOM MZM I+ FT interferometer is limited by the largest achievable group AWG λ/2 BS 0 PBS delay g;max to 1= 2g;maxc= 2ng;maxL. Since I- 0 λ - ng;max can be very large when a suitable slow-light medium /2 I ' =I -I Laser out + - is used, the spectral resolution of the slow-light FT inter- ferometer can be enhanced by the significant factor of 0 FIG. 2 (color online). Experimental setup of the FT interfer- ng;max with respect to that of a conventional setup. ometer using a rubidium vapor cell as the slow-light medium. Alternatively, for a specified spectral resolution , the 0 device size can be decreased by a factor of ng;max. In many real situations, however, a slow-light medium vacuum. Note that, in practice, both arms also contain introduces some frequency-dependent loss. In a tunable other nondispersive media such as beam splitters and air. slow-light medium of the sort we consider, the ratio be- tween the absorption coefficient and the relative group However, the optical path lengths contributed from these 0 0 0 media are assumed to be balanced between the two arms index ng at each frequency 0 remains constant as ng and therefore are not shown in Eq. (2). When the two arms is scanned. Thus, for an input of an infinitely sharp spectral 0 0 are balanced such that n L n L, one can rewrite line at 0 , the magnitude of the output Iout will 2 2 0 0 Eq. (2) as follows: decrease exponentially as ng (i.e., g) becomes larger. Z The Fourier transform of such a decay will result in a 1 0 0 0 ik ng =0L 2 Lorentzian-shape spectral line centered at 0 with I I je j d out; in 1 0 4 an effective linewidth of c=4ng. Thus, the overall spec- Z 0 0 0 1 1 ng tral resolution near 0 is given by Iin Iin cosk Ld: (3) 2 2 0 0 0 c c 0 0 max 0 ; 0 : (7) By subtracting the two outputs, and approximating k by k0, 2ng;maxL 2ng one obtains the following relation: The total spectral range of such a FT interferometer is Z 0 0 0 0 0 ng given by c= 2ngL where ng is the step size of the Iout Iout; ÿ Iout;ÿ Iin cos2 Ld; (4) 0 c change in ng. Note that our slow-light FT interferometer 0 does not require any moving arms, which is advantageous where Iout is the modified output that can be directly in certain situations in which vibration and translation measured by using a balanced homodyne detection method errors of a moving arm could introduce measurement [17]. Note that, for a pulse with center frequency near 0, errors and decrease the SNR. the relative delay between the two arms of the interfer- The theory of Eqs. (1)–(6) can be extended to the more ometer is given by general case in which the slow-light medium has an arbi- n L n L L n0 L trary frequency dependence of the refractive index near 0 g 2 2 g 0 0 0 ÿ n ÿ n : (5) in the form of n n n = f , where f g c c g 0 c c 0 g 0 describes the normalized dispersion function near 0.In We assume that the incident field contains only fre- such a case, one can replace 0 by f 0 in Eqs. (2)–(6) and quency components that are larger than 0, as is in the obtain the following inverse FT relation: case of the experiment shown below. In this way, one can Z 1 0 obtain the following inverse Fourier transform relation: 0 0 i2f g 0 0 Ioutg 0 Re Iin 0e d ; (8) 1 Z 1 0 0 0 0 I g Iin 0 cos2 gd out where g 0 is the group delay of a pulse centered at 0. 1 Z Note that g 0 can be determined from the group delay of 1 0 0 i2 g 0 0 Re Iin 0 e d ; (6) a pulse centered at any known frequency through 1 0 0 0 0 the relation g 0ng 0g 0 =ng 0 .
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