Spatial dispersion for diffraction grating based optical

Spatial Dispersion for Diffraction Grating based Optical Systems Ali Zahid1, Bo Dai1*, Bin Sheng1, Ruijin Hong1, Qi Wang1, Dawei Zhang1, Xu Wang 2 1. Engineering Research Center of Optical Instrument and System, the Ministry of Education, Shanghai Key Laboratory of Modern Optical System, University of Shanghai for Science and Technology, Shanghai, 200093, China. 2. The Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, U.K. ABSTRACT

Diffraction gratings are key components in many applications including pulse compression and stretch, optical imaging, spectral encoding and decoding and optical filtering. In this paper, spatial dispersion of two typical diffraction grating-based optical systems, single-grating system and grating-pair system, are thoroughly studied. The single-grating system consists of a diffraction grating to disperse the quasi-monochromatic lights and a convex lens to make the lights propagate in parallel and focused. In the grating–pair system, a pair of diffraction gratings is used to disperse the collimated lights in parallel. The spatial dispersion law for the two systems is developed and summarized. By investigating the spatial dispersion, the two systems are compared and discussed in detail. Keywords: Diffraction gratings, Dispersion. 1. INTRODUCTION Diffraction grating is a popular optical component that is used to spatially disperse light. There are many different applications using diffractive gratings. In these applications, there are two common configurations to employ diffraction gratings, which are shown in Fig. 1. The first configuration, referred to as “single-grating system” hereinafter, is to simply use a diffraction grating and a convex lens to disperse and focus a collimated light onto a target, as shown in Fig. 1(a). After the convex lens, the dispersed quasi-monochromatic lights propagate in parallel with each other and are focused at back focal point. The single-grating system is widely used in the applications of optical spectral encoding and decoding [1, 2], pulse shaping [3, 4], optical imaging [5–8] and optical filtering [9, 10]. The other configuration, referred to as “grating-pair system” hereinafter, is to disperse light by using a pair of diffraction gratings, as shown in Fig. 1(b). In this configuration, the dispersed quasi-monochromatic lights are parallel to each other and each light component is collimated. The grating-pair system is commonly used for pulse compression and stretch [11–14], chirped pulse amplification [15–17], and optical imaging [18, 19]. The dispersion properties of diffraction grating based optical systems are key factors in the applications. In the single-grating system, spatial dispersion was derived according to geometrical optics in [4]. The grating-pair system presents spatial dispersion in a different manner comparing to the single-grating system. In the two systems, the dispersed quasi-monochromatic lights propagate in parallel, but the each light is kept collimated in the grating-pair system, while the lights are focused in the single-grating system. The distinct properties of the spatial dispersion make the two systems suitable for different applications.

International Conference on Optoelectronics and Microelectronics Technology and Application, edited by Yangyuan Wang, Guofan Jin, Proc. of SPIE Vol. 10244, 1024423 · © 2017 SPIE · CCC code: 0277-786X/17/$18 · doi: 10.1117/12.2263412 Proc. of SPIE Vol. 10244 1024423-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 01/15/2017 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx

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Fig. 1. Sch hematic diaggram of (a) single-gratin s ng system annd (b) gratingg-pair system. BS S: beam splittter. t paper, th he propertiess of spatial dispersion d foor the both diffraction d grrating based optical In this systems are investigateed thoroughlly. The spatiial dispersionn of the gratting-pair sysstem is reveaaled for the first tim me. By comp paring the diispersion prooperties, the distinction of o the two syystems is disscussed in detail. 2. THEOR RETICAL MODEL M 2.1 The sinngle-grating system The schem matic diagraam of the single-gratin s g system iss illustrated in Fig. 1(aa). An inpuut light illuminatess on a diffrraction gratiing with an incident anngle, α, via a beam splitter. The light l is dispersed by b the diffraction gratingg into lots off quasi-monoochromatic lights l with diffracted d anggles, β. The relatioon between th he incident angle a and diffracted anggle is mλ sin (α ) + sin (β ) = (1) d

where m iss the order off diffraction, λ is the wavvelength of the light, and d is the grating periodd. In the following analysis, a m= =1 is consideered. Then, thhe dispersed d lights passs through a convex lens, which haas the focal length of f, and is focused onn a reflectorr. The reflector can bee a mirror, a spatial ligght modulatoor or other optical componentts with a speecific patternn on the refleection side for fo a particullar purpose. A mirror with total reflection is i considereed in the theeoretical moddel. The disspersed lightts are reflectted back aloong the same opticcal path and output o via thhe beam splitter. The spaatial disperssion of the single-gratinng system for f light witth a certain wavelengthh, λ, in contrast to that with the peak waveelength, λ0, iss given in [44]. (λ − λ0 ) f ΔxS (λ ) = (2) d cos(β 0 ) where β0 iss the diffractted angle of the light witth the peak wavelength. w After thhe convex len ns, the focussed beam waaist on the taarget can be expressed e ass

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cos (β ) f λ (3) cos (α ) π W in The dispersion of the single-grating system is proportional to the focal length of the lens and mainly affected by the input angle and grating period. W Out =

2.2 The grating-pair system In the grating-pair system, a pair of diffraction gratings is placed in parallel. The two gratings have the same structure. An input light is incident on the first grating with an incident angle of α via a beam splitter and dispersed by the diffraction grating. The diffracted angle for the dispersed quasimonochromatic light is β. Then, the dispersed quasi-monochromatic lights are incident onto the second diffraction grating. Since the two diffraction gratings are in parallel, the incident angle of the each light onto the second grating is same as the diffracted angle from the first grating, i.e. β. According to Eq. (1), all the quasi-monochromatic lights are diffracted by the second diffraction grating with the diffracted angle of α, which is irrelevant to the wavelength. Thus, the dispersed lights propagate in parallel after the second grating. Next, the reflector reflects the light back along the same optical path as the input. Finally, the light is output via the beam splitter. By the geometrical optics, the spatial dispersion of the grating-pair system can be derived as ΔxP (λ ) = cos(α )L[tan(β ) − tan(β0 )]

(4)

where L is the perpendicular distance between the two gratings. The incident angle and the diffracted angle into and from the first grating are same as the diffracted angle and incident angle from and into the second grating. Therefore, the change of the beam waist of the input light resulted from the first grating can be compensated during the diffraction of the second grating. Consequently, the dispersed lights are collimated and the beam waists are same as that of the input light. 3. SPATIAL DISPERSION The both systems are capable of spatially dispersing a light in parallel, but in different manner. In the single-grating system, the dispersed light is focused, while the light in the grating pair system remains collimated as the input. The system parameters have influence over the spatial dispersion. According to Eq. (2) and Eq. (4), the spatial dispersion is affected by the incident angle, α, the groove density of the grating (the reciprocal of the grating period), 1/d, the focal length of the convex lens in the single-grating system, f, and the perpendicular distance between the two gratings in the grating-pair system, L. Fig. 2 illustrates the spatial dispersion of the both systems for the light of 20 nm bandwidth with the centre wavelength at 1550 nm. In the calculation, the groove density of the grating is 800 lines/mm and the incident angle is 30 degrees. The light is dispersed linearly. The range of the spatial dispersion becomes wide with the increase of the focal length and the grating separation. The dispersed quasi-monochromatic lights are obliged to converge in parallel by the convex lens in the single-grating system and the second grating in the grating-pair system and no further spatial dispersion occurs after that. Besides, the change of spatial dispersion is in linear proportion to the change of the focal length and the grating separation. It is straightforward to control the spatial dispersion by selecting a proper convex lens and adjusting the position of the second grating.


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Fig. 2. Thee influence of o the focal length l and thhe grating seeparation ovver the spatial disspersion in the (a) singgle-grating system andd (b) gratingg-pair system. Since the t spatial dispersion d prresents in ann almost lineear fashion over o the whole bandwiddth, the rate of the spatial disp persion versuus the bandw width is connsidered. Figg. 3 shows thhe influencee of the g and the incidentt angle over the spatial dispersion d off the both syystems. groove dennsity of the grating The focal length and the grating separation are a 100 mm m. In generaal, the groovve density iss a key parameter for diffractiion gratingss. A high-grroove-densityy grating caan widely seeparate two quasimatic lights, resulting inn a wide rangge of spatial dispersion. In other worrds, it also means m a monochrom high resollution of th he spectrum m, because two quasi-monochrom matic lights with veryy close wavelengthhs can be diistinguished when they are separateed widely. Inn the both systems, s the spatial dispersion increases with w the groove density. Especiallyy in the singgle-grating system, s the groove t incident angle affectts the spatiaal dispersionn of the density plaays an important role. Inn addition, the both system ms. The deccrement of the t incident angle leads to an increm ment of the spatial disppersion. When the groove den nsity is highh, the variaation of the incident anngle dramatiically changges the dispersion.. (b)

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o the groovve density annd the incideent angle ovver the Fig. 3. Thee influence of spatial disspersion in the (a) singgle-grating system andd (b) gratingg-pair system. Thus, to t obtain hig gh spatial dispersion, it requires a convex lens with w a long focal lengthh in the single-gratting system, a large sepaaration of thhe gratings inn the gratingg-pair system m, and the gratings g


of high groove density and a small incident angle in the both systems. Furthermore, when the focal length and the grating separation are same and other system parameters are identical, comparing with the single-grating system, the grating-pair system can achieve a high spatial dispersion. 4. CONCLUSION The spatial dispersion law of the single-grating and grating-pair systems are deeply studied. The theoretical models of the dispersion are given to analyse the systems. The system characteristics are revealed. In the both systems, the dispersed quasi-monochromatic lights propagate in parallel, but the each light presents in a different manner. The lights are focused in the single-grating system while those are collimated as the input in the grating-pair system. Comparing with the single-grating system, the grating-pair system is capable of spatially dispersing the light components in a relatively wider range. Thus, the two systems can be well designed to fulfil different requirements of field-ofview in the optical imaging. The dispersion law provides a powerful theoretical tool for understanding/designing the diffraction grating based systems. Moreover, according to the study, the two systems present the diversity of the dispersion properties, which makes the systems complementary in the various applications. Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this manuscript Acknowledgments The research work is supported by National Science Instrument (2013YQ16043903), Pujiang Project of Shanghai Science and Technology

Important

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