Non-collinear upconversion of infrared light

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1 Non-collinear pconversion of infrared light Christian Pedersen, Qi H, Lasse Høgstedt, Peter Tidemand-Lichtenberg, and Jeppe Seidelin Dam * DTU Fotonik, Frederiksborgvej 399, 4000 Roskilde, Denmark * jdam@fotonik.dt.dk Abstract: Two dimensional mid-infrared pconversion imaging provides niqe spectral and spatial information showing good potential for midinfrared spectroscopy and hyperspectral imaging. However, to extract spectral or spatial information from the pconverted images an elaborate model is needed, which incldes non-collinear interaction. We derive here a general theory providing the far field of the pconverted light when two arbitrary fields interact inside a nonlinear crystal. Theoretical predictions are experimentally verified for incoherent radiation and sbseqently applied to previosly pblished data with good agreement. 014 Optical Society of America OCIS codes: ( ) Infrared imaging; (190.70) Upconversion; ( ) Spectroscopy, infrared. References and links 1. P. R. Griffiths and J. A. de Haseth, Forier Transform infrared Spectrometry nd Ed (Wiley, 007).. S. Wartewig and R. H. H. Nebert, Pharmacetical applications of mid-ir and raman spectroscopy, Adv. Drg Deliver. Rev. 57(8), (005) N. B. Colthp, L. H. Daly, and S. E. Wiberley, Introdction to Infrared and Raman Spectroscopy 3rd ed. (Academic 1990). 4. T. L. Williams, Thermal Imaging Cameras (CRC, 009). 5. J. E. Midwinter, Image conversion from 1.6m to the visible in lithim niobate, Appl. Phys. Lett. 1(3), (1968), doi: / J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, Room-temperatre mid-infrared single-photon spectral imaging, Natre Photon. 6, 788 (01) Q. Zho, K. Hang, H. Pan, E. W, and H. Zheng, Ultrasensitive mid-infrared p-conversion imaging af fewphoton level, Appl. Phys. Lett. 10(4), (013), doi: / J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, High-resoltion two-dimensional image pconversion of incoherent light, Opt. Lett. 35(), (010), doi: /ol C. Pedersen, E. Karamehmedović, J. S. Dam, and P. Tidemand-Lichtenberg, Enhanced D-image pconversion sing solid-state lasers, Opt. Express 17(3), (009), J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, Theory for pconversion of incoherent images, Opt. Express 0(), (01), N. Bloembergen and P. S. Pershan, Light waves at the bondary of nonlinear media, Phys. Rev. 18(), (196), 1. D. A. Kleinman, Theory of optical parametric noise, Phys. Rev. 174(3), (1968), A. H. Firester, Upconversion: Part III, J. Appl. Phys. 41(), (1970). 14. A. Gavrielides, P. Peterson, and D. Cardimona, Diffractive imaging in three-wave interactions, J. Appl. Phys. 6(7), (1987). 15. B. E. A. Saleh and M. C. Teich, Fndamentals of Photonics nd Ed (Wiley, 007). 16. J. W. Goodman, Introdction to Forier Optics 3 rd Ed (Roberts & Company, 005). 17. Q. H, J. Seidelin Dam, C. Pedersen, and P. Tidemand-Lichtenberg, High-resoltion mid-ir spectrometer based on freqency pconversion, Opt. Lett. 37(4), (01), L. Høgstedt, O. B. Jensen, J. S. Dam, C. Pedersen, and P. Tidemand-Lichtenberg, 500 nm continos wave tnable single-freqency mid-ir light sorce for C-H spectroscopy, Laser Phys. (11), (01), doi: /s x The commercial HITEMP database from Spectral calclator - high resoltion gas spectra, (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 807

2 1. Introdction Highly sensitive mid-infrared (mid-ir) imaging and spectroscopy is emerging as an attractive alternative to its more well-known visible or near-infrared conterparts within fndamental band gas analysis, spectral identification and qantification of complex molecles [1 3]. Mid-IR detectors exist in different forms [4], bt they are all limited by an navoidable dark noise de to thermal radiation originating from the finite temperatre of the detector itself [4]. Ths, high sensitivity mid-ir detectors need to be cooled to redce noise. This is in strong contrast to visible (VIS) or near infrared (NIR) detectors where thermal radiation (dark noise) is mch less prononced at room temperatre. An attractive alternative to cryogenically cooled mid-ir detection and imaging is roomtemperatre freqency pconversion followed by detection in the VIS or NIR region [5 7]. () This method relies on a two stage procedre. First, sm freqency generation (SFG) in a χ nonlinear crystal, converting the incoming mid-ir light into a shorter wavelength interval well below the wavelength range of thermal radiation emitted by room temperatre objects. Second, NIR/VIS detection sing highly sensitive silicon detectors or CCD cameras. With this approach, single photon mid-ir imaging can be accomplished [6]. Even high resoltion images can be pconverted [8]. However, to nderstand and exploit the attractive featres of pconversion, a more elaborate model of the image formation is needed. This is the primary scope of the present paper. In [9,10] a model was developed for pconversion of coherent and incoherent radiation respectively, assming small angles of incidence for the mid-ir light, i.e. nearly collinear interaction. In this work, the far field soltion of the pconverted field is determined for arbitrary inpt field and pmp beam respectively in the ndepleted regime. The far field restriction imposed here leads to a particlar simple description; however, if needed, near field soltions can be calclated as well. The important case of non-collinear interaction between a Gassian pmp beam and an incident plane wave at a large angle of incidence is considered thoroghly. This work can also be considered as an extension of [11] where SFG of three plane waves were treated [1], where spontaneosly parametric down conversion was considered assming plane inpt waves and a Gassian pmp beam, or [13, 14] which provide a detailed analysis of an arbitrary incident inpt field pconverted by a plane wave pmp beam. We derive a theoretical expression for the pconverted image intensity distribtion and spectral resoltion, as a reslt of the finite spot size of the pmp beam, i.e. the acceptance parameters of the nonlinear image pconversion process. The model is experimentally verified and fond to be in good agreement with measrements. Finally, previos pblished reslts will be discssed and modeled sing the developed theory.. Theory The standard treatment of SFG fond in the nonlinear optics literatre, sch as [15] relies on three copled nonlinear differential Eqs. from which the detailed interaction of the three plane waves can be calclated inclding depletion. This stdy focses on -dimensional pconversion imaging of faint objects where depletion is irrelevant; however, to describe imaging adeqately a model inclding arbitrary inpt fields is needed inclding non-collinear interaction of the incident beams. Frthermore, we will mainly be interested in farfield soltions, althogh the sggested approach can be expanded to cover near field soltions of the pconverted field as well. Walk-off in the inpt fields is implicitly inclded in the model. However, the developed model does not specifically inclde walk-off of the pconverted field, bt it can be extended to consider a constant walk-off by applying the walk-off in the opposite direction to both the inpt fields. With aforementioned assmptions, a particlarly simple pconversion model can be developed. We adopt a general approach based on the far field soltion to the Helmholtz Eq. in free space inclding a nonlinear sorce term [14] rather than a on a copled set of differential Eqs. This approach, which shares common traits with (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 808

3 scattering theory, provides a simple direct treatment exploiting reslts fond in the general electromagnetic theory, e.g [16]. Figre 1 shows the SFG process inside a nonlinear medim, ik1 r V with two arbitrary monochromatic inpt fields expressed as par-axial waves, E1 ( r) e ik r and E ( r) e with anglar freqency, ω 1 and ω respectively, generating a third plane ik3 r SFG wave, E3 ( r) e in the far field. k 1 and k can be envisioned as spatial carrier freqencies specifying the average direction of the two incident fields respectively. Fig. 1. General layot of sm freqency generation illstrating two arbitrary inpt fields inside a second order nonlinear material, V, generating a plane wave at the sm freqency in the far field. The free space, far field soltion to Helmholtz Eq. inclding nonlinear sorce terms can be expressed as [14,15], Eq. (1); exp( ik3 r ) E3( r) E3( r, k3) = S3( r' ) exp ( ik3 r' ) dr' 4π r (1) V where V is the volme of the nonlinear material, and r is the far field observation point, r = r and k 3 is the wave vector of the plane pconverted far field parallel to r, see Fig. 1. The sorce term, S3 ( r ), for a second order nonlinear interaction process is given by [15], Eq. (); P d ω 3 ( 1 ) S3( r ) eff E 1( r ) E( r ) e i k + k r = = () t c where ω3 = ω1 + ω is the anglar freqency of the pconverted field, 3 ( ) the second order nonlinear response of the two incident electrical fields, E1 ( r ) and E ( ) E r indced via r, c is the speed of light in vacm and deff is the effective second order nonlinear coefficient. We observe from Eq. (1), that we indeed have a plane wave in the far field in the direction of k 3. Two distinct assmptions are introdced in Eq. (1), the first is the 1st Born approximation and the second is that only far field soltions are considered, i.e. that the observation point is far away relative to the dimensions of the volme, V. The far-field (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 809

4 approximation implies that k 3 is the same for all points in the crystal. Hence, the approach can be applied even withot assmption of niform n 3. If needed, the far-field approximation can be removed at this point sing a more general soltion to Helmholtz Eq [15,16], ths providing the near field soltion in integral form. Inserting Eq. () into (1) provides a compact and simple expression for the plane SFG far field, E3( r, k 3) specified by the wave vector, k 3 and distance, r from the observation point; The Forier transforms, { } { ( )} { ( )} ik3r d eff ω3 e 3(, 3) = 1 c 4π r E r k F E r F E r F in Eq. (3) are evalated with the spatial Δ k freqencies, inserted, where Δk k3 k k1. In conclsion, the soltion to Helmholtz π Eq. in the far field is the Forier transformation of the nonlinear indced field inside the volme of the nonlinear material. Note that this approach generalizes to other nonlinear processes like for-wave mixing. Note that Δ k in general has three non-zero components and may point in any direction; in the special case of plane wave interaction inside a nonlinear crystal of infinite transverse dimensions, the transverse components of Δ k vanish [1], ths Δ k becomes a scalar qantity. A proper explanation of this simplifying assmption is often lacking in literatre..1 A Gassian pmp beam and plane wave pconversion Eqation (3) can be evalated analytically in the important case of a Gassian pmp field and an incident plane wave, see also Fig.. This simple case is chosen de to its relevance for the sbseqent sections. The pmp field is assmed to propagate along the z-axis, having a Gassian beam profile with beam waist, w 0 at z = 0, wavelength, λ and wave nmber, k [1], Eq. (4); (3) a x + y E( r) exp( ikzz) = exp( ikzz) exp, z 1 i z + w0 1 i z + R z R (4) kzw0 where z R = and a is the field amplitde of the linearly polarized pmp laser. The incident field is given as a plane wave. Eqation (5) ses the notation shown in Fig. 1 to explicitly emphasize the carrier wave vector component of the incident field, Eq. (5); E r exp ik r = a exp ik r, (5) ( ) ( ) ( ) where a 1 is the electric field amplitde and k 1 is the corresponding wave vector. Inserting Eq. (4) and Eq. (5) into Eq. (3) yields; ( Δ kx +Δky ) w 0 ( Δ kx +Δky ) ik3r d eff ωπ 3 w0l e l 4 E( r, Δkx, Δky, Δ kz) = a 1ae Sinc Δkz c 4π r k z (6) (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 8030

5 From Eq. (6) it is seen( Δkx, Δky ) relate to the deviation from conservation of transverse momentm, i.e. the point spread fnction in the imaging process. We note that this reslt corresponds well with [1] where the special case of spontaneos parametric down conversion was analyzed. Also note that the Goy phase shift is inclded, known to slightly shift the optimal phase match condition for focsed beams. Object f f 1 x Image k 1 k k 3 Δk θ 3 θ 1 0 -l/ l/ z Fig.. Illstration of the non-linear crystal, the object and image planes respectively and the three wave-vectors involved in the non-collinear interaction. In a practical realization the far field condition of the pconverted beam can be met by inclding a Forier transforming lens with focal length, f 1. See Fig.. Similarly, a first Forier transforming lens, with focal length f, transforms points in the object plane into hereto niqe angles inside the nonlinear crystal. Incident polychromatic light from a real object can then be modeled as a plrality of plane waves. In the following one sch incident plane wave is considered. For the incoherent case, the intensity profile at the image plane is obtained from Eq. (6) replacing E fields with intensities and spectral radiance [16], transforming anglar coordinates to positions, and finally impose proper correspondence to earlier pblished qasi-collinear theory [10] sing a proper scaling factor. The reslt is Eq. (7); I image ( r, λ ) 8π d eff Gass 3 = nnn 1 3ε0cλ3 f wk 0 3 r r r 3 1 (, ) 0 3 ( ) 3 r wk n f l k Lobject r λ1 e sinc k,, z r r (7) Δ dr π n3 f 1 k n3 f 1 λ3f1 r = x, y is the coordinates of a point in the image plane and r = r, λ f where ( ) where r is a coordinate in the object plane. f is the focal length of the f lens Forier transforming the object plane to the nonlinear crystal. kz ( r, r ) k ( r, r ) k cos( θ ) k k cos( θ ) Δ z = 3 1 i, where i l P Δ is calclated from θ and θ are the internal crystal angles of r r the incident and pconverted field, θi andθ respectively. ε 0 is the vacm nf 1 nf 3 1 permeability. Lobject ( r, λ1 ) is the spectral radiance of the object and PGass is the power of the Gassian pmp beam. Eqation (7) expresses image pconversion inclding non-collinear interaction needed for describing the image formation. The expression incldes diffraction of both the pconverted field and the laser beam. The point where the non-collinearity becomes 1 (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 8031

6 important is related to the angle of the pconverted field, θ, i.e. the radial position in the image plane, so that qasi-collinear performance is maintained for pconverted internal angles significantly smaller than ~ w0 l. Eqation (6) and Eq. (7) provide the basic Eqs. for calclation of the acceptance parameters. The actal evalation of Eq. (7) generally reqires a nmerical comptation. In the following section examples of sch calclations are presented and compared against experimental data. 3. Comparison to experimental data In this section, the developed model, i.e. Eqation (7), is first compared against experimental reslts and sbseqently applied to previosly pblished experimental data as presented in [17]. To illstrate the impact of a finite pmp beam size on the pconverted intensity distribtion, a nmerical calclation of the pconverted intensity sing a 180 µm pmp beam size is compared to that of a very large pmp beam diameter. The calclation is performed for fixed wavelengths, λ 1 =.937µm, λ = 1.064µm and λ 3 = 0.781µm varying the phase match condition throgh se of different poling periods. In order to redce the nmerical x, z. The normalized intensity complexity, the model is redced to two dimensions ( ) distribtion from the object is modelled as constant in r, i.e. niform light emission. In the following calclations, all angles will be presented as internal crystal far field angles, θ i.e. not inclding refraction of the light at the crystals interface. The ble graphs in Fig. 3(a) shows the normalized intensity for a very large Gassian pmp spot size at 11 different phase match conditions, whereas the red graphs shows the corresponding intensities for a 180 µm Gassian pmp beam. The graph in Fig. 3(b) shows the decreasing peak intensity for increasing angles when sing a 180 µm pmp beam spot size, i.e. it represents how the monochromatic peak intensity decreases for larger phase matched angles. It is important to note that the shape of the crve is independent of the choice of wavelengths, w 0, and L. In all circmstances the crve drops to abot 0.5 at the point θ = w0 L as indicated in Fig. 3(b). Normalized intensity a Θ Normalized intensity Θ w 0 L b Θ Fig. 3. Theoretical calclation of of non-collinear pconversion of monochromatic incoherent light. a) The ble crves shows the normalized intensity crves for a very large laser beam diameter for a selection of phase matched angles; θ = 0.046, 0.187, 0.6, 0.318, 0.365, 0.406, 0.444, 0.478, 0.509, 0.538, and degrees respectively. The red crves shows a similar set of crves sing a pmp beam radis of 180 µm. b) This crve shows the decreased peak intensity vales cased by the finite size of the mixing beam as a fnction of noncollinear pconverted angle. To calclate the spectral response, we now assme that the spectral radiance from the object is constant in λ 1 and r. The phase matched pconverted wavelength is still λ 1 =.937 µm, corresponding to an pconverted wavelength λ 3 = µm. The reslt is shown in Fig. 4, where each crve corresponds to the same phase matched angles sed in Fig. 3, again showing decreasing peak conversion efficiency for increasing angle. (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 803

7 1.0 Normalized intensity Λ 1 µm Fig. 4. Plot of spectral acceptance for the same angles θ sed in Fig. 4(a). The highest crve shows the sitation for the smallest angle, i.e. θ. As in Fig. 3, the absolte intensity peaks decrease for increasing θ. For easy comparison, the calclation ses the same phase matched wavelength. It is apparent that the decrease in peak intensity is accompanied with a corresponding increase in wavelength acceptance. Figre 4 demonstrates that the spectral acceptance bandwidth increases for larger angles. This is a direct conseqence of the finite pmp beam radis in the pconversion process. Note that increasing the pmp beam radis conteracts the increased acceptance bandwidth withot decreasing the nmber of pconverted photons in case of incoherent light emission from the object [10, 1]. However, the laser beam size is ltimately limited by the nonlinear crystal apertre, particlarly relevant when sing periodically poled crystals. Ths for incoherent pconversion, high spectral (and spatial) resoltion can be obtained with a large pmp beam diameter. 3.1 spatially incoherent single freqency inpt In order to test the developed theory, an experimental setp was established, see Fig. 5. The pconversion setp, described in detail in [17], ses a 0 mm long PP:LN crystal for smfreqency mixing with a single-freqency 1064 nm pmp beam with a beam size of 180 µm in the nonlinear crystal. The pconversion system is illminated by spatially incoherent bt monochromatic mid-ir light emitted from a single freqency mid-ir laser [18] scattered by a rotating diffser. Rotating diffser f 1 f f 3 f 4 f 5 Ge filter PP:LN Bandpass filter Single-freqency IR laser Filter & Imaging lenses LD Nd:YVO 4 Single-freqency converter Filter & Imaging lenses Upconverted Imaging Fig. 5. System for detection of incoherent, monochromatic light sing freqency p-conversion and a NIR/VIS CCD camera. Figre 6(a) displays two raw data frames recorded with the CCD camera at different crystal temperatres. Upconverted incoherent single freqency inpt light will appear as a single niqe ring pattern with varying radii as fnction of the temperatre of the nonlinear crystal, i.e. changing the phase match condition [6]. Figre 6(b) shows the radial cross section crves of all the data frames recorded in the measrement series (5 frames of 100 ms exposre were averaged for each crve), corresponding to the graphs in Fig. 3, each (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 8033

8 compared against theoretical crve sing Eq. (7) (red graphs). Note that the phase match crves in Fig. 6(b) have been normalized to nity de to nwanted spatial inhomogeneity in the light distribtion related to an imperfect diffser. The spatial inhomogeneity can be observed as a non-niform intensity pattern in the pconverted ring in Fig. 6(a). As a conseqence of the normalization, the decreasing peak intensity with angle, shown in Fig. 3(b), is not observable. Ths the anglar dependency only displays itself throgh the anglar acceptance. The measred crves are averaged from several images sing the center of the ring pattern as the origin. To qantify the predictive power of the presented theory, the correlation coefficient, C of the experimental (ble graphs) and theoretical data (red graphs) is calclated. C is calclated according to Eq. (8): Imeas ( θ ) Itheory ( θ ) θdθ C = (8) I θ I θ θ dθ I θ I θ θ dθ ( ) ( ) ( ) ( ) meas meas theory theory Note that Eq. (8) is expressed in polar coordinates. In Fig. 6(c), the correlation coefficient, C between the actal measred vales and those calclated from sing collinear theory [10] (red dots) and the here presented non-collinear theory (ble dots) respectively, is plotted as a fnction of the phase match angle. It is clear that the proposed non-collinear theory is sperior to the theory proposed in [10], particlarly as the angle, θ increases. Fig. 6. Experimental comparison to theoretical prediction. a) Raw data recorded by the CCD camera for two different temperatres of the nonlinear crystal. b) Cross sections of the ring patterns as fnction of θ for different crystal temperatres (ble crves) plotted with theoretical predictions (red crves). The inpt wavelength was.937 µm throghot the measrement series. c) The correlation coefficient of the 11 data series comparing collinear theory (red dots) and non-collinear theory (ble dots). 3. Qasi-single freqency inpt The developed theory for pconversion of incoherent qasi-single freqency light can also be applied to an inpt containing several freqencies. In work by H et al. [17], the emission lines of hot water vapor were stdied based on pconversion followed by post filtering sing a high finesse Fabry-Perot interferometer. In [17] the observed intensity decrease for increasing image angle was compensated sing an empiric fnction rather than from theoretical considerations. With the here developed theoretical model, proper data correction can be applied. The reslt is shown in Fig. 7. (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 8034

9 Fig. 7. The green crve shows the raw data from [17]. Post processing the green crve sing the presented theory for non-collinear pconversion provides the red crve. The spectra are compared to a modelled spectra sing [19] shown as black crve (0. nm resoltion) and ble crve (0.4 nm resoltion). The red crve resembles the modelled spectrm better as expected. The here presented non-collinear theory is seen to partly explain the decrease in intensity when moving to larger image angles, i.e. wavelengths in this case. Regarding the spectral resoltion, there is good agreement for the innermost part of the spectrm (arond.91 µm). However, for the longer wavelengths a deteriorating spectral resoltion is observed. This we believe to be cased by imperfect Fabry-Perot mirrors. The reslt is a decreased finesse for the off-center image locations, and ths redced spectral resoltion for the longer wavelengths. This can be observed by the resoltion of the measrement series is matching the calclated spectrm at 0. nm resoltion is matching experiments near.91 µm, whereas near.95 a better match is fond when comparing to 0.4 nm resoltion. A more detailed discssion of the decrease in spectral resoltion observed in [17] is not central to this work. However, spectral artifacts in the measred spectrm when compared to the theoretically compted spectrm [19] can now be explained. These spectral artifacts appear as spectral signatres in wrong places for longer wavelengths, e.g. clearly visible at arond.937 µm (marked by a circle in Fig. 7). From Fig. 4 we see that a free spectral range (FSR) of 9 nm is jst enogh to ensre monochromatic performance in the center of the image (highest black crve in Fig. 4), bt for longer wavelengths, at off-center positions in the image, this is not the case (lowest black crve in Fig. 4). This cases the improper appearance of spectral artifacts shifted by 1 FSR. 4. Conclsion In this work, a detailed theory is presented for the pconverted far field sing two arbitrary inpt fields assming no depletion. Directions are given how to inclde near field soltions as well. Eqation (3) is a main theoretical reslt, whereas the soltion in Eq. (7) enables simple comptation for the incoherent case of a plane wave mixing with a Gassian pmp beam. The general theory developed here can be sed to optimize the design of pconversion systems for ftre applications, and for improved post processing of measred data. In particlar it is noted that for the incoherent case a large pmp beam diameter favors both the spatial and the spectral resoltion at no cost in efficiency. The analytical model was experimentally verified showing good agreement with measred acceptance parameters for non-collinear interaction. The correction given by the roll-off crve (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 8035

10 shown in Fig. 3(b) was applied to previos pblished experimental data improving the agreement with SpectralCalc simlations withot the need for empirically correction as sed in [17]. Acknowledgment We thank Peter John Rodrigo for an interesting discssion on the theory presented here. (C) 014 OSA 17 November 014 Vol., No. 3 DOI: /OE OPTICS EXPRESS 8036