Pang et al. Vol. 29, No. 6 / June 2012 / J. Opt. Soc. Am. A 989 Generalized Gouy phae for focued partially coherent light and it implication for interferometry Xiaoyan Pang, 1 David G. Ficher, 2 and Taco D. Vier 1,3, * 1 Department of Electrical Engineering, Delft Univerity of Technology, Delft, The Netherland 2 Reearch and Technology Directorate, NASA Glenn Reearch Center, Cleveland, Ohio 44135, USA 3 Department of Phyic and Atronomy, VU Univerity, Amterdam, The Netherland *Correponding author: T.D.Vier@tudelft.nl Received March 16, 2012; accepted March 27, 2012; poted March 30, 2012 (Doc. ID 164864); publihed May 23, 2012 The Gouy phae, ometime called the phae anomaly, i the remarkable effect that in the region of focu a converging wave field undergoe a rapid phae change by an amount of π, compared to the phae of a plane wave of the ame frequency. Thi phenomenon play a crucial role in any application where field are focued, uch a optical coherence tomography, mode election in laer reonator, and interference microcopy. However, when the field i patially partially coherent, a i often the cae, it phae i a random quantity. When uch a field i focued, the Gouy phae i therefore undefined. The correlation propertie of partially coherent field are decribed by their o-called pectral degree of coherence. We demontrate that thi coherence function doe exhibit a generalized Gouy phae. It precie behavior in the focal region depend on the tranvere coherence length. We how that thi effect influence the fringe pacing in interference experiment in a nontrivial manner. 2012 Optical Society of America OCIS code: 030.1640, 050.1960, 180.3170, 120.3940. In 1890 Gouy found that the phae of a monochromatic, diffracted converging wave, compared to that of a plane wave of the ame frequency, undergoe a rapid change of 180 near the geometric focu [1 4]. Since then many further obervation of thi o-called phae anomaly have been reported and many different explanation for it origin have been uggeted (ee [5] and the reference therein). The Gouy phae i of great importance becaue of it role in, for example, metrology [6], laer mode converion [7], coherence tomography [8], the tuning of laer cavitie [9], higher order harmonic generation [10], terahertz time-domain pectrocopy [11], and nanooptic [12]. Under many practical circumtance, light i not monochromatic, but rather partially coherent. Example are light that i produced by a multimode laer or light that ha traveled through the atmophere or biological tiue. In thoe cae, the phae of the wave field i a random quantity. Therefore, when a partially coherent field i focued (a decribed in [13 20]), the Gouy phae i undefined; i.e., it ha no phyical meaning. In the pace-frequency domain, a partially coherent optical field i characterized by two-point correlation function, uch a the cro-pectral denity or it normalized verion, the pectral degree of coherence [21]. Thee complex-valued function have a phae that i typically welldefined. A we will demontrate for a broad cla of partially coherent field, the phae of both correlation function how a generalized phae anomaly, which reduce to the claical Gouy phae in the coherent limit. Furthermore, thi generalized Gouy phae affect the interference of highly focued field, in microcopy for example, altering the fringe pacing compared to that of a coherent field. Conider firt a converging, monochromatic field of frequency ω that i exiting a circular aperture with radiu a in a plane creen (ee Fig. 1). The origin O of the coordinate ytem coincide with the geometrical focu. The amplitude of the field i U 0 r 0 ; ω, r 0 being the poition vector of a point Q in the aperture. The field at a point P r in the focal region i, according to the Huygen Frenel principle [4, Chap. 8.2], given by the following expreion: U r; ω i λ Z U 0 r 0 ; ω exp ik d 2 r 0 ; (1) where the integration extend over the pherical wave front S that fill the aperture, jr r 0 j denote the ditance QP, and k 2π λ i the wavenumber aociated with frequency ω. A periodic time-dependent factor exp iωt i uppreed. The Gouy phae δ z of a focued, monochromatic field at an axial point r 0; 0;z i defined a the difference between the argument (or phae ) of the field U z; ω and that of a plane wave (or a pherical wave) of the ame frequency, i.e., One can how that [4, Sec. 8.8.4] δ z arg U z; ω kz: (2) δ 0 π 2: (3) Furthermore, the Gouy phae ha the ymmetry property δ z δ z π: (4) An example of the behavior of the Gouy phae near focu i hown in Fig. 2. The dicontinuitie by an amount of π occur at the zero (or phae ingularitie) of the field. The lope of the curve through the point z 0 i explained by the obervation 1084-7529/12/060989-05$15.00/0 2012 Optical Society of America
990 J. Opt. Soc. Am. A / Vol. 29, No. 6 / June 2012 Pang et al. 2a S Fig. 1. by Linfoot and Wolf [3] that near focu the wavefront are eparated by a ditance λ 1 a 2 4f 2. Thi implie that the effective phae of the field, compared to that of a plane wave, lag by an amount of kza 2 4f 2. For the choice of parameter in Fig. 2 thi tranlate into a lope of 0.621 radian per micrometer, a i indeed oberved. For a partially coherent wave field one mut conider, intead of the amplitude U 0 r 0 ; ω, the cro-pectral denity function [21, Sec. 2.4.4] of the field at two point Q 1 r 0 1 and Q 2 r 0 2, namely, W 0 r 0 1 ; r0 2 ; ω hu 0 r 0 1 ; ω U 0 r 0 2 ; ω i: (5) Here the angular bracket denote the average, taken over a tatitical enemble of monochromatic realization fu 0 r 0 exp iωt g [21, Sec. 4.7] and the aterik denote the complex conjugate. The cro-pectral denity of the focued field W r 1 ; r 2 ; ω hu r 1 ; ω U r 2 ; ω i (6) i, according to Eq. (1) and (6), given by the following formula: W r 1 ; r 2 ; ω 1 λ 2 ZZ S. Q(r') f W 0 r 0 ; r 00 ; ω exp ik 2 1 1 2 d 2 r 0 d 2 r 00 ;. Illutration of the notation. P(r). O z (7) contant indicating the effective tranvere pectral coherence length of the field. A normalized meaure of the trength of the field correlation at a pair of point P 1 r 1, P 2 r 2 in the focal region i given by the pectral degree of coherence [21, Sec. 4.3.2], which i defined a μ r 1 ; r 2 W r 1; r 2 p ; (10) S r 1 S r 2 with the pectral denity ( intenity at frequency ω ) S r i at poition r i given by the diagonal element of the cropectral denity, i.e., S r i W r i ; r i ; i 1; 2: (11) Becaue S r i i real-valued and never zero for Gauian Schell-model field [16], the argument (or phae ) of the pectral degree of coherence and that of the cro-pectral denity function are identical. We retrict our analyi to pair of point on the z axi, i.e., r 1 0; 0;z 1, r 2 0; 0;z 2. On making ue of the Debye approximation [4, Sec. 8.8.1], one can then derive for the cropectral denity the following expreion (ee [22] for detail): 2πA 2 W z 1 ;z 2 exp ρ 02 ρ 002 2σ 2 λf 0 0 expfik z 1 1 ρ 02 2f 2 z 2 1 ρ 002 2f 2 g ρ 0 ρ 0 I 0 σ 2 ρ 0 ρ 00 dρ 0 dρ 00 ; (12) where I 0 denote the modified Beel function of order zero. Let u now define a generalized Gouy phae a δ μ z 1 ;z 2 arg W z 1 ;z 2 kz 1 kz 2 : (13) where 1 jr 1 r 0 j, and 2 jr 2 r 00 j. To implify the notation, we omit the dependence of the variou quantitie on the frequency ω from now on. We aume that the field in the aperture i a Gauian Schell-model field with uniform intenity A 2 [21, Sec. 5.3.2]; i.e., W 0 r 0 ; r 00 W 0 ρ 0 ; ρ 00 (8) A 2 exp ρ 00 ρ 0 2 2σ 2 ; (9) where ρ x; y i the two-dimenional tranvere vector that pecifie the poition of a point Q on S and σ i a poitive Fig. 2. (Color online) Claical Gouy phae δ z along the optical axi for a determinitic (i.e., fully coherent) wave field. In thi example, a 1 cm, f 2 cm, and λ 0.6328 μm. Here the ubcript μ indicate that thi definition pertain to the phae of the correlation function, i.e., that of the pectral degree of coherence or, equivalently, that of the cro-pectral denity. The reference phae kz 1 and kz 2 are thoe of a plane wave of frequency ω kc, with c the peed of light, at poition z 1 and z 2, repectively. In contrat to the claical Gouy phae, definition (13) involve the phae of a two-point correlation function rather than that of a determinitic wave field that only depend on a ingle patial variable. In addition, two reference phae are taken into account intead of one. Let u take the firt obervation point at the origin O; i.e., z 1 0. Then Eq. (12) reduce to δ μ 0;z 2 arg 0 0 exp ρ 02 ρ 002 2σ 2 expf ik z 2 ρ 002 2f 2 g I 0 ρ 0 ρ 00 σ 2 ρ 0 ρ 00 dρ 0 dρ 00 : (14) Example of the generalized Gouy phae are hown in Fig. 3 for different value of the tranvere coherence length σ. It i een that δ μ 0;z 2 exhibit an anomalou phae behavior that i quite imilar to that of determinitic field, with the phae near focu undergoing a rapid phae change of π. In addition, the generalized Gouy phae obey the following relation:
Pang et al. Vol. 29, No. 6 / June 2012 / J. Opt. Soc. Am. A 991 On making ue of Eq. (3) and (17) and etting z 1 0, we ee that the generalized Gouy phae Eq. (13) reduce to δ μ 0;z 2 arg U z 2 kz 2 π 2; (18) which i, apart from an inconequential contant, the claical definition, Eq. (2). Furthermore, in the coherent limit σ Eq. (14) can be olved analytically and we obtain the reult that near z 0 δ μ 0;z 2 kz 2 a 2 4f 2 ; (19) which i identical to the Gouy phae behavior of determinitic wave a dicued in connection with Fig. 2.In[5] it wa dicued how the phyical origin of the claical Gouy phae can be explained by a tationary phae argument. A imilar explanation hold for the generalized Gouy phae. It i well known that the fringe pacing in interference microcopy i typically irregular, and it depend on both the numerical aperture (NA) and the apodization [23,24]. It ha alo recently been etablihed that the patial coherence of the incident field play a role, although it treatment ha been empirical to date. To quantitatively invetigate the effect of patial coherence on interference fringe pacing (and, ultimately, on interference metrology) and the role that the generalized Gouy phae play, we conider the Linnik microcope [6]. In uch a two-beam configuration, the field at two different poition along the z axi are combined, producing a fringe pattern ( interferogram ). On making ue of Eq. (10) we can write the pectral denity of thi uperpoition a Fig. 3. (Color online) Generalized Gouy phae δ μ 0;z 2 of a partially coherent field for different value of the tranvere coherence length of the field in the aperture, namely, σ 0.5 cm 1, 2, and 3 cm. In all example, the aperture radiu a 1 cm, the focal length f 2 cm, and the wavelength λ 0.6328 μm. δ μ 0; 0 0; (15) δ μ 0;z 2 δ μ 0; z 2 0; (16) which are the tatitical analog of Eq. (3) and (4) for the determinitic cae. In fact, apart from a π 2 offet, which can be traced back to the prefactor i in Eq. (1), they are identical. On the other hand, there are ome triking difference. For intance, the modulation depth of the generalized Gouy phae i dependent on the tranvere coherence length of the incident field. It i mall for incoherent field and increae in ize a the coherence length i increaed. In addition, the generalized Gouy phae ha region of both poitive and negative lope, unlike the coherent cae for which the lope i alway negative. The implication of thi for interference experiment will be dicued hortly. That the claical phae anomaly i a pecial cae of the generalized Gouy phae follow from conideration of a determinitic wave field. For uch a field, the enemble average reduce to a ingle realization, and the cro-pectral denity of Eq. (6) factorize into the form W z 1 ;z 2 U z 1 U z 2 : (17) I z ju 0 U z j 2 ; (20) p S 0 S z 2 S 0 S z Re μ 0;z ; (21) which i commonly known a the pectral interference law [21, Sec. 4.3]. It i clear from Eq. (21) that for an interferogram, in which the pectral denity of the uperpoition i recorded a a function of the ditance z, the pacing of the enuing fringe i determined by both S z, the pectral denity, and μ 0;z, the pectral degree of coherence of the field. In our model, the latter i characterized by a ingle parameter, namely the tranvere coherence length σ of the field in the aperture. A wa een in Fig. 3, thi parameter ha a ignificant influence on the phae behavior of the pectral degree of coherence near focu. For low-na field, S z i a lowly varying function compared to μ 0;z, which varie inuoidally on the cale of the wavelength. For high-na field, however, S z change much fater and the maxima of the interference term in Eq. (21) are no longer coincident with thoe of Re μ 0;z. To quantify the effect of the tate of coherence of the incident field on the interference proce, we have computed the pacing of the fringe for three cae, each with the ame (relatively high) NA and varying degree of patial coherence: σ a 0.5, σ a 1, and σ a 50. The reult are lited in Table 1 for the firt 11 fringe. A can be een, in all three cae, the pacing of the firt everal fringe, which are primarily dictated by μ 0;z, are larger than the free-pace wavelength. Thi increaed pacing wa dicued earlier for the coherent cae, and it i due to the behavior of the Gouy
992 J. Opt. Soc. Am. A / Vol. 29, No. 6 / June 2012 Pang et al. Table 1. Fringe Spacing for Three Value of the Tranvere Coherence Length σ a # σ a 0.5 σ a 1 σ a 50 1 0.6675 0.6702 0.6730 2 0.6671 0.6699 0.6730 3 0.6662 0.6695 0.6730 4 0.6642 0.6683 0.6729 5 0.6602 0.6657 0.6724 6 0.6509 0.6582 0.6702 7 0.6284 0.6311 0.6560 8 0.5944 0.5579 0.4647 9 0.6044 0.5926 0.5975 10 0.6368 0.6465 0.6652 11 0.6519 0.6601 0.6704 a In all cae, the aperture radiu a 1 cm, the focal length f 2 cm, and the free-pace wavelength λ 0.6328 μm. phae. Accordingly, if the fringe pacing were due olely to μ 0;z, in the coherent cae we would expect them to be identical except when the region between the correponding intenity maxima contain a phae dicontinuity of the Gouy phae. That thi i not the cae i due to the fact that the pectral denity S z modulate the pectral degree of coherence [in Eq. (21)] and diplace additional maxima in the neighborhood of the dicontinuitie. By contrat, for the partially coherent cae, a greater number of fringe are inherently affected near the phae jump of the generalized Gouy phae. Thi i becaue the tranition at the jump i more gradual (i.e., not a true dicontinuity). Furthermore, a the field become le coherent, the ize of the jump (i.e., the modulation depth) decreae and the tranition near the jump become moother. Therefore, the fringe pacing i highly irregular in all three cae, with the maximum fringe diplacement (#8) occurring for the coherent cae (σ a 50) and the maximum fringe variation (greater number of affected fringe) and mallet fringe diplacement occurring for the leat coherent cae. The maximum fringe diplacement, given by the eighth fringe in each cae, are 0.5944, 0.5579, and 0.4647, from leat coherent to mot coherent. In Fig. 4 and 5, we have plotted the interferogram correponding to the firt and third cae in Table 1 (σ a 0.5 and σ a 50). It i een in Fig. 4 that the fringe pacing of the fully coherent field (dahed red curve) i initially omewhat larger than that of the partially coherent field (olid blue curve). However, Fig. 5 how that for larger value of z 2 the fringe of the fully coherent field move cloer together and the maxima of the Fig. 4. (Color online) Interferogram for a coherent field σ a 50 (dahed red curve) and for a partially coherent field σ a 0.5 (olid blue curve). In both cae, a 1 cm, f 2 cm and λ 0.6328 μm. Fig. 5. (Color online) Same a Fig. 4, but for larger value of the axial poition z 2. fringe pattern go from trailing the partially coherent cae to leading it. Thi tranition occur around z 2 4.5 μm, which i preciely the point where the lope of the generalized Gouy phae change from being negative to being poitive (ee the top panel of Fig. 3). Near z 2 6.0 μm, the ign of thi lope change again and the fringe pacing of the partially coherent field again become maller than that of the fully coherent field. The lope of the claical Gouy phae (a hown in Fig. 2) i, apart from the dicontinuitie at the axial phae ingularitie, alway negative. Therefore, uch an effect doe not occur for coherent field. In concluion, we have defined a generalized Gouy phae for partially coherent field. In contrat to it traditional counterpart, thi phae pertain to the pectral degree of coherence, a two-point correlation function, rather than to the phae of a determinitic wave field that depend only on a ingle point. It wa hown that the claical phae anomaly i a pecial cae of the generalized Gouy phae. The generalized Gouy phae wa examined numerically and analytically for the broad cla of Gauian-correlated field. It wa demontrated that our finding have important implication for metrology with partially coherent field. REFERENCES 1. L. G. Gouy, Sur une propriété nouvelle de onde lumineue, C.R. Acad. Sci. 110, 1251 1253 (1890). 2. L. G. Gouy, Sur la propagation anomale de onde, Ann. Chim. Phy. 24, 145 213 (1891). 3. E. H. Linfoot and E. Wolf, Phae ditribution near focu in an aberration-free diffraction image, Proc. Phy. Soc. B 69, 823 832 (1956). 4. M. Born and E. Wolf, Principle of Optic, 7th (expanded) ed. (Cambridge Univerity, 1999). 5. T. D. Vier and E. Wolf, The origin of the Gouy phae anomaly and it generalization to atigmatic wavefield, Opt. Commun. 283, 3371 3375 (2010). 6. G. S. Kino and T. R. Korle, Confocal Scanning Optical Microcopy and Related Imaging Sytem (Academic, 1996). 7. M. W. Beijerbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Atigmatic laer mode converter and tranfer of orbital angular momentum, Opt. Commun. 96, 123 132 (1993). 8. G. Lamouche, M. L. Dufour, B. Gauthier, and J.-P. Monchalin, Gouy phae anomaly in optical coherence tomography, Opt. Commun. 239, 297 301 (2004). 9. T. Klaaen, A. Hoogeboom, M. P. van Exter, and J. P. Woerdman, Gouy phae of nonparaxial eigenmode in a folded reonator, J. Opt. Soc. Am. A 21, 1689 1693 (2004). 10. F. Lindner, W. Stremme, M. G. Schätzel, F. Grabon, G. G. Paulu, H. Walther, R. Hartmann, and L. Strüder, High-order harmonic generation at a repetition rate of 100 khz, Phy. Rev. A 68, 013814 (2003).
Pang et al. Vol. 29, No. 6 / June 2012 / J. Opt. Soc. Am. A 993 11. J. F. Federici, R. L. Wample, D. Rodriguez, and S. Mukherjee, Application of terahertz Gouy phae hift from curved urface for etimation of crop yield, Appl. Opt. 48, 1382 1388 (2009). 12. M.-S. Kim, T. Scharf, S. Mühlig, C. Rocktuhl, and H. P. Herzig, Gouy phae anomaly in photonic nanojet, Appl. Phy. Lett. 98, 191114 (2011). 13. W. Wang, A. T. Friberg, and E. Wolf, Focuing of partially coherent light in ytem of large Frenel number, J. Opt. Soc. Am. A 14, 491 496 (1997). 14. A. T. Friberg, T. D. Vier, W. Wang, and E. Wolf, Focal hift of converging diffracted wave of any tate of patial coherence, Opt. Commun. 196, 1 7 (2001). 15. B. Lü, B. Zhang, and B. Cai, Focuing of a Gauian Schellmodel beam through a circular len, J. Mod. Opt. 42, 289 298 (1995). 16. T. D. Vier, G. Gbur, and E. Wolf, Effect of the tate of coherence on the three-dimenional pectral intenity ditribution near focu, Opt. Commun. 213, 13 19 (2002). 17. G. Gbur and T. D. Vier, Can coherence effect produce a local minimum of intenity at focu? Opt. Lett. 28, 1627 1629 (2003). 18. T. van Dijk, G. Gbur, and T. D. Vier, Shaping the focal intenity ditribution uing patial coherence, J. Opt. Soc. Am. A 25, 575 581 (2008). 19. S. B. Raghunathan, T. van Dijk, E. J. G. Peterman, and T. D. Vier, Experimental demontration of an intenity minimum at the focu of a laer beam created by patial coherence: application to the optical trapping of dielectric particle, Opt. Lett. 35, 4166 4168 (2010). 20. G. Gbur and T. D. Vier, The tructure of partially coherent field, in Progre in Optic, Vol. 55, E. Wolf, ed. (Elevier, 2010), pp. 285 341. 21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optic (Cambridge Univerity, 1995). 22. D. G. Ficher and T. D. Vier, Spatial correlation propertie of focued partially coherent light, J. Opt. Soc. Am. A 21, 2097 2102 (2004). 23. J. T. Foley and E. Wolf, Wave-front pacing in the focal region of high-numerical-aperture ytem, Opt. Lett. 30, 1312 1314 (2005). 24. K. Creath, Calibration of numerical aperture effect in interferometric microcope objective, Appl. Opt. 28, 3333 3338 (1989).