Fractional Gouy phase - OSA Publishing

6 downloads 0 Views 349KB Size Report
2Department of Diagnostic Radiology, National University of Singapore, 5 Lower Kent Ridge Road, Singapore. 119074, Singapore. *Corresponding author: ...
June 15, 2008 / Vol. 33, No. 12 / OPTICS LETTERS

1363

Fractional Gouy phase Elijah Y. S. Yew1 and Colin J. R. Sheppard2,* 1

Division of BioEngineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore 2 Department of Diagnostic Radiology, National University of Singapore, 5 Lower Kent Ridge Road, Singapore 119074, Singapore *Corresponding author: [email protected] Received February 15, 2008; revised May 5, 2008; accepted May 7, 2008; posted May 21, 2008 (Doc. ID 92754); published June 13, 2008 In general, the total Gouy phase shift has the form n␲, where n need not be an integer. As a result of the Fourier transforming property of a lens, the Gouy phase is found to be related to the types of discontinuities at the upper or lower range of the pupil function Q共c兲 resulting from the asymptotic order of the Fourier transform. The sign of the Gouy phase is also related to the slope of the pupil function. The oscillations of the Gouy phase shift arise from the strength of the nondominant discontinuity. © 2008 Optical Society of America OCIS codes: 260.0260, 350.5030.

The Gouy phase [1] has remained a mystical and intriguing topic since it was first described. Several papers have attempted to explain it based on classical or quantum theories [2–6]. The phase shift is related to the spacing of phase fronts in a focused beam [7] and is hence practically important in optical cavities [4,8] and interferometry [9,10]. It also plays a role in the properties of second-harmonic generation by focused beams [11–13]. For the time-frequency case, the frequency response is the Fourier transform of the impulse response. If the impulse response is square integrable and one sided, this is sufficient to ensure that the frequency response is analytic in the upper half of the complex-frequency plane, and the real and imaginary parts of the frequency response related through the Hilbert transform [14]. For the spatial case of an axially symmetric aperture, in polar coordinates the radius is positive only, and a simple Fourier transform relationship between the axial amplitude and the aperture function Q共c兲 exists [15]. Again the axial amplitude is analytic if Q共c兲 is square integrable. This property has been used for phase retrieval [16]. For Laguerre–Gaussian beams it is known that the Gouy phase is 共2p + l + 1兲 ␲ / 2 [17]. This can be explained as the asymptotic behavior of a Fourier transform is related to the order of the discontinuity [18], in this case at zero radius. For the one-dimensional case of a hard-edged aperture of finite support, the axial amplitude is in agreement with the classical resolution limit. For one-dimensional Hermite–Gaussian beams, the fall-off of the Gaussian is fast enough to ensure that the axial amplitude is analytic. In the absence of evanescent waves, the angular spectrum is necessarily of finite support for a nonparaxial system, thus allowing the phase to be directly retrieved [19]. Our conclusion is that the Gouy phase is a direct result of the discontinuities in the angular spectrum. Recently there has been interest in the behavior of beams in the nonparaxial regime, including the effects of inhomogeneous polarization. For a lens illuminated with linear polarization, the field on axis in the focal region of a lens is also linearly polarized and is given by [20], 0146-9592/08/121363-3/$15.00

E共z兲 =







A共␪兲exp共ikz cos ␪兲d␪.

共1兲

Putting c = cos ␪, this can be written as E共z兲 =



cos ␤

Q共c兲exp共ikzc兲dc,

cos ␣

共2兲

where Q共c兲 depends on the apodization of the lens and cos ␤ is taken as 1 or 0.5 for a clear aperture or a central obstruction, respectively. If Q共c兲 is a continuous function, then the asymptotic behavior of E共z兲 is determined by the discontinuities at the limits of the integral. For simple examples we have Q共c兲 = 冑c共1 + c兲/2,

共3兲

Q共c兲 = 共1 + c兲/2,

共4兲

Q共c兲 = 共1 + c兲2/4

共5兲

for the cases of an aplanatic system (A), a system satisfying the Herschel condition (H), or the mixed dipole (crossed electric and magnetic dipole) fields

Fig. 1. (Color online) Plot of Q共c兲 for A, H, and MD: (a) NA 1, no central obstruction; (b) NA 0.95, with a central obstruction (see text); (c) NA 0.95, no central obstruction; and (d) NA 0.95 with a central obstruction (see text). © 2008 Optical Society of America

1364

OPTICS LETTERS / Vol. 33, No. 12 / June 15, 2008

Fig. 2. (Color online) Phase shift for (1) A, (2) H, and (3) MD: (a) Gouy phase shift with NA 1, no central obstruction; (b) Gouy phase shift with NA 1, with central obstruction (see text); (c) Gouy phase shift NA 0.95, with no central obstruction; and (d) Gouy phase shift with NA 0.95, with central obstruction (see text).

Fig. 3. (Color online) Plot of Q(c) for (1) AU, (2) AL, (3) HU, and (4) HL: (a) NA 1, no central obstruction; (b) NA 1, with a central obstruction (see text); (c) NA 0.95, no central obstruction; and (d) NA 0.95, with a central obstruction (see text).

(MD), respectively [21]. These functions are plotted in Fig. 1 for systems of numerical aperture (NA) 1 and 0.95, with and without a central obscuration. It is seen that these all exhibit a discontinuity at each boundary, with the greater value occurring at the upper value of c. For the aplanatic case with NA= 1, the lower boundary exhibits a discontinuity in slope. The asymptotic phase behavior is determined by the dominant discontinuity, with the other discontinuity, being a smaller phasor, resulting in the phase oscillating about a limiting value. In Fig. 2 the total Gouy phase shift through focus is −␲. The phase oscillations are weaker for the aplanatic case, as expected from the form of the pupil function. The behavior is more varied for the currently topical case of a lens illuminated with radial polarization [22]. The field on axis is now purely longitudinal, with

phase is 共m + 1兲␲, where m can now take non-rational values. Thus the order of the discontinuity can take non-rational values. The total Gouy phase shift is determined by the behavior of the pupil function at its discontinuities and not on the complete shape of the pupil function. Thus the Gouy phase shift of a Laguerre–Gauss beam is not a direct effect of the angular phase variation in the angular spectrum but of the limiting form of the pupil function near the axis. This can be seen from the fact that a central obscuration, however small, completely alters the Gouy phase behavior. On the other hand, the behavior of the phase near the focus does depend on the shape of the pupil function. At the focus we can write from Eq. (2) for the gradient of the phase ␸ and the gradient of the excess phase ⌽ = kz − ␸, assuming Q共c兲 is real,

Q共c兲 = 冑c共1 + c2兲,

共6兲

Q共c兲 = 冑c共1 − c2兲,

共7兲

Q共c兲 = 冑共1 − c2兲,

共8兲

Q共c兲 = 1 − c2 ,

共9兲

for the cases of aplanatic uniform (AU), a truncated (overfilled) aplanatic Laguerre (AL), Herschel uniform (HU), and Herschel–Laguerre (HL) [23]. The functions are illustrated in Fig. 3 and the Gouy phase in Fig. 4. For some cases the dominant discontinuity is at the lower boundary, and the Gouy phase is positive. In general there is a Gouy phase shift of ±n␲, where n is the order of the discontinuity. It is 23 for a corner that changes its slope from vertical to horizontal (Type I) as seen in Figs. 3 and 4. Another type of discontinuity in slope is of order 2 [18] (Type II). The results are summarized in Table 1. Eqs. (7)–(9) are all of the form Q共c兲 = cm共1 − c2兲n .

共10兲

For a NA of 1, if we take n = 1 so that the lower boundary is dominant if m ⬍ 1, we find that the Gouy

d␸ dz

=

kq1 q0

,

共11a兲

Fig. 4. (Color online) Gouy phase shift for (1) AU, (2) AL, (3) HU, and (4) HL: (a) Gouy phase shift for NA 1, no central obstruction; (b) Gouy phase for NA 1 with a central obstruction (see text); (c) Gouy phase shift for NA 0.95, no central obstruction; (d) Gouy phase shift for NA 0.95 with central obstruction (see text).

June 15, 2008 / Vol. 33, No. 12 / OPTICS LETTERS Table 1. Relationship between the Type of Discontinuity and the Gouy Phase Shifta Discontinuity Function Slope (Type I) Slope (Type II)

Order

Gouy Phasea

s−1

±␲ ± 3␲ / 2 ±2␲

s

−3 / 2

s−2

a

The sign of the Gouy phase shift is dependent on the location of the dominant discontinuity.

d⌽ dz

=k

冉 冊 q0 − q1 q0

,

共11b兲

where q0 and q1 are the zero and first moments of the pupil Q共c兲, respectively, so that 共q1 / q0兲 is the value of vc (the speed of light) at the center of gravity of the pupil. The phase gradient along the axis is related to the group velocity, vg = 关共vcq0兲 / q1兴. It is well known that the group velocity exhibits regions where it is greater or smaller than vc along the axis for an aplanatic lens of high NA [24,25]. In conclusion, the Gouy phase shift is related to the discontinuity of the pupil function and has the form n␲, where n need not be an integer (nor even rational). The sign of the Gouy phase depends on the location of the dominant discontinuity. The results indicate that Gouy phase engineering is possible and will be useful in harmonic generation [11–13], particle delivery [26], acceleration [27], chemical reactions [28], and superresolution [29]. References 1. L. G. Gouy, C. R. Acad. Sci. 110, 1251 (1890). 2. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. London B 69, 823 (1956). 3. S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001). 4. T. Ackemann, W. Grosse-Nobis, and G. I. Lippi, Opt. Commun. 189, 5 (2001). 5. Q. Zhan, Opt. Commun. 242, 351 (2004). 6. P. Hariharan and P. A. Robinson, J. Mol. Spectrosc. 43, 219 (1996).

1365

7. J. T. Foley and E. Wolf, Opt. Lett. 30, 1312 (2005). 8. C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998). 9. F. R. Tolman and J. G. Wood, J. Sci. Instrum. 33, 236 (1956). 10. C. J. R. Sheppard and K. G. Larkin, Appl. Opt. 34, 4731 (1995). 11. J. Mertz and L. Moreaux, Opt. Commun. 196, 325 (2001). 12. J. X. Cheng and X. S. Xie, J. Opt. Soc. Am. B 19, 1604 (2002). 13. S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, J. Opt. Soc. Am. B 23, 2134 (2006). 14. E. C. Titchmarsh, Introduction to Fourier Integrals (Oxford U. Press, 1948), 2nd ed., pp. 119–131. 15. C. J. R. Sheppard and Z. S. Hegedus, J. Opt. Soc. Am. A 5, 643 (1988). 16. H. J. Matthews, D. K. Hamilton, and C. J. R. Sheppard, J. Mol. Spectrosc. 36, 233 (1989). 17. H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966). 18. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1965), p. 253. 19. K. G. Larkin and C. J. R. Sheppard, J. Opt. Soc. Am. A 16, 1838 (1999). 20. B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959). 21. C. J. R. Sheppard and K. G. Larkin, J. Mol. Spectrosc. 41, 1495 (1994). 22. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Opt. Commun. 179, 1 (2000). 23. C. J. R. Sheppard and E. Y. S. Yew, Opt. Lett. 33, 497 (2008). 24. C. J. R. Sheppard and H. J. Matthews, J. Opt. Soc. Am. A 4, 1354 (1987). 25. L. J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 277 (2000). 26. Y. Q. Zhao, Q. W. Zhan, Y. L. Zhang, and Y. P. Li, Opt. Lett. 30, 848 (2005). 27. C. Varin and M. Piche, J. Opt. Soc. Am. A 23, 2027 (2006). 28. V. J. Barge, Z. Hu, J. Willig, and R. J. Gordon, Phys. Rev. Lett. 97, 263001 (1006). 29. A. I. Whiting, A. F. Abouraddy, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, Opt. Express 11, 1714 (2003).