Analysis of the propagation dynamics andGouy phase of Airy beams using the fast

Fresnel transform algorithm

Don M. Cottrell, Jeffrey A. Davis,* Cassidy A. Berg, and Christopher Li FreemanDepartment of Physics, San Diego State University, San Diego, California 92182-1233, USA

*Corresponding author: jeffrey.davis@mail.sdsu.edu

Received 6 November 2013; revised 28 January 2014; accepted 17 February 2014;posted 25 February 2014 (Doc. ID 200773); published 27 March 2014

There is great interest in Airy beams because they appear to propagate in a curved path. These beams areusually generated by inserting a cubic phase mask onto the input plane of a Fourier transform system.Here, we utilize a fast Fresnel diffraction algorithm to easily derive both the propagation dynamics andthe Gouy phase shift for these beams. The trajectories of these beams can be modified by addingadditional linear and quadratic phase terms onto the cubic phase mask. Finally, we have rewrittenthe equations regarding the propagating Airy beams completely in laboratory coordinates for use byexperimentalists. Experimental results are included. We expect that these results will be of greatimportance in applications of Airy beams. © 2014 Optical Society of AmericaOCIS codes: (050.1940) Diffraction; (350.5500) Propagation.http://dx.doi.org/10.1364/AO.53.002112

1. Introduction

Accelerating Airy and parabolic beams have at-tracted a great deal of interest because they appearto travel with an invariant shape in a curved pathwith ballistic dynamics similar to objects underthe influence of gravity [1–7]. Recently, radial varia-tions of these beams have been created that produceabruptly focusing beams, where the energy densityincreases suddenly near the focus point [8–12].

These beams are usually generated by encoding acubic phase pattern onto the input plane of a Fouriertransform system. The propagation of the beam isthen followed in the region past the output planeof the Fourier transform system. In effect, we arestudying the Fresnel diffraction of the Airy beamas it propagates from this output plane. However,the mathematics of the propagation of these beamscan be difficult [13] and become more challenging

when additional optical elements are inserted intothe optical train.

In some recent work [14], we have been exploringan elegant algorithm for describing the Fresneldiffraction process. Sypek [15] named this algorithmthe “convolution approach,” while other authors[16,17] referred to it as the “angular spectrum ap-proach.” This algorithm uses two Fourier transformoperations with a lens operation (quadratic chirp)inserted between them. Much of our interest con-cerns the increase in computational speed providedby the discrete fast Fourier transform (DFFT) oper-ation for Fresnel diffraction computations. We used aray matrix approach [14] for establishing the dis-tance ranges under which this algorithm would bevalid when using a computational DFFT algorithm.

However, we realized that this algorithm isextremely powerful and can be applied to theoreticalanalyses in addition to computational ones. In thiswork, we apply the algorithm to the case of acceler-ated Airy beams.We show that it easily allows for thederivation of the propagation dynamics and Gouyphase shift [18,19] of these beams. We also realized

1559-128X/14/102112-05$15.00/0© 2014 Optical Society of America

2112 APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014

that this formalism could be applied to moreadvanced investigations of these beams, where weintroduce additional linear and quadratic phaseterms onto the cubic phase term that representsthe Fourier transform of the accelerated Airy beam.As a consequence, we generate expressions for theinfluence of these terms on the trajectory of theaccelerated Airy beams. Since the radially symmetricmasks used in abruptly focusing Airy beams can bewritten with the same notation, we find that we canapply this formalism to these kinds of beams as well.

Finally, we have rewritten the equations regardingthe propagating Airy beams completely in laboratorycoordinates for use by experimentalists.

2. Review of the Fast Fresnel Algorithm

We begin with a brief review of the Fresnel diffrac-tion algorithm and use a one-dimensional notationfor convenience. We begin with an input functiong�u�. The Fresnel diffraction pattern gF�u; z� formedat a distance z from the input screen is given [15–17]by

gF�u; z� � I−1�Z��ξ; z�G�ξ�� � I−1�Z��ξ; z�Ifg�u�g�:(1)

This approach of Eq. (1) can be easily understoodas follows. First, the Fourier transform G�ξ� of theinput function g�u� is multiplied by a paraxial propa-gation transfer function. We identify this term as aconverging lens function represented as Z��ξ; z�,whose focal length FZ is related to the propagationdistance z [14]. Finally, the inverse Fourier transformof the product is performed. We will discuss thisnotation in much more detail in Section 5. For com-putational purposes, this algorithm is much fasterthan the Fresnel diffraction algorithm because itinvolves two fast Fourier transform operations.

We can apply this algorithm to the Fresnel diffrac-tion of the Airy beam from the output plane of theoptical Fourier transform system to the viewingplane located at distance z from this focal plane.

3. Application of the Fast Fresnel DiffractionAlgorithm to Airy Beams

We begin with the recognition that the Airy beam isgiven by the Fourier transform of the cubic phasefunction [1–3] as

Ai�u� �Z

∞

−∞exp�iaξ3� exp�i2πuξ�dξ: (2)

Here, we neglect a number of constants for simplicitythat affect the amplitude and spatial scale of thebeam, but do not affect the results of this work.

However, this expression is very useful for exam-ining the Fresnel diffraction propagation of the Airybeam function because the Fourier transform of theAiry beam function that is used in Eq. (1) is given byIfg�u�g � IfAi�u�g � exp�iaξ3�. As before, u and ξ arethe Fourier transform variables. The variable a is

related to the extent of the cubic phase shift overthe input plane and will be discussed in terms ofthe experimental system parameters in Section 5.

Now we can examine the Fresnel diffraction of thisAiry beam through a distance z. The converging lensin Eq. (1) is given [14] as Z��ξ; z� � exp�−iπξ2∕λFZ� �exp�−ibξ2�, where b � π∕λFZ.

Now we insert the expressions for the Fouriertransform of the Airy beam and the lens function intoEq. (1) to obtain

AiF�u; z� �Z

∞

−∞exp�i�aξ3 − bξ2�� exp�i2πuξ�dξ: (3)

The key to this derivation is to form a cubic term inthe exponent by adding and subtracting terms. Thiscan be simplified to obtain the following:

AiF�u; z� � exp�ia�b3a

�3� Z

∞

−∞exp

�ia�ξ −

�b3a

�3��

× exp�i2πξu� exp�−i3aξ

�b3a

�2�dξ: (4)

Because the integral extends from −∞ to∞, we candefine a new variable as η � ξ − b∕3a. The integralcan now be rewritten more simply as

AiF�u; z� � exp�i�2πbu3a

�− i2a

�b3a

�3�Z

∞

−∞exp�iaη3�

× exp�i2πη

�u −

b2

6πa

��dη: (5)

The integral portion of Eq. (5) represents theFourier transform of a cubic phase [as in Eq. (2)]and yields an Airy beam whose origin varies withpropagation distance as

AiF�u; z� � exp�i�2πbu3a

�− i2a

�b3a

�3�Ai

�u −

b2

6πa

�:

(6)

This expression has two components. The secondterm shows that the center of the Airy beam deflectsquadratically with propagation distance as

u � b2

6πa: (7)

The first term is identified as the Gouy phase andis usually interpreted as the phase of the beamrelative to a plane wave. Although it is useful ininterferometric or remote-sensing applications, it iscommonly seen in the output of laser resonators,where the different TEMlmq modes have oscillationfrequencies depending on the laser cavity configura-tion [20].

In the case of Airy beams, the authors of [18] havedefined the Gouy phase as the phase of the beamrelative to a diverging cylindrical wave. We would

1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS 2113

expect that the Gouy phase shift would be most use-ful when measured along the path of maximumintensity for the Airy beam as given by Eq. (7). Usingthis assumption, the Gouy phase is given byEq. (8) as

ΦG � b3

27a2 : (8)

These results show that the propagation dynamicsand the Gouy phase shift of the accelerating Airybeams are easily derived using the fast Fresnelalgorithm.

4. Extension to Ballistic Dynamics and Inclusion ofLens Functions onto the Cubic Phase Term

In developing this work, we found that we could addadditional terms to the cubic phase term and foundinteresting consequences. Here, we add a linearphase term as exp�i2πu0ξ�, where δ � 1∕u0 is theperiod of the linear phase term and a quadratic lensterm as exp � �icξ2� � exp�−iπξ2∕λF�. Here, (F > 0)represents a converging lens onto the cubic phaseterm and c � π∕λF. As a result, Eq. (3) can be rewrit-ten as

AiF�u; z� �Z

∞

−∞exp�i�aξ3 − bξ2 − cξ2��

× exp�if2π�u� u0�ξgdξ: (9)

By combining terms and defining B � b� c �π∕λFZ � π∕λF and U � u� u0, we can obtain anequivalent version to Eq. (3) as

AiF�u; z� �Z

∞

−∞exp�i�aξ3 − Bξ2�� exp�i2πUξ�dξ: (10)

We follow the previous steps to obtain a cubic termin the exponent, and define a new variable asη � ξ − B∕3a. As a result, we obtain a variation ofEq. (6) that produces an Airy beam whose origin isshifted as below:

AiF�u; z� � exp�i�2πBu3a

�− i2a

�B3a

�3�Ai

�U −

B2

6πa

�:

(11)

As with Eq. (6), this expression has two compo-nents. The displacement of the propagating Airybeam is now given by

U � u − u0 � B2

6πa: (12)

The Gouy phase in Eq. (11) can again be examinedalong the propagation path of the Airy beam as givenin Eq. (12) and gives a new result as

ΦG � B3

27a2 . (13)

Equation (12) correctly shows that the lateral pathof the Airy beam will be offset by the linear phaseterm. This has been experimentally demonstrated[12,21], and we will neglect this term for the remain-der of this work.

However, both the propagation dynamics and theGouy phase shift for the Airy beam are affected bythe external lens function as shown by the variableB � b� c � π∕λFZ � π∕λF. These results agree withthose using a ray matrix analysis [12] reportedearlier.

5. Experimental and Theoretical Results

Before discussing our results, the equations must beconverted into a laboratory frame of reference. Thisimportant process has not been discussed to ourknowledge except in [12].

We first clarify that the variable u in Eq. (1) mea-sures the translational distance away from the originof the optical system in the propagation of the Airybeam, while the variable ξ represents the transversedimension in the Fourier transform plane (where thecubic phase term is encoded).

The translational variable u can be defined interms of the laboratory output plane as

u � sin θ∕λ � x∕λf FT: (14)

Here, f FT represents the focal length of the lens inthe Fourier transform system and x represents thetransverse shift of the Airy beam from the opticalaxis in the laboratory frame.

Next, we showed in previous work [Eq. (14)] thatthe focal length FZ of the converging lens functionin the Fresnel diffraction algorithm in Eq. (1) is re-lated to the propagation distance z and the focallength of the Fourier transform lens f FT by

FZ � f 2FT∕z: (15)

Finally we normalize the cubic phase over thewidth of the spatial light modulator (SLM) so thatthe maximum value of the phase is defined as βradians at the edge of the SLM. This edge is defined,where ξ � NΔ∕2. Here,N is the number of pixels andΔ is the pixel size. Consequently, we define

a � β∕�NΔ∕2�3 � 8β∕�NΔ�3: (16)

Combining all of these definitions, Eq. (12) can berewritten as

x � π�NΔ�348λf 3FTβ

�z� f 2FT

F

�2

: (17)

This expression shows that the origin of theaccelerating beam is shifted by a displacement of

2114 APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014

z � f 2FT∕F, where F can be either a converging or adiverging lens.

This approach represents a third technique foraltering the ballistic dynamics of an acceleratingbeam. The authors of Ref. [3] adjusted the trajectoryby translating the position of the Fourier lens, whilethe authors of Refs. [5,7] adjusted the position of thecubic phase on the SLM. In this newest approach, wesimply program a lens function onto the cubic phasepattern.

The experimental system has been well docu-mented [3,6]. The cubic phase is encoded onto theinput plane of an optical Fourier transform system,and the Airy beam is formed in the back focal plane.For the experiments, we used two-dimensional cubicphase masks as reported earlier [5,6]. This causesthe beam to be accelerated in both the x and y direc-tions. Linearly polarized light from a He–Ne laser isspatially filtered, expanded, and collimated. Thephase patterns are encoded onto a CRL ModelXGA-3 liquid crystal display having 1074 × 768pixels with a pixel spacing of Δ � 18 μm. The deviceis set up for phase operation using polarizationeigenvectors [22]. Our Fourier transform lens hasa focal length of f FT � 52.66 cm and is located a focallength away from the cubic phase pattern encodedonto the SLM. The output patterns were recordedin the region of the Fourier plane located a focallength from the Fourier lens with a Sony modelXC-37 having 491 × 384 pixels on an 8.86 mm ×6.6 mm sensor.

As stated earlier, a very detailed explanation of theexperimental setup including several variations hasbeen reported in [6]. We also do not show images ofthe Airy beams because these have also been welldocumented. Our emphasis is on the trajectories ofthe beams, particularly when a lens is encoded ontothe cubic phase pattern.

Figure 1 shows a comparison of the theoreticaland experimental data using Eq. (17). In our

experiments, we used a value of β � 300 rad and dif-ferent values for the added converging lens functionof F � 160 cm and F � 240 cm. Experimental re-sults agree extremely well with theory and show thatthe focus point can be adjusted in the axial direction.

Finally we examined the theoretical evolution ofthe Gouy phase shift along the path of the Airy beam.When written in our laboratory coordinates, theGouy phase shift of Eq. (13) is given by

ΦG � B3

27a2 � �πz∕λf 2FT � π∕λF�327�8β∕�NΔ�3�2 : (18)

Figure 2 shows a comparison of the Gouy phaseusing Eq. (18). Here, we again used a value ofβ � 300 rad and different values for the added lensfunction of F � 160 cm and F � 240 cm. Here, wesee that the phase shift varies with the focal lengthof the lens. However, we did not perform experimen-tal measurements of this.

6. Conclusion

We have used an extremely elegant Fresnel diffrac-tion algorithm to derive the propagating characteris-tics of Accelerating Airy beams. This approach allowsus to incorporate additional phase patterns includinggratings and lenses onto the cubic phase required foran Airy beam that affect the ballistic dynamics of theAiry beam. We have also rewritten the equationsin terms of the experimental laboratory variables.Experimental results agree with theory. These re-sults would be valid for all of the kinds of accelerat-ing beams that are currently being studied, includingabruptly autofocusing beams.

References1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite

energy Airy beams,” Opt. Lett. 32, 979–981 (2007).2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N.

Christodoulides, “Observation of accelerating Airy beams,”Phys. Rev. Lett. 99, 213901 (2007).

Fig. 1. Comparison of theory and experiment for cases in whichthe Fourier transform lens has a focal length of 52.65 cm,β � 300 rad, and data are shown for no external lens, and for casesin which the focal length of the external lens is F � 160 cm andF � 240 cm.

Fig. 2. Computed values of the Gouy phase along the path of theaccelerating Airy beam using the parameters pertinent to ourexperimental system. Again, the Fourier transform lens has a focallength of 52.65 cm and β � 300 rad. Values are shown for noexternal lens and for the cases where the focal length of the exter-nal lens is F � 160 cm and F � 240 cm.

1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS 2115

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N.Christodoulides, “Ballistic dynamics of Airy beams,”Opt. Lett.33, 207–209 (2008).

4. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33,1678–1680 (2008).

5. J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell,“Observation of accelerating parabolic beams,” Opt. Express16, 12866–12871 (2008).

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley,and D. M. Cottrell, “Generation of accelerating Airy and accel-erating parabolic beams using phase-only patterns,” Appl.Opt. 48, 3171–3177 (2009).

7. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen,“Optimal control of the ballistic motion of Airy beams,” Opt.Lett. 35, 2260–2263 (2010).

8. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofo-cusing waves,” Opt. Lett. 35, 4045–4047 (2010).

9. D. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S.Tzortzakis, “Observation of abruptly autofocusing waves,”Opt. Lett. 36, 1842–1844 (2011).

10. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides,“Pre-engineered abruptly autofocusing beams,” Opt. Lett.36, 1890–1892 (2011).

11. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N.Christodoulides, and Z. Chen, “Fourier-space generation ofabruptly autofocusing beams and optical bottle beams,”Opt. Lett. 36, 3675–3677 (2011).

12. J. A. Davis, D. M. Cottrell, and D. Sand, “Abruptly autofocus-ing vortex beams,” Opt. Express 20, 13302–13310 (2012).

13. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,”Am. J. Phys. 47, 264–267 (1979).

14. J. A. Davis and D. M. Cottrell, “Ray matrix analysis of the fastFresnel transform with applications towards liquid crystaldisplays,” Appl. Opt. 51, 644–650 (2012).

15. M. Sypek, “Light propagation in the Fresnel region.New numerical approach,” Opt. Commun. 116, 43–48(1995).

16. D. Mendlovic, A. Zalevsky, and N. Konforti, “Computation con-siderations and fast algorithms for calculating the diffractionintegral,” J. Mod. Opt. 44, 407–414 (1997).

17. D.Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F.Marinho,“Fast algorithms for free-space diffraction patterns calcula-tion,” Opt. Commun. 164, 233–245 (1999).

18. X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airybeams,” Opt. Lett. 36, 2492–2494 (2011).

19. C. J. Zapata-Rodriguez, D. Pastor, and J. J. Miret,“Considerations on the electromagnetic flow in Airy beamsbased on the Gouy phase,” Opt. Express 20, 23553–23558(2012).

20. A. Yariv, Optical Electronics in Modern Communications, 5thed. (Oxford University, 1997), Chap. 4.6.

21. X. Z. Wang, Q. Li, and Q. Wang, “Arbitrary scanning of theAiry beams using additional phase grating with cubic phasemask,” Appl. Opt. 51, 6726–6731 (2012).

22. J. A. Davis, B. A. Slovick, C. S. Tuvey, and D.M. Cottrell, “Highdiffraction efficiency from one- and two-dimensional Nyquistfrequency binary-phase gratings,” Appl. Opt. 47, 2829–2834(2008).

2116 APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014