Page 1

Vectorial self-diffraction effect

in optically Kerr medium

Bing Gu,1Fan Ye,1Kai Lou,1Yongnan Li,1

Jing Chen,1and Hui-Tian Wang1,2,3,∗

1MOE Key Laboratory of Weak Light Nonlinear Photonics and School of Physics,

Nankai University, Tianjin 300071, China

2National Laboratory of Solid State Microstructures, Nanjing University,

Nanjing 210093, China

3htwang@nju.edu.cn

∗htwang@nankai.edu.cn

Abstract:

of a cylindrical vector field passing though an optically thin Kerr medium.

Theoretically, we obtain the analytical expression of the focal field of the

cylindrical vector field with arbitrary integer topological charge based on

the Fourier transform under the weak-focusing condition. Considering

the additional nonlinear phase shift photoinduced by a self-focusing

medium, we simulate the far-field vectorial self-diffraction patterns of

the cylindrical vector field using the Huygens-Fresnel diffraction integral

method. Experimentally, we observe the vectorial self-diffraction rings of

the femtosecond-pulsed radially polarized field and high-order cylindrical

vector field in carbon disulfide, which is in good agreement with the

theoretical simulations. Our results benefit the understanding of the related

spatial self-phase modulation effects of the vector light fields, such as

spatial solitons, self-trapping, and self-guided propagation.

We investigate the far-field vectorial self-diffraction behavior

© 2011 Optical Society of America

OCIS codes: (190.3270) Kerr effect; (190.4420) Nonlinear optics, transverse effects in;

(050.1940) Diffraction; (260.7120) Ultrafast phenomena.

References and links

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Opt. Express 17, 10970–10975 (2009).

1. Introduction

Interaction of intense laser with matter changes the refractive index of a material and could

result in a lot of spatial self-phase modulation effects, such as spatial solitons, self-trapping,

and self-guided propagation. Among these effects, the self-diffraction effect has attracted ex-

tensively interest since Callen et al. [1] observed the far-field annular intensity distribution of a

Gaussian laser beam passing through carbon disulfide in 1967. Afterwards, the far-field annu-

lar self-diffraction patterns were found in many materials, such as nematic liquid crystals [2],

azo-doped polymer film [3], absorbing self-defocusing media [4], carbon nanotube solutions

[5], hybrid materials [6], and photopolymer [7]. At the same time, the formation and evolution

of the self-diffraction patterns were extensively investigated [8–11]. Besides, the underlying

mechanisms of the novel self-diffraction phenomena, including photothermal effect [1], non-

local optical nonlinearities [12], and thermal nonlinearity with gravitational effect [5], have

been exploited. Up to now, most of investigations were devoted to exploring self-diffraction

behaviors of linearly polarized Gaussian beams. Correspondingly, the scalar self-diffraction

phenomena were widely studied.

Recently, cylindrical vector fields, which have the inhomogeneous distribution of states of

polarization in the cross-section of field, have become a subject of rapidly growing interest, due

to the unique features compared with homogeneously polarized fields and novel applications

in various realms [13]. Under the vector field excitation, many novel nonlinear optical effects,

suchassecond-harmonic generation [14],third-harmonicgeneration [15],andself-focusingdy-

namics [16], have been studied. Consequently, one may expect the appearance of novel effects

induced by the vector light fields.

In the present article, for the first time to our knowledge, we investigate the far-field vecto-

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rial self-diffraction patterns of a cylindrical vector field passing though an optically thin Kerr

medium. We theoretically obtain the focal field of the cylindrical vector field with arbitrary in-

teger topological charge and simulate the far-field vectorial self-diffraction patterns induced by

a self-focusing medium. We experimentally observe the vectorial self-diffraction rings of the

femtosecond-pulsed cylindrical vector fields in carbon disulfide. The experimental observations

are in good agreement with the theoretical simulations.

2. Theory

In general, a cylindrical vector field can be written as [13]

E(ρ,ϕ) = A(ρ)P(ϕ) = A(ρ)[cos(mϕ +ϕ0)ˆ ex+sin(mϕ +ϕ0)ˆ ey],

where ρ and ϕ are the polar radius and azimuthal angle in the polar coordinate system, respec-

tively. Here P(ϕ) is the unit vector describing the distribution of the states of polarization of

the vector field, m is the topological charge, and ϕ0is the initial phase [17]. ˆ exand ˆ eyare the

unit vectors in the Cartesian coordinate system, respectively. Interestingly, two extreme cases

of vector fields describing by Eq. (1) are the radially and azimuthally polarized vector fields

when m = 1 with ϕ0= 0 and π/2, respectively. In the case of m = 0, Eq. (1) describes the

horizontal and vertical linearly-polarized fields, for ϕ0= 0 and ϕ0= π/2, respectively. A(ρ)

stands for the amplitude distribution in the cross-section of the cylindrical vector field. Under

the uniform-intensity illumination, we have A(ρ) = A0within the region of 0 ≤ ρ ≤ a, where

a is the radius of the cylindrical vector field.

The focused field distribution of the cylindrical vector field by a lens with a focal length

of f under the weak-focusing condition can be written, according to the Fourier transform, as

follows

(1)

Ef(r,ψ) = A0

?a

0

ρdρ

?2π

0

P(ϕ)exp

?

−jkρr

f

cos(ϕ −ψ)

?

dϕ,

(2)

where k = 2π/λ is the wave vector and λ is the wavelength of the used laser in free space. A

Cartesian system (x?,y?) and a corresponding polar coordinate system (r,ψ) are attached in the

rear focal plane of the lens. After integrating Eq. (2) over ϕ for an integer m ≥ 0, we yield the

focused field distribution

Ef(r,ψ) = Bm(r)?cos(mψ +ϕ0)ˆ ex? +sin(mψ +ϕ0)ˆ ey??,

?πr

Here ω0= λ f/2a is the beam waist, E0denotes the electric field amplitude at the focus,1F2[·]

is the generalized hypergeometric function, and Amis a normalized constant obtained by the

condition of (|Ef|2/|E0|2)max= 1. The first five coefficients of Amare listed in Table 1. Setting

m = 0 in Eq. (3), the field distribution has the well-known Airy spot profile that describes the

focused field of the top-hat beam [18].

(3)

where

Bm(r) = E0

j3mAm

(m+2)m!2ω0

?m

1F2

?

m

2+1,

?m

2+2,m+1

?

;−

?πr

2ω0

?2?

.

(4)

Interestingly, the focused field profile of the cylindrical vector field is the so-called doughnut

light field with the central dark spot and a single outer bright ring, as shown in Fig. 1. This

doughnut field has found some exciting applications such as optical trapping [19] and manipu-

lating nanoobjects [20]. In addition, the radius of the doughnut field increases as the topological

charge m of the cylindrical vector field increases.

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Table 1. Coefficients Amfor different topological charges m.

m

Am

0

2

12345

3.931735.226566.386607.478318.52707

Normalized Intensity

Normalized Intensity

Normalized Intensity

Normalized Intensity

-100 10

0.0

0.5

1.0

?r/?

-100 10

0.0

0.5

1.0

?r/?

-100 10

0.0

0.5

1.0

?r/?

-10010

0.0

0.5

1.0

?r/?

m=0

m=1

m=2 m=3

0 00

0

Fig. 1. Intensity patterns (top row) and the intensity profiles along the diameter (lower row)

of the focused vector fields with different topological charges m.

For the vector field with a temporal Gaussian pulse profile, the peak intensity at the focus

can be written as I0m= 2√πln2ε/ω2

width at half maximum of the pulse duration.

As described by Eq. (3), the focused vector field belongs to a kind of local linearly polarized

vector field. For the sake of simplicity, we consider only that an optically thin isotropic Kerr

sample (its thickness is much less than the Rayleigh length of the light field) is located at the

focal plane. Correspondingly, the field Ee(r,ψ) at the exit plane of the sample can be given as

0A2

mτF, where ε is the incident energy and τFis the full

Ee(r,ψ) = Ef(r,ψ)exp[jkn2|Ef(r,ψ)|2L],

(5)

where n2is the third-order nonlinear refraction index, and L is the sample thickness. We now

define the peak nonlinear refractive phase shift at the focus, as ΔΦm= kn2I0mL.

Based on the Huygens-Fresnel diffraction integral method, we obtain the field distribution in

the far-field observation plane attached to a Cartesian system (x??,y??) and a corresponding polar

coordinate system (ra,φ), which has a distance d from the exit plane of the sample

Ea(ra,φ) =kj3m−1exp(jkd)

d

exp

?jkr2

a

2d

??cos(mφ +ϕ0)ˆ ex?? +sin(mφ +ϕ0)ˆ ey???

?jkr2

×

?∞

0

Bm(r)exp[jkn2|Bm(r)|2L]exp

2d

?

Jm

?krra

d

?

rdr.

(6)

whereJm(·)istheBesselfunctionofthefirstkindofmthorder.Intheabovetheoreticalanalysis,

we only consider the optical field under the steady-state condition. The temporal profile of the

laser pluses have been omitted. In fact, we can easily extend the steady-state results to transient

effects induced by a pulse train by using the time-averaging approximation. For the cylindrical

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vector field with a temporally Gaussian pulses, we yield the average nonlinear refractive phase

shift at the focus as ?ΔΦm? = ΔΦm/√2.

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

No Polarizer

Polarizer

m=0

m=1

m=2 m=3

Fig. 2. Far-field intensity patterns without (top row) and with (middle row) a vertical po-

larizer (lower row) of the vector fields with different topological charges m by taking

ΔΦm= π, ϕ0= 0, λ = 804 nm, ω0= 65 μm, d = 180 mm, and ΔΦm= π. The bottom

row gives the intensity profiles along the diameter of the far-field intensity patterns shown

in the top row.

To investigate the characteristics of the far-field intensity of the vector field when the non-

linear sample is located at the rear focal plane of the lens, we take the parameters as ϕ0= 0,

λ = 804 nm, ω0= 65 μm, and d = 180 mm. The top row of Fig. 2 presents the numerical sim-

ulations of the far-field patterns of the vector fields for different topological charges m at a fixed

value ΔΦm=π. For the sake of comparison, the far-field pattern of the scalar linearly-polarized

top-hat beam (i.e., the case of m=0) is also shown in the first column of Fig. 2. Compared with

the single ring structures in the far-field patterns of the vector fields in the absence of the non-

linearity, as shown in Fig. 1, the far-field patterns induced the nonlinearity exhibit the multiple

concentric ring structures, originating from the refractive-index changes self-induced by the

nonlinearity. To identify the polarization features of the far-field multiple ring patterns induced

by the nonlinearity, a vertical polarizer is used, and the corresponding patterns are displayed in

the middle row of Fig. 2. To more clearly show the the properties of the far-field multiple ring

patterns, the bottom row also shows the corresponding intensity profiles along the horizontal

center lines of the intensity patterns in the top (or middle) row. For the case of the scalar top-hat

beam, as shown in the first column, we find the complete extinction. For the case of vector

fields, the far-field intensity patterns induced by the nonlinearity behind the vertical polarizer

appear the radial extinction and exhibit the the radially-modulated fan-shaped structures. More-

over, the number of the extinction directions is the same as the topological charge m. Evidently,

such a extinction nature is the same input vector fields as reported in Ref. 21. The results im-

ply that the nonlinearity has no influence on the distributions of states of polarization for the

vector fields whereas influences on the far-field intensity distributions. The phenomena can be

understood as follows. The focused vector field induces a change in the refractive index of the

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-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

No Polarizer

Polarizer

??2 = 0.5?

??2 = 1.0?

??2 = 1.5?

??2 = 2.0?

Fig. 3. Theoretically simulated far-field intensity patterns without (top row) and with (mid-

dle row) a vertical polarizer of the vector fields with m = 2 at different nonlinear phase

shifts ΔΦ2, by taking ϕ0= 0, λ = 804 nm, ω0= 65 μm, and d = 180 mm. The bottom

row is the intensity profiles along the diameter of the far-field intensity patterns shown in

the top row.

sample by Δn(r) = γI(r). As a result, the different locations of the field cross-section in the

radial direction experience the different nonlinear phase shifts of ΔΦ(r) = kn2I(r)L, resulting

in the self-phase modulation. If the phase difference between the two locations in the far-field

plane is ΔΦ(r) = hπ, h being an even or odd integer, the constructive or destructive interfer-

ence takes place, respectively, giving rise to the appearance of self-diffraction patterns with the

multiple concentric ring structures.

It is interesting to investigate the dependence of the far-field intensity distributions of the

vector fields passing through a thin Kerr medium on the nonlinear phase shift. As an example,

Fig.3illustratesthefar-fieldpatternsofthevectorfieldwithm=2andϕ0=0atdifferentvalues

of ΔΦ2, under the conditions of λ = 804 nm, ω0= 65 μm, and d = 180 mm. In principle, the

change of the nonlinear phase shift can originate from the different nonlinearity (i.e. different

nonlinearmedium)foragivenlaserintensityorthedifferentlaserintensityforagivennonlinear

medium and both. One can see that there occurs always a dark spot in the central zones of the

far-field patterns and its size is almost independent of the nonlinear phase shift ΔΦ2. As ΔΦ2

increases, the number of bright diffraction rings around the dark spot increases and the more

light energy is diffracted into the outer rings.

3. Experiment

It is completely different from the illumination with homogeneously polarized field that a cylin-

drical vector field results in a vectorial self-diffraction effect, as described in Eq. (6) and illus-

trated in Figs. 2 and 3. In what follows, we experimentally verify this effect by performing the

femtosecond-pulsed cylindrical vector fields in carbon disulfide. The experimental arrangement

is illustrated in Fig. 4. The light source used in our experiments is a Ti:sapphire laser (Coherent

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?

?

2a

Lens

E(?, ?)

f

d

Sample

Polarizer

Ea(ra, ?)

Ef (r, ?) Ee(r, ?)

Fig. 4. Experimental scheme for investigating the self-diffraction behaviors of the fem-

tosecond vector fields.

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

Without the nonlinear sample

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

With the nonlinear sample

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

a

a

No Polarizer

Polarizer

No Polarizer

Polarizer

Fig. 5. Experimentally observed far-field intensity patterns (top row) and theoretically sim-

ulated far-field intensity patterns (middle row) of the femtosecond vector field with m = 1

and ϕ0= 0. The solid (dotted) lines in the bottom row give the corresponding intensity

profiles alone the horizontal center line of the intensity patterns shown in the top (middle)

row. The former two columns and the latter two columns are the cases of without and with

the nonlinear sample at the focal plane, respectively.

Inc.). Based on the principle of the wavefront reconstruction and by using the configuration in

Refs. [17, 21], we generated the femtosecond-pulsed cylindrical vector fields for different topo-

logical charges m with the fixed pulse energy of ε = 1.3 μJ, the pulse duration of τF? 70 fs,

and the repetition rate of 1 kHz at the central wavelength of λ = 804 nm. In addition, it should

be emphasized that the generated femtosecond-pulsed cylindrical vector field has a top-hat spa-

tial distribution with a diameter of 2a = 3.7 mm and a near-Gaussian temporal profile. The

cylindrical vector field was focused by a lens with a focal length of f = 300 mm, producing the

beam waist of ω0= 65 μm at the focus (the Rayleigh length z0= 16.5 mm). As the nonlinear

medium, the carbon disulfide was contained in 5 mm thick quartz cell at room temperature and

standardatmosphere. Thewallthicknessofthequartz cellwas1mm.Thesample waslocated at

the focal plane. A detector (Beamview, Coherent Inc.) was placed at the observation plane with

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a distance of d = 180 mm from the sample to probe the transmitted intensity distribution. Con-

sidering the interface losses of the light energy, we determine the optical intensities for vector

field with m = 1 and 2 within the solution as I01= 74 and I02= 42 GW/cm2, respectively.

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

Without the nonlinear sample

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

With the nonlinear sample

-202

0.0

0.5

1.0

Normalized Intensity

r (mm)

a

a

a

No Polarizer

Polarizer

No Polarizer

Polarizer

Fig. 6. Experimentally observed far-field intensity patterns (top row) and theoretically sim-

ulated far-field intensity patterns (middle row) of the femtosecond vector field with m = 2

and ϕ0= 0. The solid (dotted) lines in the bottom row give the corresponding intensity

profiles alone the horizontal center line of the intensity patterns shown in the top (middle)

row. The former two columns and the latter two columns are the cases of without and with

the nonlinear sample at the focal plane, respectively.

The far-field intensity patterns for the radially polarized field (namely, m = 1 and ϕ0= 0)

without and with the nonlinear sample at the focal plane are shown in the middle row of Fig. 5.

To identify the vectorial properties of the far-field field, a horizontal polarizer was used. Cor-

respondingly, the results are also displayed in Fig. 5. Most importantly, as shown in Fig. 5, we

experimentally detected the vectorial self-diffraction ring, for the first time to our knowledge.

For a high-order cylindrical vector field with m = 2 and ϕ0= 0, the corresponding experimen-

tal results are displayed in the middle row of Fig. 6. It should be pointed out that the saturated

signals exist in the observed experimental results due to the energy saturation of the detector.

As is well known, carbon disulfide and the quartz wall exhibit the isotropic self-focusing ef-

fectwithanonlinearrefractiveindexofn2=3×10−6cm2/GWandnq=3.3×10−7cm2/GWat

around 800 nm under the femtosecond pulse excitation [22, 23], respectively. Under our exper-

imental condition, the average nonlinear refractive phase shifts for the cylindrical vector fields

with m = 1 and 2 arising from the carbon disulfide were estimated to be ?ΔΦ? = 1.93π and

1.10π, respectively. The phase shifts originating from the quartz walls were so small, ∼ 4%,

that it should not induce a significant change of the far-field intensity. Hence, the contribution of

the quartz walls was not taken into consideration in the analysis. Using Eq. (6) and the known

experimental parameters, we simulate the far-field self-diffraction patterns without and with

the nonlinear sample at the focal plane, as shown in the lower rows of Figs. 5 and 6. Clearly,

the theoretical simulations are consistent with the experimental observations, implying that our

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theoretical analysis is reasonable and could gain an insight on the underlying mechanisms for

the observed vectorial self-diffraction effect. It should be noted that the discrepancy between

the theory and experiment in the periphery of the self-diffraction patterns is apparent. This dif-

ference is anticipated for the following reason. A monochromatic field is considered for a single

defined wavelength and a unique peak intensity in the theoretical simulations; whereas a fem-

tosecond pulse train at a central wavelength with a spectral bandwidth of tens-of-nanometers is

used in the measurements. Furthermore, the peak intensity considerably changes over the pulse

temporal envelope.

Both the theoretical and experimental results demonstrate that the far-field of the focused

cylindrical vector field remains its polarized feature without the disturbance of optical nonlin-

earity. Besides, the far-field self-diffraction field induced by the Kerr nonlinearity also holds

the cylindrical vector polarized feature and exhibits the multiple diffraction ring structures. Our

results should benefit the understanding of the related spatial self-phase modulation effects of

the vector light fields.

4.Conclusion

In summary, we have theoretically and experimentally investigated the vectorial self-diffraction

effect of a cylindrical vector field passing though a self-focusing medium. We have presented

the analytical focal field of the cylindrical vector field with arbitrary integer topological charge

and obtained the far-field vectorial self-diffraction patterns using the Huygens-Fresnel diffrac-

tion integral method. Moreover, we have observed the femtosecond-pulsed cylindrical vector

field induced the vectorial self-diffraction ring from carbon disulfide, which is in good agree-

ment with the theoretical simulations.

Acknowledgments

This work was supported by the National Basic Research Program of China (Grant No.

2012CB921900), the National Science Foundation of China (Grants No. 11174160 and No.

11174157),andtheProgramforNewCenturyExcellentTalentsinUniversity(NCET-10-0503).

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(C) 2012 OSA

Received 6 Oct 2011; revised 17 Nov 2011; accepted 3 Dec 2011; published 19 Dec 2011

2 January 2012 / Vol. 20, No. 1 / OPTICS EXPRESS 157