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PHYSICAL REVIEW A 79, 053811 2009

Diffraction-induced laser pulse splitting in a linear photonic crystal
V. A. Bushuev, B. I. Mantsyzov,* and A. A. Skorynin
Department of Physics, M. V. Lomonosov Moscow State University, Moscow 119991, Russia Received 12 January 2009; published 6 May 2009 We demonstrate analytically a linear optical property of photonic crystals--diffraction-induced incident optical pulse splitting in two pulses propagating with different group velocities in a linear photonic crystal. The reason of this phenomenon is in spatially inhomogeneous field localization within the photonic crystal in case of the Bragg diffraction at the Laue scheme. The field of the fast first pulse is mainly localized within low refractive index layers, whereas the slow second pulse field is mostly in high refractive index layers. Changing optical properties of either high-index or low-index layers of periodical multilayer structure, it is possible to control parameters of each propagating pulse separately. The distance between two transmitted and two diffractively reflected output pulses can be controlled by varying the crystal thickness and modulation depth of the refractive index. DOI: 10.1103/PhysRevA.79.053811 PACS number s : 42.25.Fx, 42.25.Bs, 42.70.Qs, 78.67.Pt

I. INTRODUCTION

Short laser pulse propagation dynamics and its stability and decay in linear and nonlinear media are studied extensively in modern optics and laser physics 1 . Usually a process of optical solitary wave decay on their constituents is investigated in the case of nonlinear light-matter interaction. It is well known that the phenomena of high-order soliton decay are induced by high-order nonlinear effects 2,3 , as well as by third-order dispersion 4 in optical fibers. When a large area coherent optical pulse propagates in a resonant two-level medium, it separates into train of steady state 2 pulses of self-induced transparency 5 . In a resonant photonic crystal RPC , which is formed by the periodically distributed thin layers of two-level atoms in uniform host dielectric medium 6 , nonlinear diffraction-induced laser pulse splitting arises at nonlinear Borrmann effect 7 . In this case radiation propagates in the structure under condition of the Laue transmission scheme 8 of the Bragg diffraction and incoming optical pulse decays near the RPC boundary on two pulses, which propagate in resonantly absorbing structure without absorption and with different group velocities. The first pulse linearly interacts with the structure because it is formed by two coupled Borrmann modes and its wave nodes are localized on resonant layers of RPC. The second pulse propagates as slow nonlinear Laue soliton of self-induced transparency because it consists of two antiBorrmann modes with the field maxima on the resonant layers and it strongly interacts with two-level atoms 7 . Although the RPC is a convenient model to study theoretically nonlinear optical phenomena in photonic crystals, experimental work with the RPC still is a difficult problem until now and so nonlinear diffraction pulse splitting still has not been observed experimentally. Here a linear effect of diffraction-induced pulse splitting DIPS in a linear photonic crystal PC at the Laue scheme of the Bragg diffraction has been predicted theoretically. In the simplest one-dimensional 1D case, a PC is a multilayer

periodic structure Fig. 1 . By means of analytical solution of a boundary problem for 1D PC with low refractive index contrast, we show that an input picosecond laser pulse brakes up within the structure on two pulses propagating with different group velocities. The DIPS effect is explained by spatially inhomogeneous light localization in the PC at the Laue geometry of diffraction. One pulse field is mainly localized within low-index layers of the structure, but another pulse field is localized in the high-index layers. Therefore parameters of these pulses can be changed separately by means of changing the high-index or low-index layer optical properties. Such structure of field localization in the PC bulk is caused by wave diffraction. In the PC, each spectral component of a pulse incident at the Bragg angle with respect to the PC layers splits up into a coherent superposition of two transmitted and two diffracted diffractively reflected waves. These waves propagate in the PC with two different effective refractive indices and, as a result, with two different velocities. With an increase in the propagation depth in the PC bulk, these waves are divided into two pulses, propagating

*bmantsyzov@gmail.com
1050-2947/2009/79 5 /053811 5

FIG. 1. Color online Schematic diagram of doubling of the number of laser pulses caused by splitting of an incident pulse within PC at the Bragg diffraction in the Laue geometry; T and R are transmitted and diffractively reflected pairs of pulses at the PC output. ©2009 The American Physical Society

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PHYSICAL REVIEW A 79, 053811 2009

with different group velocities. Each pulse is a wave packet consisting of pairs of coupled waves for each spectral component: one transmitted and one diffracted waves moving with the same velocity. At the PC output, a pair of transmitted T and a pair of diffractively reflected R pulses are emitted Fig. 1 . The time interval between the pulses emerging from the PC is proportional to the crystal thickness and the Fourier component of the difference between the refractive indices n x = n2 - n1 of the PC layers. Calculations show that to observe the DIPS effect for the pulse with radiation wavelength 0 500 ­ 1000 nm and duration 0 0.1 ­ 10 ps, one should use PC with the period d 0, the refraction index modulation depth n 0.1 ­ 0.3, and the thickness L 0.1 ­ 3 cm.
II. OPTICAL PULSE PROPAGATION UNDER LAUE SCHEME OF DIFFRACTION

Ein r, t = e

in -

A

in

exp ikr - i t d ,

4

where the frequency spectrum of the pulse envelope has the form A
in

= 1/2
-

Ain t exp i t dt .

The field E r , t in the PC will also be represented as a spectral expansion, E r, t =
-

E r,

exp - i t d .

5

Consider a linear PC composed of periodically layers with thicknesses d1 and d2 and refractive and n2. The layers are oriented perpendicularly surface Fig. 1 , and the period of the structure is A light pulse in the form of a planar wave packet on a PC along a fixed direction s0 at an angle to to the surface, Ein r, t = einAin t - /c exp ik0r - i
0

alternating indices n1 to the PC d = d 1 + d 2. is incident the normal

For definiteness, we will restrict ourselves to consideration of a PC with a small modulation depth of the refractive index 0 ne and the light diffraction will be considered in the two-wave approximation, which implies existence of two strong coupled waves in a crystal: transmitted E0 and diffracted Eh . In this case, the field E r , for each spectral component in Eq. 5 can be written as E r, =E
0

exp iq0r + E

h

exp iqhr ,

6

t,

1

where envelope of the wave packet Ain is generally a complex slowly varying function; c is the speed of light in vacuum; 0 is the central signal frequency; k0 = k0 = 0 / c =2 / 0, where 0 is the central wavelength, k0x = k0 sin , and k0z = k0 cos ; and = rs0 = x sin + z cos , where the x axis is directed along the PC surface and the z axis is directed into the bulk of the crystal along the normal to the surface. The dependence of the refractive index on the transverse coordinate x is represented in the form n x = ne + n x , 2

where qh = q0 + h; h is the reciprocal lattice vector and h =2 / d. In our symmetric case, hx =-h and hz =0. The field amplitudes E0,h and the wave vectors q0 and qh in Eq. 6 are found from Eq. 3 and the boundary conditions. From the condition of continuity of the tangential projections of wave vectors in vacuum and in the medium, the projection of q0 on the x axis is q0x = kx = k sin . The projection of q0 on the z axis is q0z = k 0 + , where 0 = n2 e - sin2 1/2 and the quantity , which is to be determined, describes the change in the wave vector q0 along the normal to the surface, caused by the Bragg diffraction. The permittivity x in the two-wave approximation has the form x=
0

+

h

exp - ihx +

-h

exp ihx ,

7

where ne = n1d1 + n2d2 / d = n2 + 0 is the average refractive index, d is the period of the structure, 0 = n1 - n2, and = d1 / d. In the layers with the thickness d1, the function n x = 0 1- and in the layers with thickness d2, n x =- 0 . The complex electric field in a PC composed of optically isotropic layers with small n x obeys the wave equation E r, t - c
-2

where 0, h, and -h are the Fourier components of the permittivity, determined by the relation
d g

= 1/d
0

x exp igx dx ,

8

where g =0, h, and -h. Hence, taking into account the relation x = ne + n x 2 n2 +2ne n x , we find that 0 = n2, e e
h

x

2

E r, t / t2 =0 ,

=i n

3
-h

e0

/ /

1 - exp i2 1 - exp - i2

, .

where x = n2 x is the complex permittivity and 2 2 = / x + 2 / z2 is the Laplace operator. Following Ref. 9 we will represent field 1 of the incident wave packet as a spectral expansion, i.e., in the form of a set of plane monochromatic waves with spectral ampli, frequencies = 0 + , and wave vectors k tudes Ain = / c,

=- i n

e0

Obviously, in a medium without absorption h = -h. Let us now substitute Eqs. 5 ­ 7 into the wave equation 3 and equate the expressions with identical exponentials. As a result, we obtain the following system of equations for the scalar amplitudes of the transmitted E0 and diffracted Eh waves:

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PHYSICAL REVIEW A 79, 053811 2009

2 2
0

0

E0 - C -

-hE h

=0 , 9

Eg r,

=A

in

Bg r,

,

14

Eh - C hE0 =0 ,

where g =0, h; B0 and Bh are the amplitude coefficients of transmission and diffractive reflection, respectively, Bg r, =
j=1,2

where the polarization factor C = 1 and C = cos 2 for s- and p-polarized radiations, respectively. System 9 is obtained on the assumption that 0, which is valid in the case of weak modulation of refractive index 2 . The quantity in the second equation in Eq. 9 determines deviation of the and the angle of incidence from the exact frequency diffraction condition = 0, where = k2 - k + h
2

Fgj exp iqgxx + iq0jz z .

15

Here, q0x = kx , qhx = kx - h, F01 =-2R2kz/D, F02 =2R1kz/D,
2 0z

Fhj = R jF0 j ,
1 0z

/k2 = h 2kx - h /k2 .

10

D = R1 kz + q

- R2 kz + q

.

16

System 9 has the same form as the system of equations of the dynamic diffraction theory in the x-ray optics 8 , which was obtained for a plane monochromatic wave, where the parameter is determined by the deviation of the angle of incidence of a wave from the Bragg angle. In our case, system 9 describes the diffraction of a pulse with an arbiis determined trary time dependence, and the parameter both by the detuning of the radiation frequency from the so-called Bragg frequency and by the deviation from the Bragg angle. Let us determine the Bragg angle = B for radiation with the central frequency 0 using the relation 2 0 / c sin B = h or, what is the same, sin B = 0 / 2d. Let now the spectral component of field 4 with the frequency = 0 + , where , where 0, be incident on a PC at the angle = B + B. Then, the projection of the wave vector kx in Eq. . Substituting this 10 has the form kx = k0 + / c sin B + relation into Eq. 10 , we obtain the following expression for ,: the diffraction parameter = 2 sin 2
B

The final expressions for the fields Eg r , t at a depth z at an instant t have the following integral form: Eg r, t =
-

A

in

Bg r,

exp - i t d .

17

These fields can also be written as wave packets, similarly to Eq. 1 , Eg r, t = Ag r, t exp ikgxx + ik
00

z-i

0

t.

18

Here, k0x = k0 sin , khx = k0x - h, and the envelopes of the wave packets have the form Ag r, t =
-

A

in j=1,2

F

gj

exp i

j

d,

j

,

= k jz +

/c x sin +

0

z-

t,

19

+

/

0

tan

B

.

11

The condition for nontriviality of the solution to system 9 leads to a square equation with respect to the desired quantity , whose roots have the form
1,2

where j are the complex phases of the waves in the case of an absorbing medium. The intensities of the transmitted g =0 and diffracted g = h pulses at an arbitrary depth of the PC are IT z , t = E0 z , t 2 and IR z , t = Eh z , t 2, respectively.
III. DIFFRACTION-INDUCED SPLITTING OF AN OPTICAL PULSE

= 1/4

0

2

+4C

2

h -h

1/2

.

12

It follows from the first equation in Eq. 9 that the amplitudes of the transmitted and diffracted waves in a PC are related by a simple expression Ehj = R jE0 j, where R j =2 0 j / C -h j =1,2 . Taking into account the specular reflection from the PC input surface with an amplitude Er, the boundary conditions for continuity of the electric and magnetic fields in the plane z = 0 for the s polarization have the form Ain + Ar = E01 + E02 , kz Ain - Ar = q01 E01 + q02 E02 , z z R1E01 + R2E02 =0 , 13

where q0jz = k 0 + j and the quantities j are determined by relations 11 and 12 . As a result, we derive from Eqs. 6 and 13 the following expression for the spectral components of the fields in Eq. 5 at an arbitrary point in the PC bulk:

Consider the dynamics of propagation and splitting in PC of incident pulse with the Gaussian envelope Ain t = exp - t / 0 2 and with the width 0 Fig. 2 . It follows from Eqs. 12 , 15 , and 16 that the region of strong diffractive interaction of radiation with a PC is determined by 2C h . If a pulse is incident on a PC at an the relation exact Bragg angle =0 , the spectral range of diffractive reflection and transmission is determined by the so-called 2 spectral Bragg width B = 0 h / 2 sin B. This width increases with a decrease in the Bragg angle and with an increase in the modulation depth 0 of the PC refractive index, reaching a maximum at = 0.5. In order to use the two-wave approximation correctly, the pulse spectral width 0 =2 / 0 is assumed to be smaller than the spectral Bragg width, i.e., 0 B. Figure 2 shows spatial-temporal dynamics of evolution of transmitted and diffracted fields in the PC under exact Bragg condition calculated by Eq. 17 . For convenience, the plots are shown as functions of the coordinate tz = t - z 0 / c in dynamical frame of reference which is rigidly related to the transmitted pulse in an homogeneous

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.

PHYSICAL REVIEW A 79, 053811 2009
1.6

2

(a)

50,00

0.5

1

45,45

40,91

36,36

Z(mm)

31,82

27,27

1

22,73

18,18

mj

0.25 0

1.5

3 2

IT(z,tz)
0

13,64

9,091

4,545

0

-0.4

0 tz(ps)

0.4

1.4 -1
B
50,00

0 (rel. units)

1

.

2
Z(mm)

(b)

45,45

0.5 0.25 0

40,91

36,36

31,82

27,27

FIG. 4. Color online Frequency dependences of the effective refractive indices 1 m1, 2 m2, and 3 Re 0 for a PC with the period d =1 m, = 0.5, n1 = 1.5, and 0 = 0.1; the wavelength 0 = 0.8 m.

1

22,73

18,18

IR(z,tz)
0
-0.4 0 tz(ps) 0.4

13,64

9,091

4,545

0

FIG. 2. Color online Spatial-temporal dynamics of the incident pulse evolution inside a PC, one pulse splits on two pulses. Intensities of a transmitted IT z , tz = E0 z , tz 2 and b diffracted IR z , tz = Eh z , tz 2 waves calculated by expression 17 arb. units . The parameters of the incident pulse and PC are as follows: m, = 0.5, n1 = 1.5, 0 = 0.1, and d =3 m. 0 = 0.1 ps, 0 = 0.8

medium with a maximum at tz = 0. During some time, the pulse propagates inside PC as a single "refracted" pulse. It means that all four waves, two transmitted E01,2 and two diffracted Eh1,2, exist in any point of the PC simultaneously. The superposition of these waves leads to periodical total transfer of energy from the transmitted to the diffracted waves and vice versa. This pendular solution represents in Fig. 3. The extinction depth , at which complete transfer occurs, is obtained from Eq. 15 , = 0 0 / 2C h . As it follows from Eqs. 14 and 15 , the one pair of transmitted E01 and diffracted Eh1 coupled waves and second
0.06

pair E02 and Eh2 propagate in the PC with different effective refractive indices m j =Re 0 + j Fig. 4 and, hence, with different velocities. The wave velocity difference gives rise to the splitting of the propagating pulse into two pulses with different group velocities Fig. 2 . The first pulse is formed by the waves E01 and Eh1, whereas the second one is formed by waves E02 and Eh2. We call this phenomenon the diffraction-induced pulse splitting. In Fig. 5, the space distribution of the field amplitude inside a PC in the case of the DIPS has been shown. It can be seen that the field has a strong spatially inhomogeneous localization. The fields of different pulses are localized in different space areas in the PC. The field of the first propagating pulse is mainly localized in low-index layers, therefore the pulse is faster, while the second pulse field is in the high-index layers and this pulse moves slower. Actually, these two pulses propagate in effectively different optical media and, consequently, the parameters of each pulse amplitude, width, velocity, and polarization are independently determined by optical properties material dispersion, nonlinearity, anisotropy, refraction index sign, and so on of either low-index or high-index layers. For instance, if the high-index layers are chosen as nonlinear but the low-index layers are linear, the slow pulse will demonstrate the properties of nonlinear wave, whereas the fast pulse will be a linear one. This effect has been de1.5
0,4000 0,3500 0,3000

(a)

(b)

70,00

0.7

0.4

61,25

0.04
z(mm)

52,50

0,2500
43,75

35,00

0.02

IT(z,tz)

0.35

Z (mm)

1.4

0,2000

0.2

0,1500

26,25

IR(z,tz)

0,1000

17,50

0,05000

8,750

0

0

0

0

0

-0.3

0 0.3 tz(ps)

-0.3

0.3 0 tz(ps)

1.3

n(x)

2.403

X (mm)

2.406

2.409

FIG. 3. Color online Initial stage of pulse evolution represented in Fig. 2. The superposition of four waves E01,2 and Eh1,2 leads to so-called pendular solution when energy is periodically completely transferred from a the transmitted IT z , tz to b the diffracted IR z , tz waves and vice versa arb. units .

FIG. 5. Color E0 x , z + Eh x , z the pulse splitting line at the bottom index profile n x .

online Spatial distribution of the arb. units inside a PC at the point has occurred, calculated in Eq. 17 of the figure shows schematically

field module of time when . The stepped the refraction

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PHYSICAL REVIEW A 79, 053811 2009

IT (arb. units)

1 0.2 4 0.1 0.0 3 2 1 2 3

(a)

4

-0.5 1 2

0.0

tz (ps)

0.5

1.0

0.2

(b)
3 4 4

0.1

1

0.0

0

2

t z (ps)

4

6

FIG. 6. Color online Dynamics of the transmitted pulse intensity IT z , tz at PC thicknesses z = 1 0.05, 2 0.2, 3 0.4, and 4 0.8 mm. The parameters of the incident pulse and PC are as follows: 0 = 0.1 ps, 0 = 0.8 m, = 0.5, n1 = 1.5, and 0 = 0.1; the period d = a 3 and b 0.8 m. The dashed curves 1 are related to the diffracted pulse at z = 0.05 mm.

scribed earlier for the special case of ultrathin resonant nonlinear layers in a RPC 7 . Our calculations show also that if material dispersion of the low-index layers, for example, is larger in comparison with high-index material of a linear thick PC, then the fast pulse will be greatly broader than the slow one. Let us now discuss the conditions necessary for DIPS. It is easy to estimate the distance z0 at which split pulses will be spaced by a time interval 2 0. If the exact Bragg condition = 0 is satisfied, it follows from Eq. 12 that z0 =2c 0 0 / C h . This distance decreases with a decrease in the pulse width and with an increase in the modulation depth of the refractive index in the PC. At the same time, the distance z0 should not strongly exceed the photoabsorption length La =1 / 2k0 Im 0 . It is noteworthy that the depth z0, measured in units of the extinction depth , is determined only by the pulse wavelength and width: z0 / =4c 0 / 0. For example, for the pulse width 0 = 0.1 ps, 0 = 0.8 m, d =1 m, = 0.5, n1 = 1.5, the modulation parameter 0 = 0.1, = 6.3 m, and the pulse becomes noticeably split in a fairly thin PC: z0 = 0.93 mm. Figure 6 shows the temporal dynamics of pulse transmission through different cross sec-

tions z of a PC. It can be seen that at small thicknesses z z0 the pulse is not yet split; however, with an increase in z DIPS occurs and the time interval between the split pulses increases with the distance z. Outside of the structure, i.e., in homogenous medium, the earlier coupled transmitted and diffracted waves of either pulse will propagate independently and, as result, the incident pulse will split by PC on four pulses: two transmitted and two diffracted Figs. 1 and 6 . In view of the symmetry = 0, the amplitudes and shapes of transof the problem at mitted and reflected pulses completely coincide at z z0. Due to the features of the effective frequency dispersion of , the pulse propagation velocities the refractive indices m j differently depend on the period d, i.e., on the angle of incidence B. For example, at almost normal incidence B = 7.7°; Fig. 6 a , one of the pair of pulses advances beyond the pulse in a continuous medium, while the second pulse lags behind. With an increase in the angle of incidence B = 30°; Fig. 6 b both pulses propagate faster than a pulse in a homogeneous medium. In this case, the first pulse stronger decreases in amplitude and broadens in time in comparison with the second pulse. It should be noted that the frequency dispersion dn1,2 / d of the PC refractive indices, which is disregarded here, is generally smaller by more than 1 order of magnitude than the dispersion dm j / d of the effective refractive indices and does not affect the pulse propagation in thin PCs.
IV. CONCLUSION

IT (arb. units)

The diffraction-induced picosecond laser pulse splitting in a linear PC under condition of the Bragg diffraction in the Laue geometry, which we have described, is a linear effect. The DIPS is independent of the field intensity and can be implemented even for weak pulse. Therefore, even a simple linear thin 1D PC can be used as miniature-type device to transform arbitrary power laser pulses into double pulses with controlling delay, amplitudes, durations, and polarizations, which could be useful, for instance, for pump-probe spectroscopy and laser physics. Moreover, the DIPS effect allows us to double the laser pulse repetition rate.
ACKNOWLEDGMENT

This work was supported in part by the Russian Foundation for Basic Research under Grant No. 09-02-00786-a.

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6 B. I. Mantsyzov and R. N. Kuz'min, Sov. Phys. JETP 64, 37 1986 . 7 B. I. Mantsyzov, Opt. Commun. 189, 275 2001 . 8 Z. G. Pinsker, Dynamical Scattering of X-rays in Crystals, Springer Series in Solid-State Science Vol. 3 Springer, Berlin, 1977 . 9 V. A. Bushuev, J. Synchrotron Radiat. 15, 495 2008 .

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