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J. Opt. Soc. Am. B / Vol. 23, No. 11 / November 2006
P. P. Banerjee and G. Nehmetallah
Linear and nonlinear propagation in negative index materials Partha P. Banerjee and Georges Nehmetallah Department of Electrical and Computer Engineering, University of Dayton, Dayton, Ohio 45469 Received May 24, 2006; accepted July 10, 2006; posted July 14, 2006 (Doc. ID 71274) We analyze linear propagation in negative index materials by starting from a dispersion relation and by deriving the underlying partial differential equation. Transfer functions for propagation are derived in temporal and spatial frequency domains for unidirectional baseband and modulated pulse propagation, as well as for beam propagation. Gaussian beam propagation is analyzed and reconciled with the ray transfer matrix approach as applied to propagation in negative index materials. Nonlinear extensions of the linear partial differential equation are made by incorporating quadratic and cubic terms, and baseband and envelope solitary wave solutions are determined. The conditions for envelope solitary wave solutions are compared with those for the standard nonlinear Schrodinger equation in a positive index material. © 2006 Optical Society of America OCIS codes: 160.4670, 190.4400, 060.5530, 190.5530.
1. INTRODUCTION Negative index materials have attracted a great deal of recent interest due to their unconventional characteristics with respect to wave propagation through such media, and the behavior of waves across interfaces between a conventional positive index material and a negative index material.1–3 While negative index materials may not commonly be found in nature, researchers have fabricated artificial negative index materials that show the typical characteristics of negative phase velocity and positive group velocity over a range of frequencies.4,5 The simulation of modulated pulses in linear and nonlinear negative index media has been performed recently by Scalora et al.6,7 on the basis of a dispersion relation for a Drude medium. In the latter work, a generalized nonlinear Schrödinger (NLS) equation was derived to describe the propagation of ultrashort pulses through nonlinear negative index media.7 Approximately two decades ago, a unified theory of wave propagation in linear and nonlinear media was developed, starting from the knowledge of the dispersion relation to derive the underlying linear wave equation and with suitable modifications to incorporate the nonlinearity.8 The nonlinearity was modeled using a simple kinematic wave equation picture of the nonlinear modification to the phase velocity of a wave. The developed model was used to derive an exhaustive list of commonly occurring nonlinear wave equations such as the Korteweg–deVries (KdV) equation, the nonlinear Klein– Gordon (NKG) equation, and the NLS equation, in one and higher spatial dimensions, and particular solitary wave or soliton solutions of these equations were found. In this paper we apply the concept of starting from the dispersion relation as the basis for modeling wave propagation for negative index materials as well, and derive the underlying partial differential equation for wave propagation in such a medium. Using the simplest possible dispersion relation, a partial differential equation (PDE) 0740-3224/06/112348-8/$15.00
modeling wave propagation in such media is developed. The linear characteristics of wave propagation in negative index media are first verified. A transfer function approach to wave propagation such as commonly done in Fourier optics is also developed, and spatial and temporal transfer functions describing propagation in negative index media are used to study the evolution of various initial profiles. This approach also yields the transfer function for propagation of beams in negative index media. This, in turn, also leads to the definition of the q parameter of a Gaussian beam propagating in such a medium and establishes the laws of q transformation. The implications are reconciled with a transfer matrix approach for rays developed for negative index materials and applied to the q transformation of Gaussian beams. Also a nonlinear extension of the linear partial differential equation is performed by heuristically adding quadratic and cubic nonlinear terms as in the NKG equation, and baseband as well as envelope solitary wave solutions are derived. The characteristics of the solutions, such as the amplitude, width, and group and phase velocities that result as a balance between nonlinearity and dispersion are examined. For envelope solitary waves, the solutions are compared with the well-known solutions of the NKG equation. It is shown that the envelope solitary wave solutions derived require anomalous dispersion and a negative sign of the nonlinearity in an intensity-dependent dispersion relation.
2. LINEAR DISPERSION RELATION General dispersion relations can be expressed in the form P共 , k兲 = 0 where is the angular frequency and k is the propagation constant.8 The corresponding PDE for the wave 共z , t兲 can be found by substituting and k by the operators © 2006 Optical Society of America
P. P. Banerjee and G. Nehmetallah
→ − j / t,
Vol. 23, No. 11 / November 2006 / J. Opt. Soc. Am. B
k → j / z
共1兲
for 1D propagation along z. We will use the simplest type of dispersion relation, which can be expressed in either of the forms
= W共k兲,
k = K共兲.
共2兲
For 共z , t兲 to be real, it follows that the dispersion relations above must be odd functions of k and , respectively. This has been shown in Ref. 8, but will be repeated here for the benefit of readers. By writing 共z , t兲 as a continuum of dispersive plane waves
共z,t兲 =
1 2
冕
⬁
⌿共兲exp兵j关t − K共兲z兴其d ,
共3兲
−⬁
it follows that
* 共z,t兲 =
1 2 1
=
2
冕 冕
⬁
⌿ * 共兲exp兵− j关t − K共兲z兴其d
−⬁ ⬁
⌿ * 共− 兲exp兵j关t + K共− 兲z兴其d , 共4兲
p共 0兲 = while
g共 0兲 =
3. DISPERSION RELATION FOR NEGATIVE INDEX MATERIALS AND UNDERLYING PARTIAL DIFFERENTIAL EQUATION The simplest model dispersion relation that can be used for a negative index material can be expressed in the form
= W共k兲 = −
C k
,
C ⬎ 0,
共5兲
which is an odd function of k, and is plotted schematically in Fig. 1. For
= 0 ⬎ 0,
k = − k0 = − C/0 ⬍ 0,
共6兲
so that from Eqs. (5) and (6),
k
= − 02/C ⬍ 0,
d dk
共7兲
0
冏 冏 冏冏 C
=
0
k2
= 02/C ⬎ 0.
共8兲
0
Thus the phase velocity is negative, and the group velocity is positive, which defines a negative index medium.3 The phase velocity is equal in magnitude and opposite in sign to the group velocity for the chosen dispersion relation. Note that the dispersion relation above is consistent with the propagation characteristics of a lumped circuit transmission line in the high-pass configuration, where the series element is a capacitor instead of an inductor, and the shunt element is an inductor instead of a capacitor.9 It has been shown that this model of the transmission line behaves as a waveguide filled with a negative index material.3 The PDE for 共z , t兲 can be found by substituting Eq. (1) in Eq. (5) to give
−⬁
upon replacing → −. For real 共z , t兲, we have, upon comparing Eqs. (3) and (4), that K共兲 = −K共−兲, implying that K共兲 is an odd function of . In a similar way, it follows that W共k兲 must be an odd function of k.
冏冏
2349
2 z t
+ C = 0.
共9兲
Example 1. Initial condition at z = 0: 共z = 0 , t兲 = f共t兲. The solution to Eq. (9) can be found by using standard transform methods. For instance, upon taking the Fourier transform with respect to t, we obtain d⌿ dz
= j共C/兲⌿,
共10兲
which has the solution ⌿共z, 兲 = ⌿共z = 0, 兲exp关j共C/兲z兴 = F共兲exp关j共C/兲z兴, 共11兲 ⬁ 共z , t兲exp共−jt兲dt ⌿共z , 兲 = Ft / 兵共z , t兲其 = 兰−⬁
is the where Fourier transform of 共z , t兲 with respect to t. One can alternately define a transfer function of the propagation in the negative index material as H共z, 兲 =
⌿共z, 兲 ⌿共z = 0, 兲
= exp关j共C/兲z兴.
共12兲
Using the concept of the transfer function above, the propagation of a modulated Gaussian pulse 共z = 0 , t兲 = f共t兲 = exp−共t / 兲2cos共0t兲 is shown in Fig. 2. It is seen that at z0 = 2, the modulated pulse is delayed and arrives at approximately t0 = z0 / g = Cz0 / 02. In our example, for the choice of parameters, = 0.5, 0 = 20, C = 200, and t0 = 1, as seen from Fig. 2(d) one monitors the peak of the modulated pulse. That the phase velocity is negative can be determined by carefully examining a movie of the propagation of the modulated pulse. It is readily seen that setting = 0 + ⍀ in Eq. (12) and using Eq. (6), Eq. (8) leads to H共z, 兲 = exp共jk0z兲exp共− j⍀z/g兲, Fig. 1.
Dispersion relation of the form = −C / k, where C = 1.
共13兲
implying that the modulated packet has a phase velocity that is negative and a group velocity that is positive.
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J. Opt. Soc. Am. B / Vol. 23, No. 11 / November 2006
P. P. Banerjee and G. Nehmetallah
Fig. 2. Propagation of a modulated Gaussian pulse. (a) Initial pulse: 共z = 0 , t兲 = f共t兲 = exp−共t / 兲2cos共0t兲 , = 0.5, 0 = 20, (b) Fourier transform of the initial pulse, (c) Fourier transform of the real part of H for z0 = 2 , C = 200, (d) initial pulse after propagation at distance z = 2.
Example 2. Initial condition at t = 0: / 共z , t = 0兲 = g共z兲. This time we take the Fourier transform with respect to z, which we define as ⌿共kz,t兲 = Fz兵共z,t兲其 =
冕
⬁
共z,t兲exp共+ jkzz兲dz
−⬁
to derive ⌿共kz,t兲 = ⌿共kz,t = 0兲H共kz,t兲 = G共kz兲exp关− j共C/kz兲t兴. 共14兲 Note that the Fourier transform with respect to z defined above is consistent with the Fourier transform defined before with respect to t. By the term “Fourier transform,” we will, in this paper, imply the transformation of a signal in time or space to its corresponding temporal or spatial frequency domain. Similarly, by the term “inverse transform,” we will imply the transformation of the spectrum of the signal back to its time or space domain. Assuming 共z , t = 0兲 = g共z兲 = u共z兲 where u共z兲 denotes the step function, we can show that the inverse transform of Eq. (14) can be expressed, after some algebra (see Appendix A), as
共z,t兲 =
1
冕
⬁
0
1 kz
冉
sin
C kz
冊
t + kzz dkz .
共15兲
Comparing this with standard integrals, we find
共z,t兲 = J0共2冑Czt兲u共t兲,
共16兲
where u共t兲 denotes the unit step function. The solution is plotted in Fig. 3 for increasing values of t. The simulations show that the zeroes of the Bessel function move to-
Fig. 3. input.
Output function 共z , t兲 = J0共2冑Czt兲u共t兲 due to a step
ward the left with increasing time, reminiscent of phase fronts traveling to the left since the phase velocity is negative. Also note that the total integrated energy over the range of z plotted monotonically decreases. This is due to the spatial filtering action of the medium whose transfer function is given by H共kz , t兲 = exp关−j共C / kz兲t兴. Incidentally, using a similar recipe, one can also predict the behavior of a step function in time with propagation. In this case, one would need to start from the transfer function H共z , 兲 = exp关j共C / 兲z兴 derived above.
4. BEAM PROPAGATION THROUGH NEGATIVE INDEX MATERIALS Using the extension of the dispersion relation (5), and the operator formalism, we can derive the PDE for envelopes of beams as follows. First, note that from Eq. (5),
P. P. Banerjee and G. Nehmetallah
Vol. 23, No. 11 / November 2006 / J. Opt. Soc. Am. B
1 k = 冑k2x + k2y + kz2 ⬇ kz + 共k2x + k2y 兲kz−1 ⬅ − C−1 , 共17兲 2 or, equivalently,
冋
1
kz2 +
共k2x + k2y 兲
2
册
⬅ − Ckz .
共18兲
Now replacing , kx , ky , kz by their respective operators8
→ − j / t,kx → j / x,ky → j / y,kz → j / z, 共19兲 and operating on 共x , y , z , t兲, we derive the PDE
3
+
z t 2
1 3 2 x t 2
+
1 3 2 y t 2
+C
z
= 0.
共20兲
h−共x,y,z兲 = − j
冋
C 20z
exp j
C 2 0z
册
共x2 + y2兲 .
共26兲
It is interesting to examine the implications of the negative sign in the argument of the exponential of the transfer function for the negative index material. Note that for a positive index material, the transfer function (25) can be used to explain the spreading of a Gaussian beam due to diffraction. In fact, a Gaussian beam initially of waist w0 acquires positive (diverging) phase curvature and spreads after traveling a distance z0 in a conventional medium. The complex envelope of the Gaussian can be expressed as
e = j
k0+w02 2q共z0兲
exp关− jk0+共x2 + y2兲/2q共z0兲兴,
q共z0兲 = z0 + j
k0+w02 2
,
共27兲
Now, writing
共x,y,z,t兲 = Re兵e共x,y,z兲exp关j共0t + k0z兲兴其,
k0 = C/0 , 共21兲
in Eq. (20) and equating the coefficients of exp关j共0t + k0z兲兴, we obtain the paraxial wave equation for beam propagation in a negative index medium as
e z
=j
冉
0 2 e 2C
x2
+
2 e y2
冊
.
共22兲
Proceeding in exactly the same way as that for propagation in a conventional medium, one can define the transfer function for propagation by taking the Fourier transform of (22) with respect to x, y and solving the ODE for the spectrum10 ⌿e共kx,ky,z兲 = Fx,y兵e共x,y,z兲其 =
冕冕 ⬁
⬁
−⬁
−⬁
e共x,y,z兲exp关j共kxx + kyy兲兴dxdy. 共23兲
This yields H−共kx,ky,z兲 =
2351
⌿e共kx,ky,z兲 ⌿e共kx,ky,z = 0兲
= exp关− j共0/2C兲共k2x + k2y 兲z兴, 共24兲 11
which is in agreement with the findings of Tassin et al. We would like to point out that the important difference between the transfer function for propagation in a negative index material as compared to a conventional medium is that the sign on the argument of the exponential is negative for the case of the negative index material. For a conventional positive index material,10 H+共kx,ky,z兲 =
⌿e共kx,ky,z兲 ⌿e共kx,ky,z = 0兲
and can be taken as the general expression for a Gaussian with increasing width and diverging wavefronts. If such a complex Gaussian is introduced into a negative index material at z = 0, we can use the transfer function to determine its shape after propagation through an arbitrary distance z in the medium. Note that corresponding to Eq. (27), ⌿e共kx,ky,z = 0兲 = w02expj共k2x + k2y 兲q共z0兲/2k0+.
Using Eq. (24) and the definition of q共z0兲 from Eq. (27), we find that after a distance of propagation z = zf =
Cz0
0k0+
共25兲 where k0+ denotes the propagation constant in the positive index medium. The corresponding impulse response is given by the inverse transform of Eq. (24):
共29兲
,
the spectrum of the beam will be purely real, which corresponds to the spectrum of a Gaussian with plane wavefronts and of waist wf = w0, under paraxial approximation, indicating a focusing of the Gaussian beam. This proves that the negative index material can act as a focusing medium for a diverging Gaussian beam. The wave picture developed here can be corroborated with a corresponding ray matrix approach to Gaussian beam propagation in a negative index medium as developed in Appendix B. Alternatively, consider a Gaussian with initially plane wavefronts at z = 0 in a negative index medium:
e共x,y,z = 0兲 = exp− 共x2 + y2兲/w02.
共30兲
Then ⌿e共kx , ky , z = 0兲 = w02exp−共k2x + k2y 兲w02 / 4 and using Eq. (24), ⌿e共kx,ky,z兲 = w02exp− j共k2x + k2y 兲q0/2C,
共31兲
where we have defined a q parameter of the Gaussian beam propagating in a negative index material as q−共z兲 = z − j
= exp关j共k2x + k2y 兲z/2k0+兴,
共28兲
Cw02 20
.
共32兲
Note that the definition of the q of the Gaussian beam in the negative index medium is consistent with the usual definition of the q in a positive index medium. In the latter case, the q is defined [see Eq. (27)] so as to have the expression for the spectrum of the Gaussian to be of the
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J. Opt. Soc. Am. B / Vol. 23, No. 11 / November 2006
P. P. Banerjee and G. Nehmetallah
same form as that of the transfer function for propagation, with z replaced by q. A comparison of Eq. (31) with Eq. (24) shows that the same is true for the negative index medium with q defined as in Eq. (32). Furthermore, as in the positive index medium, the second term in Eq. (32) contains the information about the propagation constant in the medium, which is negative in the case of a negative index medium (see also Appendix B). The inverse transform of Eq. (31) yields, analogous to Eq. (26),
e共x,y,z兲 = − j
冋
Cw02
exp j
2 0q
册
C
共x2 + y2兲 .
2 0q
共33兲
Upon substituting Eq. (32) into Eq. (33) and separating into magnitude and phase, it is possible to determine the variation of the width and radius of curvature of the Gaussian during propagation in the negative index medium, along with the on-axis amplitude and the Guoy phase. In fact, the exponential part in Eq. (33) gives the information about the width and radius of curvature and can be re-expressed as
冋
exp j
C 2 0q
共x2 + y2兲
再
= exp
−C
册
2
w02/202
2关z2 + 共Cw02/20兲2兴
再
⫻exp j
共x + y 兲 2
共C/0兲z 2关z2 + 共Cw02/20兲2兴
2
冎
2z2 + 共Cw02/20兲2 C2w02/202
,
z t
冎
共x2 + y2兲 .
R共z兲 = −
A. Base Band Solitary Waves The nonlinear extension of the linear PDE [Eq. (9)] can be achieved by heuristically adding a quadratically nonlinear term, similar to that done for the nonlinear extension of the Klein–Gordon equation.8 We remark that Eq. (9) by itself is not the Klein–Gordon type, since the latter contains a wave operator rather than the mixed derivative; however, both equations are of second order. The mixed derivative in our case automatically follows from the chosen dispersion relation (the dispersion relation for the Klein–Gordon equation is like the waveguide dispersion). As we will mention in Subsection 5.B in connection with envelope solitary waves, any dispersion relation is appropriate to model propagation in negative index materials as long as its Taylor series expansion around the operating or carrier frequency has the properties of propagation in negative index materials, viz., negative phase velocity and positive group velocity. Thus, the quadratically nonlinear PDE in a negative index material modeled by the dispersion relation (5) becomes
2
共34兲
Note that the width w共z兲 monotonically increases in this case, just like propagation in a conventional positive index medium. The radius of curvature R共z兲, on the other hand, has a sign opposite to that for a positive index medium. This means that forward propagation gives converging wavefronts. With propagation, the phase fronts move from the right to the left, as shown in Fig. 4. The variations of w共z兲 and R共z兲 are given as
w2共z兲 =
5. NONLINEAR EXTENSION OF THE DISPERSION RELATION FOR NEGATIVE INDEX MATERIALS: UNDERLYING PARTIAL DIFFERENTIAL EQUATION AND SOLITARY WAVE SOLUTIONS
z2 + 共Cw02/20兲2 z
+ C + D2 = 0.
共36兲
In our quest for baseband solitary wave solutions, we move to a traveling frame of reference = z − t. Incorporating this in Eq. (36), we get −
2 2
+ C + D2 = 0.
共37兲
Following Korpel,12 a solution of Eq. (37) can be expressed in the form = A sech2共兲. A direct substitution into Eq. (37) yields 2A2sech2共兲3 sech2共兲 − 2 + CA sech2共兲 + DA2sech4共兲 = 0, which leads to
.
共35兲
= C/42,
Thus the solution of Eq. (36) becomes
=−
Fig. 4. Changes in wavefront radius with propagation distance. Solid line indicates n ⬍ 0; dotted line n ⬎ 0.
3C 2D
共38兲
A = − 3C/2D.
冋冉
sech2 z −
C 42
t
冊册
.
共39兲
The velocity of the solitary wave is positive, as a result of balance between dispersion and nonlinearity. Also, the amplitude is positive (negative) for a negative (positive) nonlinearity, respectively. Note that a taller solitary wave has a larger velocity, similar to the conventional KdV soliton. However, unlike the KdV soliton, the amplitude and the width are not directly related to each other. In our case, a solitary wave with a larger amplitude and width has a larger velocity.
P. P. Banerjee and G. Nehmetallah
Vol. 23, No. 11 / November 2006 / J. Opt. Soc. Am. B
B. Envelope Solitary Waves The nonlinear extension of Eq. (9) can also be achieved by heuristically adding a cubic nonlinear term, as in the Klein–Gordon equation.8 For the traditional NKG equation with cubic nonlinearity, it has been shown that both baseband and envelope solitary wave solutions exist. For the latter case, the solution of the envelope is similar to that of the NLS equation.8,13 Upon incorporating a cubic nonlinearity term, Eq. (9) becomes
= 冑A1 =
=
z t
共40兲
We remark, in passing, that a baseband solitary wave solution of this equation can be expressed in the form of a sech function in a manner similar to that derived in the previous subsection. To find envelope solitary wave solutions, we first assume to be as defined in Eq. (21), and substitute it in Eq. (40) to get
2 e z t
冉
+ j 0
e
+ k0
z
冊
e t
3
+ D2e *e = 0. 4
−
2
+ j共0 − k0兲
e
3 +
4
D2e *e
共42兲
= 0.
Now, upon substituting e = a共兲exp关j共兲兴 in Eq. (42) and separating the real and imaginary parts we get 2
Re:−
d a d2
+ va
冋
Im: 2
冉 冊 d
2
− 共0 − k0兲a
d
da d d d
+a
d 2 d2
册
冉 冊 d d
+ 共 0 − k 0 兲
3
3
+ Da = 0, 4
da d
= 0.
d
=
冉
1 0 2
C −
0
冊
.
d
2
= A 1a + A 3a 3 ,
where A1 =
冉
3 0 4
C −
0
冊
4
0
C −
,
0
A=
2
=2
A1
−2
−
2
3D
A3
.
C+
冑3
A=
0
2 0
冑 冉冑 3
2
− D 2
3
0 + C
冊
共46兲
.
If we substitute the value of from Eq. (46) into Eq. (44) and integrate with respect to we get
=−
=−
冉
1 0 2
C −
冉冑
1 2 2
3
0
z−
冊冉
z−
0 2/冑3 + C/0
20/冑3
2/冑3 + C/0
t
冊
冊
共47兲
t .
The final solution of Eq. (40) is
= Re
冦冑
2 0
D −3
冋冉
⫻ exp j
2
冉冑
2 3
0 + C
0/2 1 + C冑3/共20兲
冊
冤冢
02
sech z −
t−
冑3
z
冊册
2
冑3
t
0 + C
冣冥 冧
exp关j共0t + k0z兲兴 .
共48兲
共43b兲
From Eq. (48) we can find the revised group and phase velocities as
g⬘ =
共44兲
共45兲
2
,
.
Following Korpel,12 assume that the solution of Eq. (45) is a solitary solution of the form: a = A sech共兲. Upon substituting this solution in Eq. (45) we get
冋
0 2/冑3 + C/0
0 1 + ⬘p = −
3D A3 =
冊 冑 冊 冑
C −
共43a兲
If we put Eq. (44) into Eq. (43a) we get d 2a
D
,
2
After multiplying Eq. (43b) with a and integrating with respect to we get d
0
2 0
02
共41兲
Next, setting e共z , t兲 = e共z − t兲 = e共兲, Eq. (41) becomes
2 e
−
冉
We can also calculate A and as function of only and arrive at
2
+ C + D3 = 0.
2
冑 冉
=
冑3
2353
C
0
冉
⬇
02 C
1 2 + C冑3/共0兲 1
1−
C冑3/共0兲
冊
册
⬅ g ,
⬇−
02 C
共49a兲
⬅ p ,
共49b兲
for Ⰶ k0. Note that the change in the group and phase velocities is due to the effect of the balance between nonlinearity and dispersion, as in the case of soliton solutions of the NLS equation. Note also that for A to be real, the nonlinearity coefficient D has to be negative. The intensity dependent dispersion relation8,14 can be found by substituting = Re兵a exp j共t − kz兲其 into Eq. (40) to get
= − C/k − 3Da2/共4k兲,
共50兲
and expressing the dispersion around the operating frequency and wavenumber 共0 , −k0兲 as a Taylor series expansion. After some algebra, this becomes
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J. Opt. Soc. Am. B / Vol. 23, No. 11 / November 2006
− 0 = 共/ k兲
冏
1 2 2 0共k + k0兲 + 共 / k 兲 2
+ 共/ a2兲
冏
0a
2
冏
2
2 0共k + k0兲
,
+ C = 0.
z t 共51兲
where
共52兲
For D ⬍ 0 , 共 / a2兲兩0 ⬍ 0; also note that for the chosen dispersion relation (see Fig. 1), 共2 / k2兲兩0 ⬎ 0, which are the conditions for the solitary wave solutions in our negative index medium. Note that the general dispersion relation (51) expressed in a Taylor series can model any negative index material in general as long as the group velocity 共 / k兲兩0 ⬎ 0. It is automatically understood that the phase velocity is negative from the nature of the Taylor series expansion around 0 , −k0 ; 0 ⬎ 0 , k0 ⬎ 0. We remark that for the conventional NLS equation bright soliton solutions are obtained for similar conditions: 共2 / k2兲兩0 ⬎ 0 [implying anomalous dispersion 共2k / 2兲兩0 ⬅ 2 ⬍ 0]13 and n2 ⬀ −共 / a2兲兩0 ⬎ 0 where n2 represents the nonlinear refractive index coefficient.8
共A1兲
Putting an initial condition 共z , t = 0兲 = u共z兲 = 21 关1 + sgn共z兲兴, its Fourier transform is
冋
1
⌿共kz,t = 0兲 =
共/ k兲兩0 = 02/C,共2/ k2兲兩0 = 203/C2,共/ a2兲兩0 = 3D0/4C.
P. P. Banerjee and G. Nehmetallah
2
␦共kz兲 + j
2 kz
册
.
Taking the Fourier transform with respect to z of Eq. (A1), we arrive at ⌿共kz,t兲 = ⌿共kz,t = 0兲exp共− jC/kzt兲 1 =
2
冋
␦共kz兲 + j
2 kz
册
exp共− jC/kzt兲.
共A2兲
Taking the inverse Fourier transform of Eq. (A2) we get
共z,t兲 =
冕 冋 ⬁
1 2
1
−⬁
⫻exp共− jkzz兲dkz =
2
j 2
␦共kz兲 + j
冕
⬁
册
2 kz
exp共− jCt/kz兲
1
−⬁
kz
exp共− jCt/kz − jkzz兲dkz . 共A3兲
6. CONCLUSION We have analyzed linear and nonlinear propagation in negative index materials by first starting from a simple dispersion relation and by deriving the underlying partial differential equation. The dispersion relation we used modeled a transmission line with modified distributed circuit parameters, but the technique can be adapted to any dispersion relation. Illustrative examples based on a transfer function approach show the propagation of modulated pulses through such a medium and its step response. Similar to the transfer function approach to diffraction, we have also developed transfer functions describing beam propagation in a negative index medium. Gaussian beam propagation is analyzed, and the results reconciled with a ray transfer matrix approach, as applied to propagation in negative index materials. We have also made nonlinear extensions of the linear partial differential equation by incorporating quadratic and cubic terms, similar to what is done for the NKG equation. With quadratic and cubic terms added, baseband solitary wave solutions have been found. Furthermore, using a cubic nonlinearity, we have derived the nonlinear PDE for the envelope and found envelope solitary wave solutions, whose mathematical form is similar to the soliton solutions of the NLS equation. We have shown that solitary wave solutions require anomalous dispersion and a negative sign of the nonlinearity coefficient 共 / a2兲兩0.
APPENDIX A: TEMPORAL EVOLUTION OF A UNIT STEP FUNCTION IN A NEGATIVE INDEX MEDIUM From the dispersion relation for a negative index material we obtain the PDE
Setting kz = −kz⬘ in Eq. (A3) we get
共z,t兲 = −
j 2
冕
⬁
1 kz
−⬁
exp共jCt/kz + jkzz兲dkz .
共A4兲
From Eqs. (A3) and (A4) we get
共z,t兲 =
1
冕
⬁
0
1 kz
sin共Ct/kz + kzz兲dkz .
共A5兲
From Ref. 15 we get
再
for
z ⬎ 0,
共z,t兲 = J0共2冑Czt兲,
z ⬍ 0,
共z,t兲 = 0.
共A6兲
APPENDIX B: RAY TRANSFER MATRIX APPROACH TO PROPAGATION IN LINEAR NEGATIVE INDEX MEDIUM Using the definition of the ray transfer matrix ABCD as
冉 冊冋 xout
out
=
A
B
C
D
册冉 冊 xin
in
,
共B1兲
where 共x , 兲T represent the ray position and angle coordinates, one can derive the translation matrix T for propagation z through negative index material as T=
冉 冊 1
z
0
1
,
共B2兲
˜ from air (refractive index 1) and the refraction matrix R to the negative index material (refractive index −兩n兩) as
P. P. Banerjee and G. Nehmetallah
˜= R
冤
Vol. 23, No. 11 / November 2006 / J. Opt. Soc. Am. B
1
0
兩n兩 + 1 1
−
−
兩n兩 R
冥
1 , 兩n兩
共B3兲
where R is the radius of curvature of the interface. For a plane interface R → ⬁, the refraction matrix from air to the negative index material reduces to11 ˜= R
冋
1
0
0
− 1/兩n兩
册
.
˜⬘= R
共B4兲
5.
2. 3. 4.
6.
共B6兲 7.
the q parameter is again purely imaginary, implying focusing of the initially diverging beam at a distance zf = 兩n 兩 z0, which is in agreement with Eq. (29). Also, upon incorporating the definition of the q parameter and noting that the propagation constant in the negative index material is −k0, it follows that wf = w0, as derived earlier by using wave theory. Incidentally, using the transfer matrix and q transformation method, it can also be readily shown that if the Gaussian beam traverses a slab of the negative index material of thickness d, and re-emerges in air, it can refocus a distance zf⬘ = d/兩n兩 − z0
共B7兲
behind the negative index medium, in agreement with the inferences from Pendry.1 In calculating this, the refrac˜ ⬘ from the negative index medium to air tion matrix R
0
− 兩n兩
册
共B8兲
REFERENCES
q共z兲 = − 兩n兩q0 + 共z − 兩n兩z0兲.
z = zf = 兩n兩z0
0
Corresponding author P. Banerjee can be reached by e-mail at
[email protected].
1.
Thus at
1
needs to be used as well for transformations of the q parameter. The result [Eq. (B7)] is also in agreement with the results of calculations of conjugate planes of the negative material (lens).11
For a Gaussian beam with plane wavefronts and hence a q parameter q0 = j共k0+w02 / 2兲 at a distance z0 in air in front of a semi-infinite negative index material of refractive index −兩n兩, one can use the bilinear transformation q⬘ = Aq + B / Cq + D to track the q parameter q共z兲 of the beam a distance z inside the medium.10 Using straightforward algebra, this becomes 共B5兲
冋
2355
8. 9. 10. 11.
12. 13. 14. 15.
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