Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

Chapter

6 Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems Dong Yang, Shiva Kumar, and Hao Wang

Contents

1. Introduction 2. Configuration of a Time-Lens–based Optical Signal Processing System 3. Wavelength Division Demultiplexer 4. Dispersion Compensator 5. Optical Implementation of Orthogonal Frequency-Division Multiplexing Using Time Lenses 6. Conclusions Acknowledgment Appendix A Appendix B Appendix C References

203 206 211 216 219 226 227 227 228 229 231

1. INTRODUCTION The analogy between the spatial diffraction and temporal dispersion has been known for years (Kolner, 1994; Papoulis, 1994; Van Howe & Xu, 2006). In spatial domain, an optical wave propagating in free space diverges due to diffraction. As an analog, in the temporal domain, an optical pulse propagating in a dispersive medium broadens due to dispersion. This spacetime duality can also be extended to lenses. The conventional space lens produces quadratic phase modulation on the transverse profile of the input Department of Electrical and Computer Engineering, McMaster University, Canada Advances in Imaging and Electron Physics, Volume 158, ISSN 1076-5670, DOI: 10.1016/S1076-5670(09)00011-1. c 2009 Elsevier Inc. All rights reserved. Copyright

203

204

Dong Yang et al. (1)

β 21

(2)

β 21

(1)

β 22

Time lens 1 f1

(2)

β 22

Time lens 2 f1

Temporal filter

f2

f2

FIGURE 1 Scheme of a typical 4-f system. PM1 = phase modulator 1; PM2 = phase modulator 2. (Based on Yang et al. (2008).)

beam and its analog, time lens, simply applies a quadratic phase modulation to a temporal optical waveform. On this basis, several temporal analogs to spatial systems based on thin lenses have been created and many real-time optical signal processing applications, including temporal imaging, pulse compression, and temporal filtering based on time-lens systems have been proposed (Azana & Muriel, 2000; Berger, Levit, Atkins, & Fischer, 2000; Kolner & Nazarathy, 1989; Lohmann & Mendlovic, 1992). The temporal filtering technique was first proposed by Lohmann and Mendlovic (1992). In their pioneering work, a temporal filter was introduced in a 4-f configuration consisting of time lenses. In the spatial domain, a conventional lens produces the Fourier transform (FT) at the back focal plane of an optical signal placed at the front focal plane, which is known as a 2-f configuration or 2-f subsystem. The spatial filter placed at the back focal plane modifies the signal spectrum, and a subsequent 2-f subsystem provides the Fourier transform of the modified signal spectrum, which returns the signal to the spatial domain with spatial inversion. There exists an exact analogy between spatial filtering and temporal filtering techniques (Lohmann & Mendlovic, 1992). In the case of temporal filtering, the spatial lens is replaced by a time lens (which is nothing but a phase modulator), and spatial diffraction is substituted with secondorder dispersion. Yang, Kumar and Wang (2008) discuss a modified 4-f system consisting of two time lenses (Figure 1). In this 4-f configuration, the subsystem T1 provides the Fourier/inverse FT of the input signal. The signs of chirp and dispersion coefficients are reversed in the 2-f subsystem T2 after the temporal filter. In contrast, Lohmann and Mendlovic (1992) proposed the 4-f configuration in which the signs of the dispersion coefficients of the 2-f subsystems are identical. This implies that the second 2-f subsystem (Lohmann & Mendlovic, 1992) provides the FT of the modified signal spectrum leading to a time-reversed bit pattern, whereas in the approach used by Yang et al. (2008), it provides the inverse Fourier transform (IFT) so that the output bit pattern is not reversed in time. The approach proposed

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

205

by Yang et al. (2008) has no spatial analog since the sign of spatial diffraction cannot be changed (Goodman, 1996, chap. 4). Based on the 4-f configuration consisting of time lenses, three applications have been numerically implemented. One of them is a tunable wavelength division demultiplexer (Yang et al., 2008). The wavelength division multiplexing (WDM) demultiplexer is realized using the temporal band pass filter in a 4-f system. The temporal filter is realized using an amplitude modulator, and the channel to be demultiplexed can be dynamically changed by changing the input voltage to the amplitude modulator. The passband of the temporal filter is chosen to be at the central frequency of the desired channel with a suitable bandwidth such that only the signal carried by the channel to be demultiplexed passes through it with the least attenuation. The wavelength division multiplexed signal passes through the 2-f subsystem T1 (see Figure 1) and its FT in the time domain is obtained. After the temporal filter, the signals in any other undesired channels are blocked and the signal in the desired channel then passes through the 2-f subsystem T2, which finally produces the demultiplexed signal as its original bit sequence by the exact IFT. Another application of the 4-f temporal filtering scheme is a higherorder dispersion compensator (Yang et al., 2008). The temporal filter in this application is realized by a phase modulator. To compensate for fiber dispersion, the time domain transfer function of the phase modulator has the same form as the frequency domain transfer function of fiber but the signs of dispersion coefficients are opposite. At the temporal filter, the Fourier transformed input signal is multiplied by the time domain transfer function of the phase modulator so that fiber dispersion–induced phase shift is canceled out. Finally, an implementation of orthogonal frequency-division multiplexing (OFDM) in the optical domain using Fourier transforming properties of time lenses is discussed (Kumar & Yang, 2008). The first 2-f subsystem provides the IFT of the input signal that carries the information. This input signal is obtained by the optical/electrical time division multiplexing of several channels. The kernel of the IFT is of the form exp (i2π f t) and therefore, IFT operation can be imagined as the multiplication of the signal samples of several channels by the optical subcarriers. These subcarriers are orthogonal and therefore, the original signal can be obtained by a Fourier transformer at the receiver, which acts as a demultiplexer. The temporal filter between T1 and T2 is characterized as the transfer function of a fiber-optic link with nonlinearity. The second 2-f subsystem provides the FT of the output from the fiber link so that the convolution of the input optical signal and the fiber transfer function in time domain is converted into the product of the FT of both, but still in time domain. The output of the Fourier transformer is the product of the original input signal (input signal of T1) and the phase correction due to the fiber transfer

206

Dong Yang et al.

function. The photodetector responds only to the intensity of the optical signal and therefore, the deleterious effects introduced by the fiber that appear as the phase correction are removed by the photodetector. This occurs because the fiber transfer function due to dispersive effects is of the type exp [iβ( f )]. In the absence of fiber nonlinearity, optical carriers are orthogonal and the Fourier transformer at the receiver demultiplexes the subcarriers without introducing any penalty. However, strong nonlinear effects can destroy the orthogonality of the optical subcarriers. Nevertheless, the simulation in Kumar and Yang (2008) shows that the time-lens–based optical OFDM system scheme has good tolerance to the fiber nonlinearity.

2. CONFIGURATION OF A TIME-LENS–BASED OPTICAL SIGNAL PROCESSING SYSTEM Figure 1 shows the modified 4-f system based on time lenses (Yang et al., 2008). This 4-f system consists of two cascaded 2-f subsystems T1 and T2, each of which contains a time lens and two segments of single-mode fibers (SMFs) that are symmetrically placed on both sides of the time lens. A time lens is a temporal analog of a space lens and it is realized by an electro-optic phase modulator that can generate quadratic phase modulation. The signal spectrum can be modified using a temporal filter. The temporal filter can be realized using an amplitude and/or phase modulator. The transfer function of the temporal filter can be changed by changing the input voltage to the amplitude and/or phase modulator. For the proper operation of this system, two phase modulators and the temporal filter should be properly synchronized. If Tk is the absolute time at the phase modulator k, k = 1, 2, then they are related by T2 = T1 + f 1 /υg1 + f 2 /υg2 ,

(1)

where υg1 and υg2 are the group speeds of the fibers after the time lens 1 and just before the time lens 2, respectively. f 1 and f 2 are the fiber lengths as shown in Figure 1. This delay between the driving voltages of the phase modulators can be achieved using a microwave delay line. Similarly, the absolute time T f at the temporal filter is related to T1 by T f = T1 + f 1 /υg1 ,

(2)

where f 1 is the length of the SMF in T1. Propagation of the optical signal in a system consisting of a dispersive SMF and the time lens (such as the 2-f subsystem T1 in Figure 1) results in the FT of the input signal at the length 2 f 1 . We use the following definitions

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

207

of FT pairs: U˜ (ω) = F{u(t)} = u(t) = F −1 {U˜ (ω)} =

1 2π

Z

+∞

u(t) exp(iωt)dt −∞ Z +∞

U˜ (ω) exp(−iωt)dω,

(3) (4)

−∞

where u(t) is a temporal function and U˜ (ω) is its FT. The time lens is implemented using an optical phase modulator whose time domain transfer function is given by   h j (t) = exp iC j t 2 ,

j = 1, 2,

(5)

where C j is the chirp coefficient of the phase modulator j in the 2-f subsystem T j , j = 1, 2, and t is the time variable in a reference frame that moves with an optical pulse. Using Eq. (5) in Eq. (3), we obtain the FT of the transfer function h j (t): p ω2 H˜ j (ω) = iπ/C j exp −i 4C j

! ,

j = 1, 2.

(6)

Suppose that the field envelope of the input signal is u(t, 0), and the (k) (k) corresponding FT is U˜ (ω, 0). Let β21 and β22 be the dispersion coefficients of the first and second fibers in the subsystem Tk , k = 1, 2, respectively, and f k be the length of the SMF in Tk , k = 1, 2. Before the time lens of T1 (z = f 1− ), the temporal signal is (Agrawal, 1997) 1 u(t, f 1− ) = 2π

Z

+∞

−∞

U˜ (ω, 0) exp



 i (1) 2 β f 1 ω − iωt dω. 2 21

(7)

The behavior of the time lens is described as follows: u(t, f 1+ ) = u(t, f 1− ) h 1 (t).

(8)

Because the product in the time domain becomes a convolution in spectral domain, taking the FT of Eq. (8), we obtain U˜ (ω, f 1+ ) = F {u(t, f 1− )h 1 (t)} Z ∞
1 = U˜ ω − ω0 , f 1− H˜ 1 (ω0 )dω0 , 2π −∞

(9)

208

Dong Yang et al.

where U˜ (ω, f 1− ) is the FT of u(t, f 1− ). Hence, at the end of the first 2-f subsystem T1, we obtain 1 u(t, 2 f 1 ) = 2π

Z

∞

U˜ (ω, f 1+ ) exp



−∞

 i (1) 2 β f 1 ω − iωt dω. 2 22

(10)

Substituting Eq. (9) into Eq. (10), by choosing the focal length as (Lohmann & Mendlovic, 1992)   (1) f1 = 1 2β22 C1 ,

(11)

and after some algebra (see Appendix A), we obtain the following: √ iπ/C1 ˜ U u(t, 2 f 1 ) = (1) 2π β22 f 1

t (1)

β22 f 1

! , 0 exp (−iφres ) ,

(12)

where the residual phase term is given by 

φres

 (1) (1) β + β 21 22  2  1 =  (1) −  t . 2 (1) β22 f 1 2 β22 f1 

(13)

Let us consider the case when the temporal filter is absent. This can be divided into two cases. (1)

(1)

(1)

Case 1: β22 = β21 = β2 In this case, φres = 0 and

√ iπ/C1 ˜ U u(t, 2 f 1 ) = (1) 2π β2 f 1

t (1)

β2 f 1

! ,0 .

(14)

.  (2) (2) (2) (2) Choosing β21 = β22 = β2 and f 2 = 1 2β22 C2 in the 2-f subsystem T2 and noting that     (1) (1) (1) F{U˜ (t/β2 f 1 , 0)} = 2π β2 f 1 u −β2 f 1 ω, 0 ,

(15)

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

209

we finally obtain (see Appendix B) (1)

u(t, 2 f 1 + 2 f 2 ) = u −

β2 f 1 (2)

β2 f 2

! t, 0 ,

(16)

where 2 f 1 + 2 f 2 is the total length of the time-lens system. For the 4-f configuration proposed by Lohmann and Mendlovic (1992), the magnification factor is defined as (1)

M =−

β f1 C2 = − 2(2) . C1 β f2

(17)

2

    (1) (2) If sgn β2 = sgn β2 , from Eq. (16) it follows that M is negative. Defining a positive stretching factor m = |M|, Eqs. (17) and (16) can be rewritten as (1)

(2)

β2 f 1 = mβ2 f 2

and C2 = mC1

(18)

and u (t, 2 f 1 + 2 f 2 ) = u (−mt, 0) ,

(19)

which shows that T2 provides the scaled FT of its input and then leads to an inverted image of the signal input of the 4-f configuration (Lohmann & Mendlovic, 1992). If M = −1, T2 provides the exact FT and this leads to the reversal of the bit sequence within a frame, which requires additional signal processing in the optical/electrical domain to recover the original bit sequence. In Yang et al. (2008), the 4-f of Lohmann and Mendlovic (1992)   system   (1)

is reconfigured. Suppose sgn β2 rewritten as (1)

(2)

(2)

= −sgn β2

β2 f 1 = −mβ2 f 2

, Eqs. (17) and (16) can be

and C2 = −mC1

(20)

and u(t, 2 f 1 + 2 f 2 ) = u(mt, 0),

(21)

which shows that the 2-f subsystem T2 provides the scaled IFT so that the signal at the end of the 4-f system is not time-reversed. If M = 1, the output signal is identical to the input signal. We note that in spatial optical

210

Dong Yang et al.

TABLE 1 Comparison of fiber dispersions and phase coefficients for different time-lens systems Parameters

PM1

PM2

T1

Function

T2

C1

C2

(1) β21

(1) β22

(2) β21

(2) β22

Case 1

+ − + −

+ − − +

+ − + −

+ − + −

+ − − +

+ − − +

Time reversal Time reversal No time reversal No time reversal

Case 2

+ −

− +

− +

+ −

− +

+ −

No time reversal No time reversal

Based on Yang et al. (2008).

signal processing, it is not possible to obtain both direct and inverse Fourier transformation since the sign of diffraction cannot be changed (Goodman, 1996, chap. 4). Table 1 lists the signs of fiber dispersions and chirp coefficients of subsystems T1 and T2 to produce signals with and without time reversal. (1) (1) Case 2: β22 = −β21 In this case, φres = exp −i

!

t2

(22)

(1)

β22 f 1

and √ iπ/C1 ˜ U u(t, 2 f 1 ) = (1) 2π β22 f 1

t (1)

β22 f 1

! , 0 exp −i

t2

!

(1)

β22 f 1

.

(23)

If we configure the 2-f subsystem T2 with reversed parameters, (2)

(1)

β21 f 2 = −β22 f 1 ,

C2 = −C1 ,

(2)

(1)

β22 f 2 = −β21 f 1 ,

(24)

then we find that the input signal of T1 can be exactly recovered at the end of T2 (see Appendix C), u(t, 2 f 1 + 2 f 2 ) = u(t).

(25)

Lines 5 and 6 in Table 1 present two possible configurations of Case 2. The (2) (1) result of Eq. (25) has a simple physical explanation. When β21 f 2 = −β22 f 1 , the accumulated dispersion of the second fiber of T1 is compensated by that

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

211

of the first fiber of T2, leading to unity transfer function. After that, since chirp coefficients C1 and C2 are of opposite sign, they cancel each other too, making the transfer function from PM1 to PM2 (see Figure 1) to unity. Finally, the accumulated dispersion of the first fiber of T1 is compensated by that (1) (2) of the second fiber of T2 when β22 f 2 = −β21 f 1 . Thus, the total transfer function of the 4-f system is unity. By inserting a temporal filter between two 2-f subsystems (see Figure 1), we can easily fulfill various kinds of optical signal processing in the time domain. The important advantage of the time-lens–based temporal filtering technique is that the transfer function of the temporal filter can be dynamically altered by changing the input voltage to the amplitude/phase modulator and therefore, this technique could have potential applications for switching and multiplexing in optical networks. Next, we provide two examples of optical signal processing based on this time-lens temporal filtering technique to show the potential advantages of this temporal filtering technique. One is a tunable wavelength division demultiplexer and the other is a higher-order fiber dispersion compensator.

3. WAVELENGTH DIVISION DEMULTIPLEXER For fiber-optic networks, tunable optical filters are desirable so that center wavelengths of the channels to be added or dropped at a node can be dynamically changed. Tunable optical filters are typically implemented using directional couplers or Bragg gratings. Yang et al. (2008) discuss a temporal filtering technique for the implementation of a tunable optical filter. As an example, let us consider the case of a 2-channel WDM system at 40 Gb/s channel with a channel separation of 200 GHz. Let the input to the 4-f system be the superposition of two channels as shown in Figure 2b. Here we ignore the impairments caused by the fiber-optic transmission and assume that the input to the 4-f system is the same as the transmitted multiplexed signal. In this example, we have simulated a random bit pattern consisting of 16 bits in each channel. The bit “1” is represented by a Gaussian pulse of width 12.5 ps. We assume that the bit rate is 40 Gb/s and therefore the signal bandwidth for each channel is approximately 80 GHz (Returnto-zero (RZ) signal with duty cycle of 0.5). Thus, the channel separation of 200 GHz is wide enough to avoid channel interference. Assume that channel 1 is centered at the optical carrier frequency ω0 and channel 2 is centered at ω0 + 21ω, where 21ω is the channel separation. A band pass filter that is also centered at ω0 and with double-side bandwidth of 21ω can allow channel 1 to transmit while it blocks the channel 2. In this case, the temporal band pass filter is realized using an amplitude modulator—for instance, an

212

Dong Yang et al. (b) 8

Channel 1 Power (mW)

1

0 –200

–100

(a)

0 Time (ps)

100

0 –200

M

200

4

U

0 Time (ps)

200

100

Power (mW)

CH2

200

0 –200

200

1

H(t)

q(t,2f1–)H(t)

–80

0 80 Time (ps) H(t)

Temporal Filter

T2 1

0 200

Transmission

Power (mW)

X 1

–100

100

CH1

|q(t,2f1– )|2

2 Power (mW)

0 Time (ps)

(c)

Channel 2 2

0 –200

–100

H(t) (Arb. units)

Power (mW)

2

q(t,2f1–) T1

q(t,0)

Demuliplexer 0 –200

–100

0 Time (ps)

100

200

(d) Demultiplexed signal

FIGURE 2 A WDM demultiplexer based on a 4-f time-lens system (M = 1). (a) Input signals from channel 1 and channel 2. (b) Multiplexed output signal. (c) Combined signals before and after the temporal filter. (d) Demultiplexed signal in channel 1. (Based on Yang et al. (2008).)

electroabsorption modulator (EAM), with a time domain transfer function H (t) = (1)

(1)



1, 0,

(1)

|t/β2 f 1 | ≤ 1ω otherwise,

(26)

(1)

where β21 = β22 = β2 is the dispersion coefficient of the SMF in the 2-f subsystem T1 and 1ω/2π = 100 GHz. In this section and in Section 4, ( j) ( j) ( j) we assume that β21 = β22 = β2 , j = 1, 2, and M = 1 unless otherwise specified. Considering the realistic implementation of the highspeed amplitude modulator, we need to choose the parameters of the 4-f system carefully. The state-of-the-art EAM operates up to a bit rate of 40 Gb/s (Choi et al., 2002; Fukano, Yamanaka, Tamura & Kondo, 2006), which implies that it can be turned on and off with a temporal separation of 25 ps. From Eq. (26), we see that the amplitude modulator should be turned on for a duration (1) 1T = 2 β2 f 1 1ω

(27)

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

213

and then it is turned off. Setting 1T ≥ 25 ps, we find that (1) β2 f 1 ≥ 20 ps2 .

(28)

Equation (28) can be satisfied using a dispersion compensation module (1) (DCM) with dispersion coefficient β2 = 123 ps2 /km and length f 1 = 1 km. From Eq. (27), we find 1T = 155 ps. To implement IFT in T2, we use a (2) standard SMF with dispersion coefficient β2 = −21 ps2 /km and length (2) (1) f 2 = 5.86 km, which leads to β2 f 2 = −β2 f 1 and M = 1. Figure 2 shows a wavelength division demultiplexer based on a 4-f timelens system. After the 2-f subsystem T1, we obtain the FT of the multiplexed signal in time domain, given by h i q(t, 2 f 1− ) = U˜ 1 (ω) + U˜ 2 (ω)

 , (1) ω=t/ β2 f 1

(29)

where U˜ 1 and U˜ 2 are the spectra of the signals for channel 1 and channel 2, respectively. Then it passes through the temporal filter defined by Eq. (26), and the output is given by h i q(t, 2 f 1+ ) = H (t) U˜ 1 (ω) + U˜ 2 (ω) = U˜ 1 (ω)

  (1) ω=t/ β2 f 1

 , (1) ω=t/ β2 f 1

(30)

and as shown in Figure 2c, the signal from channel 2 is blocked. Thus, at the input end of the 2-f subsystem T2, only the signal from channel 1 is retained. Finally, we obtain the demultiplexed signal for channel 1 at the output of the 2-f subsystem T2 as shown in Figure 2d. According to Eq. (21), it is q(t, 2 f 1 + 2 f 2 ) = u 1 (t) for m = 1, which is identical to the original input of the channel 1. As can be seen, the data in channel 1 can be successfully demultiplexed. For a practical implementation, the quadratic phase factor of Eq. (5) cannot increase indefinitely with time and therefore, a periodic time lens is introduced in Kumar (2007). It is given by

h j (t) =

+∞ X n=−∞


h 0 j t − nt f ,

(31)

214

Dong Yang et al. (a) Input 2 1

Power (mW)

0 –600

–400

–200

0

200

400

600

200

400

600

200

400

600

(b) Output, M = –1 tf

2 1 0 –600

–400

–200

0 (c) Output, M = 1

2 1 0 –600

–400

–200

0 Time (ps)

FIGURE 3 Input and output bit sequences of the WDM demultiplexer based on a 4-f time-lens system. (a) Input. (b) Output with time reversal, M = −1. (c) Output without time reversal, M = +1. Guard time tg = 0 and t f = 400 ps. (Based on Yang et al. (2008).)

where h 0 j (t) =

(   exp iC j t 2 , otherwise

|t| <

t f + tg , 2

j = 1, 2,

(32)

where t f is the frame time and tg is the guard time between the frames. Based on the periodic time lenses, we redo the above simulation with a random bit pattern consisting of 3 frames (=48 bits) in each channel. We assume the frame time t f = 400 ps and therefore, a frame consists of 16 bits. Figure 3a shows the input bit sequence of channel 1. Figures 3b and 3c show the demultiplexed signals for M = −1 (time reversal) and M = 1 (without time reversal), respectively. For the case of M = −1, we choose (1) (2) the same parameters as for the case of M = 1 except that β2 f 1 = β2 f 2 . From Figure 3b, we see that the pulses at the edges of frames are distorted, whereas the output pulses without time reversal as shown in Figure 3c suffer only some minor distortions. As discussed in Kumar (2007), the time-reversal system introduces distortion if there is no guard time between the frames because the bits at the left and right edges of a frame are imaged by the (1) neighboring (in time domain) time lenses. As β2 increases, the distortion also increases because pulses at the edges of a frame broaden more due to high dispersion of the 2-f subsystems and occupy the neighboring frames. In this modified 4-f configuration (M = +1) in Yang et al. (2008), the distortion

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

215

(a) Input 2

tg

tg

1

Power (mW)

0

–600

–400

–200

0

200

400

600

200

400

600

200

400

600

(b) Output, M = –1 tf

2 1 0

–600

–400

–200

0 (c) Output, M = 1

2 1 0

–600

–400

–200

0 Time (ps)

FIGURE 4 Input and output bit sequences of the WDM demultiplexer based on a 4-f time-lens system. (a) Input. (b) Output with time reversal, M = −1. (c) Output without time reversal, M = +1. Guard time tg = 50 ps and t f = 400 ps. (Based on Yang et al. (2008).)

is less (Figure 3c) than with M = −1 (Figure 3b). This is probably because the distortion introduced by the first 2-f system is suppressed by that due to the second 2-f system since the signs of the dispersions are reversed. These distortions can be avoided by introducing a guard time between frames, as shown in Figure 4. Figure 4a shows the input bit sequence with a guard time of 50 ps. Figures 4b and 4c show the output signals with and without time reversal, respectively. From Figures 4b and 4c, we find that the distortion is effectively eliminated by adding the guard time between frames. The advantage of this scheme is that the channels to be demultiplexed at a node can be dynamically reconfigured. For example, if channel 1 has to be blocked instead of channel 2, the transmittivity of the amplitude modulator [Eq. (26)] can be dynamically changed to

H (t) =

  

1,

 

0,

t (1) − 21ω ≤ 1ω β f1 2 otherwise

(33)

such that channel 1 is in the stop band of the temporal filter and channel 2 is in the passband.

216

Dong Yang et al.

4. DISPERSION COMPENSATOR The following example is the implementation of a higher-order fiber dispersion compensator based on the temporal filtering technique. As the bit rate of a single channel increases, its spectrum broadens and the higher-order dispersion leads to pulse distortion in such a transmission system that limits the system performance. Various dispersion compensation techniques have been developed. The most commercially advanced technique is negativedispersion fiber-based dispersion compensation, but the disadvantage is that, in practice, it is hard to design and manufacture a dispersioncompensating fiber with negative dispersion slope due to its higher sensitivity to waveguide profile fluctuations (Srikant, 2001). Yang et al. (2008) discuss a dispersion compensation technique based on the time-lens–based temporal filtering technique that can compensate for any order of fiber dispersion in fiber-optic transmission systems. In this application, the temporal filter is realized using a phase modulator instead of an amplitude modulator and its time domain transfer function is given by X ωn d n β H (t) = exp −i L n! dωn n≥2

!

,

(34)

(1) ω=t/(β2 f 1 )

where d n β/dωn is the nth order dispersion. To realize the transfer function given by Eq. (34), an arbitrary waveform generator (AWG) is required to drive the phase modulator. Considering that the bit sequence is processed frame by frame, if the frame time is t f , then for each frame, the frequency resolution is given by 1f =

1 . tf

(35)

Before using the temporal filter we obtain the FT of the input signal in the   (1) ˜ time domain as U t/β2 f 1 . Therefore, the corresponding time resolution is given by (1) 1t = β2 f 1 (2π 1 f ) .

(36)

The maximum achievable bandwidth, Bmax , of an AWG using the current technology is ∼12.5 GHz (Kondratko et al., 2005). Therefore, we obtain 1t ≥

1 Bmax

= 80 ps.

(37)

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

217

Using Eqs. (35) and (36) in Eq. (37), we obtain the following constraint on the parameters of the time-lens system: (1) β2 f 1 ≥

tf . 2π Bmax

(38)

From Eq. (38), we see that if the frame is too wide, the fiber with higher dispersion coefficient is required when the bandwidth of AWG is fixed. However, on the other hand, if the frame is too narrow, synchronization of various modulators should be done at a higher frame rate (=1/t f ). Usually synchronization is done by extracting the clock from the signal (Nakazawa, Yamada, Kubota & Suzuki, 1991) and synchronization is easier at lower frame rates. From Eq. (38), we see that if the bandwidth of the AWG is too small, we need a fiber with large dispersion. We choose dispersion coefficient (1) β2 = 123 ps2 /km and the frame time t f = 100 ps. From Eq. (38), we find f 1 = 10.4 km. With the above choice of frame time, the synchronization of the frame needs to be done by extracting a clock at 10 GHz (=1/t f ). Without loss of the generality, in this example only the second- and thirdorder fiber dispersion effects are taken into account. Therefore, the temporal filter is simplified to   H (t) = exp −iω2 β2 L/2 − iω3 β3 L/6

,  (1) ω=t/ β2 f 1

(39)

where β2 and β3 are the second- and third-order dispersions of the (1) transmission fiber, respectively. L is the transmission distance. β2 is the dispersion coefficient of the SMF in the 2-f subsystem T1. In the simulation, we ignore the nonlinear effect and amplifier noise. The bit “1” is represented by a Gaussian pulse with width of 25 ps, and the input bit sequence is shown in Figure 5a. We simulate a random bit pattern with 20 bits (10 frames), and the bit rate is 20 Gb/s such that there are two bits within each frame in this case. The guard time is chosen as tg = 12 ps. The second-order dispersion of the transmission fiber is β2 = −21 ps2 /km and the transmission distance is 10 km. To highlight the effect of the third-order dispersion, we set β3 = 20 ps3 /km, rather than the standard value of 0.1 ps3 /km, but it does not affect the generality of this dispersion compensation technique. After propagation along the transmission fiber with both the second- and third-order dispersions, each bit has been broadened beyond its bit interval and hence the output signal after fiber transmission is distorted as shown in Figure 5b, which is then input to the 4-f system. As analyzed previously, the 2-f subsystem T1 provides the FT of the signal input to the 4-f system in the time domain such that

218

Dong Yang et al. (b) 1.6

Power (mW)

(a) Input

Power (mW)

2

1

0

Transmission

–500

0 Time (ps)

1

0

–500

0 Time (ps)

500

500

(c)

(d) Output 2.5

Temporal Filter H(t) Power (mW)

2 T2

T1

1 Dispersion Compensator

0

–500

0 Time (ps)

500

FIGURE 5 A dispersion compensator based on a 4-f time-lens system. t f = 100 ps, tg = 12 ps, and M = 1. (a) Input signal. (b) Output signal after fiber propagation. (c) Dispersion compensator based on the time-lens system. (d) Output after the dispersion compensator. (Based on Yang et al. (2008).)

  q(t, 2 f 1− ) = U˜ (ω) exp iω2 β2 L/2 + iω3 β3 L/6

 , (1) ω=t/ β2 f 1

(40)

where U˜ (ω) is the FT of the launched bit sequence as shown in Figure 5a and the exponential part is the linear transfer function of the fiber transmission system. Thus, combining Eqs. (39) and (40), after the temporal filter we obtain q(t, 2 f 1+ ) = q(t, 2 f 1− )H (t)  = U˜ (ω) (1)

ω=t/ β2 f 1

,

(41)

and it is shown from Eq. (41) that the phase shift caused by the transmission fiber has been completely canceled out by the temporal filter. Since M = 1, (1) the 2-f subsystem T2 provides the exact IFT of q(t, 2 f 1+ ) = U˜ (t/β2 f 1 ) so that q(t, 2 f 1 + 2 f 2 ) = u(t, 0), where u(t, 0) is the input signal of the fiber transmission system. Figure 5 demonstrates the process of the dispersion compensation undertaken by the time-lens–based temporal filtering system.

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

219

(a) Input 2 1

Power (mW)

0

–500

0

500

(b) Output, t g = 12 ps 2 1 0

–500

0

500

(c) Output, t g = 24 ps 2 1 0 –600

–400

–200

0 Time (ps)

200

400

600

FIGURE 6 Input and output bit sequences of the dispersion compensator based on a 4-f time-lens system (M = 1 and t f = 100 ps). (a) Input. (b) Output with guard time tg = 12 ps. (c) Output with guard time tg = 24 ps. (Based on Yang et al. (2008).)

Figure 5d shows that there is some distortion of the output signal since the guard time (=12 ps) used in this simulation is quite small. To avoid this distortion, we use the guard time tg = 24 ps and redo the above simulation. Figure 6 shows the simulation results with tg = 12 ps and tg = 24 ps and the rest of the parameters is the same as that used in Figure 5. Figure 6a shows the input bit sequence (the same as Figure 5a). Figure 6b (the same as Figure 5d) and Figure 6c show the output signals for the cases of tg = 12 ps and tg = 24 ps, respectively. As can be seen, the distortion is more effectively suppressed in the case of tg = 24 ps compared with the case of tg = 12 ps.

5. OPTICAL IMPLEMENTATION OF ORTHOGONAL FREQUENCY-DIVISION MULTIPLEXING USING TIME LENSES Orthogonal frequency-division multiplexing is widely used in wireless communication systems and recently has drawn significant attention in optical communication (Djordjevic & Vasic, 2006; Lowery & Armstrong, 2005). OFDM is a form of multiple subcarrier modulation in which subcarriers are orthogonal. Kumar and Yang (2008) propose an optical implementation of OFDM using time lenses. Figure 7 shows an example of a fiber-optic communication system based on OFDM. Single-channel input data are converted into N parallel data paths using serial-to-parallel conversion. Let the symbol interval of the input to the inverse fast Fourier transform (IFFT) be Tblock and each of the parallel

220

Modulation

IFFT

Parallel to serial

...

Serial to parallel

...

Binary input

...

Dong Yang et al. D/A Converter

RF Upconverter

Laser

FIGURE 7 (2008).)

A/D converter

Serial to parallel

FFT

Demodulation

...

RF Downconverter

...

PD

Fiber link

...

Optical modulator

Parallel to Output serial

Block diagram of a conventional OFDM system. (Based on Kumar and Yang

Laser

Modulation

~ uin(t ') Serial input

Inverse Fourier transform

FT

fiber

u out (t )

Fourier transform

u out (t ')

Direct detection

IFT

u out (t )

Fiber link

Serial output

I(t ')

FIGURE 8 Block diagram of the time-lens–based scheme. (Based on Kumar and Yang (2008).)

data channels consists of a complex data ak , k = 1, 2, . . . , N , within each block. These signals are multiplied by subcarriers and multiplexed together using IFFT, which processes data on a block-by-block basis. The subcarriers are sinusoids with frequencies that are integer multiples of 1/Tblock , which makes them orthogonal to each other. The output of IFFT is up-converted to the radio frequency (RF) domain after parallel-to-serial and digital-toanalog (D/A) conversion. An optical carrier is intensity modulated by the RF signal consisting of an OFDM band using optical modulator. At the receiver, the RF signal is down-converted to baseband and the input complex data ak can be recovered from the output of fast-Fourier transform (FFT). The signals in adjacent blocks could interfere with each other due to fiber dispersion leading to interblock interference, which can be avoided by introducing guard intervals between blocks (Djordjevic & Vasic, 2006; Lowery & Armstrong, 2005). Figure 8 shows an optical implementation of OFDM using time lenses. Let the input optical field envelope be u˜ in (t 0 ). We

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

221

define the FT pairs as u(t ˜ ) = F[u(t); t → t ] = 0

0

Z

∞

u(t) exp(−i2π t 0 t)dt,

(42)

−∞

and u(t) = F

−1

[u(t ˜ ); t → t] = 0

0

Z

∞

u(t ˜ 0 ) exp(i2π t 0 t)dt 0 .

(43)

−∞

The IFT and FT are implemented using time lenses. Propagation of an input optical field envelope in a 2-f system consisting of dispersive elements and a phase modulator (time lens) results in either FT or IFT of the input signal at the output of the 2-f system (Jannson, 1983; Lohmann & Mendlovic, 1992). The output of the 2-f system can either be the FT or IFT depending on the signs of second-order dispersion coefficients and chirp factors of the phase modulator (Yang et al., 2008). We choose (1)

(1)

(1)

β21 f 1 = β22 f 1 = β2 f 1 , (2)

(2)

(1)

β21 f 2 = β22 f 2 = −β2 f 1 ,

(44)

so that the first 2-f subsystem provides the IFT and the second 2-f subsystem ( j) provides the direct FT. Define the S j = β2 f j , j = 1, 2 as the accumulated second-order dispersion coefficient of the 2-f subsystems T1 and T2, respectively, we have S1 = −S2 . The temporal filter in this application is characterized as a transfer function of fiber link. Using the time-lens–based 2-f system, the IFT of the input field envelope, u˜ in (t 0 ) is given by (Lohmann & Mendlovic, 1992) p −1 0 0 u IFT out (t) = F [u˜ in (t ); t → t/(2π S1 )]/ −i2π S1 , p = u in (t/(2π S1 ))/ −i2π S1 .

(45)

The input and output of the IFT (T1) are serial and therefore, there is no need for serial-to-parallel and parallel-to-serial conversion. The signal u IFT out (t) is transmitted over the fiber-optic link and at the receiver, the output of the fiber-optic link, u fiber out (t) passes through a Fourier transformer (T2) whose output is 1 0 0 u˜ FT F[u fiber out (t); t → t /(2π S2 )], out (t ) = √ −i2π S2 p 0 = u˜ fiber out (t /(2π S2 ))/ −i2π S2 .

(46)

222

Dong Yang et al.

After passing through the direct-detection optical receiver, the photodiode current is given by 0 2 I (t 0 ) = |u˜ FT out (t )| ,

(47)

assuming unity responsivity of the photodiode. Consider the optical OFDM signal u IFT out (t) propagating in a fiber whose linear transfer function is (Agrawal, 1997) H f ( f ) = exp[iθ ( f )L],

(48)

θ ( f ) = β2 (2π f )2 /2 + β3 (2π f )3 /6 + β4 (2π f )4 /24 + . . . ,

(49)

where

L is the fiber length, and βn is the nth-order dispersion coefficient. In the absence of noise and nonlinearity, the fiber output is IFT u fiber out (t) = u out (t) ∗ h f (t),

(50)

where ∗ denotes convolution and h f (t) = F −1 [H f ( f )] is the fiber impulse response function. Convolution in the time domain becomes a product in frequency domain and therefore, after the Fourier transformer (see Figure 8), the output signal can be written as 1 0 0 0 u˜ FT F{u IFT out (t ) = √ out (t); t → t /(2π S2 )}F{h f (t); t → t /(2π S2 )}. (51) −i2π S2 Using Eqs. (45) and (48), we obtain 0 0 0 u FT out (t ) = u˜ in (t ) exp[i Lθ (t /(2π S2 ))].

(52)

From Eq. (47), the photocurrent is I (t 0 ) = |u˜ in (t 0 )|2 .

(53)

Thus, we see that the effect of fiber dispersion is a mere phase shift [Eq. (52)] in this system and therefore, dispersive effects completely disappear after direct detection and the photocurrent is directly proportional to the input power. So far we have assumed that the aperture of the time lens is infinite. To process the input signal block-by-block basis, periodic time lenses with finite aperture should be introduced (Kumar, 2007; Yang et al., 2008). In this

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

223

case, the phase modulator multiplies the incident optical field envelope by a function ∞ X

h(t) =

h 0 (t − nTofdm ),

(54)

n=−∞ 2

h 0 (t) = exp(iαt ) for |t| ≤ Tblock/2 = 0, elsewhere

(55)

and input signal is of the form u˜ in (t 0 ) =

∞ X

n=−∞ f n (t 0 )

u n (t 0 ) = = 0,

u n (t 0 − nTofdm ),

(56)

for |t 0 | ≤ Tblock /2 elsewhere

(57)

where f n (t 0 ) is an arbitrary input signal in the block n and Tofdm is the total width of the OFDM signal. The difference Tofdm − Tblock is the guard time introduced between consecutive blocks to ensure that interblock interference is not significant. Consider the input signal u n (t 0 ), which is limited to the interval [−Tblock /2, Tblock /2]. Let the highest-frequency component of the signal u n (t 0 ) be f max . After passing through the IFT, the corresponding signal should be confined within the interval [−Tblock /2, Tblock /2]. Using Eq. (45), the above condition leads to |S1 | ≤ Tblock /(2 f max ),

(58)

which gives an upper bound for |S1 |. There are a few differences between the conventional OFDM and the scheme proposed by Kumar and Yang (2008) (see Figure 8). First, the conventional OFDM uses the discrete FT, whereas the scheme of Figure 8 uses continuous FT. Second, conventional OFDM requires high-speed digital signal processors to compute FFT and IFFT and computational time increases as the aggregate bit rate becomes higher. In contrast, the FT and IFT operations of Figure 8 are almost instantaneous except for the small propagation delays in the dispersive elements. This time-lens–based scheme should not be considered a method to compensate for second-order dispersion of the transmission fiber. This is because the dispersive elements of the FT and IFT should have both positive and negative second-order dispersion coefficients. This technique can be used to compensate for thirdand higher-order dispersion coefficients. To illustrate this point, numerical simulations of a dispersion-managed transmission system is carried out

224

Dong Yang et al. 3.5 3

With FT&IFT input Without FT &IFT

Power (mW)

2.5 2 1.5 1 0.5 0 10.4

10.41

10.42 Time (ns)

10.43

10.44

FIGURE 9 Plot of input and output powers vs time. The solid line denotes output power without using FT and IFT. The dotted line shows input power and the crosses show output power with FT and IFT. (Based on Kumar and Yang (2008).)

using the following parameters: bit rate = 320 Gb/s, second- (β2 ), third(β3 ) and fourth- (β4 ) order dispersion of the transmission fiber (SMF) are −21.6 ps2 /km, 0.05 ps3 /km, and 1.69 × 10−4 ps4 /km, respectively, and total transmission distance = 400 km. The commercially available dispersioncompensating module (DCM) has a zero dispersion slope. Therefore, we choose β2 = 128 ps2 /km, β3 = 0 and β4 = 0 for the DCM. The nonlinear coefficients of SMF and DCM are set to zero. Fiber losses are 0.2 dB/km and 0.5 dB/km for SMF and DCM, respectively. Fiber losses are fully compensated by amplifiers that are placed 80 km apart. The amplifiers are assumed to be noiseless to see the impact of dispersion clearly. The secondorder dispersion of the SMF is fully compensated by the inline DCM placed just before each amplifier. u n (t 0 ) is a random bit sequence consisting of Gaussian pulses with FWHM of 1.56 ps, Tblock = 6.4 ns, Tofdm = 7.2 ns, and S1 = 397 ps2 . The dotted line, crosses and solid line in Figure 9 show the input power(|u in (t 0 )|2 ), output power (I (t 0 )) with FT and IFT, and output power without FT and IFT, respectively. In Figure 9, the amplifiers are assumed to be noiseless to see the impact of dispersion clearly. Four blocks of data are simulated and Figure 9 shows a part of the data within a block. As can be seen, there is a significant distortion due to β3 and β4 when FT and IFT are not used, and higher-order dispersive effects can be suppressed using the scheme of Figure 8. The transmission distance can further be increased without distortion by increasing Tblock . In Figure 10, we compare the bit error rate (BER) of the time-lens–based scheme (see Figure 8) and the system without using FT and IFT. In this simulation, amplifier noise is turned on and 4 million bits are used at the input; all the other parameters are kept the same as in Figure 9, except Tblock is changing. Optical signal-noise ratio (OSNR) is calculated using a bandwidth

225

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems 0 –0.5 No OFDM

–1

T block= 3.84 ns T block= 6.4 ns

log10 BER

–1.5 –2 –2.5 –3 –3.5 –4 –4.5 27.5

28

28.5

29

29.5 30 OSNR (dB)

30.5

31

31.5

32

FIGURE 10 Comparison of BER between OFDM-based systems with different TOFDM and the conventional OOK system. (Based on Kumar and Yang (2008).) 4.5

With FT&IFT input

4

Power (mW)

3.5 3 2.5 2 1.5 1 0.5 0

22.42

22.44 22.46 Time (ns)

22.48

22.5

FIGURE 11 Nonlinear performance of the time-lens–based scheme. (Based on Kumar and Yang (2008).)

of 0.1 nm. In the absence of FT and IFT, from Figure 10 we see that BER is nearly independent of OSNR, indicating that degradation is mainly due to higher-order dispersion. As OFDM symbol interval increases, BER reduces, since larger Tblock corresponds to lower subchannel bit rates and therefore higher dispersion tolerance. Figure 11 shows the evolution of pulses in the presence of fiber nonlinearity. The parameters are same as in Figure 8 except that nonlinear coefficients of SMF and DCF are 1.25 W−1 km−1 and 5 W−1 km−1 , respectively, and the launch peak power is 4 mW. As can be seen, there is a pulse distortion due to intrachannel nonlinear impairments, which should be expected

226

Dong Yang et al.

since this time-lens–based scheme does not compensate for nonlinear impairments. Note that the FT and IFT should be properly synchronized to ensure that they process the data on a block-by-block basis. However, the synchronization needs to be done only at the OFDM symbol rate (=1/Tofdm ), which is much lower than the aggregate bit rate.

6. CONCLUSIONS Various issues in the design of a time-lens–based optical signal processing system are discussed. The 2-f subsystem T2 is chosen to be anti-symmetric (2) (1) with respect to the 2-f subsystem T1, that is, β2 = −β2 and C2 = −C1 so that the 2-f subsystem T2 provides the exact inverse Fourier transform, and as a result the bit sequence at the output is not time-reversed. In contrast, as for the 4-f configuration in previous work (Lohmann & Mendlovic, 1992), the sign of dispersion of the fiber in the 2-f subsystem T2 is identical to that in T1 and therefore, T2 provides the FT of its input, which leads to a timereversed output bit pattern and therefore, additional signal processing in the optical/electrical domain to recover the original bit sequence is required. We have discussed three applications using the time-lens–based optical signal processing scheme. First, a possible implementation of a tunable wavelength division demultiplexer is demonstrated. The temporal filter in this example is an amplitude modulator. The advantage of this timelens–based wavelength division multiplexing demultiplexer is that the channels to be demultiplexed at a node of an optical network can be dynamically chosen by changing the passbands of the temporal filter. Second, a temporal filtering technique for the implementation of a higherorder fiber dispersion compensator is discussed. In this case, the temporal filter is realized by a phase modulator. The transfer function of the filter can be expressed as a summation of several independent terms, each of which corresponds to the amount of the phase shift caused by the certain order of fiber dispersion. This dispersion compensation technique can flexibly compensate for any order of fiber dispersion by simply modifying the transfer function of the temporal filter. Finally, an implementation of orthogonal frequency-division multiplexing in the optical domain using Fourier transforming properties of time lenses is discussed. In this application, the first 2-f subsystem (T1) takes role of an inverse Fourier transformer and the second 2-f subsystem (T2) works as a Fourier transformer. The temporal filter between T1 and T2 is characterized as the transfer function of a fiber-optic link with nonlinearity. The output of the Fourier transformer is the original input signal of IFT (T1) multiplied by a phase factor due to fiber dispersive effects. Since the photocurrent is proportional to the absolute square of the optical field, dispersive effects are eliminated using this approach.

227

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

ACKNOWLEDGMENT The authors thank the National Science and Engineering Research Council of Canada for research grant support.

APPENDIX A The output signal at the end of 2-f subsystem T1 is given by 1 u(t, 2 f 1 ) = 2π

+∞

Z

U˜ (ω, f 1+ ) exp



−∞

 i (1) 2 β f 1 ω − iωt dω, 2 22

(59)

where 1 U˜ (ω, f 1+ ) = 2π

∞

Z

U˜ (ω − , f 1− ) H˜ 1 ()d.

(60)

−∞

Substituting Eq. (60) into Eq. (59), we obtain u(t, 2 f 1 ) =



1 2π

2 Z

+∞ Z ∞

U˜ (ω − , f 1− ) H˜ 1 ()   i (1) β22 f 1 ω2 − iωt ddω. × exp 2 −∞

−∞

(61)

From Eqs. (7) and (6), we have U˜ (ω, f 1− ) = U˜ (ω, 0) exp



i (1) β f 1 ω2 2 21

 (62)

and p ω2 H1 (ω) = iπ/C1 exp −i 4C1

! .

(63)

Let ω0 = ω −  in Eq. (61). Inserting Eqs. (62) and (63) into Eq. (61) and rearranging terms, we obtain u(t, 2 f 1 ) =



1 2π

2 s

iπ C1

Z

∞

φ(ω0 )U˜ (ω0 , 0) exp

−∞

 i (1) 02 0 × exp β f 1 ω − iω t dω0 , 2 22



i (1) 2 β f 1 ω0 2 21





(64)

228

Dong Yang et al.

where φ(ω ) = 0

Z

+∞

−∞

i (1) 2 exp β22 f 1 2 − i 2 4C1

!

  (1) exp iβ22 f 1 ω0  − it d. (65)

The 2 terms appearing in the argument of the exponent in Eq. (65) are (1) eliminated if we select the chirp coefficient C1 , dispersion β22 , and the fiber length f 1 are related by C1 =

1 (1) 2β22 f 1

.

(66)

Hence, Eq. (65) becomes Z

+∞

  (1) exp iβ22 f 1 ω0  − it d −∞   (1) = 2π δ t − β22 f 1 ω0 ,

φ(ω0 ) =

(67)

where δ(·) is a delta function. Inserting Eq. (67) into Eq. (64), we finally obtain ! √ t iπ/C1 ˜ U u(t, 2 f 1 ) = ,0 (1) (1) β22 f 1 2π β22 f 1   

  (1) (1) β + β 21 22    1 × exp −it 2  (1) −   . 2 (1) β22 f 1 2 β22 f1

(68)

APPENDIX B (2)

(2)

If β21 = β22 in T2, following the same derivation as in Appendix A, we obtain ! √ iπ/C2 ˜ t U u(t, 2 f 1 + 2 f 2 ) = , 2 f1 (69) (2) (2) β22 f 2 2π β22 f 2 where U˜

t (2)

β22 f 2

! , 2 f1

= F {u(t, 2 f 1 )}|ω=t/β (2) f . 22

2

(70)

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

229

From Eqs. (14) and (15), we have √ iπ/C1 ˜ U u(t, 2 f 1 ) = (1) 2π β22 f 1

t (1)

β22 f 1

! ,0

(71)

and ( F U˜

!)

t (1)

β22 f 1

,0

    (1) (1) = 2π β22 f 1 u −β22 f 1 ω, 0 .

(72)

With the help of Eqs. (71) and (72) and noting that (1)

1/C1 = 2β22 f 1

(73)

(2) 2β22 f 2 ,

(74)

1/C2 = we obtain

q ! (1) (2) (1) i β22 f 1 β22 f 2 β22 f 1 u(t, 2 f 1 + 2 f 2 ) = u − (2) t, 0 . (1) β22 f 2 β22 f 1

(75)

Ignoring the trivial constant in Eq. (75), we finally obtain (1)

u(t, 2 f 1 + 2 f 2 ) = u −

β22 f 1 (2)

β22 f 2

! t, 0 .

(76)

APPENDIX C For Case 2, from Eq. (23), we have √ iπ/C1 ˜ U u(t, 2 f 1 ) = (1) 2π β22 f 1

t (1)

β22 f 1

! , 0 exp −i

t2 (1)

β22 f 1

! .

(77)

The FT of Eq. (23) is given by F {u(t, 2 f 1 )} = U˜ (ω, 2 f 1 ) .

(78)

230

Dong Yang et al.

After propagating in the first single-mode fiber of T2, we obtain U˜ (ω, 2 f 1 + f 2− ) = U˜ (ω, 2 f 1 ) exp



 i (2) β21 f 2 ω2 , 2

(79)

where U˜ (ω, 2 f 1 + f 2− ) is the Fourier transform of the signal right before the time lens 2. Noting that U˜ (ω, 2 f 1 ) = U˜ (ω, f 1+ ) exp



 i (1) 2 β f1ω , 2 22 (2)

(80) (1)

substituting Eq. (80) into Eq. (79) and choosing β21 f 2 = −β22 f 1 , we obtain U˜ (ω, 2 f 1 + f 2− ) = U˜ (ω, f 1+ ) ,

(81)

u (t, 2 f 1 + f 2− ) = u (t, f 1+ ) .

(82)

which leads to

After the time lens 2, we obtain   u (t, 2 f 1 + f 2+ ) = u (t, 2 f 1 + f 2− ) exp iC2 t 2 .

(83)

The first time lens introduces a quadratic phase factor   u (t, f 1+ ) = u (t, f 1− ) exp iC1 t 2 .

(84)

If C2 = −C1 , inserting Eqs. (82) and (84) into Eq. (83), we obtain u (t, 2 f 1 + f 2+ ) = u (t, f 1− ) .

(85)

U˜ (ω, 2 f 1 + f 2+ ) = U˜ (ω, f 1− ) .

(86)

From Eq. (85), we have

After propagating in the second single-mode fiber of T2, we obtain U˜ (ω, 2 f 1 + 2 f 2 ) = U˜ (ω, 2 f 1 + f 2+ ) exp



 i (2) 2 β f2ω . 2 22

(87)

231

Temporal Filtering Technique Using Time Lenses for Optical Transmission Systems

Also note that U˜ (ω, f 1− ) = U˜ (ω, 0) exp



 i (1) β21 f 1 ω2 . 2 (2)

(88) (1)

Substituting Eqs. (86) and (88) to Eq. (87) and choosing β22 f 2 = −β21 f 1 , we obtain U˜ (ω, 2 f 1 + 2 f 2 ) = U˜ (ω, 0)

(89)

u (t, 2 f 1 + 2 f 2 ) = u (t, 0) ,

(90)

and

where u (t, 2 f 1 + 2 f 2 ) is the output signal of T2 and u (t, 0) is the input signal of T1. In summary, for the Case 2, if we choose (2)

(1)

(2)

(1)

β21 f 2 = −β22 f 1 , C2 = −C1 , β22 f 2 = −β21 f 1 ,

(91) (92) (93)

then at the end of the 4-f time-lens system, we can exactly replicate the input signal without time reversal.

REFERENCES Agrawal, G. P. (1997). Fiber-optic communication systems. In Series in microwave and optical engineering (2nd ed.) (p. 49). New York: Wiley. Azana, J., & Muriel, M. A. (2000). Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings. IEEE Journal of Quantum Electronics, 36(5), 517–526. Berger, N. K., Levit, B., Atkins, S., & Fischer, B. (2000). Time-lens-based spectral analysis of optical pulses by electrooptic phase modulation. Electronic Letters, 36(19), 1644–1646. Choi, W.-J., Bond, A. E., Kim, J., Zhang, J., Jambunathan, R., Foulk, H., et al. (2002). Low insertion loss and low dispersion penalty InGaAsp quantum-well high-speed electroabsorption modulator for 40-Gb/s very-short-reach, long-reach, and long-haul applications. Journal of Lightwave Technology, 20(12), 2052–2056. Djordjevic, I. B., & Vasic, B. (2006). Orthogonal frequency division multiplexing for high speed optical transmission. Optics Express, 14(9), 3767–3775. Fukano, H., Yamanaka, T., Tamura, M., & Kondo, Y. (2006). Very-low-driving-voltage electroabsorption modulators operating at 40 Gb/s. Journal of Lightwave Technology, 24(5), 2219–2224. Goodman, J. W. (1996). Fresnel and Fraunhofer diffractions. In Introduction to Fourier optics (2nd ed.) (pp. 63–95). San Francisco: McGraw-Hill. Jannson, T. (1983). Real time Fourier transformation in dispersive optical fibers. Optics Express, 8(4), 232–234.

232

Dong Yang et al.

Kolner, B. H. (1994). Generalization of the concepts of focal length and f-number to space and time. Journal of the Optical Society of America A, 11(12), 3229–3234. Kolner, B. H., & Nazarathy, M. (1989). Temporal imaging with a time lens. Optics Letters, 14(12), 630–632. Kondratko, P. K., Leven, A., Chen, Y.-K., Lin, J., Koc, U.-V., Tu, K.-Y., et al. (2005). 12.5ghz optically sampled interference-based photonic arbitrary waveform generator. IEEE Phontonics Technology Letters, 17(12), 2727–2729. Kumar, S. (2007). Compensation of third-order dispersion using time reversal in optical transmission systems. Optics Letters, 32(4), 346–348. Kumar, S., & Yang, D. (2008). Optical implementation of orthogonal frequency-division multiplexing using time lenes. Optics Letters, 33(17), 2002–2004. Lohmann, A. W., & Mendlovic, D. (1992). Temporal filtering with time lenses. Applied Optics, 31(29), 6212–6219. Lowery, A. J., & Armstrong, J. (2005). 10 Gb/s multimode fiber link using power efficient orthogonal frequency division multiplexing. Optics Express, 13(25), 10003–10009. Nakazawa, M., Yamada, E., Kubota, H., & Suzuki, K. (1991). 10 Gbit/s soliton data transmission over one million kilometres. Electroic Letters, 27(14), 1270–1272. Papoulis, A. (1994). Pulse compression, fiber communications, and diffraction: a unified approach. Journal of the Optical Society of America A, 11(1), 3–13. Srikant, V. 2001, Broadband dispersion and dispersion slope compensation in high bit rate and ultra long haul systems. In Optical Fiber Communication Conference (OFC), 2001 OSA Technical Digest Series, Optical Society of America. Van Howe, J., & Xu, C. (2006). Ultrafast optical signal processing based upon space-time dualities. Journal of Lightwave Technology, 24(7), 2649–2662. Yang, D., Kumar, S., & Wang, H. (2008). Temporal filtering for optical communication systems. Optics Communications, 281(2), 238–247.