Time-frequency conversion based on linear time lens for measuring arbitrary waveform

Optik 127 (2016) 738–741

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Optik journal homepage: www.elsevier.de/ijleo

Time-frequency conversion based on linear time lens for measuring arbitrary waveform夽 Liu Xingyun a,b , Lu Chimei a , Sun Junqiang b,∗ , Liu Hongri a a b

College of Physics and Electronic Science, Hubei Normal University, Huangshi 435002, PR China Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, PR China

a r t i c l e

i n f o

Article history: Received 18 December 2014 Accepted 10 October 2015 Keywords: Time lens All-optical signal processing Nonlinear optics Ultrafast information processing

a b s t r a c t We propose a new technique to realize an optical time-frequency conversion system for ultrafast temporal processing that is based on linear time lens. The demonstrated time lens produces more than 25,000 of phase shift. Time-frequency conversion systems are realized by the combination of temporal quadratic phase modulation and group-velocity dispersion. We simulate the results with a time resolution of 27 fs over a time window of 1 ns, representing a large enough time-bandwidth product. Crown Copyright © 2015 Published by Elsevier GmbH. All rights reserved.

1. Introduction Space-time duality is based on the analogy between the equations that describe the paraxial diffraction of light beams in space and the first-order temporal dispersion of optical pulses in a dielectric [1–3]. The duality can also be extended to consider imaging lenses: the use of quadratic phase modulation on a temporal waveform is analogous to the action of a spatial lens on the transverse profile of a spatial beam [4,5]. Pulse compression [6], timing-jitter reduction [7], time magnification [8], time cloaking [9] have been demonstrated using this concept. The effects of diffraction and spatial lenses on a beam of light are equivalent to the effects of dispersion and time lenses on a pulse of light. The timefrequency conversion process is best understood by noting the analogy between a temporal optical system manipulating pulses of light and a spatial optical system manipulating beams of light. Here we investigate a new regime in the interaction between optical pulses and time lenses. The optical pulse to be measured travels through one focal time of dispersion, supplied by the standard single-mode fiber, then is phase modulated by the time lens with a quadratic time phase shift. We know from Fourier optics that the field distributions at the front focal plane and output plane of the spatial lens are related by a Fourier transform, and we will show below that the same relation holds for a time lens. Also the output

夽 Supported by the Science and Technology Programme of Education Department of Hubei Province under Grant No. D20132505. ∗ Corresponding author. Tel.: +86 15629598206. E-mail address: [email protected] (S. Junqiang). http://dx.doi.org/10.1016/j.ijleo.2015.10.029 0030-4026/Crown Copyright © 2015 Published by Elsevier GmbH. All rights reserved.

field distribution and its spectral frequency distribution are related by a Fourier transform [10,11]. We also provide a simulation experimental demonstration of the phenomenon using a linear time lens. Among other potential applications, spectro-temporal imaging can be applied for the measurement of the intensity temporal profile of ultrashort optical pulses using a conventional spectrum analyzer [12,13]. In contrast with other approaches, this method provides a fast, direct (singleshot), and unambiguous measurement of the temporal waveform, which is important in some relevant engineering applications such as modern high-speed communications. In this letter, we derive the conditions for achieving time-frequency conversion using a single time lens as well as the expressions governing this operation. 2. Principle of arbitrary waveform measurement The fundamental principle behind temporal imaging is the analogy between the electric field propagation behavior in the cases of paraxial diffraction and narrow-band dispersion. In paraxial diffraction, the spatial envelope profile E(x, y, z) of a monochromatic beam, E(x, y, z, t) = E(x, y, z)exp(i(ω0 t − kz)), propagates according to [2]

∂E i =− 2k ∂z



2

2

∂ E ∂ E + ∂ x 2 ∂ y2



.

(1)

This is easily solved in the transverse spatial frequency domain where the spectrum, ε(kx , ky , z) = F{E(x, y, z)}, acquires a quadratic spectral phase upon propagation



ε(kx , ky , z) = ε(kx , ky , 0) exp

iz(kx2 + ky2 ) 2k



.

(2)

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For the narrow-band dispersion problem, we consider the evolution of a pulse envelope A(z0 ) for the plane wave profile, A(z, t) = A(z0 )exp(i(ω0 t − ˇz)). The spectrum ε(z, ω) consists of the baseband spectrum A(z, ω) of the envelope convolved up to the carrier ω0 , i.e., ε(z, ω) = A(z, ˝) exp(−iˇz), where ˝ = ω − ω0 . About the carrier at frequency ω0 , each spectral component represents a plane wave with the propagation constant ˇ(ω), which can be expanded to second order in a Taylor series [8] ˇ(ω) = ˇ0 + (ω − ω0 )ˇ + (ω − ω0 )2 Where



dˇ(ω)  dω 

ˇ =

ˇ0 = ˇ(ω0 ),



d2 ˇ(ω)  ˇ = dω2  

ω=ω0

= ω=ω0

d = dω



ˇ , 2

1

vg (ω0 )

1 vg (ω)

   

(3)

and

Fig. 1. The temporal profile of the phase and its derivative for ideal time lens.

(4) ω=ω0

Time

Here ω0 is used to represent a general carrier frequency. In a parametric temporal imaging system, the carrier will change due to frequency conversion in the time lens and subsequent dispersive delay lines must be evaluated at the new carrier frequency. By transforming to a traveling-wave coordinate system



 = (t − t0 ) −

z − z0 vg (ω0 )

∂A = 2 ∂



2

∂ A ∂ 2

 = z − z0 ,

and



,

(6)

 i

A(, ˝) = A(0, ˝) exp −

2

˝

2



,

(7)

where,  = ˇ , ˝ is a baseband Fourier spectral component. Any mechanism that produces a phase modulation that is quadratic phase shift in time can be considered as a time-domain analog to a space lens. Kolner showed that the ideal time lens has a quadratic phase versus time [3]:



−i

2 2f



.

(8)

Fig. 1 shows the concept of time lens with parabolic phase curve and linear chirp curve. An ideal lens imparts a quadratic phase ϕf () to the signal such that ϕf () = − 2 /2f , where f is the focal groupdelay dispersion (GDD) associated with the time lens and is equal to the inverse of the second derivation of the phase. The definitions of the input dispersion and time lens in a timefrequency conversion system are given in Table 1. The time-frequency conversion system operation is determined by following the input field E0 (0, ), an optical carrier at frequency ω0 modulated by the pulse envelope u0 (0, ), E0 (0, ) = u0 (0, )e

iω0 

1

Input dispersion

G1 (1 , ) =

Time lens

H() = exp

2 i 1



2

exp

i

2 2 1

Frequency





G1 (1 , ˝) = exp



 −i 2 

H(˝) =



f

−i

 ˝2



1

2

 −2 if exp

 ˝2

i



f

2

(5)

Again, easily solved in the frequency domain, the spectrum of the pulse envelope A(, ˝) = F{A(, )} is described independent of the carrier frequency and evolves during propagation according to [4]

H() = exp





where t0 and z0 are references in real time-space, defining the center of the waveform at  = 0 in the traveling-wave system, and vg (ω0 ) is the group velocity at the carrier frequency ω0 , the governing equation for the evolution of the envelope A(, ) is of the same form as that for paraxial diffraction (1) iˇ

Table 1 Equation describing the elements of a time-frequency conversion system.

(9)

Firstly, the effect of the dispersion through a standard singlemode fiber with the propagation constant ˇ(ω) distorts the input

Fig. 2. Diagram of the proposed configurations for measuring optical pulses based on linear time Lens. SSMF = standard single-mode fiber, OSA = optic spectrum analyzer.

pulse envelope u0 (0, ) into u1 ( 1 ,   ):



∞

u1 (1 ,   ) =

u0 (0, )G1 (1 ,  −   )d

−∞

=





1

u0 (0, ) exp

2 i1 



∞

i

−∞

G1 (1 ,  −  ) =





1 2 i1

exp

( −   )2 21

( −   )2 i 21

 d,

(10)

 .

(11)

The propagation constant ˇ(ω) of the dispersive medium can be approximated by a three term Taylor series Eq. (3). A sketch of the experimental setup for a time lens measurement is shown in Fig. 2. The pulse u1 ( 1 ,   ) is then quadratic phase modulated by the time lens. Eq. (8) showed that the ideal time lens has a quadratic phase versus time. After passing through the dispersive medium and the optical quadratic phase modulator, the output pulse u2 ( 1 + ,   ) is related to the input pulse u0 (0, ) by u2 (1 + ,   ) = u1 (1 ,   )H(  ) =

=



1



2 i1 1



2 i1

 

exp

∞

u0 (0, ) exp −∞

−i

exp

i

 i

2 2f

2 2

2 21

 





∞

u0 (0, ) exp −∞

1 1 −  1 f



exp

−i



  1

i

(  − )2 21

 d

 d.

(12)

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L. Xingyun et al. / Optik 127 (2016) 738–741

Fig. 3. The temporal waveform of the input signal (left) and frequency spectrum from the input signal (right).

Fig. 4. The temporal waveform of the output signal (left) and frequency spectrum from the output signal (right).

If the amount of dispersion is adjusted such that the object pulse u0 (0, ) is at the front focal plane of the time lens, that is, 1 = f , then the quadratic phase term outside the integral disappears. Under this condition, the form of Eq. (12) shows that the output pulse u2 ( 1 + ,   ) is an inverse Fourier transform of the input pulse u0 (0, ) multiplied by a quadratic phase factor:





u2 (1 + ,  ) =



i2 −1 F u0 (0, ) exp f



i 2 2f

 .

(13)

  =− ω f

Finally, the power at the output of the time-frequency conversion system, from (13) becomes



 2 2     u2 (1 + ,   )2 = 2  F−1 u0 (0, ) exp i      2 f

f

.

(14)

 =− ω f

The power spectrum measured by the optical spectrum analyzer as the optical Fourier transform is the pulse spectrum centered at the carrier frequency ω0 . Therefore, the input pulse intensity is found by measuring the power spectrum of the output pulse and scaling the frequency and wavelength axis to a time axis using

 = −f ω,

 =

2 cf 0

.

(15)

3. Analysis of the measurement system based on linear time lens An input optical pulse measurement by the optic oscilloscope and optic spectrum analyzer is Fig. 3. An output optical pulse measurement by the optic oscilloscope and optic spectrum analyzer behind time lens is Fig. 4. The output spectrum and wavelength convert to time domain using the scale factor −3.9 ps/nm given by Eq. (15). This asymmetric two peak pulse cannot be seen by intensity autocorrelation, which, by definition, must symmetric.

The directly measured width for two peak pulse using this timefrequency conversion has a ∼11.9 ps, while the computed width is ∼10 ps. The important performance measure of this system is the temporal resolution and temporal field of view. Just as a spatial lens has a diffraction limited spot size arising from its finite aperture [14], the time lens has a dispersion limited pulse width because of its finite time window. The time window with the measurement pulse duration is defined by the quadratic phase shift versus time of the time lens, as shown in Fig. 1.

4. Conclusions We demonstrate all-optical temporal measurement of arbitrary waveform by time-frequency conversion system, based on ideal time lens with the quadratic phase shift versus time, achieving a large enough time-bandwidth product (temporal resolution ∼27 fs and measurement time window ∼1 ns). The quadratic phase shift and the temporal duration from time lens define the focal length and the numerical aperture of the time lens, respectively. Hence, when detected on a suitable optical spectrum analyzer, the output pulse of time-frequency conversion system in the frequency domain for the optical Fourier transform, represents the temporal profile of the input arbitrary waveform. The addition of a standard spectrum analyzer after the time lens provides direct measurement of the unknown pulse profile. The method is simple and direct accessible with slower conventional technology in real time. It is suitable for single-shot real-time ultrashort pulse measurement. References [1] [2] [3] [4] [5]

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