- Email: [email protected]
Time-to-frequency conversion using a single time lens
Optics Communications 217 (2003) 205–209 www.elsevier.com/locate/optcom
Time-to-frequency conversion using a single time lens ~a * Jose Azan Photonic Systems Group, Department of Electrical and Computer Engineering, McGill University, 3480 University Street, Montreal, Que., Canada H3A 2A7 Received 30 September 2002; accepted 9 January 2003
Abstract We introduce and formalize a time-frequency dualism between first-order dispersion and quadratic phase modulation (time lens). As a particular case of this dualism, we demonstrate that when a time lens operates on an input optical pulse, this pulse can enter a regime which is the frequency-domain equivalent of the temporal Fraunhofer regime (real-time Fourier transformation by dispersion). In this new regime, which we will refer to as spectral Fraunhofer regime, the input pulse amplitude is mapped from the time domain into the frequency domain (time-to-frequency conversion). Here we derive the conditions for achieving time-to-frequency conversion using a single time lens as well as the expressions governing this operation. Besides its intrinsic physical interest, time-to-frequency conversion can be used for measuring ultrafast temporal waveforms with a spectrum analyzer. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Space-time duality is based on the analogy between the equations that describe the paraxial diffraction of beams in space and the first-order temporal dispersion of optical pulses in a dielectric [1–11]. 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 thin lens on the transverse profile of a spatial beam [1–7]. The quadratic phase modulation (time lens) can be practically implemented using an electro-optic phase modulator [1,3,4]; by mixing the original pulse with a chirped
*
Tel.: +1-514-398-2266; fax: +1-514-398-3127. E-mail address: [email protected].
pulse in a nonlinear crystal (sum-frequency generation) [2,5]; or by means of cross-phase modulation of the original pulse with an intense pump pulse in a nonlinear fiber [6]. By appropriately combining dispersive devices and time lenses, one can create new systems for the processing and control of temporal optical signals (equivalent of spatial signal processing systems based on diffraction and imaging lenses) [1–11]. Examples of such systems include temporal imaging systems [1,2], time-to-frequency converters [3–6], real-time Fourier transformers (frequency-to-time converters) [7–10] and repetition-rate multipliers [11]. In this paper we show that there exists an interesting time-frequency dualism between first-order temporal dispersion processes and time lenses. In particular, the action of a time lens in the time domain [frequency domain] is equivalent to the
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01113-1
206
J. Aza~na / Optics Communications 217 (2003) 205–209
effect of first-order dispersion in the frequency domain [time domain]. This concept, which was previously introduced in the spatial optics context (diffraction versus imaging lens) [12], provides a powerful and intuitive way for the analysis and synthesis of optical temporal processors based on dispersion and time lenses. To illustrate the potential of the dispersion-time lens dualism, here we theoretically investigate the dual regime of the well-known temporal Fraunhofer regime (realtime Fourier transformation, RTFT, by dispersion). RTFT can be observed when an optical pulse is temporally stretched by first-order dispersion [8–10]. For sufficiently high dispersion, the temporal envelope of the dispersed pulse is proportional to the Fourier transform of the original pulse envelope (frequency-to-time conversion) [8]. RTFT has been widely applied for the analysis and processing of optical pulses [9,10]. Here we demonstrate that when a time lens operates on an optical pulse, this pulse can enter a regime which is the frequency-domain dual of the RTFT operation, such that the input pulse amplitude is mapped from the time domain into the frequency domain (time-to-frequency conversion). We will refer to this new regime as spectral Fraunhofer regime. In this work, we derive the conditions for achieving time-to-frequency conversion using a single time lens as well as the expressions governing this operation. Time-to-frequency conversion can be used for the measurement of the intensity temporal profile of ultrashort optical pulses by means of a spectrum analyzer. As compared with conventional autocorrelation techniques, the proposed method provides a direct and unambiguous measurement of the exact pulse intensity temporal profile. Time-to-frequency conversion has been previously demonstrated by using a time lens preceded by a suitable dispersive device [3–6]. Conventional time-to-frequency conversion systems are subject to the equivalent of aperture limitations. In particular, if the input pulse is too short (broadband), it is stretched in excess by the dispersion preceding the time lens and overfills the finite time aperture (usable time window [1]) of the time lens. This effect drastically limits the achievable temporal resolution, i.e., the shortest pulse that can be processed with the system. Our proposal simplifies
the design and implementation of time-to-frequency converters since it is based on the use of a single time lens (i.e., the preceding dispersive device is not required). It also alleviates the limitations in resolution associated to the use of the dispersive device preceding the time lens. First, we introduce and formalize the dispersion-time lens dualism shown schematically in Fig. 1. We will assume the optical signals to be spectrally centered at the optical frequency x0 . In the following development, we will work with the complex envelope of the involved signals and we will ignore the average delay introduced by the components under analysis (dispersion device or time lens). A dispersive medium can be described as a phase-only filter with a frequency transfer function [8] € x =2x2 Þ; H ðxÞ / expðjUðxÞÞ / expðj½U
ð1Þ
where x ¼ xopt x0 , xopt being the optical fre€ x ¼ ½o2 UðxÞ=ox2 quency variable, and U x¼0 is the first-order dispersion coefficient of the medium. The related temporal impulse response can be obtained as the inverse Fourier transform of H ðxÞ € x t2 Þ [8]. The reand results hðtÞ / expðj½1=2U sponse bðtÞ of the medium to an input signal aðtÞ is given simply by the convolution of this input signal with the filter impulse response, i.e., bðtÞ ¼ aðtÞ hðtÞ. This can also be described in the frequency domain as a product (modulation) BðxÞ ¼ AðxÞH ðxÞ, where BðxÞ and AðxÞ are the Fourier transforms of bðtÞ and aðtÞ, respectively.
Fig. 1. Schematic of the dispersion – time lens dualism. Only magnitude of the signals is represented.
J. Aza~na / Optics Communications 217 (2003) 205–209
207
On the other hand, a time lens is a phase-only modulator with a modulation function [1] mðtÞ ¼ expðj/ðtÞÞ / expðj½/€t =2t2 Þ; ð2Þ 2 2 € where /t ¼ ½o /ðtÞ=ot t¼0 . The output dðtÞ from the time lens to an input pulse cðtÞ is the product of this input signal and the modulation function mðtÞ, i.e., dðtÞ ¼ cðtÞmðtÞ. In the frequency domain, this can be described as a convolution DðxÞ ¼ CðxÞ MðxÞ, where DðxÞ and CðxÞ are the Fourier transforms of dðtÞ and cðtÞ, respectively and MðxÞ is the Fourier transform of the timelens modulation function mðtÞ, MðxÞ / expðj ½1=2/€t x2 Þ. By directly comparing the mechanism of first-order dispersion and time lens, we observe that the action of a time lens on an optical pulse in the time domain is analogous to the effect of dispersion in the frequency domain, i.e., in both cases the operation is a phase modulation by a quadraticphase function [modulation functions in Eqs. (1) and (2), respectively]. Equivalently, the action of a time lens in the frequency domain is analogous to the effect of first-order dispersion in the time domain (convolution by a quadratic-phase function). This means that first-order dispersion and the time lens are dual operations in the transformed domains. From Eqs. (1) and (2), the dualism can be mathematically formalized using the following relations (first column for dispersion and second column for time lens) x½t () t½x;
ð3aÞ
€ x () /€t : U
ð3bÞ
By extension, to every temporal optical system comprising dispersive processes and time-lenses [1–10], there corresponds a dual system that implements identical operations in the transformed domain. The dual system is formed by replacing each time lens [dispersive device] in the original scheme by the equivalent dispersive device [time lens] (according to relations in (3a) and (3b)). To illustrate the introduced dualism, we investigate the dual regime of the well-known temporal Fraunhofer regime. Fig. 2 shows a schematic of the dualism under analysis. The temporal Fraunhofer regime can be observed when an optical pulse is temporally stretched by first-order dis-
Fig. 2. Dual Fraunhofer regimes: (a) frequency-to-time conversion using dispersion and (b) time-to-frequency conversion using a single time lens.
€x persion [8–10]. If the dispersion coefficient U verifies the condition [8] Dt2 ; ð4Þ 8p Dt being the total duration of the unstretched optical pulse, then the output pulse envelope bðtÞ is proportional to the spectrum of the input pulse € x Þj. AðxÞ [see Fig. 2(a)], i.e., jbðtÞj / jAðx ¼ t=U Inequality (4) has been usually referred to as temporal Fraunhofer condition since it is the timedomain analog of the well-known Fraunhofer condition in the problem of spatial diffraction. Let us now analyze the interaction between an input optical pulse cðtÞ ½CðxÞ and a time lens of phase parameter /€t and phase modulation function mðtÞ ½MðxÞ (according to Eq. (2)). For our purposes, it is more convenient to analyze this interaction in the frequency domain. As defined above, the spectrum of the output pulse from the time lens DðxÞ can be calculated as € xj jU
DðxÞ ¼ CðxÞ MðxÞ Z 2 / CðXÞ expðj½1=2/€t ½x X Þ dX Dx Z / MðxÞ CðXÞ expðj½1=2/€t X2 Þ Dx
expðj½1=/€t xXÞ dX;
ð5Þ
where Dx is the total bandwidth of the input pulse cðtÞ½CðxÞ. If this bandwidth is sufficiently narrow so that
208
J. Aza~na / Optics Communications 217 (2003) 205–209
Dx2 ; ð6Þ 8p then the phase factor expðj½1=2/€t X2 Þ within the last integral in Eq. (5) can be neglected, since ð1=2/€t ÞX2 < ð1=2/€t ÞðDx=2Þ2 p. In this case, Eq. (5) can be approximated by Z CðXÞ expðj½1=/€t xXÞ dX DðxÞ / MðxÞ
j/€t j
Dx
¼ MðxÞcðt ¼ x=/€t Þ:
ð7Þ
Note that the last integral has been solved by considering expðj½1=/€t xXÞ as the kernel of a Fourier transformation. Eq. (7) indicates that under the conditions of inequality (6), the spectrum of the output pulse DðxÞ is, within a phase factor ½MðxÞ, proportional to the temporal waveform of the input pulse cðtÞ, evaluated at the instant t ¼ x=/€t . In other words, under the conditions of inequality (6), the energy spectrum of the output pulse is an image of the input temporal waveform, i.e., an efficient time-to-frequency conversion is achieved [see Fig. 2(b)]. Note that inequality (6) is the frequency-domain dual of the temporal Fraunhofer condition in Eq. (4). In keeping with the spirit of the dispersion-time lens dualism, we will refer to this inequality as spectral Fraunhofer condition. To the best of our knowledge, the spectral Fraunhofer regime is described here for the first time. In contrast with the temporal Fraunhofer regime, the space-domain counterpart of the spectral Fraunhofer regime (space-to-angular spectrum conversion by the action of an imaging lens) has not been previously described either. As mentioned previously, time-to-frequency conversion can be used for determining the intensity temporal profile of an ultrashort optical pulse by directly measuring the spectrum of the output pulse from the time lens. The measured frequency variable x and the time variable t are related by t ¼ x=/€t [Eq. (7)]. Eq. (6) constitutes the key equation when designing a time lens as a time-tofrequency converter (condition (6) ensures that the measured spectrum corresponds to the input temporal waveform). In order to check that the time lens verifies condition (6), only a previous estimation of the input pulse bandwidth Dx is required. Note that the time lens must also exhibit a time
aperture longer than the pulses to be measured. It is also important to mention that in our theoretical treatment no assumptions on the phase of the pulses have been made. This means that the proposed technique will also work with chirped pulses, independently of the pulse chirp profile. The temporal resolution dt of the measurement will directly depend on the resolution of the spectrum analyzer dx and according with Eq. (7), it can be estimated as dt dx=j/€t j. This means that the bigger the magnitude of the phase factor of the time lens, the better the temporal resolution of the measurement will be. Note that this requirement is inherently compatible with condition (6). The introduced concept is illustrated by means of numerical simulations. Fig. 3(a) shows the temporal waveform (average optical intensity) of the input signal used in our simulations and Fig. 3(b) shows the corresponding energy spectrum. The input signal consists of two consecutive and partially overlapped transform-limited Gaussian pulses of different intensity: each individual pulse has a full-width-half-maximum (FWHM) – timewidth of 1 ps and the two pulses are separated by 5 ps. The total bandwidth of this input signal is estimated to be Dx 2p 800 GHz. We simulated an ideal time lens process, i.e., an ideal quadratic phase modulation process. According to Eq. (6), in order to ensure an efficient time-to-frequency mapping, the phase factor of the time lens must
Fig. 3. Simulations of the action of an ideal time lens on an optical pulse. (a) Temporal waveform (average optical intensity) and (b) energy spectrum of the input pulse to the time lens.
J. Aza~na / Optics Communications 217 (2003) 205–209
satisfy that j/€t j Dx2 =8p 2p 16 104 GHz2 . Specifically, the phase factor of the time lens used in the simulations is fixed to j/€t j 2p 106 GHz2 . Note that this value can be achieved with current technology. In particular, phase factors up to /€t 2p 6 106 GHz2 have been reported [2]. Figs. 4(a) and (b) show the temporal waveform and energy spectrum (solid curve) of the output signal from the time lens, respectively. As expected, the output energy spectrum is an image of the input pulse temporal shape (note that the time lens process does not affect the temporal pulse shape). For comparative purposes, the temporal pulse waveform is represented in the same plot as the spectrum (with dashed curve), using the timeto-frequency scale change defined in Eq. (7). Slight deviations between the input temporal shape and the output spectrum can be observed. These deviations are related with the fact that the phase factor of the time lens j/€t j verifies the condition (6) but with less than an order of magnitude of difference with respect to the value at the right of the inequality. In summary, we have introduced and formalized a time-frequency dualism between dispersion and time lens. According to this dualism, to every temporal optical system comprising dispersive processes and time-lenses, there corresponds a
209
dual system that implements identical operations in the transformed (Fourier) domain. This concept should prove to be very useful for the synthesis of new systems for temporal optical signal processing operations. As a particular example, here we have theoretically demonstrated that when a time lens operates on an optical pulse, it can enter a regime which is the frequency-domain equivalent of the temporal Fraunhofer regime. Within this new regime, the temporal shape of the original pulse is mapped into the pulse spectrum by the action of the time lens (time-to-frequency conversion). Besides its inherent physical interest, this operation could be applied for measuring ultrafast temporal pulse shapes with a spectrum analyzer. Acknowledgements The author gratefully acknowledges Prof. Brian H. Kolner for insightful discussions on the topic of time lens. The author also thanks N. Gryspolakis and T. Zambelis for their help with the preparation of this manuscript. This research was supported by a Postdoctoral Fellowship from the Ministerio de Educaci on y Cultura (Government of Spain) and by the Natural Sciences and Engineering Research Council (Canada). References
Fig. 4. Simulations of the action of an ideal time lens on an optical pulse. (a) Temporal waveform (average optical intensity) and (b) energy spectrum of the output pulse from the time lens. Dashed curve in the plot of the spectrum represents the input temporal waveform, using the time-to-frequency scale change in Eq. (7), t ¼ x=/€t .
[1] B.H. Kolner, IEEE J. Quantum Electron. 30 (1994) 1951. [2] C.V. Bennett, B.H. Kolner, Opt. Lett. 24 (1999) 783. [3] M.T. Kauffman, W.C. Banyai, A.A. Godil, D.M. Bloom, Appl. Phys. Lett. 64 (1994) 270. [4] N.K. Berger, B. Levit, S. Atkins, B. Fischer, Electron. Lett. 36 (2000) 1644. [5] E. Arons, E.N. Leith, A.-C. Tien, R. Wagner, Appl. Opt. 36 (1997) 2603. [6] L.K.H. Mouradian, F. Louradour, V. Messager, A. Barthelemy, C. Froehly, IEEE J. Quantum Electron. 36 (2000) 795. [7] T. Jannson, Opt. Lett. 8 (1983) 232. [8] J. Aza~ na, M.A. Muriel, IEEE J. Quantum Electron. 36 (2000) 517. [9] H. Hakimi, F. Hakimi, K.L. Hall, K.A. Rauschenbach, IEEE Photon. Technol. Lett. 11 (1999) 1048. [10] P.C. Chou, H.A. Haus, J.F. Brennan III, Opt. Lett. 25 (2000) 524. [11] J. Aza~ na, M.A. Muriel, Opt. Lett. 24 (1999) 1672. [12] A. Papoulis, J. Opt. Soc. Am. 58 (1968) 653.
Time-to-frequency conversion using a single time lens ~a * Jose Azan Photonic Systems Group, Department of Electrical and Computer Engineering, McGill University, 3480 University Street, Montreal, Que., Canada H3A 2A7 Received 30 September 2002; accepted 9 January 2003
Abstract We introduce and formalize a time-frequency dualism between first-order dispersion and quadratic phase modulation (time lens). As a particular case of this dualism, we demonstrate that when a time lens operates on an input optical pulse, this pulse can enter a regime which is the frequency-domain equivalent of the temporal Fraunhofer regime (real-time Fourier transformation by dispersion). In this new regime, which we will refer to as spectral Fraunhofer regime, the input pulse amplitude is mapped from the time domain into the frequency domain (time-to-frequency conversion). Here we derive the conditions for achieving time-to-frequency conversion using a single time lens as well as the expressions governing this operation. Besides its intrinsic physical interest, time-to-frequency conversion can be used for measuring ultrafast temporal waveforms with a spectrum analyzer. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Space-time duality is based on the analogy between the equations that describe the paraxial diffraction of beams in space and the first-order temporal dispersion of optical pulses in a dielectric [1–11]. 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 thin lens on the transverse profile of a spatial beam [1–7]. The quadratic phase modulation (time lens) can be practically implemented using an electro-optic phase modulator [1,3,4]; by mixing the original pulse with a chirped
*
Tel.: +1-514-398-2266; fax: +1-514-398-3127. E-mail address: [email protected].
pulse in a nonlinear crystal (sum-frequency generation) [2,5]; or by means of cross-phase modulation of the original pulse with an intense pump pulse in a nonlinear fiber [6]. By appropriately combining dispersive devices and time lenses, one can create new systems for the processing and control of temporal optical signals (equivalent of spatial signal processing systems based on diffraction and imaging lenses) [1–11]. Examples of such systems include temporal imaging systems [1,2], time-to-frequency converters [3–6], real-time Fourier transformers (frequency-to-time converters) [7–10] and repetition-rate multipliers [11]. In this paper we show that there exists an interesting time-frequency dualism between first-order temporal dispersion processes and time lenses. In particular, the action of a time lens in the time domain [frequency domain] is equivalent to the
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01113-1
206
J. Aza~na / Optics Communications 217 (2003) 205–209
effect of first-order dispersion in the frequency domain [time domain]. This concept, which was previously introduced in the spatial optics context (diffraction versus imaging lens) [12], provides a powerful and intuitive way for the analysis and synthesis of optical temporal processors based on dispersion and time lenses. To illustrate the potential of the dispersion-time lens dualism, here we theoretically investigate the dual regime of the well-known temporal Fraunhofer regime (realtime Fourier transformation, RTFT, by dispersion). RTFT can be observed when an optical pulse is temporally stretched by first-order dispersion [8–10]. For sufficiently high dispersion, the temporal envelope of the dispersed pulse is proportional to the Fourier transform of the original pulse envelope (frequency-to-time conversion) [8]. RTFT has been widely applied for the analysis and processing of optical pulses [9,10]. Here we demonstrate that when a time lens operates on an optical pulse, this pulse can enter a regime which is the frequency-domain dual of the RTFT operation, such that the input pulse amplitude is mapped from the time domain into the frequency domain (time-to-frequency conversion). We will refer to this new regime as spectral Fraunhofer regime. In this work, we derive the conditions for achieving time-to-frequency conversion using a single time lens as well as the expressions governing this operation. Time-to-frequency conversion can be used for the measurement of the intensity temporal profile of ultrashort optical pulses by means of a spectrum analyzer. As compared with conventional autocorrelation techniques, the proposed method provides a direct and unambiguous measurement of the exact pulse intensity temporal profile. Time-to-frequency conversion has been previously demonstrated by using a time lens preceded by a suitable dispersive device [3–6]. Conventional time-to-frequency conversion systems are subject to the equivalent of aperture limitations. In particular, if the input pulse is too short (broadband), it is stretched in excess by the dispersion preceding the time lens and overfills the finite time aperture (usable time window [1]) of the time lens. This effect drastically limits the achievable temporal resolution, i.e., the shortest pulse that can be processed with the system. Our proposal simplifies
the design and implementation of time-to-frequency converters since it is based on the use of a single time lens (i.e., the preceding dispersive device is not required). It also alleviates the limitations in resolution associated to the use of the dispersive device preceding the time lens. First, we introduce and formalize the dispersion-time lens dualism shown schematically in Fig. 1. We will assume the optical signals to be spectrally centered at the optical frequency x0 . In the following development, we will work with the complex envelope of the involved signals and we will ignore the average delay introduced by the components under analysis (dispersion device or time lens). A dispersive medium can be described as a phase-only filter with a frequency transfer function [8] € x =2x2 Þ; H ðxÞ / expðjUðxÞÞ / expðj½U
ð1Þ
where x ¼ xopt x0 , xopt being the optical fre€ x ¼ ½o2 UðxÞ=ox2 quency variable, and U x¼0 is the first-order dispersion coefficient of the medium. The related temporal impulse response can be obtained as the inverse Fourier transform of H ðxÞ € x t2 Þ [8]. The reand results hðtÞ / expðj½1=2U sponse bðtÞ of the medium to an input signal aðtÞ is given simply by the convolution of this input signal with the filter impulse response, i.e., bðtÞ ¼ aðtÞ hðtÞ. This can also be described in the frequency domain as a product (modulation) BðxÞ ¼ AðxÞH ðxÞ, where BðxÞ and AðxÞ are the Fourier transforms of bðtÞ and aðtÞ, respectively.
Fig. 1. Schematic of the dispersion – time lens dualism. Only magnitude of the signals is represented.
J. Aza~na / Optics Communications 217 (2003) 205–209
207
On the other hand, a time lens is a phase-only modulator with a modulation function [1] mðtÞ ¼ expðj/ðtÞÞ / expðj½/€t =2t2 Þ; ð2Þ 2 2 € where /t ¼ ½o /ðtÞ=ot t¼0 . The output dðtÞ from the time lens to an input pulse cðtÞ is the product of this input signal and the modulation function mðtÞ, i.e., dðtÞ ¼ cðtÞmðtÞ. In the frequency domain, this can be described as a convolution DðxÞ ¼ CðxÞ MðxÞ, where DðxÞ and CðxÞ are the Fourier transforms of dðtÞ and cðtÞ, respectively and MðxÞ is the Fourier transform of the timelens modulation function mðtÞ, MðxÞ / expðj ½1=2/€t x2 Þ. By directly comparing the mechanism of first-order dispersion and time lens, we observe that the action of a time lens on an optical pulse in the time domain is analogous to the effect of dispersion in the frequency domain, i.e., in both cases the operation is a phase modulation by a quadraticphase function [modulation functions in Eqs. (1) and (2), respectively]. Equivalently, the action of a time lens in the frequency domain is analogous to the effect of first-order dispersion in the time domain (convolution by a quadratic-phase function). This means that first-order dispersion and the time lens are dual operations in the transformed domains. From Eqs. (1) and (2), the dualism can be mathematically formalized using the following relations (first column for dispersion and second column for time lens) x½t () t½x;
ð3aÞ
€ x () /€t : U
ð3bÞ
By extension, to every temporal optical system comprising dispersive processes and time-lenses [1–10], there corresponds a dual system that implements identical operations in the transformed domain. The dual system is formed by replacing each time lens [dispersive device] in the original scheme by the equivalent dispersive device [time lens] (according to relations in (3a) and (3b)). To illustrate the introduced dualism, we investigate the dual regime of the well-known temporal Fraunhofer regime. Fig. 2 shows a schematic of the dualism under analysis. The temporal Fraunhofer regime can be observed when an optical pulse is temporally stretched by first-order dis-
Fig. 2. Dual Fraunhofer regimes: (a) frequency-to-time conversion using dispersion and (b) time-to-frequency conversion using a single time lens.
€x persion [8–10]. If the dispersion coefficient U verifies the condition [8] Dt2 ; ð4Þ 8p Dt being the total duration of the unstretched optical pulse, then the output pulse envelope bðtÞ is proportional to the spectrum of the input pulse € x Þj. AðxÞ [see Fig. 2(a)], i.e., jbðtÞj / jAðx ¼ t=U Inequality (4) has been usually referred to as temporal Fraunhofer condition since it is the timedomain analog of the well-known Fraunhofer condition in the problem of spatial diffraction. Let us now analyze the interaction between an input optical pulse cðtÞ ½CðxÞ and a time lens of phase parameter /€t and phase modulation function mðtÞ ½MðxÞ (according to Eq. (2)). For our purposes, it is more convenient to analyze this interaction in the frequency domain. As defined above, the spectrum of the output pulse from the time lens DðxÞ can be calculated as € xj jU
DðxÞ ¼ CðxÞ MðxÞ Z 2 / CðXÞ expðj½1=2/€t ½x X Þ dX Dx Z / MðxÞ CðXÞ expðj½1=2/€t X2 Þ Dx
expðj½1=/€t xXÞ dX;
ð5Þ
where Dx is the total bandwidth of the input pulse cðtÞ½CðxÞ. If this bandwidth is sufficiently narrow so that
208
J. Aza~na / Optics Communications 217 (2003) 205–209
Dx2 ; ð6Þ 8p then the phase factor expðj½1=2/€t X2 Þ within the last integral in Eq. (5) can be neglected, since ð1=2/€t ÞX2 < ð1=2/€t ÞðDx=2Þ2 p. In this case, Eq. (5) can be approximated by Z CðXÞ expðj½1=/€t xXÞ dX DðxÞ / MðxÞ
j/€t j
Dx
¼ MðxÞcðt ¼ x=/€t Þ:
ð7Þ
Note that the last integral has been solved by considering expðj½1=/€t xXÞ as the kernel of a Fourier transformation. Eq. (7) indicates that under the conditions of inequality (6), the spectrum of the output pulse DðxÞ is, within a phase factor ½MðxÞ, proportional to the temporal waveform of the input pulse cðtÞ, evaluated at the instant t ¼ x=/€t . In other words, under the conditions of inequality (6), the energy spectrum of the output pulse is an image of the input temporal waveform, i.e., an efficient time-to-frequency conversion is achieved [see Fig. 2(b)]. Note that inequality (6) is the frequency-domain dual of the temporal Fraunhofer condition in Eq. (4). In keeping with the spirit of the dispersion-time lens dualism, we will refer to this inequality as spectral Fraunhofer condition. To the best of our knowledge, the spectral Fraunhofer regime is described here for the first time. In contrast with the temporal Fraunhofer regime, the space-domain counterpart of the spectral Fraunhofer regime (space-to-angular spectrum conversion by the action of an imaging lens) has not been previously described either. As mentioned previously, time-to-frequency conversion can be used for determining the intensity temporal profile of an ultrashort optical pulse by directly measuring the spectrum of the output pulse from the time lens. The measured frequency variable x and the time variable t are related by t ¼ x=/€t [Eq. (7)]. Eq. (6) constitutes the key equation when designing a time lens as a time-tofrequency converter (condition (6) ensures that the measured spectrum corresponds to the input temporal waveform). In order to check that the time lens verifies condition (6), only a previous estimation of the input pulse bandwidth Dx is required. Note that the time lens must also exhibit a time
aperture longer than the pulses to be measured. It is also important to mention that in our theoretical treatment no assumptions on the phase of the pulses have been made. This means that the proposed technique will also work with chirped pulses, independently of the pulse chirp profile. The temporal resolution dt of the measurement will directly depend on the resolution of the spectrum analyzer dx and according with Eq. (7), it can be estimated as dt dx=j/€t j. This means that the bigger the magnitude of the phase factor of the time lens, the better the temporal resolution of the measurement will be. Note that this requirement is inherently compatible with condition (6). The introduced concept is illustrated by means of numerical simulations. Fig. 3(a) shows the temporal waveform (average optical intensity) of the input signal used in our simulations and Fig. 3(b) shows the corresponding energy spectrum. The input signal consists of two consecutive and partially overlapped transform-limited Gaussian pulses of different intensity: each individual pulse has a full-width-half-maximum (FWHM) – timewidth of 1 ps and the two pulses are separated by 5 ps. The total bandwidth of this input signal is estimated to be Dx 2p 800 GHz. We simulated an ideal time lens process, i.e., an ideal quadratic phase modulation process. According to Eq. (6), in order to ensure an efficient time-to-frequency mapping, the phase factor of the time lens must
Fig. 3. Simulations of the action of an ideal time lens on an optical pulse. (a) Temporal waveform (average optical intensity) and (b) energy spectrum of the input pulse to the time lens.
J. Aza~na / Optics Communications 217 (2003) 205–209
satisfy that j/€t j Dx2 =8p 2p 16 104 GHz2 . Specifically, the phase factor of the time lens used in the simulations is fixed to j/€t j 2p 106 GHz2 . Note that this value can be achieved with current technology. In particular, phase factors up to /€t 2p 6 106 GHz2 have been reported [2]. Figs. 4(a) and (b) show the temporal waveform and energy spectrum (solid curve) of the output signal from the time lens, respectively. As expected, the output energy spectrum is an image of the input pulse temporal shape (note that the time lens process does not affect the temporal pulse shape). For comparative purposes, the temporal pulse waveform is represented in the same plot as the spectrum (with dashed curve), using the timeto-frequency scale change defined in Eq. (7). Slight deviations between the input temporal shape and the output spectrum can be observed. These deviations are related with the fact that the phase factor of the time lens j/€t j verifies the condition (6) but with less than an order of magnitude of difference with respect to the value at the right of the inequality. In summary, we have introduced and formalized a time-frequency dualism between dispersion and time lens. According to this dualism, to every temporal optical system comprising dispersive processes and time-lenses, there corresponds a
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dual system that implements identical operations in the transformed (Fourier) domain. This concept should prove to be very useful for the synthesis of new systems for temporal optical signal processing operations. As a particular example, here we have theoretically demonstrated that when a time lens operates on an optical pulse, it can enter a regime which is the frequency-domain equivalent of the temporal Fraunhofer regime. Within this new regime, the temporal shape of the original pulse is mapped into the pulse spectrum by the action of the time lens (time-to-frequency conversion). Besides its inherent physical interest, this operation could be applied for measuring ultrafast temporal pulse shapes with a spectrum analyzer. Acknowledgements The author gratefully acknowledges Prof. Brian H. Kolner for insightful discussions on the topic of time lens. The author also thanks N. Gryspolakis and T. Zambelis for their help with the preparation of this manuscript. This research was supported by a Postdoctoral Fellowship from the Ministerio de Educaci on y Cultura (Government of Spain) and by the Natural Sciences and Engineering Research Council (Canada). References
Fig. 4. Simulations of the action of an ideal time lens on an optical pulse. (a) Temporal waveform (average optical intensity) and (b) energy spectrum of the output pulse from the time lens. Dashed curve in the plot of the spectrum represents the input temporal waveform, using the time-to-frequency scale change in Eq. (7), t ¼ x=/€t .
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