Temporal imaging with a time lens

A temporal imaging system is presented consisting of a dispersive input path, a phase modulator producing a phase modulation substantially equal to A+Bt.sup.2, and an output dispersive path. This temporal imaging system can be combined with other temporal lenses to image input signals in the same manner that spatial lenses can be used to image light from spatial sources. In particular, this temporal imaging system can be used to expand, compress and or invert input temporal signals.

BACKGROUND OF THE INVENTION 
This invention relates in general to pulse compressors and relates more 
particularly to an optical system that can expand and compress optical 
pulses while substantially retaining the temporal profile of the pulse. 
In the figures, the first digit of a reference numeral indicates the first 
figure in which is presented the element indicated by that reference 
numeral. 
In a conventional optical pulse compressor like that illustrated in FIG. 1, 
a frequency sweep is imparted to a travelling wave pulse of light by a 
phase modulation mechanism so that the frequency at the trailing end of 
the pulse is higher or lower than at the leading end of the pulse. This 
swept frequency process is referred to as a "chirp" because an audio pulse 
of comparable shape sounds like the chirp of a bird. When this pulse is 
transmitted through a dispersive optical element in which the frequency 
components at the leading edge of the pulse travel slower than the 
frequency components at the trailing edge of the pulse, the trailing end 
of the pulse compresses toward the leading edge of the pulse producing a 
pulse of reduced width and increased amplitude. 
In one class of embodiments of such pulse compressors, the frequency chirp 
is imparted to the input pulse by self-phase modulation in an optical 
fiber (see, for example, D. Grischkowsky and A. C. Balant, Appl. Phys. 
Lett. 41, 1 (1982)). In another class of embodiments, the frequency chirp 
is imparted by electro-optic phase modulation (see, for example, D. 
Grischkowsky, Appl. Phys. Lett. 25, 566 (1974); or B. H. Kolner, Appl. 
Phys. Lett. 52, 1122 (1988)). In either case, a quadratic or nearly 
quadratic time-varying phase shift across the temporal envelope of the 
pulse results. After the pulse is chirped, it passes through a dispersive 
delay line such as a diffraction grating-pair which produces temporal 
compression of the pulse (See, for example, Edmond B. Treacy, "Optical 
Pulse Compression With Diffraction Gratings", IEEE Journal of Quantum 
Electronics, Vol. QE-5, No. 9, September 1969). 
SUMMARY OF THE INVENTION 
In accordance with the illustrated embodiment, a pulse compressor is 
presented that operates on a temporal pulse in a manner analogous to the 
operation of an optical imaging system. This pulse compressor can 
therefore be viewed conceptually as a temporal imaging system that 
utilizes at least one temporal lens and at least two dispersive paths. 
This temporal imaging system is developed in analogy to a spatial imaging 
system. 
Such a temporal imaging system provides a number of useful advantages. Just 
as a spatial optical imaging system can produce an image that is larger or 
smaller than the optical object, this temporal imaging system can be used 
to compress or expand an optical pulse. When this temporal imaging system 
is used to produce a temporal image compressed in time, it functions as a 
pulse compressor that converts an input pulse having the same shape as the 
input pulse but having a reduced temporal scale. Thus, particular optical 
waveforms could be prepared on a long time scale to allow accurate 
preparation of the waveform and then it can be compressed for applications 
in coherent spectroscopy and nonlinear pulse propagation experiments. 
Optimum long pulse shapes could be prepared for high power amplification 
and subsequent compression. For communications applications, data streams 
could be encoded at nominal rates and compressed and multiplexed for high 
density optical communications. As in other types of pulse compressors, 
this temporal imaging system can be used to produce narrower temporal 
pulses of increased amplitude as well as to produce pulses having steeper 
transitions. These steepened transitions are useful in high speed 
switching and these narrower pulses are useful in high speed sampling. 
When this temporal imaging system is used to produce a temporal image 
expanded in time, it functions as a pulse expander that is analogous to an 
optical microscope. It can be used to expand ultrafast optical phenomena 
to a time scale that is accessible to conventional high-speed photodiodes. 
This temporal imaging system could extend the range of direct optical 
measurements to a regime that is now only accessible with streak cameras 
or nonlinear optical techniques. 
In a streak camera, an optical pulse is directed onto the cathode of a 
cathode ray tube. The carrier frequency of the optical pulse is high 
enough that the photons have sufficient energy to emit electrons by 
photoelectric emission. As a result of this, an electron beam from the 
cathode is produced that has the same temporal variation as the temporal 
variation of the optical pulse. This electron beam is scanned across the 
face of the cathode ray tube, producing a streak that has the same spatial 
variation in intensity as the temporal variation in intensity of the 
optical beam. The spatial variation of the streak intensity is measured, 
thereby measuring the temporal variation of the optical pulse. 
Unfortunately, a streak camera has a temporal resolution limit of about 5 
picoseconds and is limited to light of frequency greater than the work 
function of the cathode divided by Planck's constant. 
Examples of pulse detectors utilizing nonlinear optical techniques are 
presented in Chapter 3 of "Ultrashort Light Pulses", Springer Verlag, 1st 
Edition, Volume 18, edited by S. L. Shapiro. Such techniques include 
second harmonic autocorrelation techniques and sum-frequency 
cross-correlation techniques. 
In the second harmonic autocorrelation technique, an optical pulse is split 
into two beams that are injected along paths oriented relative to the 
crystal axes of a crystal such that an output beam is formed that has a 
component proportional to the product of these two pulses. The delay of 
the pulse in one beam relative to the pulse in the other beam is varied, 
thereby producing an output signal that varies in time as the 
autocorrelation function of this pulse. Such a pulse detector is useful in 
analyzing pulses that are generally Gaussian in shape, but are not useful 
in determining the temporal profile of more complicated pulses. 
In the sum-frequency cross-correlation pulse detectors, a first beam 
contains a relatively wide pulse of complicated temporal profile and the 
other contains a much narrower substantially Gaussian shaped pulse. As the 
relative delay between these pulses is varied, the temporal profile of the 
first pulse in the first beam is determined as the cross-correlation 
between these two pulses. 
Just as a spatial imaging system can produce an inverted image, the 
temporal imaging system can also produce an inverted image. In the 
temporal case of an inverted image, the leading edge of the input pulse 
becomes the trailing edge of the output pulse. Such temporal inversion can 
be useful in signal processing applications such as convolution where time 
reversed waveforms are needed.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
The temporal imaging system is developed in analogy to a spatial imaging 
system. An optical spatial imaging system generally consists of a set of N 
spatial lenses SL.sub.1, . . . ,SLN, an input spatial path SP.sub.1, an 
output spatial path SP.sub.N+1 and N-1 spatial paths SP.sub.2, . . . 
,SP.sub.N, each between a pair of adjacent lenses. Such a system is shown 
in FIG. 2. A temporal imaging system can therefore be produced if temporal 
analogs of the above optical lenses and spatial paths can be produced. 
In the temporal imaging system presented herein, it is recognized that 
dispersion in a temporal path is the temporal analog of diffraction in an 
optical spatial path. It is recognized that a dispersive temporal path is 
the temporal analog of an optical spatial path diffraction. It is also 
recognized that a set of N+1 dispersive temporal paths can be combined 
with a set of N temporal lenses to produce a temporal imaging system. In 
the following, the temporal analog of a spatial optical lens is first 
discussed and then the temporal analog of the spatial paths is discussed. 
Spatial Optical Lens 
That an optical phase modulator can serve as a temporal lens to compress or 
expand an optical pulse is analogously illustrated by reference to FIG. 3. 
In FIG. 3 is shown a spatial lens 30 having a first spherical surface 31 
of radius R.sub.1, a second spherical surface 32 of radius R.sub.2 and an 
index of refraction n. In the paraxial approximation, the optical rays are 
treated as if they pass through the lens along a path substantially 
parallel to the axis 33 of the lens. The presence of this lens increases 
the optical pathlength of a paraxial ray at a distance r from axis 33 of 
the lens by an amount (n-1).multidot.{.DELTA..sub.1 (r)+.DELTA..sub.2 (r)} 
which, to lowest order in r is equal to r.sup.2 /2f where f is defined to 
be equal to {(n-1).multidot.(R.sub.2.sup.-1 -R.sub.1.sup.-1)}.sup.-1 and 
is called the focal length of the lens (a similar derivation is presented 
in J. W. Goodman, "Introduction to Fourier Optics"). Thus, in the paraxial 
approximation, to lowest order in f, a lens introduces to an optical wave 
of wavenumber k an additional phase .DELTA..phi.=k.multidot.r.sup.2 /2f. 
Thus, a spatial lens can be viewed as a phase modulator that modulates the 
phase of a ray at distance r from axis 33 of the lens by an amount that 
varies approximately quadratically with r. 
Temporal Lens 
This quadratic phase variation in the spatial domain as a function of r can 
be mirrored in the temporal domain by use of a phase modulator that 
produces a phase modulation substantially equal to .theta.(t)=A+Bt.sup.2 
for some constants A and B (i.e., the output signal v.sub.out (t) is equal 
to v.sub.in (t)e.sup.i.theta.(t) where v.sub.in (t) is the input signal to 
the phase modulator). Such phase modulation can be approximated by timing 
the modulation signal that drives the phase modulator such that the 
temporal pulse to be imaged is centered over an extremum of the phase 
modulation. The modulation signal can be any shape that has such extremum, 
provided that the shape is predominantly quadratic over the duration of 
the optical pulse. A sinusoidal modulation signal is particularly easy to 
generate and is therefore a useful choice. This temporal lens will 
function as a positive lens (i.e., a converging lens) or a negative lens 
(i.e., a diverging lens) depending on whether said extremum is a minimum 
or a maximum of the phase modulation. 
The input temporal signal has the general form u.sub.in (t)e.sup.i.omega.t 
where .omega. is the angular frequency of the optical carrier signal and 
u.sub.in (t) is the modulation function of the carrier signal. This 
modulation function is also referred to as the "envelope function" of the 
optical pulse input to the temporal imaging system. The term A introduces 
a constant phase shift that does not affect the envelope function. Thus, 
such temporal modulation is an analog of the corresponding spatial imaging 
system in the paraxial ray limit. 
As shown in texts on Fourier optics, such as the text "Introduction To 
Fourier Optics" by J. W. Goodman, the optical paths in the above spatial 
imaging system are governed by the mathematics of optical diffraction. It 
is well known that the mathematical equations for spatial diffraction are 
analogous to the mathematical equations for temporal dispersion (see, for 
example, S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, 
R. V. Khoklov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 
(1968)). There is a correspondence between the time variable in the 
dispersion problem and the transverse space variable in the diffraction 
problem. 
Spatial Optical Paths 
To see this correspondence, it will be shown that the envelope function of 
a temporal pulse obeys substantially the same differential equation as 
does a spatial wave. For a charge free medium, Maxwell's equations are 
EQU .gradient..times.E=-.differential.B/.differential.t 
.gradient..multidot.B=0(1) 
EQU .gradient..times.B=.differential.D/.differential.t 
.gradient..multidot.D=0(2) 
where E is the electric field vector, B is the magnetic field vector and D 
is the dielectric displacement vector equal to E (where is the 
dielectric constant). From standard vector calculus, these two equations 
imply that 
EQU .gradient..sup.2 E=.mu..sub.0 .differential..sup.2 D/.differential.t.sup.2( 
3) 
For a monochromatic optical signal of angular frequency co in the paraxial 
ray limit, E has the form u(x,y,z)e.sup.i(kz-.theta.t) where k.sup.2 
=.mu..sub.0 (.omega.).omega..sup.2 and u is an envelope function for the 
optical pulse. In this situation, equation (3) reduces to 
EQU .gradient..sup.2 u=2ik.differential.u/.differential.z (4) 
The paraxial approximation also assumes that the term .differential..sup.2 
u/.differential.z.sup.2 is negligible in equation (4) so that the wave 
equation (3) reduces to the functional form of a 2-dimensional diffusion 
equation (4) in which the time parameter of a diffusion equation is 
replaced by the parameter z and in which the two spatial variables are x 
and y. Since this equation is linear in u, we can add together solutions 
for different frequencies so that this result is not limited only to 
monochromatic fields. It will now be shown that the dispersion problem 
also has the same functional form. 
Temporal path in dispersive medium 
A temporal pulse can be decomposed into a linear sum of monochromatic 
signals. The phenomenon of dispersion results in different propagation 
velocities for these various Fourier components. If we solve equation (3) 
for each plane wave Fourier component, we can sum together the whole 
spectrum with each corresponding propagation constant to construct the 
real time pulse. If we limit our analysis to z-directed plane waves, we 
let 
EQU E(z,t)=u(z)e.sup.i.omega.t (5) 
and thus 
EQU D(z,t)= (.omega.)u(z)e.sup.i.omega.t (6) 
When this form is used in equation (3), that equation becomes 
EQU .differential..sup.2 u/.differential.z.sup.2 =-.omega..sup.2 
(.omega.)u(z)(3') 
which has the solution 
EQU u(z)=u.sub.0 e.sup.i.beta.z (7) 
where .beta. is the frequency-dependent wave number: 
EQU .beta..sup.2 (.omega.)=.mu..sub.0 (.omega.).omega..sup.2. (8) 
From equations (3') (7), we see that the function u(z) satisfies the 
differential equation 
EQU .differential.u(z,.omega.)/.differential.z=-i.beta.(.omega.)u(z,.omega.)(9) 
for a particular angular frequency .omega.. 
A temporal pulse consists of a slowly varying envelope function times a 
carrier travelling wave signal. Equivalently, this means that u(.omega.) 
is nonzero except in a narrow range about the carrier wave angular 
frequency .omega..sub.0. Therefore, in equation (9), .beta.(.omega.) can 
be expanded to second order in a power series about .omega..sub.0 to give 
EQU .differential.u(z,.omega.)/.differential.z=-i{.beta..sub.0 +.beta..sub.1 
(.omega.-.omega..sub.0)+.beta..sub.2 (.omega.-.omega..sub.0).sup.2 
}u(z,.omega.) (10) 
where .beta..sub.k .ident.(1/k!).differential..sup.k 
.beta./.differential..omega..sup.k evaluated at .omega.=.omega..sub.0. The 
temporal Fourier transform of this gives 
EQU (.differential.u(z,t)/.differential.z+v.sub.g.sup.-1 
.multidot..differential.u(z,t)/.differential.t)=i.beta..sub.2 
.multidot..differential..sup.2 u(z,t)/.differential.t.sup.2(11) 
where v.sub.g is the group velocity of the pulse and is equal to 
.beta..sub.1.sup.-1. This equation can be further simplified by 
transformation to the travelling wave coordinates .tau..ident.t-z/v.sub.g 
and z. In this coordinate system, equation (11) becomes 
EQU .differential..sup.2 u/.differential..tau..sup.2 =(i.beta..sub.2).sup.-1 
.differential.u/.differential.z (12) 
Equation (12) has the same functional form as equation (4) so this temporal 
pulse travelling in a dispersive medium satisfies substantially the same 
form of equation as spatial transmission of a wave with associated 
diffraction. Thus, the functional behavior of the temporal pulse through a 
dispersive medium corresponds to the functional behavior of a spatial beam 
along a spatial path. In equation (12), the travelling wave coordinate 96 
is analogous to the lateral parameters x and y of equation (4). The 
parameter z plays the same role in both cases. 
FIG. 4 illustrates a temporal imaging system and is analogous to the 
spatial imaging system of FIG. 2. This temporal imaging system consists of 
N+1 temporal paths TP.sub.1, . . . ,TP.sub.N+1 and temporal lenses 
TL.sub.1, . . . ,TL.sub.N. 
Because of this correspondence, a temporal imaging system generally 
consists of a set of N temporal lenses TL.sub.1, . . . ,TL.sub.N, an input 
signal path TP.sub.1, an output temporal path TP.sub.N+1 and N-1 temporal 
paths TP.sub.2, . . . ,TP.sub.N, each between a pair of adjacent temporal 
lenses. Because of the functional behavior between the spatial and 
temporal cases, the temporal imaging system exhibits the same 
magnification as the spatial imaging system. For example, for a single 
lens spatial imaging system, the magnification M is equal to -S.sub.2 
/S.sub.1 where S.sub.1 is the distance from the object O to the spatial 
lens and S.sub.2 is the distance from the spatial I to the image. S.sub.1 
and S.sub.2 satisfy the lens equation 1/f=1/S.sub.1 +1/S.sub.2 where f is 
the focal length of the lens. Under conventional sign conventions, S.sub.1 
and S.sub.2 are each positive for real objects and images and are each 
negative for virtual objects and images. When M is negative, this just 
indicates that the image has been inverted. 
In the temporal domain, the inversion of the image means that the leading 
edge of the input pulse becomes the trailing edge of the output pulse. 
Such inversion can be used in signal processing applications, such as 
convolution, where time reverse waveforms are needed. 
In the above analysis, the paraxial approximation was utilized for the 
spatial imaging system and, in the temporal imaging system, .beta. was 
expanded only to second order in .omega.-.omega..sub.0 and the modulation 
signal was also expanded only to second order in time about the time of an 
extremum point of that signal. If higher order terms are retained, then 
various types of aberration arise just as they do in the spatial imaging 
case. Thus, such aberrations should be small in the same way they must be 
small in the spatial imaging case. When greater clarity of imaging is 
required, these aberrations can be corrected in a way fully analogous to 
the spatial imaging case. 
FIGS. 2 and 4 illustrate analogous spatial and temporal imaging systems in 
which the spatial and temporal lenses are collinear, but, just as there 
are noncollinear spatial imaging systems, there can also be noncollinear 
temporal imaging systems. Indeed, the equivalence between the spatial and 
temporal lenses and interconnecting signal paths means that there are 
temporal imaging systems analogous to each of the spatial imaging systems 
utilizing just lenses and interconnecting paths. 
In the above discussion of the spatial optical lens, it was indicated that 
to the lowest nonzero order in the transverse distance r from the optical 
axis of the lens, the spatial optical lens introduces additional phase 
variation of the form k.multidot.r/2f where k is the wavenumber of the 
optical wave and where f is defined to be equal to 
{(n-1).multidot.(R.sub.2.sup.-1 -R.sub.1.sup.-1)}.sup.-1 and is called the 
focal length of the lens. Similarly, in the above discussion of the 
temporal lens, it is indicated that the temporal phase modulator 
introduces a phase substantially equal to A+Bt.sup.2. The A term 
introduces a constant phase that does not affect the shape of the pulse, 
but instead introduces a phase shift into the carrier signal on which the 
pulse is carried. The Bt.sup.2 term is therefore analogous to the 
k.multidot.r.sup.2 /2f term in the spatial lens case and shows that the 
temporal lens has an effective temporal focal length f.sub.t equal to 
.omega./2B where .omega. is the angular frequency of the carrier signal. 
Just as a simple spatial lens satisfies the optics equations: 
EQU 1/s.sub.1 +1s.sub.2 =1/f (13) 
and 
EQU M=-s.sub.2 /s.sub.1 (14) 
(where s.sub.1 is the distance from the object to the lens and s.sub.2 is 
the distance from the lens to the image), so also does the temporal pulse 
satisfy analogous equations 
EQU 1/t.sub.1 +1/t.sub.2 =1/f.sub.t (15) 
and 
EQU M.sub.t =-t.sub.2 /t.sub.1 (16) 
where M.sub.t is the temporal magnification factor, where t.sub.k (for k=1, 
2) is equal to 2.omega..sub.0 .multidot.z.sub.k 
.multidot..beta..sub.2.sup.(k), z.sub.k is the spatial length of the kth 
dispersive path TP.sub.k, and .beta..sub.2.sup.(k) is equal to one half of 
the second derivative of the wavenumber of the carrier signal in the kth 
dispersive medium evaluated at the frequency .omega..sub.0 of the carrier 
signal. In FIG. 5, an input pulse 51 passing through temporal length 
t.sub.1 of dispersive path TP.sub.1, temporal lens TL.sub.1, and temporal 
length t.sub.2 of dispersive path TP.sub.2, has a wider pulse width than 
the resulting output pulse 52, indicating that the absolute value of the 
magnification is less than 1 in that system. For the single temporal lens 
of FIG. 5, the magnification of the imaged pulse is negative so that the 
leading edge of input pulse 51 becomes the trailing edge of output pulse 
52. Just as a plurality of lenses can be used in tandem to image an object 
field, so too can a plurality of temporal lenses be utilized in tandem to 
temporally image an input temporal waveform. The dispersive paths and 
focal times must be chosen to produce an imaged pulse at the output O.