Vol. 11, No. 12/December 1994/J. Opt. Soc. Am. A 3229

Generalization of the concepts of focallength and f-number to space and time

Brian H. Kolner

Department of Electrical Engineering, University of California, Los Angeles, 405 Hilgard Avenue,Los Angeles, California 90024-1594

Received March 23, 1994; revised manuscript received July 29, 1994; accepted July 29, 1994

One can exploit the duality between the parabolic equations of diffraction and dispersion to create the time-domain equivalent of a spatial imaging system. A key element in this temporal imaging system is the timelens (quadratic phase modulator) with attributes analogous to the focal length and the f-number of conven-tional space lenses. These temporal analogs are analyzed and compared with their spatial counterparts ina generalized framework, from which we conclude that chirp rate and fractional bandwidth are the definingquantities in both systems.

1. INTRODUCTION

In introductory courses on optics we learn that the focallength of a lens is the distance to which a collimated beamwill be reduced to a small spot, or we associate it withan imaging condition prescribing the distances betweenobject, lens, and image. In more advanced courses wefind that the origin of the focal length lies in the rela-tive strength of the phase transformation produced by athin lens in the paraxial approximation. Similarly, wethink of the f-number as describing the ratio of the fo-cal length to the useful aperture of the lens. Recently,there has been interest in the similarity between the prob-lems of diffraction and dispersion,1 and this has led to atime-domain analog of spatial imaging called (naturally)temporal imaging.2 ,3 In this context there arise conceptsanalogous to those of focal length and f-number that per-tain to the time domain. It seems prudent, then, to con-sider what equivalent forms the temporal focal length andf-number take and what physical significance can be as-cribed to them. For simplicity, we treat space and timelenses as ideal quadratic phase transformers wherein weuse the focal length (or focal time) to scale the strengthof the phase transformation. The consistency of this ap-proach results in the conclusion that the concepts of fo-cal length and f-number are describable solely in termsof the chirp rate of the instantaneous (spatial or tempo-ral) Fourier frequency and the fractional bandwidth of thecomposite Fourier spectra.

2. BACKGROUND

The principle of temporal imaging is based on the analogybetween the equations that describe the diffraction ofbeams in space and the dispersion of pulses in dielectrics.Both problems are described by parabolic differentialequations with imaginary coefficients and therefore admitwave solutions. When cast as an initial-value problem,the evolution of the fields is found by first Fourier trans-forming the initial field distribution, then multiplyingthe spectrum by a phase function that is quadratic in theFourier variable and linear in the propagation distance,and finally applying an inverse Fourier transformation. 3

A lens, on the other hand, produces a quadratic phasetransformation in the real-space coordinates. The time-domain analog of this operation is simply a quadraticphase modulation in time. This could be realized, forexample, with an electro-optic modulator driven with a si-nusoid (Fig. 1). As the sinusoidal field propagates downthe electro-optic crystal, any optical wave copropagat-ing with this field and confined to a region of a cusp willreceive a quadratic phase modulation with modulation in-dex proportional to the electric-field strength, the electro-optic tensor element(s), and the interaction length.4

When quadratic phase modulation is followed by alinear dispersive delay line, we have the elements ofa pulse compressor, which is the temporal analog of alens focusing a beam.5 This configuration is the basisof many optical pulse compression schemes that use thepassive action of self-phase modulation6 or the active ap-proach with electro-optic phase modulators as discussedabove.7 `9 However, if the phase modulation is also pre-ceded by a dispersive delay line, the three elements of dis-persion + quadratic phase modulation + dispersion canbe adjusted to mimic exactly in the time domain theprocess of imaging in the spatial domain.2

It is most advantageous to analyze temporal imagingsystems with the shorthand and the jargon of spatialimaging systems. This approach results in the temporalequivalents of focal length, f-number, the imaging equa-tion, magnification ratios, and other familiar concepts. Itis the goal of this paper to take a closer look at two ofthese, focal length and f-number, to try to gain a deeperappreciation for their significance and their relationshipsto each other. As we shall see below, it is the relativeFourier spectral bandwidths, both temporal and spatial,that link these concepts.

3. FOCAL LENGTH OF A SPACE LENS

In the paraxial approximation we usually consider a thinlens as producing a phase transformation

tl(x, y) exp(-ikonAo)exp i o (x2 + y2)] (1)

0740-3232/94/123229-06$06.00 © 1994 Optical Society of America

Brian H. Kolner

3230 J. Opt. Soc. Am. A/Vol. 11, No. 12/December 1994

4 V(xt)- ~~~" 1_

IEO PHASE MODULATOR

interpretation that offers another point of view. Recallthat, for time waveforms, we define an instantaneous an-gular frequency cwi as the time derivative of the phase.In a similar way we can define an instantaneous spatialfrequency or wave number as the transverse derivative ofthe spatial phase:

E(x,t)

Fig. 1. Prototypical time lens realized with a traveling-waveelectro-optic (EO) modulator. An optical pulse E(O, t) enteringthe modulator at x = 0 and confined to a cusp of the driving fieldobtains a quadratic phase modulation after passage through themodulator.

[assuming exp(+iwot) time dependence], where AO is themaximum lens thickness, ko = 2/Ao is the free-spacepropagation constant, and f is the focal length definedin terms of the physical properties of the lens and themedium in which it is immersed.' 0 When the medium isvacuum, we have

r ~~1

ki = d kxtd ~X = _

(strictly speaking, the instantaneous transverse wavenumber ki is related to the instantaneous spatial fre-quency fi according to ki = 2irfi).

For our simple case of a quadratic phase the instan-taneous wave number is a linear function of space andpasses through zero on the lens axis (Fig. 3). It is therate of change or slope of this function that is the impor-tant quantity. Differentiating once more, we have

dki d 2 0ax ko

- dx2 fwhich might be called the chirp rate of the instantaneousspatial frequency and is directly linked to the focal length.We may now redefine the focal length in terms of this

Since the quadratic phase terms in tl(x, y) can be sepa-rated into the product of two phase functions,

tl(x, y) = exp(-ikonAo)exp[i&.(x)]exp[ik(y)], (3)

it will be convenient to consider only the function

kox 2

0XX)= 2f (4)x

as representative of the lens action in either coordinate.The effect of the phase function (1) can be viewed most

easily in the complex plane. Figure 2 displays the com-plex electric field amplitude in the transverse dimen-sion for a Gaussian beam immediately before and aftera thin lens. For simplicity, we have assumed that thebeam waist is located at the input plane of the lens, andthus there is no initial phase curvature [Fig. 2(a)]. Im-mediately past the lens the electric field has acquired aquadratic phase profile while maintaining its Gaussianamplitude [Fig. 2(b)].

So far, we have described the focal length only in termsof the physical attributes of the lens. It would be usefulto have a more general description of this concept thatwould apply to other quadratic phase operations as well.To this end, let us write a generalized phase function interms of a Maclaurin series,

d I + 2 d 2 Iq ok~ x)= k o(0) x I x= o '2! d x x=0

(a)

Im

x

(b)

Fig. 2. Complex Gaussian electric field in one dimension (a) be-fore and (b) after passage through a thin lens that produces thequadratic phase transformation.

(5)

and then equate the right-hand side to the specific phasefunction (1) term by term. We immediately find that

(6)

Now this expression for focal length can be interpreted asthe ratio of the wave number to the phase curvature intro-duced by the lens. However, there is a slightly different

Fig. 3. Instantaneous transverse wave number ki produced bya thin lens in the paraxial approximation.

E(0,t)Sir

(7)

(8)

Re

Im

Brian H. Kolner

kZ)

f = 2 ko X2d 0,/d

(n - -R, R2

Vol. 11, No. 12/December 1994/J. Opt. Soc. Am. A 3231

chirp rate. Rearranging Eq. (8), we obtain

f ko (9)= dki/dx

This is the expression for the generalized focal length.Its utility will become evident when we construct a simi-lar analysis in the time domain. Its limitations should,however, be obvious at once, since our analysis was re-stricted to second-order curvature.

4. FOCAL LENGTH OF A TIME LENS

Any mechanism that produces a phase modulation thatis quadratic in time can be considered as a time-domainanalog to a space lens (thus a time lens). A practicalexample of such a mechanism would be an electro-opticphase modulator driven with a sinusoidal waveform asshown in Fig. 1. The net phase shift imparted to anoptical waveform traveling with the modulating electricfield is an accumulated effect and can be calculated fromthe integral formula

Now we seek a more general expression for the focaltime that will encompass any quadratic phase mecha-nism. Following an identical approach to that outlinedin Section 3, we can deduce a general form for fT by onceagain expanding the phase function, this time in a Taylorseries:

- d~~~ (t - to)2 d2,0(t) = o(to) + (t to) d 2! dt2

(15)

Since the time origin is arbitrary, we can set to = 0 andthen compare the phase in Eq. (13) term by term withEq. (15) and conclude that

IT = dtd2 0/dt2

(16)

We can again invoke the concept of instantaneous fre-quency (now in the time domain), but here we must alsoinclude the optical carrier frequency

k(z, t) = coot - koz - ° z An(z', t)dz'c

= ot - koz - r(z, t), (10)

where An(z', t) is the traveling-wave perturbation in theindex of refraction produced by the electro-optic effect.For the case of perfect velocity matching between theoptical and modulating electric fields, and assuming thatthe optical waveform is confined to a cusp of the field, theretardation F(z, t) can be written as

r(z, t) ro_ ( t)2 ] (11)

where com is the frequency of the modulating field andro is the peak phase retardation, which depends on themagnitude of the modulating field, the electro-optic coeffi-cient, and the interaction length. Thus the electro-optictime lens produces a phase transformation

[o Fot) 2 1Hi(t) = exp(-iro)expi (2 (12)

It will be advantageous to generalize this expressionto allow for any process that produces a quadratic phasemodulation. In keeping with the spirit of the space-timeduality, let us write the phase transformation for a gen-eralized time lens as

H1(t) = exp(-iFo)exp(i 2o )

0i g ao + dk (17)

which simply adds an offset. Differentiating again andsubstituting into Eq. (16), we can redefine the focal timein terms of the chirp rate dj/dt:

fT 0) 0dcojdt (18)

Now the characteristics of the time lens can be viewedon an equal basis with those of the space lens. In Fig. 4we plot the instantaneous angular frequency wi versustime to highlight the connection between the chirp rateand the focal time.

The fundamental relationship that is common to spaceand time is the following: In the quadratic approxima-tion the focal length (time) of a lens is given by the ratio ofthe characteristic frequency to the chirp rate of the modu-lated variable.

5. PHYSICAL NATURE OF THEFOCAL TIME

In spatial imaging systems the concept of the focal lengthof a lens plays a dual role. It provides a measure of thedegree of quadratic phase transformation, and it also cor-responds to the distance to which a collimated beam is re-

wit(13)

where &jo is the optical carrier frequency and fT is thefocal time. We have deliberately chosen this form forthe phase function in order to mimic its spatial counter-part (1). The focal time fT plays the same role as thatof a scaling factor on the quadratic phase. Compar-ing Eqs. (12) and (13), we see that, for the electro-optictime lens,

fT = 2 * (14)roCOM2

Aw

1.

__-- --- / SLPE

/ I T

/ ~~~~~~~~~~~~~~~I_ _y ~~~~~~~~~~~~~~~~~~~~~~~~~~~~I

z ~~I _

Wot

/ k t - t

Fig. 4. Instantaneous temporal frequency wo produced by a timelens in the quadratic approximation.

l S ll S tl

Brian H. Kolner

I

3232 J. Opt. Soc. Am. A/Vol. 11, No. 12/December 1994

medium suffers a delay e/vg = Tg (the group delay), andtherefore

d2 ,/ d (g' drgdco2 dcw Vg dt

is the group-delay dispersion. Substituting Eq. (22) intoEq. (20), we find that

p [ 2 w2 1

(b)FREQUENCY TIME FREQUENCY

DOMAIN DOMAIN DOMAIN

Fig. 5. Temporal imaging system as the time-domain analog ofconventional spatial imaging. (a) Temporal imaging configura-tion. Input and output dispersions (shown here as diffractiongrating pairs) play the role of free-space diffraction, while aquadratic phase modulator acts as a lens in the time domain.(b) Mathematical effects on waveforms in temporal imaging.

duced to its smallest size. The question then arises, "Isthere an analogous dual interpretation for a time lens?"To address this, we consider the temporal imaging systemshown in Fig. 5. The legitimacy of this system in per-forming temporal imaging lies in the mathematical analy-sis of the cascaded processes of dispersion + quadraticphase modulation + dispersion being the exact dual of thespatial processes diffraction + quadratic phase transfor-mation + diffraction.2 3 The equivalent to the lens lawis the temporal imaging condition

1d2

eld0

1 _o+ d2p2 fT

2 dc 2(19)

where t is the propagation distance through a mediumwith normalized dispersion d2,3/dco2 . The producted 2fl/d(02 is therefore the net dispersion.

One can find the focal time fT by assuming that thetemporal equivalent to a plane wave is incident uponthe imaging system. This waveform is simply an infinitesinusoid and can be produced by a pulse propagatingthrough infinite dispersion, just as a true plane wave iscreated by a point source at infinity. Thus, in Eq. (19),we set 6 = x and find that

d2 32 _ fT62 -- ~~~~~~(20)dc 2 (90

It remains now to interpret carefully the dispersion termon the left-hand side of Eq. (20).

We know from elementary considerations that, for amedium with propagation constant 8 (co), the group veloc-ity is defined according to

drg fT,

which is easily solved to yield

fT := - ( )9

(23)

(24)

Thus we see that the focal time for the mean frequencyto = too is equal to the negative of the group delay throughthe dispersive medium following the time lens. The ori-gin of the minus goes back to the fundamental differencebetween the parabolic equations governing diffraction anddispersion.3

6. f-NUMBER

Traditionally, the f-number is regarded as the ratio of thefocal length to the aperture of a lens, f# f/Ax, whereAx is the lens diameter on the x axis. If we use thegeneralized focal length (9) in this expression, we have

ft= f _ koAx Ax(dki/dx)

(25)

Now, in the denominator, we recognize the form of theincrement

Ak = Ax di-dx

(26)

That is, for a given spatial chirp dki/dx, the total instan-taneous spatial bandwidth is given by the product of thechirp rate and the spatial increment (or aperture) Ax.Thus we can write the f-number in terms of the inverseof the fractional spatial bandwidth:

(27)f#= ko.Ok

Now this form for the f-number may appear to haveno advantages over our traditional definition (25) untilwe consider the equivalent form for the time lens. Ifwe define the temporal aperture as At and consider thetemporal f-number to be defined as the ratio of the focaltime to the aperture, then we have

# = fT = _ _ _T-At - At(dwi/dt) (28)

(21) Again, the denominator defines the increment

deiAw" =At dtd2p _ d 1dws2 dco Vg)

is frequently referred to as the group-velocity dispersion.Now, a pulse that propagates a distance e through this

from which we conclude that the temporal f -number canbe written as

fT - A, (30)

INPUTDISPERSION

TIME LENS OUTPUTDISPERSION

(22)

d,8 1

dco Vg

and (29)

Brian H. Kolner

exp [i W.,r2 ] exp ib d 202,,��

2f, I- 2 dW2 I

Vol. 11, No. 12/December 1994/J. Opt. Soc. Am. A 3233

We can now generalize our view of the concept of thef-number: In both the spatial and temporal cases thef-number is given by the inverse of the fractional band-width introduced by the lens.

7. DISCUSSION

What advantage does this new point of view bring? Thisis most easily answered with respect to time lenses andtemporal imaging systems. Since temporal imaging is arelatively new concept, we often look for simplificationsand shortcuts in understanding or analyzing imaging sys-tems. Certainly, an important measure of any imagingsystem (spatial or temporal) is its resolution capability.It can be shown3 that the resolution of any temporal imag-ing system is given by the Fourier transform of its tempo-ral aperture function, as one would expect. For the caseof a time microscope with a rectangular aperture functionthe minimum resolvable temporal feature is given by

arm. = Tof# = To o = A 7' A

(31)

where To is the period of the optical carrier. To obtainresolution approaching 1 optical cycle requires a very lowf-number (i.e., a fast lens) and implies that the time lensmust induce a very large bandwidth on the optical wave-form under study, approximately 500 THz in the nearinfrared. The spatial analog of this result is familiar:higher resolution is obtained with lenses of large numeri-cal aperture or, equivalently, large spatial bandwidth.

Let us consider an example of a realistic time micro-scope using an electro-optic phase modulator as the timelens (Fig. 1). Quadratic phase modulation will be ob-tained over only a fraction of the driving period. Thisresults in a useful aperture time of approximately At =

T12-g = 1/corn, where is the modulation frequency.9

Using this and Eq. (14) in our original definition of tem-poral f-number yields

fT At o0 , (32)

and thus

,ri = To _ = 1To-l Fom =TrO (3

ate the extraordinary requirements on an electro-opticdevice. We are currently developing such a time lensbased on upconversion with a linearly chirped pump inour laboratory.'

8. SUMMARY

From the origins of the space-time duality between dif-fraction and dispersion a time lens is seen to be a quad-ratic time phase modulator for electromagnetic waves.The properties of time lenses can be characterized ina manner entirely analogous to the characterization oftheir spatial counterparts. The familiar concepts of f-number and focal length have been developed in the timedomain and share common underlying physical attributeswith space lenses. We have shown that for both spaceand time lenses the focal length is given by the ratio ofthe mean Fourier frequency to the chirp rate. Also, thef-number is understood to be the inverse of the fractionalFourier bandwidth. These results are summarized in therelations

SPACE

dki/dxf#= o

lAk

TIMEco0fT = doil/dt

fT A-0 (34)

Finally, we have shown that the focal time of a timelens is equal in magnitude to the group delay through thedispersive medium following the lens when the system isadjusted as a pulse compressor. This result correspondsnicely to the focal length of a space lens being the distanceto which a collimated beam is focused to a spot.

ACKNOWLEDGMENTS

The author gratefully acknowledges R. P. Scott and C. V.Bennett for stimulating discussions and insightful ques-tions. This study was supported in part by the NationalScience Foundation through grant ECS 9110678 and bythe David and Lucile Packard Foundation, to which theauthor is particularly grateful.

Brian H. Kolner's telephone number, 310-206-9202;fax number, 310-206-8495; e-mail address, kolner~ee.ucla.edu.

One can also obtain this result by noting that the band-width produced by a quadratic phase modulator with anaperture At = 1/corn is Af = Fofrn. Using this in Eq. (31)gives the same result.

We can now prescribe the performance of our electro-optic time lens for a given resolution. Assume that wewish to resolve temporal features with a resolution 8,ril =100 fs. If we can construct a phase modulator that oper-ates at f = 50 GHz, then we must obtain a peak phasedeviation F0 = 200 rad. This is a nontrivial requirementfor an electro-optic phase modulator with currently avail-able materials, but there are indications that such a de-vice may be realizable in the near future.

Other mechanisms for inducing a quadratic phasemodulation on the waveform under study are certainlylegitimate candidates for time lenses and may allevi-

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Brian H. Kolner

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