The pinhole time camera

Brian H. Kolner Vol. 14, No. 12 /December 1997 /J. Opt. Soc. Am. A 3349

The pinhole time camera

Brian H. Kolner

Department of Applied Science, 228 Walker Hall, University of California, Davis, Davis, California 95616

Received February 3, 1997; revised manuscript received July 14, 1997; accepted July 15, 1997

A pinhole time camera is proposed and analyzed. This instrument is the temporal analog of a conventionalpinhole camera and is formed by a succession of dispersion, a time gate, and more dispersion. It is capable ofmagnifying or demagnifying time waveforms while preserving their envelope profile. The system require-ments and the performance of the pinhole time camera in terms of magnification, dispersion, aperture size, andimpulse response are examined. © 1997 Optical Society of America [S0740-3232(97)02612-4]

1. INTRODUCTIONThe pinhole camera or camera obscura is perhaps the old-est and simplest imaging device.1 Devoid of complex re-fractive surfaces, it remains even today a useful tool forspecial applications such as imaging x-rays2,3 or observ-ing solar eclipses. The chief attributes of the instrumentare its low cost, freedom from linear distortion, wide an-gular field of view, and enormous depth of field.4 On theother hand, it suffers from very low light gathering powerand relatively poor resolution.

The imaging performance of the pinhole camera can beunderstood as a trade-off between the limits imposed bythe size of the aperture. Decreasing the aperture size toobtain a sharper image eventually causes blurring due todiffraction effects. Increasing the aperture size resultsin a geometrical projection of the aperture onto the imageplane, obscuring a sharp image. Between these two lim-its exists an optimum diameter that balances the two ef-fects. The criteria for the best point-spread function andits evaluation have been analyzed over time with the re-sult that the optimum aperture diameter dOPT ' A3ll,where l is the distance from the pinhole to the imageplane.5–8

Owing to an appealing analogy between the equationsof diffraction and dispersion, the mathematical analysisof spatial imaging can be developed for time-domainwaveforms as well, and from this emerges the concept oftemporal imaging.9–14 A temporal imaging system worksby passing a short optical waveform through a region ofdispersion, which pulls the Fourier frequency componentsapart in a manner similar to the diffractive effects on abeam. Then the waveform undergoes a quadratic phasemodulation similar to the action of a thin lens on thetransverse spatial profile of a beam. Finally, the wave-form passes through more dispersion, which acts as thelens-to-object distance in the spatial system. If the inputand output dispersions are chosen correctly with respectto the degree of phase modulation, a ‘‘temporal image’’ iscreated—that is, an output waveform with amplitude pro-file essentially the same as the input amplitude profilebut stretched or compressed in time.10,15,16 The analo-gies are so strong that a complete system of nomenclaturecan be derived for temporal imaging systems that in-

0740-3232/97/123349-09$10.00 ©

cludes the temporal equivalents of focal length, f-number,magnification, and the imaging equation.17,18

In the temporal imaging system, the quadratic phasemodulation plays the role of a lens and is therefore calleda time lens. As in the case of a space lens of finite aper-ture, a time lens cannot produce this modulation indefi-nitely, and thus a time window (or aperture) is estab-lished in which the modulation occurs. Let the durationof this window be called Dt. It can be shown that theresolution dt of a temporal imaging system depends onthe temporal f-number (fT

]) according to dt 5 TfT] , where

T is the optical period and

fT] 5

v0

Dt~d2f/dt2!, (1)

where v0 is the optical carrier frequency and d2f/dt2 isthe quadratic phase curvature.17,18 Clearly, a lowf-number is obtained by producing a strong quadraticphase modulation for a long duration.

Now consider reducing the aperture time Dt to a dura-tion so short that little or no phase modulation occurswithin Dt. The time lens is now superfluous, and the sys-tem becomes the temporal equivalent of the pinhole cam-era (Fig. 1). Can such a pinhole time camera make tem-poral images without a lens? In this paper I propose thatsuch an instrument is feasible and, using linear systemstechniques, I analyze the performance in terms of aper-ture size, magnification, dispersion, and impulse re-sponse. As we will see, the time-bandwidth product ofpulses in the input waveform plays an important role inthe system resolution.

2. IMPULSE RESPONSEThe pinhole time camera is a linear system in the sensethat there is a linear relationship between the input andoutput field amplitudes. Thus we can use the concepts ofsuperposition and impulse response to characterize thesystem’s output given an arbitrary input waveform. Inthis section we derive the impulse response in terms ofthe three relevant parameters: input dispersion, outputdispersion, and the shape of the temporal aperture or pin-hole. To obtain the optimum aperture size for a given

1997 Optical Society of America

3350 J. Opt. Soc. Am. A/Vol. 14, No. 12 /December 1997 Brian H. Kolner

magnification, we will derive the conditions under whichthe impulse response is minimized.

We begin by finding a general solution for the disper-sive part of the system. We then mathematically cascadeinput dispersion, temporal shutter, and output disper-sion, using an impulse as the input stimulus. The result-ing function then represents the impulse response.

To simplify the analysis, we assume an electric fieldcomposed of infinite plane waves with a slowly varyingenvelope of the form

E~x, y, z, t ! 5 A~z, t !exp@i~v0t 2 b~v0!z !#, (2)

where v0 is the optical carrier frequency and b(v0) is themidband value of the propagation constant b(v). Theenvelope A(z, t) is the waveform that we wish to imagewith the time camera.

When the field described by Eq. (2) propagates througha dispersive medium, the spectral components of the en-velope function A(z, t) will pull apart according to thefrequency dependence of b(v). Provided that the band-width of this spectrum is small enough that a Taylor se-ries expansion of b(v) to second order is valid, then usingthe traveling-wave coordinates

t 5 ~t 2 t0! 2 S z 2 z0

vgD , (3a)

Fig. 1. (a) Schematic diagram of the conventional pinhole cam-era. Object Io(x, y) placed a distance do from an aperture isprojected as an image Ii(x/M, y/M) at a distance di behind theaperture (M [ di /do 5 magnification). (b) Temporal analog:the pinhole time camera. Arbitrary object waveform Io(t)propagates through input dispersion j1d2b1 /dv2. A shutteracts as a temporal aperture and allows a small burst of the dis-persed input waveform to pass through the output dispersionj2d2b2 /dv2, where it forms a temporal image Ii(t/M) of the ob-ject @M [ (j2d2b2 /dv2)/(j1d2b1 /dv2)#. Features such asmagnification and inversion (local time reversal) are common toboth space and time.

j 5 z 2 z0 , (3b)

where t0 and z0 are arbitrary reference points and vg5 (dv/db) is the group velocity, the envelope functionA(j, t) satisfies the differential equation19

]A~j, t!

]j5

i

2

d2b

dv2

]2A~j, t!

]t 2 . (4)

The solution to this equation can be found by first per-forming a temporal Fourier transform and solving for theevolution of the spectrum A(j, v). We find that the inti-tial spectrum A(0, v) is multiplied by a quadratic phasefunction that manifests the dispersion. Transformingback to the time domain gives the total solution for theenvelope function

A~j, t! 51

2pE

2`

`

A~0, v!

3 expS 2ij

2

d2b

dv2 v2D exp~ivt!dv. (5)

Figure 2(a) shows a schematic representation of a wave-form undergoing dispersion in the normal (z, t) space–time coordinate system and the (j, t) traveling-wave coor-dinate system. In the traveling-wave coordinate system,the time axis t travels with the group velocity of thewaveform.

With the prescription for the effects of dispersion on awaveform envelope given in Eq. (5), we need only add atime gate or shutter as the temporal pinhole followed byadditional dispersion to complete an imaging system.

Fig. 2. (a) Space–time diagram showing dispersion of a propa-gating pulse envelope in the z,t plane with traveling-wave coor-dinates j,t. (b) Impulse response in the traveling-wave coordi-nate system.


Table 1. Time- and Frequency-Domain Functions Corresponding to Input and Output Dispersion and theShutter in a Pinhole Time Camera

Physical Effect

Function

Time Frequency

Inputdispersion G1~j1 , t! 5

1

A4piaexpSi t 2

4aD G 1~j1 , v! 5 exp~2iav2!, a 5j1

2

d2b1

dv2

Outputdispersion G2~j2 , t! 5

1

A4pibexpSi t 2

4bD G 2~j2 , v! 5 exp~2ibv2!, b 5j2

2

d2b2

dv2

Shutter P~t! 5 expS2 t 2

4mD P ~v! 5 A4pm exp~2mv2!, m 5~Dt!2

16 ln 2

Since the system is linear in the field amplitude, we mayassume that the output takes the form of a superpositionintegral

A~j, t! 5 E2`

`

h~t; t0!A~0, t0!dt0 , (6)

where h(t; t0) is the response of the system at time t toan impulse applied at time t0 . A(0, t0) is an arbitraryinput waveform and serves as a weighting function. Ourgoal is to find h(t; t0). Figure 2(b) displays the conceptof an impulse response in the traveling-wave coordinatesystem. [An interesting feature of this concept is that al-though there is no violation of causality in the real space-time system (z, t), no such restriction applies in thetraveling-wave reference frame (j, t). Indeed, as thesimple pulse in Fig. 2(a) spreads due to dispersion, energyappears at progressively earlier time (2t) with respect tothe position t 5 0 where it began. This is due to the factthat the frequency-domain transfer function that oper-ates on the envelope spectrum in Eq. (5) depends on v2 istherefore not Hermitian.20 An analysis of this problemin a stationary coordinate system (z, t) would necessitateHermitian transfer functions and fields retarded by groupdelays in order to preserve causality.]

To find the impulse response of the system, we apply animpulse at time t0 and calculate the system responseh(t; t0). To simplify our analysis of the imaging system,we will designate the frequency-domain filtering functiondue to dispersion in Eq. (5) and its inverse Fourier trans-form as G (j, v) and G(j, t), respectively. Their exactforms are shown in Table 1. Since the effect of dispersionis multiplication in the frequency domain by G (j, v), thenthe fields in time are given by a convolution with G(j, t).Thus the fields at the end of the input dispersion are

A~j1 , t! 5 G1~j1 , t! * d~t 2 t0! 5 G1~j1 , t 2 t0!.(7)

The fields then pass through the shutter to which weassign the arbitrary pupil function P(t). Immediatelypast the pupil,

A~j1 1 e, t! 5 G1~j1 , t 2 t0!P~t!. (8)

Finally, after propagation through the output dispersionwe have

h~t; t0! 5 A~j2 , t! 5 G1~j1 , t 2 t0!P~t! * G2~j2 , t!,(9)

which is the system response at time t due to an impulseat time t0 . From Table 1, we can substitute the func-tions G1 and G2 and expand the convolution as follows:

h~t; t0! 5 E2`

`

G1~j1 , t8 2 t0!P~t8!G2~j2 , t 2 t8!dt8

51

4piAabE

2`

`

P~t8!

3 expF i~t8 2 t0!2

4a GexpF i~t 2 t8!2

4b Gdt8

51

4piAabexpF i

4 S t 02

a1

t 2

b D G E2`

`

P~t8!

3 expF it82

4 S 1a

11b D GexpF2it8

2 S t0

a1

t

b D Gdt8.

(10)

Substantial physical insight can be gained from Eq. (10)by considering its spatial equivalent, the point-spreadfunction. In the usual physical case the effects of diffrac-tion are the same on either side of the aperture, and anexpanding spherical wave from a point source continuesto expand after passing through the aperture [Fig. 3(a)].This corresponds in the time domain to the sign of the dis-persion being the same on either side of the time apertureP(t8). Therefore, when sgn(a) 5 sgn(b) in Eq. (10), thequadratic phase term within the integrand will nevervanish and a pulse created by the aperture P(t8) will con-tinue to spread. (Note: sgn(x) [ 11 if x . 0,21 ifx , 0).

The other possibility is that the signs of the dispersionterms a and b are opposite. The (hypothetical) spatialanalog for this case is shown in Fig. 3(b), where the dif-fraction has the opposite sign (so to speak) and causes re-focusing of the propagating wave front. For the specialcase where a 5 2b, the integral in Eq. (10) reduces to

E2`

`

P~t8!expF2it8

2a~t0 2 t!Gdt8 5 P S t0 2 t

4pa D , (11)


which is evidently a time-reversed and magnified replicaof the Fourier transform of the aperture offset by the de-lay t0 . For a finite-size aperture, increasing the disper-sion (a and b) broadens the impulse response, owing to aneffect similar to the Abbe theory of image formation(hence the scaling coefficient 4pa). In this analogy, in-creasing the dispersion causes high- and low-frequencycomponents to miss the finite-time aperture. Clearly, ifthe size of the aperture becomes large enough, then all ofthe dispersed frequency components will pass through theaperture and the impulse response tends toward a deltafunction located at t 5 t0 .

We are therefore faced with the possibility of perform-ing imaging with either of the two combinations of disper-sion discussed above. Both signs of the dispersion beingthe same corresponds to the conventional spatial pinholecamera of Fig. 1(a). The optimum impulse response for agiven position of the image plane is determined by thesize of the aperture. As we shall see, making the aper-ture too large produces an impulse response that is the‘‘geometrical time projection’’ of the aperture while mak-ing the aperture too small results in a large impulse re-sponse that is due to dispersive spreading.

When the signs of the dispersion are opposite, we havea unique imaging situation that cannot be realized in aspatial system, since it corresponds to one medium hav-ing a negative refractive index. In the time domain, re-focusing of the impulse when a 5 2b implies that an im-age of an input waveform will be created at the point of

Fig. 3. (a) Impulse response in a conventional pinhole camera.Index of refraction on both sides of the aperture have the samesign, and thus the wave front continues to expand past the aper-ture. (b) Impulse response in a hypothetical pinhole camerawhere the index of refraction has opposite sign on either side ofthe aperture. This results in a refocusing of the impulse and isanalogous to a pinhole time camera with dispersions of oppositesign on either side of the shutter. (This is also a form of disper-sion compensation).

refocusing. This principle can be used for dispersioncompensation in optical fiber communications systems.Its application to general temporal imaging is limited,however, since the image has exactly the same scalelength as the object waveform. In other words, we can-not use this for magnification ratios other than 1:1.

We now briefly digress to discuss the notion of magni-fication. In temporal imaging systems employing timelenses, the magnification was found to be the ratio of theoutput to input dispersions, M [ 2b/a.17 The disper-sion plays the role of the propagation distances (do , di)in the spatial imaging system. That is, we may associatedo ⇔ a, di ⇔ b. It is reasonable to argue that since in areal pinhole camera magnification is also defined in termsof the ratio of the object-to-image distance, our pinholetime camera will keep the same definition, viz., M[ 2b/a. This step is further justified by once again ex-amining the impulse response [Eq. (10)]. The integralhas the form of a Fourier transform of the pupil functionwith a quadratic phase term included. The transformvariables are t8 and t. The absence of a phase term lin-ear in t8 means that the center of mass of the pupil’stransform is given by the coefficient on the integrationvariable t8 in the kernel. Thus the displacement of thecenter of the impulse response compared with its origin att0 gives us the magnification, and we can rewrite the ker-nel in Eq. (10) as

expF2it8

2 S t0

a1

t

b D G 5 expF2it8

2b~t 2 Mt0!G . (12)

A. Observation: Local Time InvarianceBefore proceeding to a convolution integral, we must es-tablish local time invariance. In a conventional spatialimaging system, absolute space invariance does not exist.However, if we reference all fields to the optical axis of theimaging system and include any magnification effects inthe output plane, then with respect to that axis there islocal space invariance and thus we can use convolutionwith an impulse response to characterize the form of animage. With a temporal imaging system, there is also noabsolute time invariance, since the opening and closing ofthe shutter establishes a preferred moment within thesystem. However, if we transform the fields to atraveling-wave coordinate system that is moving with thegroup velocity of the input waveform and synchronize theopening of the shutter to the arrival of the input wave-form, then, including any possible scale magnifications,the impulse response is locally time invariant and we canuse convolution to characterize the output image wave-form. A rigorous treatment of this point can be found inRef. 21 for spatial imaging and Ref. 17 for temporal im-aging with quadratic-phase time lenses.

For our purposes here, we can sufficiently establishtime invariance as follows: since Eq. (10) represents animpulse response, we will assume that most of the contri-bution to h(t; t0) in the output time frame will arise froma small region of time centered on the input point t0 .Thus we can relate the corresponding output point to theinput point: t0 5 t/M. The first exponential term canthen be written as


expF i

4S t0

2

a1

t2

bD G ' expF i

4 S t2

aM2 1t2

b D G5 expF it2

4b S 1 21

M D G . (13)

The quadratic phase is now independent of the position ofthe impulse. With the further substitution of a 5 2b/Mwithin the integrand, Eq. (10) becomes

h~t; t0! 51

4piAabexpF it 2

4b S 1 21M D G

3 E2`

`

P~t8!expF it82

4b~1 2 M !G

3 expF2it8

2b~t 2 Mt0!Gdt8. (14)

Finally, if we let t0 [ Mt0 , then, apart from the qua-dratic phase preceding the integrand, the impulse re-sponse depends only on the difference in time t 2 t0 , andthe superposition integral (6) can now be written as a con-volution

A~j, t! 5 E2`

`

h~t 2 t0!A~0, t0 /M !dt0 . (15)

The global quadratic phase preceding the integral in Eq.(14) does not seriously detract from local time invariance,since in most cases we will be measuring a power andtherefore all extraneous phase terms disappear.

B. Example: Gaussian ApertureFor an arbitrary aperture function P(t8), Eq. (14) givesthe general solution for the impulse response. In theconventional pinhole camera, the pupil is usually a simpleaperture with hard edges. If a similar temporal aperturecould be constructed, Eq. (14) could be solved withFresnel integrals. However, time gates with infinitelyfast rise and fall times are impossible to construct, andtherefore it is more realistic to assign a reasonable func-tional form to P(t). For this we will choose a Gaussianpupil function, which will render Eq. (14) exactly solvableand thus give us greater physical insight into the natureand limitations of the imaging system. To this end, let

P~t8! 5 expF24 ln 2 S t8

Dt D 2G , (16)

where Dt is the full width at half-maximum of the aper-ture. The impulse response can now be written as

h~t; t0! 51

4piAabexpF it 2

4b S 1 21M D G

3 E2`

`

expF24 ln 2 S t8

Dt D 2G3 expF it82

4b~1 2 M !G

3 expF2it8

2b~t 2 Mt0!Gdt8, (17)

which is the Fourier transform of a complex Gaussianwith solution

h~t; t0! 51

4piAab UF p

4 ln 2

Dt 2 2i

4b~1 2 M !G 1/2U

3 expF it 2

4b S 1 21

M D G3 expF 2~p f !2

4 ln 2

Dt 2 2i

4b~1 2 M !G . (18)

To obtain the sharpest temporal image, we must deter-mine the aperture duration that minimizes the impulseresponse width. Since the form of this response is com-plex Gaussian, it will suffice to minimize the denominatorof the real part of the exponent with respect to Dt. Theoptimum aperture size is thus calculated to be

DtOPT 5 4S ln 2 U b1 2 MU D 1/2

. (19)

This function is plotted in Fig. 4 for an arbitrary outputdispersion (b 5 1). The pole at M 5 1 corresponds tothe output dispersion exactly balancing the input disper-sion (i.e., refocusing the input impulse). Therefore thelargest possible aperture (DtOPT → `) will optimize theimpulse response, and this is in agreement with Eq. (11).One can understand the behavior for M . 1 with con-stant output dispersion b by noting that, when b is fixed,increasing the magnification requires that the input dis-persion a be reduced. This means bringing the objectwaveform closer to the aperture. The closer the object,experience suggests, the smaller the aperture must be toobtain a sharp image.

In the region 0 , M , 1, the object is receding to infi-nite time away from the aperture, and thus infinite dis-persion broadens the input impulse into a very longslowly chirped waveform. This is akin to the planarwave front generated by a point source at a great dis-tance. A finite aperture will therefore create a temporalgeometrical image in the output. When M 5 0, the inputwaveform is essentially unchirped and of infinite dura-tion, and therefore the optimum aperture size is deter-mined only by the distance to the image plane and hence

Fig. 4. Optimum temporal aperture duration DtOPT versus mag-nification M for a fixed output dispersion (b 5 1).


the amount of dispersive spreading. It is, in effect, thetemporal equivalent of the Rayleigh range.

Finally, when M , 0, which corresponds to a conven-tional spatial pinhole camera, there is no refocusing, andthe optimum aperture size depends only on the distance(time) from the input impulse to the aperture and thusthe magnification (since b is held constant).

We have now determined the optimum aperture timefor a given magnification ratio and dispersion (input oroutput). It is interesting to study next the variation inimpulse response as the aperture deviates from the opti-mum. To see the general trend, we first note that theform of the impulse response is essentially

h~t! } expF24 ln 2 S t

DtIMPD 2G , (20)

where, by comparison with Eq. (18), we make the identi-fication

DtIMP2 [ S 16b ln 2

Dt D 2

1 Dt 2~1 2 M !2. (21)

It will be convenient to normalize the impulse responsewidth to the optimum aperture size [Eq. (19)] so that

DtIMP

DtOPT5 u1 2 MuAS DtOPT

DtD 2

1 S Dt

DtOPTD 2

. (22)

A plot of this function is shown in Fig. 5. The asymptoticbehavior has a simple physical explanation. If the timeaperture Dt is made very small, the sharp gating effectwill create new high-frequency components that will rap-idly spread apart in the following dispersive medium.The first term under the radical in Eq. (22) representsthis effect. In the other limit where the aperture is madevery large, no new frequencies are created by the gatingeffect, but a section of the dispersed waveform passesthrough the aperture and continues to disperse slowly inthe following medium, beginning with a temporal profileexactly matched to the aperture shape. This creates ageometrical time projection of the aperture function.Again, the analogy to the spatial pinhole camera is aptsince a spatial aperture of suitable size will create a geo-metrical replica of itself in the image plane. The secondterm in Eq. (22) accounts for this effect. When

Fig. 5. Impulse response DtIMP normalized to optimum aper-ture time DtOPT versus normalized aperture time Dt/DtOPT forvarious magnification ratios. Note that when Dt 5 DtOPT , theimpulse response is minimized for any magnification, as ex-pected.

Dt 5 DtOPT , the effects of dispersion and geometricaltime projection are equal and Eq. (22) is minimized, asseen in Fig. 5. The minimum output pulsewidth is there-fore

DtIMP 5 u1 2 MuA2DtOPT . (23)

Again we see that for very small magnification (M ' 0),the impulse response is a slightly larger version of the ap-erture, DtIMP 5 A2DtOPT. The situation here is entirelyequivalent to the spot size one Rayleigh range from abeam waist. In both problems the spot size (pulse width)is minimized at a given propagation distance.

3. MINIMUM INPUT BANDWIDTHNow that we have established the optimum impulse re-sponse, let us consider the imaging of waveforms. First,we will see what constraints propagation in the input dis-persion imposes on the system and then explore similareffects in the output. Consider an object field spanning atotal duration of T seconds modeled simply by a pair ofpulses (Fig. 6). As the pulses propagate through the in-put dispersion (j1d2b1 /dv2), they broaden and overlap.They must both overlap at the point where the temporalshutter P(t) opens up so that information from eachpulse passes through the shutter. The spatial equivalentof this requirement is that light emanating from a pointon an object must diffract and overlap light from all otherpoints on the object by the time it reaches an aperture.In the case of extended illumination and diffuse reflectionthis is not a problem, since a rough surface scatters thelight into a large angular spectrum. However, in thetime domain, transform-limited pulses are analogous tosingle-mode collimated beams and disperse slowly owingto their minimum time-bandwidth product. Therefore,in the case of imaging pulses with the pinhole time cam-era, we must establish a criterion for the net bandwidth ofeach pulse in the input waveform so that energy fromeach of them overlaps at the shutter. Conversely, for a

Fig. 6. Propagation of the envelopes of two pulses showing theoverlap and interference required so that energy from each pulsepasses through the shutter pupil function P(t).


waveform with a given bandwidth, we can establishguidelines for possible imaging configurations and definean effective field of view T.

Assume that the input pulses are Gaussian with non-transform-limited bandwidth B and pulsewidth DtIN .Then for a propagation distance j in a dispersive mediumcharacterized by a dispersion D 5 2pjd2b/dv2 5 4pa,Eq. (6) can be used to show that the output pulsewidthDtOUT is given by

DtOUT2 5 DtIN

2 1 ~BD !2 (24)

or DtOUT ' BuDu for large spreading (recall that D can bepositive or negative). In order for two pulses T secondsapart to spread out enough so that some energy from eachpasses through the shutter, BuDu * T, provided that thepulses are symmetrically timed with respect to the open-ing of the shutter. We can now combine this require-ment with the previous results for the optimum impulseresponse. Since the magnification M links the input andoutput dispersions,

B >T

uDu5

T4puau

5T

4p UMb U, (25)

equivalently,

uau >T

4pBor ubu >

T4pB

uMu. (26)

When the dispersions, magnification, and pulse band-widths combine to satisfy relation (25), a new constraintis imposed on the impulse response. This is becausepulses with small time-bandwidth products will requirelarge dispersions to overlap, and this will necessarily in-crease the shutter duration and hence the impulse re-sponse. To see this, we assume that the imaging systemalways has the optimum aperture DtOPT [Eq. (19)], whichwhen substituted into Eq. (23) yields

DtIMP 5 4A2 ln 2 ub~1 2 M !u. (27)

Since relation (26) represents the lower bound for the dis-persion, we can substitute for ubu into Eq. (27) to arrive atthe lower bound for the impulse response:

DtIMP > 2A2 ln 2 TuM~1 2 M !

pB. (28)

To summarize, this expression gives the width of theimpulse response of a system with an optimum aperturetime, configured with enough input dispersion to get asample of all of the input pulses of bandwidth B in thefield of view T through the aperture. Increasing the dis-persions beyond the minimum required by relation (26)while maintaining their ratio (and hence M) increases theimpulse response according to Eq. (27).

4. RESOLVABLE POINTSWe have seen how the bandwidth of input pulses arelinked to the input dispersion and define a temporal fieldof view. In this section we consider the nature of the out-

put pulses and show that by allowing a minimum amountof spreading due to the finite impulse response, we cancombine all of our previous results and establish themaximum number of resolvable points, one of the mostimportant features of any imaging system.

To model an output waveform we will assume a se-quence of Gaussian pulses of width DtIN applied to the in-put and, as usual, a Gaussian aperture function. Theoutput is thus composed of Gaussian pulses that are aconvolution between the impulse response and the geo-metrical optics image of the input. Thus each outputpulse will be of the form

expS 2t 2

Dt IMP2 1 Dt GEO

2 D , (29)

where DtGEO [ MDtIN . The output pulse width,Dt OUT

2 [ Dt IMP2 1 Dt GEO

2 , is broadened by the impulseresponse, which for a pinhole imaging system in certainconfigurations may be significant. It now remains to es-tablish criteria for the various system parameters thatwill minimize the impact of this broadening.

In an ideal imaging system, any broadening of the geo-metrical projection of an input waveform is undesirable.However, relation (28) in the preceding section demon-strates that when the input dispersion requirement issatisfied, the temporal field of view T is linked to the im-pulse width DtIMP . Therefore a finite impulse responseis desirable to the extent that it maximizes the field ofview without unduly distorting the temporal image. Wecan assign a reasonable limit to the impulse response andarrive at a very useful and general result by simply allow-ing the impulse response pulsewidth to equal that createdby the geometrical image: DtIMP 5 DtGEO. Applyingthe definitions for DtIMP [Eq. (28)] and DtGEO (above) andrearranging, we have

N [ S TDtIN

D 5p

8 ln 2 U M1 2 MUBDtIN , (30)

where N is the number of resolvable pulses and BDtIN isthe time-bandwidth product of the input pulses. This ex-pression has great utility when contemplating the designof a pinhole temporal imaging system. For example, forimaging Fourier-transform-limited pulses (minimumtime-bandwidth product), in inverting systems (M , 0),moderate to large magnification ratios are desirable, andeven then, the largest attainable number of resolvablepulses will be N ' BDtIN/2. The system is thereforebest suited for imaging non-transform-limited pulses orwaveforms with very low temporal coherence such asmight be obtained from gain-switched laser diodes andsuperluminescent sources. In essence, this requirementis analogous to the spatial case of imaging objects with avery diffuse reflectance possessing a Lambertian angularcharacteristic. On the other hand, a low time-bandwidthproduct corresponds to imaging a source composed ofhighly collimated beams. Only those beams close to theaxis of the imaging system will be collected and imaged.

Figure 7 demonstrates the relationship between N andBDtIN as expressed by Eq. (30). A three-pulse sequence$1011% of unit amplitude Gaussian pulses is sent through


a pinhole time camera configured with M 5 23. Thepulses have an adjustable quadratic phase modulation sothat their bandwidths may be set independently of theirpulse widths. The magnified images in Fig. 7(b) are cal-culated for five different values of BDtIN , using Eq. (5) forthe dispersive regions and the pupil function [Eq. (16)] forthe shutter. For each sequence, the combination of pulsebandwidth B, magnification M, and total time windowT 5 2 ps sets the input and output dispersion accordingto relation (26). (In fact, a slightly reduced value of dis-persion was used to prevent energy spilling over from ad-jacent periods (utu . 1 ps), which are implicit when dis-crete Fourier transforms are used.) The shutter timewas set to DtOPT [Eq. (19)] for each case.

Several interesting features are apparent in Fig. 7(b).First, there is dramatic improvement in resolution whenthe time-bandwidth product increases. For uMu 5 3, Eq.(30) tells us that N varies from 0.86 to 13.8 over the range2 < BDtIN > 32. This is consistent with the input field,which has a pulse density corresponding to 16 per period(although only four are shown). Second, there is evi-dence of vignetting in the first pulse. This is due to thedispersion being reduced to avoid the spillover effect. In-creasing the dispersion would limit this problem at aslight cost in resolution. Finally, notice the reduction inamplitude in the imaged waveforms. The pinhole cam-

Fig. 7. (a) Input Gaussian pulse sequence to demonstrate reso-lution effects of varying time-bandwidth product BDtIN . Pulsesare on a 2/16 5 0.125-ps spacing and form a $1011% sequence.The pulsewidths are 20 fs (intensity FWHM) wide. Bandwidthsare adjusted by introducing a quadratic phase term to eachpulse. (b) Pinhole temporal images of pulses calculated for vari-ous time-bandwidth products and magnification M 5 23. Foreach value of BDtIN , dispersions a and b and shutter durationDtOPT were set for optimum resolution as defined in the text.

era is notoriously inefficient because the aperture mustremain small for good resolution. In this case, the fieldamplitude of the output waveform is down to ;10% of theinput, implying that the intensity would be reduced to;1% for the simulation shown.

5. SIMULATIONSSimulating the output waveform from a pinhole timecamera is quite straightforward. All calculations aremade on the amplitude of the field envelope. The carrierphase need not be introduced, since it was factored out inthe beginning of the analysis. Arbitrary bandwidth canbe introduced by adding additional phase nonlinearity,second order being the easiest. However, additionalphase perturbations require that the input waveform besampled more often to avoid aliasing.

The effects of input and output dispersion are dealtwith most effectively in the frequency domain with use ofEq. (5), where the quadratic dispersion appears naturallyin the phase of a spectral filter. The shutter function is asimple time window multiplied by the fields calculated af-ter the input dispersion. Figure 8 shows the evolution ofthe electric field amplitude of a waveform in thetraveling-wave coordinate system as it propagatesthrough a pinhole time camera configured with an opti-mum shutter time for a magnification M 5 23 and time-bandwidth product BDtIN 5 18. In the input region, j1 ,the three-pulse sequence $1011% disperses and causes thenecessary overlap by the time the fields reach the shutter.In the output dispersive region, j2 , energy emerges fromthe shutter and propagates until the image is formed at adistance j2 5 3j1 . Notice that the spacing betweenpulses has increased threefold and that the sequence istime reversed, as it should be for negative M. This some-

Fig. 8. Evolution of a three-pulse sequence in a pinhole timecamera. Asymmetrical $1011% pulse sequence is on a 125-fs pe-riod, as in Fig. 7. Shutter function P(t) occurs at the boundarybetween input and output dispersions (j1 , j2). MagnificationM 5 23 and time-bandwidth product BDtIN 5 18. Amplitudescaling of 4 3 is applied in the region of output dispersion forclarity.


what remarkable graphical image gives a satisfying con-nection between the operation of the pinhole time cameraand its spatial progenitor.

6. CONCLUSIONSThis paper has considered the analysis, design, and limi-tations of the temporal analog of the pinhole camera. Wehave seen that, like its spatial counterpart, the pinholetime camera has an impulse response governed by its ap-erture or shutter duration, which balances the effects ofdispersion and temporal geometrical projection. The op-timum shutter time results in an impulse response onlyslightly larger than the shutter time itself. Choosingthis optimum depends only on the dispersion and the de-sired magnification ratio M. However, as we have seen,the time-bandwidth product of object pulses bears an im-portant relation to the total input dispersion, so this char-acteristic impacts the resolution. All of these systemconstraints are embodied in Eq. (30).

The analysis was limited to considering waveforms un-dergoing second-order dispersion, which is realistic formany classes of pulses and waveforms of interest and al-lows for closed-form solutions and rapid physical insight.It is equivalent to the Fresnel approximation in spatialpropagation. For extremely wide bandwidth pulses andtemporally incoherent sources, however, a more rigorousanalysis will have to be invoked.

Applications for the pinhole time camera are somewhatlimited because of its resolution and low throughput.Modern techniques for pulse analysis such as frequency-resolved optical gating (FROG)22 are a powerful methodfor analyzing the amplitude and phase of an arbitrarypulse. The pinhole time camera may find use where thetransformation of the time scales of waveforms is impor-tant, that is, in temporal imaging of arbitrary waveforms,especially with large bandwidths. It may have its great-est utility for manipulating waveforms in spectral regionswhere quadratic phase time lenses are not available orare not sufficiently powerful to warrant their use. Here,dispersion and a simple time gate may be of great value.

ACKNOWLEDGMENTSThe author is indebted to C. V. Bennett and R. P. Scott formany insightful discussions on this topic. This work wassupported in part by the National Science Foundation un-der grant ECS 9521604 and by the David and LucilePackard Foundation.

Contact the author as follows: tel, 916-754-4370; fax,916-752-1652; e-mail, [email protected].

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The pinhole time camera