Femtosecond pulse shaping by a reflection grating in the resonance domain

Optics Communications 223 (2003) 247–254

www.elsevier.com/locate/optcom

Femtosecond pulse shaping by a reflection grating inthe resonance domain

Hiroyuki Ichikawa*, Kaoru Fukuoka1

Department of Electrical and Electronic Engineering, Ehime University, 3 Bunkyo, Matsuyama 790-8577, Japan

Received 5 March 2003; received in revised form 27 May 2003; accepted 18 June 2003

Abstract

Femtosecond pulse shaping by temporal superresolution with a single metallic diffraction grating in the resonance

domain has been numerically investigated. It is predicted that a femtosecond pulse can be compressed by more than

20%. Comparing with the previously reported pulse compression with a transmission grating, a metallic reflection

grating can avoid several problems associated with a grating substrate of a transmission one. Therefore, the proposed

pulse compression scheme is highly feasible in practical experiments. Effects of fabrication errors on the performances

and applicable pulse width range are discussed. In addition, potential problems of analysis and design in using a

multilevel grating for pulsed light are discussed.

� 2003 Elsevier B.V. All rights reserved.

PACS: 42.79.D; 42.65.R

Keywords: Diffraction gratings; Femtosecond optics

1. Introduction

Diffractive optical elements (DOEs) are now

establishing their status in modern optical systems

because of their advantages which are summarised

as generating arbitrary wavefronts and hybridising

multiple functions, as well as their thinness and

lightweight. However, their successful application

* Corresponding author. Tel.: +81899279780; fax:

+81899279792.

E-mail address: [email protected] (H. Ichikawa).1 Currently with Minolta Co., Ltd., Toyokawa, Japan.

0030-4018/$ - see front matter � 2003 Elsevier B.V. All rights reservdoi:10.1016/S0030-4018(03)01671-7

has so far been limited to systems with continuouswave light sources, and quite few systems are for

pulsed light. This is easy to understand from the

nature of a diffraction grating, which can be re-

garded as a local approximation of DOEs. The

well-known grating equation tells that a diffraction

angle is a function of the wavelength of a light

source. As the spectrum of pulsed light has finite

width, each spectral component propagates indifferent direction, when the pulsed light is dif-

fracted by a grating. This problem becomes par-

ticularly serious in gratings in the resonance

domain where the grating period is comparable

to the wavelength of the light source. We have

ed.

mail to: [email protected]

Fig. 1. Concept of temporal superresolution by a grating in the

resonance domain.

248 H. Ichikawa, K. Fukuoka / Optics Communications 223 (2003) 247–254

experimentally confirmed that the beam size of

diffracted femtosecond pulses by a grating in the

resonance domain expands too much to be utilised

any more in the latter part of the optical system [1].

This means that DOEs, at least in the resonance

domain, cannot function properly for femtosecondpulsed light with regard to the principle that the

advantages of DOEs are based on the behaviour of

higher diffraction order waves. Then, in order to

rectify this problem, we proposed to use zeroth

order wave which does not suffer beam expansion

upon diffraction [2].

This approach functions as wavelength filtering

if the grating is in the resonance domain, becausethe efficiency of each diffraction order is different

at each wavelength owing to the phenomenon

called grating anomaly [3]. Although wavelength

filtering is one of the most traditional and com-

mon functions of diffraction gratings, our ap-

proach can claim its uniqueness in that all filtered

wavelength components of a signal wave propa-

gate in the same direction, leading a possibility ofmanipulating temporal shape of an incident pulsed

wave.

As an example application, we consider femto-

second pulse compression. This is achieved in the

same way as the well known superresolution, i.e.,

filtering out the central portion of wavelength

spectrum of the pulse, which we call temporal su-

perresolution. We have already done numericalinvestigation on the performances of temporal

superresolution with dielectric transmission grat-

ings in detail and pointed out drawbacks such as

multiple reflection, spectral phase shift during

propagation and possible nonlinear optical effects

all associated with a substrate in transmission

scheme [2].

In this paper, we extend the concept of thetemporal superresolution to reflection scheme

which is more practically feasible because there

is no undesirable substrate effects mentioned

above. In addition to our main study on binary

grating structure, behaviour of multilevel reflec-

tion gratings on the incident femtosecond pulses

is also investigated, attempting to improve per-

formances of temporal superresolution whichheavily depends on the shape of grating anom-

alies.

2. Concept and diffraction problem

The concept of our proposed approach is

summarised in Fig. 1. A metallic grating is simply

illuminated by an incident femtosecond pulsedlight and reflected zeroth order wave is treated as

an output signal wave. In a linear system, a tem-

poral shape and its spectrum of a pulsed wave are

related by the Fourier transform. Thus, the spec-

trum of the incident pulsed light is modified by

temporal superresolution, if the grating is in the

resonance domain. Then, the output pulse is

compressed at the expense of efficiency and withslight increase of sidelobes.

The idea of modifying the spectrum of a pulse

itself with diffraction gratings has long been well

known in pulsed laser communities [4,5], and a

Fourier transform type 4f system [6] is commonlyused in femtosecond pulse shaping nowadays. It

consists of two gratings, two lenses and one spatial

filter, whereas our system needs only a singlegrating.

The numerical analysis employed here follows

the procedure as detailed in [2].

2.1. Fourier-transform

An incident Fourier-transform-limited pulse of

10 fs FWHM with the central wavelength k0 of 0.8lm is decomposed into spectral components withthe separation of 3.75 THz. The value must be

wider than the separation of discrete spectra of the

pulse, which corresponds to the repetition rate of

the pulsed laser. It is roughly 100 MHz for a

TE wave

TM wave

Fig. 2. FWHM pulse width in fs for a=d ¼ 0:75 and h ¼ 25�.White region denotes pulse broadening.

H. Ichikawa, K. Fukuoka / Optics Communications 223 (2003) 247–254 249

widely used mode-locked Ti:sapphire pulsed laser,

for example.

2.2. Modulation

Properties of each spectral component, i.e.,amplitude and phase of zeroth order wave upon

diffraction are evaluated with Fourier modal

method (FMM) [7], which is the most widely used

numerical solution of the electromagnetic theories

of diffraction gratings.

2.3. Inverse Fourier transform

All spectral components are synthesised to re-

construct temporal shape of an output pulse.

As a grating material, we employ gold, whose

complex refractive index at k ¼ 0:8 lm is inter-polated 0:185þ i4:84 from the data listed in [8]and assumed constant over the spectral region of

the 10 fs pulse considered for the most of examples

presented.

(a)

(b)

Fig. 3. Comparison between pulse widths and zeroth order

efficiency at the central wavelength. (a) Period dependency

(h ¼ 1:70k0). (b) Depth dependency (d ¼ 1:45k0).

3. Example performance

We investigated the quality of a compressed

pulse with three figures of merits, i.e., FWHM

temporal width, peak pulse intensity and sidelobe

intensity for wide range of grating periods d,depths h, fill factors a=d and incident angles h.Fig. 2 shows depth and period dependency of

FWHM pulse width for a=d ¼ 0:75 and h ¼ 25�,in which the darker regions denote shorter pulses,

i.e., solutions for pulse compression. Here, the

number of diffraction orders M considered in theFMM is 80. Please remember, however, that those

solutions do not automatically mean �good� usefulpulses, because the peak and sidelobe intensity

must be taken into account in addition.

Considering the principle of temporal super-

resolution, solutions for pulse compression nearly

correspond to the ones in which efficiencies of the

central wavelength approaches minimum. This is

well presented in Fig. 3, where period d and depthh dependency of pulse widths and zeroth orderefficiencies in TM wave are compared. When d andh are small, the change of pulse width and zeroth


order efficiencies are well synchronised, while for

larger d and h, the tendency is lost. This is mainlydue to phase shift between each spectral compo-

nent, which causes the shift of pulse compression

solution. This can be observed, e.g., if we draw a

line of d ¼ constant in Fig. 2 and shift it vertically.Among potential grating structures for pulse

compression, we demonstrate an example solution

in Fig. 4, where d ¼ 1:45k0, h ¼ 1:70k0,a=d ¼ 0:75, and h ¼ 25�. For this particular ex-ample, material dispersion is also considered in

addition to the non-dispersive model. For the

former case, complex refractive index of gold is

represented as the sixth degree polynomial re-gression on the wavelength using the data in [8].

Then, the obtained FWHM pulse widths, the in-

tensities of the pulse peak compared to the inci-

dent pulse intensity and sidelobe intensities to the

peak intensities are 7.6 fs, 0.20 and 0.15 for the

non-dispersive model, and 7.9 fs, 0.22 and 0.12 for

the dispersive one, respectively. As found in Fig. 4,

Fig. 4. An example solution for pulse compression. (a) Tem-

poral intensity modulation. (b) Intensity spectrum.

effects of dispersion look relatively small at least in

this material. Similar results were reported for a

transmission grating in Fig. 13 of [2]. This is un-

derstandable, because origin of the grating

anomalies is rapid phase shift to which the differ-

ence of refractive indices of two media consistingof the grating contributes more than material

dispersion unless refractive indices of the two

media are very close. Anyway, it is obvious in

Fig. 4(b) that the principle of temporal superres-

olution is working and more than 20% pulse

compression can be achieved with a mere single

diffraction grating.

As we have already described, performances ofgratings in the resonance domain are quite sensi-

tive to changes in structural parameters. This may

cause the fear of possible poor tolerances in

practical experiments, because it is not possible to

fabricate designed structure perfectly. We then

investigate the effects of errors in structure and

optical setup. FWHM pulse width is plotted

against the deviation of each parameter from thedesigned value while others are kept constant at

the designed ones in Fig. 5 and the results are

summarised in Table 1.

Among four parameters analysed, incident an-

gle is the least sensitive and there should be no

problem to control it within the tolerance range of

ordinary simple equipment, while designed period

d and depth h are not so easy to obtain accurately.However, it would be possible to achieve this level

of tolerances with high performance e-beam pat-

tern generator [9] and subsequent reactive ion-

beam etching [10].

So far, we have concentrated our interest on a

10 fs pulse, because pulse shaping of this sort of

pulse width range is challenging as mentioned in

[2]. Fig. 6(a) shows pulse width dependency ofcompression performances with the same grating

structure as in Fig. 4. For this particular grating

structure, the 10 fs is the most suitable pulse width.

This is explained in Fig. 6(b), where wavelength

dependency of the zeroth order efficiency is plot-

ted. As easily imagined, spectral widths of the

pulses longer than 10 fs seem too narrow to utilise

the sharp efficiency change around the centralwavelength in order to achieve the temporal su-

perresolution.

Table 1

Tolerances of parameters for guaranteeing increase of pulse

width within 5%

Parameters Designed Lower Upper

Period d (lm) 1.16 )0.03 +0.02

Depth h (lm) 1.36 )0.03 +0.03

Fill factor a=d 0.75 )0.02 +0.02

Incident angle h (deg) 25 )1.5 +11.3

Fig. 5. Effects of errors in parameters on FWHM pulse width. Dotted lines denote 5% increase of pulse width from designed value of

7.6 fs: (a) period; (b) depth; (c) fill factor; (d) incident angle.


4. Multilevel gratings

In this section, we introduce multilevel grating

structure for the pulse compression attempting to

improve performances, because more complicated

structures offer more design freedoms and bettercontrol of grating anomalies upon which the

temporal superresolution depends. As arbitrary

multilevel structure is difficult to optimise, a step-

wise approximation of an isosceles triangular

grating is assumed for simplicity (Fig. 7). Then, the

grating structure can be characterised by three

parameters: period d, depth h and the number of

layers L which corresponds to the number of levelsminus one.

For incident pulses of 10 fs, performances of the

output pulses are compared for various values of

L, while d ¼ 1:3k0, h ¼ 1:4k0 and h ¼ 25� are keptconstant at the optimised values for L ¼ 5. In ac-tual computation, so-called S-matrix algorithm[11] is incorporated into the FMM.

The quality of pulses can be characterised withthree parameters: FWHM pulse width, peak in-

tensity and sidelobe intensity. The change of those

parameters in increasing the number of layers is

shown in Fig. 8, where the number of diffraction

orders is fixed at M ¼ 200. The pulse shape looksconverging in increasing L, though strong fluctu-ation is observed around L ¼ 20.The effects of increasing the number of layers L

are illustrated in Fig. 9. In temporal modulation

(Fig. 9(a)), the pulse peak is becoming lower and

broader, and at the same time its preceding side-

lobe is rising to eventually form a pulse peak.

Between L ¼ 20 and L ¼ 22, the highest peak ofthe pulse jumps from the original position to the

Fig. 6. Dependency on the incident pulse width. (a) Pulse width

and intensity of the output pulse. Broad line – FWHM pulse

width; filled circles – peak intensity relative to the incident one;

open circles – sidelobes intensity relative to the peak intensity.

(b) Wavelength dependency of the zeroth order efficiency.

Horizontal lines with arrows denote spectral range corre-

sponding to the e�2 intensity.

Fig. 7. Multilevel grating structure considered. Solid line –

L ¼ 5; dotted line – L ¼ 100; d – period, h – depth.


other, i.e., the preceding one. The similar changeof the intensity spectrum is given in Fig. 9(b). The

spectrum with the optimised structure shows clear

drop in the centre. In increasing L, the peak in theshorter wavelength becomes lower first and then

the peak in the longer wavelength becomes lower.

This makes the central drop less significant and

thus the temporal superresolution less effective.

This phenomenon causes the sharp change of

the pulse width and the sidelobe intensity around

L ¼ 20 seen in Fig. 8. The main reason for thiswould be strong electric fields along the surface of

gratings which occur in multilevel metallic gratings

illuminated by TM wave [12]. Thus, the phenom-enon is not observed in superresolution in TE

polarisation.

In addition to the number of layers L, thenumber of diffraction orders M also plays impor-tant role here, because deep or multilevel grating

structure inherently requires more diffraction or-

ders to be included for accurate computation.

The change of three parameters specifying pulsequalities with the number of diffraction ordersM isplotted in Fig. 10. Here, L ¼ 20 is omitted, becausethere are strong resonance effects as described

earlier. Obviously, a grating structure with more

layers requires more diffraction orders to reach

convergence. For the optimised structure (L ¼ 5),all three parameters look converging with the va-

lue of M ¼ 100. On the other hand, for L ¼ 100, atleast M ¼ 160 or even M ¼ 200 seems necessary.The increase of L and M is a drawback both in

computer memories and computational time.

Here, actual computation time T is measured forvarious values of L and M with the FORTRANrunning on an PC with a Pentium 4 processor of

1.9 GHz with 1 GB RAM. First, the necessary

time to carry out computation in FMM is found

T / L � 10M=50: ð1ÞIn addition, here in temporal superresolution,

the condition for convergence

M ¼ 84 log Lþ 26 ð2Þmust be considered, finally yielding

T / L2:7: ð3ÞThis indicates that the computation time is in-

creasing more than parabolically to the number of

layers.

5. Concluding remarks

Temporal superresolution with a metallic grat-

ing will give good pulse compression performance

and we presented practically feasible grating so-

Fig. 9. Effects of increasing the number of layers L: (a) tem-poral modulation; (b) intensity spectra.

Fig. 8. Change of qualities of the pulse in increasing the number of layers L. M is fixed at 200.


lutions as an example. This concept is no more

limited to pulse compression, but applicable tovarious types of pulse shaping and on-axis wave-

length filtering.

The femtosecond pulse compression perfor-

mance of multilevel isosceles triangular gratings is

not so good as binary gratings, because the binary

one has extra design freedom, i.e., the fill factor.

We also have investigated several asymmetrical

triangular structures, but particularly better per-formances would not be expected and the results

are not included here. This indicates that more

complicated multilevel structures are needed, if the

grating anomalies be more suitably adjusted for

the temporal superresolution. However, there ap-

pears a serious problem in such an attempt. As the

number of layers increases, the required compu-

tational load increases and in addition more dif-

fraction orders are necessary, as described in the

previous section. As a result, detail optimisation ofthe grating structure becomes more and more im-

practical. This problem in design stage will be

conquered at the expense of accuracy, if one uses

finite-difference time-domain method instead of

FMM. Nevertheless, difficulties in fabrication still

remain. This implies that gratings with small

number of layers should be employed for pulsed

light application. If we dare use gratings withsmooth surface relief structure, the above compu-

tational problem may be relieved by another nu-

merical technique such as C method [13].

However, considering fabrication accuracy, we

would like to recommend to stick to binary grat-

ings.

Although we employ a relatively unusual con-

cept of temporal superresolution as an example ofDOEs for femtosecond pulsed light, the obtained

results concerning to multilevel structure must be

generally applicable. In order to manipulate fem-

tosecond pulses with DOEs in the resonance do-

main, the wavelength dependence of the diffraction

efficiency must be well controlled. In this respect,

multilevel structure may prone to compli-

cated resonant effects, in particular combined with

Fig. 10. Effects of the number of diffraction orders M : (a) pulsewidth; (b) peak intensity; (c) sidelobe intensity.


fabrication error in practical experiments, as de-

scribed in the previous section.

If, in addition, we need to utilise the higher

diffraction orders of DOEs with femtosecond

pulses, more natural approach such as combining

two DOEs to correct chromatic aberration [14]

would be necessary.

We have treated the grating material as non-

dispersive medium, mainly because we wished to

separate various effects for detail investigation. Itis important to include material dispersion for

further calculation, in particular with highly dis-

persive materials and in practical designing of

DOEs. Also, it should be noted that femtosecond

pulse shaping with the present temporal superres-

olution is valid provided that nonlinear optical

effects are negligible. We believe that some issues

discussed in the Comments of further study of [2]will still give useful insights.

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Femtosecond pulse shaping by a reflection grating in the resonance domain