Generation of coherent wavefronts and images inside laser resonators

(1)

HAL Id: jpa-00249005

https://hal.archives-ouvertes.fr/jpa-00249005

Submitted on 1 Jan 1993

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Generation of coherent wavefronts and images inside laser resonators

Michel Vampouille, Bernard Colombeau, Vincent Kermène, P. Martin, Claude Froehly

To cite this version:

Michel Vampouille, Bernard Colombeau, Vincent Kermène, P. Martin, Claude Froehly. Generation of coherent wavefronts and images inside laser resonators. Journal de Physique III, EDP Sciences, 1993, 3 (7), pp.1369-1380. �10.1051/jp3:1993205�. �jpa-00249005�

(2)

J. Phys. III France 3 (1993) 1369-1380 JULY 1993, PAGE 1369

Classification

Physics Abstracts 42,60D

Generation of coherent wavefronts and images inside laser resonators

M. Vampouille, B. Colombeau, V. Kermene, P. Martin and C. Froehly IRCOM, URA CNRS 356, 123 rue Albert Thomas, 87060 Limoges, France (Received 3 December 1992, accepted ²¹ April 1993)

Rdsumk. Nous ddcrivons _un _nouveau type ^de filtrage des modes d'un la~er basd _sur l'utilisation de deux filtres spatiaux compldmentaires situds _contre les miroirs plans d'un rdsonateur confocal h

lentille. Nous rapportons ^la synthkse de deux ondes particulidres dans _un laser h YAG/Nd : une

surface d'onde modulde sinusoidalement et un faisceau uniphase, de profil spatial plat et de

divergence minimale. L'dvolution temporelle du faisceau pendant la phase ^de construction est aussi rapportde. Ce procddd est un nouveau type de reconstruction de front d'onde _au mdme titre que l'holographie, mais _sans r£fdrence.

Abstract. We describe _a _new type ôf ^laser ^mode filtering using two complementary spatial filters located against the plane ^mirrors ôf _a confocal lens _resonator. With Nd/YAG _as amplifying medium, _we report ^the synthesis of two particular _waves _: _a sinusoidally ^modulated wavefront, and _a uniphase beam with _a flat spatial profile and minimal divergence. The temporal evolution of the beam under construction is also recorded, The process is _a _new type ôf ^wavefront

reconstruction, like holography, but without reference.

1. Introduction.

The _« phase problem _» _can be found in many fields diffraction optics, electron microscopy

and X-ray diffraction, stellar interferometry.. Considerable effort has been devoted to the

development of algorithms yielding ^numerical solutions. Although there have been _some

successes, the problem remains difficult and the difficulty depends on the available

information.

In the following, _we consider the classical problem of recovering the amplitude and phase ^of

a light wave when _we only know its modulus in _two Fourier planes _; and if the solution does exist, is this the only possible _one ?

From _an analytical point of view, ^this problem ^has ^been explained at length by Barakat and Newsam [1, 2] in _a mathematical review. In a simplified _paper [3], these authors in particular

(3)

say that the solution is unique if the function f and its Fourier transform FTIf) are

bidimensional and analytic, and if FTIf) vanishes outside a bounded interval. From a

numerical point of view, a lot of algorithms have been found by several authors. It is important

to note that most of these algorithms are not based _on rigorous mathematical principles and generate ^solutions fluctuating around _an average value. From _an experimental point of view, as

far _as _we know, there have been up to now few experiments which really reconstructed a

complex ^function ^from ^the knowledge of its modulus and the modulus of its Fourier transform [4].

The aim of this article is to show experimentally the build-up of wavefronts inside _a laser

using data concerning the intensity into Fourier planes. These data _are placed ⁱⁿ the _resonator

by _means of two complementary binary masks. We have decided _to describe the series of

iterative operations undergone by the light in the _resonator by referring to the Gerchberg and Saxton's iterative algorithm often used for the phase retrieval. We then report experimental

results concerning two particular waves a,sinusoidally modulated wavefront, and _a uniphase

beam with _a flat spatial profile and minimal divergence. Finally, we study the temporal evolution of the beam under construction.

This paper is concerned with the works which study the control of the transversal spatial

structure of laser modes, obtained by placing different obstacles in the cavity _one _or several

diaphragms, tightened wires, binary masks, Gaussian mirrors, _or _non linear components.. [5- l0].

2. Gerchberg-Saxton algorithm.

To understand the principle of the experiment that _we have implemented, let _us first briefly explain the algorithm derived from Gerchberg and Saxton I ]. This numerical algorithm finds the phase _~ (xi of _a complex function f(x), of which only the moduli [[f(.i)[[ and

[[FT[f(~r)][[ _are known. It works by successive perturbations on all the samples of f(x) and FT[f(x)] (see Fig. I).

We start with any phase function ~~l" _and

we build g(x)

=

[ f(x)[[ ^~~~'' with the known modulus ^[f (x) _[[. We take its Fourier transform I) and _we replace (2) the modulus of this _new function by [[F (u )Î ^which îs âlso ^known. ^We ôbtain G'(u)

=

I F (u ^I ~~l"~ _and

we take the

inverse Fourier transform of this function (3). The modulus of this _new function will then be

replaced (4) by the known modulus I f (xi_II. At each step, ^the phase term will be retained, whereas the computed modulus will be replaced by its known value. The four operations will be iterated until they yield converging solutions.

Our _new holographic principle will consist in just the analogical transposition of this

algorithmic _process.

3. Analogical transposition of the algorithm.

3. I FIRST TRANSPOSITION IN LINE (see Fig. 2). The analogical Fourier transform is achieved

by propagation of _a monochromatic _wave between front and back-focal planes of _a converging

lens. The sequence of Fourier transforms is made with _a sequence of several lenses

L, having their focal planes at the _same place. A set of 4 _n lenses is necessary ^to synthesize a

non-symmetrical complex _wave. Conservation of phase _terms and the processing of moduli

exchange after each F-T- _are achieved by two bidimensional masks, the real transparency ^of which is deduced from I fix, _y_ill and ^I F (u, vi ^[ respectively. In the algorithm, the functions

are normalized at each iteration _: in the analogical _process, the losses caused by the masks must be compensated by the gain of _one _or several amplifying materials.

(4)

N° 7 GENERATION OF COHERENT WAVEFRONTS AND IMAGES 1371

Inifial Phase

4

$

4

data (f( ~ g= (f(e)~ _~ _g'= (i'~~~

~ ~ ~

l $ _F-T- ₃ I _F.T.~~

-~

G'=(F(eJ~l _~ _data _(F(

~

~ ~

Fig, 1, Gerchberg-Saxton algorithm.

F-T. ET FT. ET ET

~ ~~(~(/~ '

i L~ Fi L, <£«pzctzd

sham)

1. F-T- ~ F-T-'-

~ 4 _n lenses required for synthesis of unsymmetrical shapes,

2. Substitution of field amplitudes (« moduli substitution »j ~ approximated by filtering through real transparencies.

3. Filtering losses

~ compensation by LASER amplification.

Fig. 2. Analogical transposition of the algorithm.

We must notice _an important difference between the algorithm and _our analogical

transposition the moduli _are _not replaced by the _exact solutions, but multiplied by the real transparency ^of ^the ^masks. ^In practice, these masks _are of binary transmission. Thus, the constraints working _on the _wave _are less restrictive than in the Gerchberg-Saxton algorithm sometimes, they _are _not sufficient _to obtain the convergence.

3.2 SECOND TRANSPOSITION usE OF RING RESONATORS. It is clear that the above sequence

of operations _may be performed inside _a ring only containing ^four converging lenses and four

plane mirrors, _see figure 3. An amplifying material (A.M.) transforms this ring into _a special ring laser cavity equipped with two pairs of binary transparency ^masks Fj and F~. ^This cavity

will be able _to synthesize complex waves of any shape presenting _no particular _symmetry. We

are presently working _on such resonators.

(5)

h'~ ~~

i~

~ L

F~

I

L fi

'

(6)

4. Experimental results.

The first light field that _we synthesized _was _a spatially sinusoidal amplitude distribution with

gr phase shifts at each half-period _p of the sine _wave.

One of the masks consists in _two small circular holes drilled through an opaque screen,

separated from each other by _a length Af/p, reciprocal to the sinus periodicity _p (A ₌ wavelength, f

= focal length of the lens). The other mask is just a sequence of opaque wires (grid) accurately located _at the positions of the intensity minimums of the sinus function to be generated (see Fig. 5). The pictures (Fig. 6) presented here _were recorded by _a spatial

beam analyser producing _a pictorial representation of the different energy levels in _a three dimensional frame. The cavity works in the free-running regime, just above the laser emission threshold. In the grid plane (Fig. 6a), _we _can _see the expected sinusoidal fringe _pattern. In the far field, (Fig. 6bl, _we _can observe _two bright _spots propagating with the angular spacing ^of

the spatial frequences _± p/Af. The presence of these _two spots demonstrates that the amplitude

in the _near field is really the expected sine _wave function, rather than, ^for instance _a (sinus)2 amplitude distribution. In this latter _case, the different phases in the _near field would keep the

same sign, thus generating three spots ^instead ^of ^the two observed.

Moreover, if _we _remove the grid, ^the ^time integrated illumination in the _near field looks like

an almost uniform background time resolved observation of the pulse demonstrates that the sinusoidal fringes exist at any time, but _as they move randomly during the free running ^laser emission, they give rise _to _a statistically constant illumination. The masks achieve the phase locking of spatial modes which _can coexist inside the laser cavity.

After these first results, we lvanted _to synthesize a square limited plane _wave (constant illumination and phase across a square plane aperture) [12-14]. Such _a laser wave is very interesting for many applications drilling, soldering, ^surface treatment, information storage..

Here, we find the _same cavity (see Fig. 7). Only the masks are different. The first _one is _a square aperture, of width _a, _cut in _a black _screen, the other _one is _a grid made of opaque wires

coinciding with the zeroes of the bidimensional sin (gra.r/Af)/(grax/Af), sin (gray/Af)/

(gray/A f) function. These wires _are engraved on the output ^mirror by a chemical process.

~r ifl~($~#%ular _apzrturas

Grid

Fig. 5. First experimental ~ynthesis ^of a sinusoidal _wave.

(7)

a)

.~ ~

~ _~ ,

''

~ ~ ~

~ ~" ~ ' ~ "'

' .

"~ _'

~ '

'

' ~

' " ~

"

'

",~~

' ~ '~~~' '~ _~ ~' ~i

~

'

M '

~ ' ~~ ~

~ (,

" ,

,

'~ ~

'~ , '~

~

~ii

~

.

" ~

~

' & ~ -' ~ ' ~

4

' ,, ' _~- _, ~

~ ~ ~

~ '"

. " ^~ ~ i .fll(/^~j"/.

[ ~ i x t

' '

~ ~/~~ic,j~~

' j ~ ~ / i

' '~ _~~.' ~ ' '

~, " ' ~~ ~'~~'~ _~ ' ~ ~ ~ ^~'

~~"

, ~ ~

~ ~ ~ ~~

'~ ~

'~

o ~ ^.

' o / _~ ~' _~

~~~ ~ ' '

'

~ ' _~~

~ ^' ~ ' ,

~ '~ ' ~ i

~,, ^)]~ ^[,[

; ^~ _{_~'}., ~

~ ~ . _. ~ ~ '

;

" ~ " ~ '

~ '~ ~

' ~ ' " $ "~ ~ '

' ~÷~~'~>

" ~~~ '

< ~ ,

~_ ^~~ ~ _i _' _~/~,'f

j~,~

"~. « ~~ ',~~~' ~' ~'~' _~~~'~'/

/ " '~

,' '~ " '

'~ '~

~r

~'i' ~ '~/S

~~~%W ' ~ ? '

< "

'

'~ _. _~ _~

,

~ ' ~" ^~ ~ ~ &' ~i ( _~~'" ⁱ 1' ^~ ^~ ^'4

Figure 8 presents ^the illumination pattern ^recorded ⁱⁿ ^the square aperture ^located ⁱⁿ ^the output ^mirror plane. As we can see, the illumination is nearly constant across the square

aperture. ^The residual fluctuations, of the order of 8 fl, _are due panially to the spectral

truncation beyond ^the grid outline. In order _to verify the phase flatness, we examine the illumination in the far field (that is, ^{in the} image plane ^of ^the grid). The illumination of figure 9

was recorded in the Fourier plane ^of the square aperture. ^We can easily recognize a

(sinX/X)~ bidimensional function with _a main lobe twice _as wide _as the side lobes. This

(8)

Mirror

' Fm"t

~lf#r

Fig. 7. Industrial application synthesis of _a square limited PLANE _wave.

~'

)M~~

~"~"0

Fig. 8. Illumination in the square plane aperture.

intensity pattern ⁱⁿ ^fact corresponds to the far field pattern ^of a plane wave truncated by a

square apenure. ^The ^number ^of pixels in the synthesized _wave is of the order of 500.

To control the quality of the reconstructed _wave, _we have recorded the diffracted patterns in different planes between the square aperture ^and ^the ^far ^field plane, recognizing the typical

Fresnel patterns ^of a square hole illuminated by a monochromatic plane _wave. Then _we

compared the illuminations in several planes after the output ^mirror ^with ^the illuminations

(9)

~ '

~.

~

~ ~

~ J

,

' ^' _' ~

J ' ,

KW. k.' ~ 'i ", "~~

'

", ." "'

.' _., _,

,~ i _'. ",~

~/ "[" 'i,"l'i'~',~ _{i_1'-}

~_" ._i io_"

'~ 'i 16'~"l~'U'(~" "'5~~~' _? _,'~

~

,

°' '+''(l'-~"'l "i'~

i' 'O',I( "' $_

', W

/.- "--I' ~'$"""' '~ '~'

#~ ~""

'' ( ".""'&j'i "~i (Q/$'°'~"ii'~ Z _''"' h' ~~~,I <'@$ '~

'. _'- i ~ '

"~ ^"~ ~ ~

/ 'j ''~'

i"'

_._ ''

." .~.

~ ""

'~

._

"~

"' ' ~ ~"~~

', 'S ~

'i ~".- i/

_I.~i' ^"' 'kC_3i,s'+, 'it'i, )$j _{_.'1}_..

.~l., _z.~.

S ~ _"i _° _"" _'." _l'~ _i°I _l'i'i

'u$ _~ 'i'i I i'i

'

~., ~

i,

,-_

("

~" (, "' ~(~.l'[ ) i'i

i T' ^~'

",

'~~~°4~' °'

' k _~''i ~' "L Cl _h

"~ # f ^' ~ ,ij'" 'Ii'

"I°°"

~

44~'~

'. j'~ f ~

-ii'- -'k, _j

' '

~ "' ~' _~'

Fig. ^9. Illumination in the far field.

obtained after a square aperture lighted by _a He/Ne laser beam (Fig. 10). The _wave synthesized by our process, propagates according to the usual diffraction laws, exactly like the _wave

diffracted by a classical hologram.

The _« hologram _» which allowed this _wave reconstruction is not of the usual type : in the system we propose, it is _a pair of binary masks introduced in the confocal laser cavity.

.14Sii

a

b

Fig. 10. Illumination patterns. al In several planes after the output synthezizer mirror. b) In several planes after _a square aperture lighted by a He-Ne laser beam.

(10)

5. Study of the temporal convergence of the process.

We _are also interested in the convergence speed of the process [15]. In other words _: how many round-trips have to be performed by light inside the cavity for stable generation of the expected spatial profile ?

The experiment reported ⁱⁿ figure I _measures the rise-time of the stable stored field. In this

figure, _we recognize the synthesizer laser with its _two masks, _one being a one dimensional slit

of width _a, the other _a grid whose black stripes coincide with the minimums of the

sin (wax/A f)/(wax/A f) slit Fourier transform function. In the free~running regime, the first effects of the mask filtering cannot be _seen, because the power levels _are _too weak. For this

reason, an additionnal mode laser (L.M.S.) drives the first by a single powerful short pulse of

measurable intensity. In the cavity of the resonator, the pulse is amplified and its spatial profile

will be reshaped by ^both ^masks. ^After _a round-trip, _a small fraction of the pulse leaves the resonator, the main part continuing oscillations inside. By this _means, _we obtain _a sequence of

measurable pulses with different spatial profiles.

'°~ ~°~~~ _~~"~ ~$~j~~~~~~~L~~~~~~~~

Hi

~°~.j~

~

cyl.Lens

~~

~

j,- 84am3Flitt«r

~

~t

Picosz~cond ~

Electronic 2"~ lti'nZl

Streak Camera

Fig. II. Temporal convergence of the process.

A temporal analysis of these one-dimensional profiles (along Y in the figure) ^with a streak

camera makes it possible to observe the convergence process in real time. A frequency

doubling crystal (KDP) placed before the streak _camera _converts the laser light ^into ^visible light, because the _camera photocathode is _not sensitive to the laser infrared radiation. By

longitudinally moving the two converging lenses~ _we _can bring different image planes to focus

on the streak _camera slit. On the graph of figure II, ^the M~ grid is imaged _on the slit _; the

spatial profile _converges towards _a sin (wax/A f)/(wax/A f) function.

On the recordings of figure 12, the venical axis Y represents ^the ^dimension parallel to the slit width and the horizontal _one represents ^time.

I) The slit plane Mi is imaged _on the streak _camera slit. As the total length ^of ^time ^which can

be analysed by the _camera is about 50ns (equal to 7 round-trips) _we carried out threee

recordings, each lasting approximately 50 _ns, in order _to observe~ with three successive laser

JOURNAL DE PHYSIQUE III T ~ W7 JULY 199~ %

(11)

y

2 40

50 ao loo

~~'~~~ ~ TIQE

~O ~O ^(ns)

(a) NEAR FIELD

y

ME

--- »

20 50 80 loo 120 (ns)

(b) _FAR _FIELD

Fig. 12. For three successive laser shot~, recordings ^of the spatial profiles of the successive pulses, emitted after each round-trip in the cavity a) in the plane of the output ^mirror Mj, b) in the far field.

shots, convergence towards the required profile. ^The profiles of the first pulses are speckled

the following _ones become smoother and smoother _; the last _ones _are almost _constant. The few

extra white _smears between the different profiles come from the noise of the image intensifier.

ii) The grid plane M~ is imaged on the streak _camera. We show three successive recordings.

The speckle of the first pulses _converges quickly towards _a (sin _{XIX )~} function whose central lobe only _was recorded, because of the _non linear responses of the _camera, of the image

intensifier and of the film. The _narrow dynamic _range of the recording process prevents a

distinct view of the side lobes of the sinc function. For the last pulses, the filtering imposed by

the grid brings about less losses, because the profile is adapted the main lobe becomes _more intense, to the detriment of the secondary lobes.