Optics Communications 246 (2005) 21–24

www.elsevier.com/locate/optcom

Noninterferometric phase calculation for paraxial beamsusing intensity distribution

Aaron Lewis a, Igor Tikhonenkov b,*

a Department of Applied Physics, School of Engineering, Hebrew University of Jerusalem Givat Ram, Jerusalem 91904, Israelb Department of Chemistry, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

Received 22 August 2004; received in revised form 15 October 2004; accepted 26 October 2004

Abstract

The noninterferometric calculation of the phase of an optical field has been investigated assuming that intensity dis-

tribution is given in a space region. The phase was treated as obeying the system of the transport of intensity equation

and an eikonal one. Under paraxial approximation the solution of the last system has been found in the form of ana-

lytical expressions for a phase gradient. It has been shown that if the intensity is known then only finite branches for a

phase gradient exists.

� 2004 Elsevier B.V. All rights reserved.

PACS: 42.15.Dp; 42.25.Bs; 42.30.RxKeywords: Phase retrieval; Transport equation; Eikonal equation

The problem of phase determination for an

optical field is one of the oldest in optical science

and it continuously attracts attention. There are

several well-known approaches developed and this

paper is devoted to noninterferometric or deter-

ministic phase calculations. This method was

introduced by Teague [1] and Streibl [2] and treats

the phase of an optical field as a solution of atransport of intensity equation (TIE) assuming

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.10.069

* Corresponding author. Tel.: +972 2 6480827; fax: +972 2

678 9573.

E-mail address: igorti@bgumail.bgu.ac.il (I. Tikhonenkov).

that intensity distribution in some 3D space region

has been obtained. Despite all its advantages [3,4]

this method contains one difficulty. As the TIE is a

partial differential equation of second order with

respect to phase then for its solution to be unique

and stable additional boundary conditions are re-

quired. In some particular cases this trouble can

be avoided [1,5]. However in the general case theboundary data for a phase or its derivatives are

necessary. Thus solving of the TIE itself is not

sufficient for constructing a self-consistent nonin-

terferometric method of phase retrieval. This letter

describes a new noninterferometric technique for

ed.

22 A. Lewis, I. Tikhonenkov / Optics Communications 246 (2005) 21–24

phase calculation, which does not require any

additional information for the phase except inten-

sity distribution. The procedure is based on an

analytical solution of a nonlinear system of partial

differential equations consisting of the TIE and aneikonal equation.

Here the case of a homogeneous and isotropic

medium is considered and derivation relies on the

formalism constructed in the work of Green and

Wolf [6]. They showed that a free electromagnetic

field can be described by a scalar complex function

U(r, t) of a position r = (x,y,z) and time t (complex

potential), which satisfies the wave equation:

DU � n2

c2o2t U ¼ 0; ð1Þ

where c is the velocity of light in vacuum and n is

the refractive index of the medium. The operatorD � o2x þ o2y þ o2z is the Laplacian.

Here only monochromatic fields will be consid-

ered so

Uðr; tÞ ¼ uðrÞ expðiðk/ðrÞ � xtÞÞ; ð2Þwhere a real modulus u = |U| is introduced; xdenotes light frequency and k = x/c is a wave num-ber; i is the imaginary unit. As to the exponent fac-

tor then here we prefer to express results in terms

of the eikonal /(r) rather then the phase k/(r) it-self. In that we follow [6] and because the wave

number k is a constant all formulae could be di-

rectly rewritten in terms of the phase by simple

rescaling.

It was shown in [6] that in a homogenous isotro-pic medium the Poynting vector P and density of

electric We and magnetic energy Wm of a mono-

chromatic field may be expressed in terms of a

modulus function and eikonal one as follows (see

Eqs. (3.20)–(3.22) in [6]):

P � u2r/; W e � u2;

W m � ðruÞ2 þ k2u2ðr/Þ2; ð3Þ

where coefficients of proportionality are defined by

the system of units used; as usual $ = (ox,oy,oz).As the light intensity I �We and u � I1/2 then

intensity measurement implies the modulus u(r)

to be known. The equations connecting u and /are derived by substitution of (2) into Eq. (1)

and then separating real and imaginary part. It

gives the system (see Eqs. (3.17) and (3.18) in [6])

ðr/Þ2 ¼ n2 þr2u

k2u; 2rur/þ ur2/ ¼ 0:

Here the first equation, under the geometrical

optic limit k ! �, becomes the eikonal equationfor rays and seeking brevity is referred in the rest

by the same term. The second equation is the

TIE. It states that divP = 0 and expresses energy

conservation for a monochromatic electromag-

netic field.

In the present letter the paraxial version of the

last system has been analyzed. Following the usual

recipe we define the paraxial eikonal u(r):

/ðrÞ ¼ uðrÞ þ z; ð4Þunder asymptotic requirements |ozu| � 1,

|ozu| � k (the axis z is the optical axis). Thus the

problem for investigation takes the form: having

known the function u(r), find a function u from

a nonlinear system of partial differential equations

r?uð Þ2 þ 2ozu ¼ n2 � 1þr2?u

k2u¼ NðrÞ; ð5aÞ

ur2?uþ 2r?ur?u ¼ �2ozu; ð5bÞ

where the function N(r) can be called a ‘‘general-

ized refractive index’’; the operator $^ = (ox,oy).

As the intensity distribution measured the function

N in (5) is known.

The Eq. (5a) is a partial differential one of first

order and it is known that the solution of such an

equation is described by the family of characteris-

tic curves or ‘‘rays’’. Denoting oxu = p, oyu = q,ozu = h we write the characteristic system accord-

ing to well known procedure [7]

dxds

¼ 2p;dyds

¼ 2q;dzds

¼ 2;

dpds

¼ oxN ;dqds

¼ oyN ;dhds

¼ ozN ; ð6Þ

where s is a ray parameter. Introducing differenti-

ation along a ray

d

ds¼ dx

dsox þ

dyds

oy þdzds

oz

¼ 2oxuox þ 2oyuoy þ 2oz;

A. Lewis, I. Tikhonenkov / Optics Communications 246 (2005) 21–24 23

Eq. (5b) can be rewritten in the form [6]

r2?u ¼ � 1

ududs

: ð7Þ

Our next step is to determine how derivatives

oxxu, oyyu, oxyu are changed along rays (6). Using

the paraxial eikonal Eq. (5a) for eliminating

z-derivatives one can derive by direct calculations

the following evolution laws

d

dsoxxu ¼ oxxN � 2 oxxuð Þ2 � 2 oxyu

� �2;

d

dsoyyu ¼ oyyN � 2 oyyu

� �2 � 2 oxyu� �2

;

d

dsoxyu ¼ oxyN � 2oxyu oxxuþ oyyu

� �:

ð8Þ

Here we notice that expressions for s-derivatives

of oxxu, oyyu, oxyu include partial derivatives of

the eikonal up to second order but not higher. It

implies that high order s-derivatives of oxxu, oyyu,oxyu also contain partial derivatives of u up to sec-

ond order. This is the specific feature of rays (6)

defined by any solution of the paraxial eikonal

Eq. (5a). The property expressed by (8) causes sys-

tem (5) to be solvable analytically. As the paraxial

TIE (5b) is an additional condition to the eikonal

Eq. (5a) then for any solution of the system (5) the

TIE holds through all space and also along anyray. Thus the TIE (7) can be considered as a

ray invariant, that is a function which remains

constant along a ray:

I1 ¼ r2?uþ 1

ududs

¼ 0: ð9Þ

Differentiation of this relation gives

In ¼dn�1

dsn�1r2

?uþ 1

ududs

� �¼ 0: ð10Þ

The last equality means that quantities In also

are ray invariants. To obtain values of oxu, oyuone needs to consider I1, I2 and I3 only. Using

(8)–(10) we have

I1 ¼ P þ Q� G; ð11aÞ

I2 ¼ ðP � QÞ2 þ 4R2 � F ; ð11bÞ

I3 ¼ ðP � QÞðoxxN � oyyNÞ þ 4RoxyN � H 0; ð11cÞ

where

F ¼ oxxN þ oyyN � 2

u2duds

� �2

þ 1

ud2uds2

;

H 0 ¼1

2

dFds

þ 4GF� �

and P = oxxu, Q = oyyu, R = oxyu, G ¼ � 1ududs.

In essence each of Eqs. (11) is a condition for

the function u. Usually adding them to the initial

equation can not gain anything because the orderof partial derivatives of u increases as the next

s-differentiation is applied. However our case pos-

sesses the principal distinction because of evolu-

tion laws (8). It results in that every invariant Incontains partial derivatives of u of order not higher

then two (it is true, for example, for I2 and I3).

Owing to this fact each of Eqs. (11) is an algebraic

equation with respect to variables oxu, oyu, oxxu,oxyu, oyyu. It allows that an analytical solution

of the system (5) can be found using a method of

degree reducing. It consists of derivatives oxu,oyu expressed in terms of two unknown functions.

Thus the maximum order of partial derivatives the

problem includes is reduced. Here one possible

version of the degree reduction formalism is de-

scribed. The scalar functions proposed for useare the squared modulus of a spatial eikonal gradi-

ent h and the angle a between $^u and the x-axis.

So the relations connecting oxu, oyu and h, a are

h2 ¼ ðoxuÞ2 þ ðoyuÞ2; oxu ¼ h cos a;

oyu ¼ h sin a: ð12Þ

Substitution of (12) into eikonal Eq. (5a) gives

ozu = (N � h2)/2. Using (7) and relations

oxxu ¼ oxh cos a� hoxa sin a;

oyyu ¼ oyh sin aþ hoya cos a;

oxyu ¼ oyh cos a� hoya sin a

¼ oxh sin aþ hoxa cos a;

we directly find a system for first order derivatives

of functions h, a

oxhþ hoya ¼ Gðh; aÞ cos a; oyh� hoxa

¼ Gðh; aÞ sin a; ð13Þ

where

24 A. Lewis, I. Tikhonenkov / Optics Communications 246 (2005) 21–24

Gðh; aÞ ¼ � 2

uozuþ h oxu cos aþ oyu sin a

� �� �:

Thus for full determination of functions a and hone needs two additional relations containing first

h,a derivatives, which invariants (11b) and (11c)

serve for. Namely we have

ðoxh� hoyaÞ2 þ ðoyhþ hoxaÞ2 ¼ F ðh; aÞ;ðoxh� hoyaÞ cos c� ðoyhþ hoxaÞ sin c ¼ Hðh; aÞ;

ð14Þwhere

H ¼ H 0 A20 þ 4B2

0

� ��1=2; A0 ¼ oxxN � oyyN ;

B0 ¼ oxyN ;

cos c ¼ A0 cos aþ 2B0 sin að Þ A20 þ 4B2

0

� ��1=2;

sin c ¼ A0 sin a� 2B0 cos að Þ A20 þ 4B2

0

� ��1=2:

Eqs. (13) and (14) can be resolved with respect

to oxh, oyh, oxa and oya

2oxh ¼ G cos aþ H cos cþ K sin c;

2oyh ¼ G sin a� H sin cþ K cos c;

2hoxa ¼ �G sin a� H sin cþ K cos c;

2hoya ¼ G cos a� H cos c� K sin c;

ð15Þ

where K = ±(F � H2)1/2. The last expressions

through consistency conditions

oyðoxhÞ ¼ oxðoyhÞ; oyðoxaÞ ¼ oxðoyaÞ ð16Þgive two nonlinear algebraic equations for deter-

mination of h and a. Relations (16) are those

equations.

The described procedure shows that solution of

the system (5) has a local character. That is the

functions oxu, oyu at any point are defined only

by values of u and its derivatives at the same point.In a mathematical sense such a solution is not un-

ique and in general one obtains not more than finite

number of branches for a phase. To choose the cor-

rect solution it is necessary to apply the energy min-

imum principle stating that the total field energy

�d3r(We + Wm) must have the minimum magnitude

for the physically correct phase solution.

In conclusion, this letter presents a new nonin-

terferometric method for phase retrieval from

intensity data given in a 3D space region. It is

based on solution of the TIE in a system with an

eikonal equation. This system connects the phaseof an optical field and its intensity, which is

assumed to be known. Here the paraxial case has

been investigated and the new mathematical for-

malism of ‘‘ray invariants’’ has been developed.

It is based on the fact that the eikonal equation

as a partial differential one of first order defines

in a space the family of its characteristic curves

(rays). As the eikonal equation and TIE constitutethe system then the TIE provides the condition,

which remains true along each ray. Thus the TIE

may be treated as the ray invariant. Differentiating

along rays generates the family of ray invariants.

The crucial finding is that invariants generated

by the TIE along eikonal equation rays contain

derivatives of a paraxial phase up to second order

and not higher. This mathematical fact impliesthat analytical expressions for components of a

phase gradient could be obtained by a solution

of a nonlinear algebraic system.

Acknowledgements

The authors acknowledge the support from Sci-ence Department of Absorption Ministry of Israel

and stimulating discussions with N. Axelrod and

R. Dekhter.

References

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Opt. Commun. 231 (2004) 53.

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[5] T.E. Gureyev, K.A. Nugent, J. Opt. Soc. Am. A 13 (1996)

1670.

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(1953) 1129.

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