Ervin Kolenovic´
Vol. 22, No. 5 / May 2005 / J. Opt. Soc. Am. A
899
Correlation between intensity and phase in monochromatic light Ervin Kolenovic´ Bremer Institut für angewandte Strahltechnik, Klagenfurter Straße 2, D-28359 Bremen, Germany Received June 29, 2004; accepted November 10, 2004; revised manuscript received November 21, 2004 Analytical expressions to describe the phase gradient of monochromatic light by means of the threedimensional intensity distribution are derived. With these formulas it is shown that the two-dimensional phase gradient in a plane can be completely determined from noninterferometric intensity measurements if the light propagates strictly in one direction. The analytical expressions are verified by means of numerical investigations on simulated speckle fields, and the results are discussed with respect to common deterministic phase retrieval approaches. © 2005 Optical Society of America OCIS codes: 030.6140, 100.5070, 120.5050.
1. INTRODUCTION A scalar monochromatic light field can be described by means of its amplitude and phase at every point in space. The amplitude of the light is usually determined by taking the square root of the intensity because this value can be directly measured. Considering the values of intensity and phase at a single point in space, the phase can vary without altering the intensity. An obvious example is that the phase of a monochromatic light field shifts linearly with time, even if the amplitude, and thus the intensity, is time independent. That makes it impossible to assign a specific phase value to the intensity, since all phase values will occur successively. Because of this it is easy to think that the intensity and the phase of a light field are totally detached from each other. This would mean that, in the absence of a reference wave, it should not be possible to obtain information about the phase of the light knowing only the intensity. However, if the spatial distributions of intensity and phase are taken into account, instead of only a single point, then there is a certain correlation between those distributions. An example of this correlation is the typical appearance of phase singularities. Phase singularities occur where the intensity is at zero, which means that the phase of the light is not defined at these points. Around phase singularities the phase is typically distributed in a corkscrewlike structure.1,2 The joint occurrence of certain phase and intensity distributions, such as around intensity minima, illustrates qualitatively a correlation between intensity and phase. Another example depicting a relation between those two values is the statistical finding that intensity maxima almost never coincide with stationary points of the phase, such as with phase minima, maxima, or saddle points.3 Independent of these examples, the correlation between intensity and phase has been investigated for more than two decades. The main goal of the investigations was to find whether the spatial phase distribution can be determined from noninterferometric intensity measure1084-7529/05/050899-8/$15.00
ments. Deducing the phase from intensity measurements, without using interferometric methods, is often referred to as phase retrieval.4 It has been shown that indeed in certain cases the phase can be determined in this way if some constraints are given as a priori information. The required constraints depend on the particular retrieval method used. In general, phase-retrieval methods can be characterized as either iterative4–6 or deterministic.7–12 While the iterative approach requires an initial guess for the phase, which is then iteratively improved, it is the aim of deterministic methods to find a direct mathematical expression for the phase. A major approach for deterministic phase retrieval has been developed by Teague8 and is based on solving the socalled transport-of-intensity equation. This equation was derived for light that propagates mainly in one direction and has been investigated for many years.13–18 However, with this approach, the phase could be determined only with further approximations, such as for intensity distributions that are constant in the lateral direction8,16 or for one-dimensional problems.14 In this paper the correlation between intensity and phase will be investigated in a deterministic way. Starting from the fundamental Helmholtz equation, analytical expressions for the absolute value and for the direction of the phase gradient are derived by means of the threedimensional intensity distribution. Based on these expressions, it will be shown that it is possible to determine the phase of monochromatic light from noninterferometric intensity measurements with a significantly lower number of restrictions than by utilizing the transport-ofintensity equation. However, as presented in the following sections, the transport-of-intensity equation is a subset of the basic equation used for deducing the direction of the phase gradient. In addition, it will be shown that the transport-of-phase equation, which was also derived by Teague,8 appears as a subset of the equation that will be used to determine the absolute value of the phase gradient. In general, the transport-of-phase equation has not been considered for solving phase-retrieval problems. © 2005 Optical Society of America
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= 0共x,y兲 + kz,
2. CORRELATION BETWEEN INTENSITY AND PHASE To find the spatial dependence between intensity and phase in monochromatic light, it is sufficient to consider the time-independent representation of the field. The light field U共r兲, with a real amplitude A共r兲 and a phase 共r兲, can then be expressed in a complex way as U共r兲 = A共r兲exp关− i共r兲兴,
共1兲
where r = 共x , y , z兲 denotes a location in three-dimensional space. This time-independent description of the light has to satisfy the Helmholtz equation ⵜ2U共r兲 + k2U共r兲 = 0.
共2兲
In this equation ⵜ represents the Nabla operator in three dimensions, and k = 2 / denotes the wave number for the wavelength . Substitution of Eq. (1) into Eq. (2) leads to ⵜ2A共r兲 + A共r兲k2 − A共r兲兩 ⵜ 共r兲兩2 − i关2 ⵜ A共r兲 · ⵜ 共r兲 + A共r兲ⵜ2共r兲兴 = 0. 共3兲 Equation (3) can be satisfied only if the real part and the imaginary part vanish independently. If both parts are set separately to zero, two new equations will arise. In Subsection 2.A the equation derived from the real part will be utilized to express the absolute value of the phase gradient by means of the intensity and its derivatives. The second equation, derived from the imaginary part, will deliver information about the direction of the phase gradient. For reasons of simplicity the explicit dependence on r will be omitted in the following sections. A. Absolute Value of the Phase Gradient If the real part of Eq. (3) is set to zero, the square of the phase gradient’s absolute value can be obtained by a simple transformation: 兩 ⵜ 兩2 = k2 +
ⵜ 2A A
共4兲
.
Expanding the left side in Cartesian coordinates and expressing the amplitude by means of the intensity I leads to
冉 冊 冉 冊 冉 冊 x
2
+
y
2
+
z
2
= k2 +
ⵜ2冑I
冑I
.
共5兲
This result shows that the absolute value of the threedimensional phase gradient can be expressed without any approximations by means of the amplitude or, alternatively, by the intensity of the field. In many cases measurements in optical metrology are performed with a planar sensor. Under certain conditions the absolute value of the two-dimensional phase gradient in this plane can also be obtained if the light field satisfies some assumptions. Let the sensor plane be defined by the x- and y axes, and let the z axis be perpendicular to this plane. If we consider light that propagates strictly into the z direction, then the phase can be written as
共6兲
where 0共x , y兲 denotes the phase in the sensor plane. This condition can be fulfilled, e.g., in a collimated laser beam or in the far field of scattered light. Substituting Eq. (6) into Eq. (5) leads to the expression 兩ⵜT兩2 ⬅
冉 冊 冉 冊 冑冉 x
2
+
2
1
=
y
2冑I
I
x2
+
2冑I y2
+
2冑I z2
冊
共7兲 for the squared absolute value of the two-dimensional phase gradient. In this equation ⵜT denotes the nabla operator in two dimensions. With Eq. (7) the phase gradient’s absolute value can be determined in the sensor plane without a reference beam by using only the intensity of the field. The right side of Eq. (7) contains partial derivatives in x , y, and z. This means that it is in general not sufficient to measure the intensity distribution in a single plane to determine the absolute value of the phase gradient exactly. However, in some cases an approximation for this value can be obtained even from a single plane measurement, as for example for the far field of scattered light. It is known that the speckles in a scattered light field are cigar shaped and that the speckle length depends on the distance from the object surface. From a certain distance the speckles are much more expanded in the z direction than in the x and y directions, since with increasing distance their length grows faster than their lateral size.19 From this it can be assumed for the far field of scattered light that
2冑I 2冑I 2冑I , Ⰷ . x2 y2 z2
共8兲
In general, this assumption is also fulfilled for any kind of collimated light. With this, Eq. (7) finally turns into 兩ⵜT兩2 ⬇
冑 冉 冑 1
I
2 I x2
+
2冑I y2
冊
.
共9兲
This relation allows one to determine the absolute value of the two-dimensional phase gradient from a single plane intensity measurement if the conditions discussed above are fulfilled. However, initial investigations on simulated speckle fields have indicated that for scattered light by means of Eq. (7), which utilizes the three-dimensional intensity distribution, significantly more accurate results will be achieved in most cases. As an example of the validity of the results in this section, the images in Fig. 2 show the squared absolute value of the two-dimensional phase gradient for the simulated speckle field in Fig. 1. The parameters for the speckle field have been chosen in a way that the far field condition in Eq. (6) is sufficiently fulfilled. The right image of Fig. 2 shows 兩ⵜT兩2 calculated by means of Eq. (7) solely from the three-dimensional intensity distribution. For comparison the left image depicts the exact squared absolute values of the gradient, evaluated from the phase map of the speckle field. Both images are scaled to the same maximum values, and they show good coincidence.
Ervin Kolenovic´
Vol. 22, No. 5 / May 2005 / J. Opt. Soc. Am. A
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Fig. 1. Phase (left) and intensity (right) of a simulated objective speckle field. The speckle field was simulated with a wavelength of 632.8 nm for an object with a circular illuminated area of 4 mm in diameter in a distance of 300 mm. The depicted area is 0.04 mm2.
Fig. 2. Squared absolute value of the phase gradient 兩ⵜT兩2 of the speckle field in Fig. 1. The left image is evaluated directly from the phase map, while the right image is calculated according to Eq. (7) solely from the intensity distribution. Both images are scaled to the same maximum values. Bright values represent high phase gradients, and dark values low ones.
It is notable that the considerations in the first part of this section have been almost identical to those used for deriving the eikonal equation, which describes the behavior of light rays in geometrical optics.20,21 In a slightly different form, Eq. (4) represents an intermediate step in the derivation of the eikonal equation. However, as an approximation in geometrical optics, it is assumed that the term ⵜ2A / A vanishes. Surprisingly, this unstudied term contains information about the phase gradient of the light field. It should also be remarked that the transport-of-phase equation8 can be derived as a subset of Eq. (5). For this to occur the second-order derivatives on the right side have to be performed explicitly on the square root of the intensity, and then the parabolic approximations, introduced by Teague, have to be applied to this result. The explicit
expressions for this can be found, e.g., in Eq. (28) and (29) in Fernández-Guasti.18 Equation (5) alone, and therefore also the transport-of-phase equation, contains information only about the absolute value of the phase gradient. This means that additional information about the direction of the phase gradient is required to recover the phase of a two-dimensional field.
B. Direction of the Phase Gradient In addition to the results of Subsection 2.A, a relation between the intensity and the direction of the twodimensional phase gradient in the sensor plane can also be derived from Eq. (3). For this task the imaginary part of the equation has to be considered. For reasons of clarity, only the crucial steps for the derivation of the desired
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result are presented below. The explicit derivation can be found in Appendix A. If the imaginary part is set to zero and the amplitude expressed by means of the intensity, the resulting equation gives ⵜ共I ⵜ 兲 = 0.
共10兲
This equation is universally valid, since it follows directly from the Helmholtz equation. It can be expanded to
x
冉 冊 冉 冊 冉 冊 I
+
x
I
y
=−
y
z
I
z
共11兲
.
Let  be the angle enclosed by the two-dimensional phase gradient and the x axis. The components of the phase gradient in the 共x , y兲 plane can then be written as
x
= 兩ⵜT兩cos ,
y
= 兩ⵜT兩sin  .
共12兲
With this result, and after the definition of two new values a=−
z
冉 冊 I
z
d = I兩ⵜT兩,
,
For the general case, however,  and its partial derivatives are not the only unknown values in the equations above. According to the definition in Eqs. (13), some knowledge about the phase is required to determine the fields a and d, which are crucial to determining . By utilizing the results of Subsection 2.A, one can express these two fields by means of the intensity distribution if the light propagates strictly into the z direction. By means of Eqs. (6) and (7), the fields a and d can then be written as
共13兲
a=−
x
共d cos 兲 +
y
共d sin 兲 = a.
 x  y
=
ln d
=−
y
−
ln d x
ln I y +
cos2  +
ln I x
ln I
sin2  −
x
sin  cos  −
ln I y
a d
sin  cos  +
sin  , a d
共15兲
cos  . 共16兲
The correctness of this intermediate result can be verified by substituting back into Eq. (14). From the two partial differential equations (15) and (16), it can already be shown that the direction of the twodimensional phase gradient is strongly correlated with the intensity distribution. Assuming that  and its partial derivatives are the only unknown values in these equations, the field of  can be determined numerically in the 共x , y兲 plane. To do so, it is necessary to know an initial value of  that can be substituted into the two equations in order to obtain the partial derivatives in the x and y directions. By means of the partial derivatives, it is possible to approximate the neighboring values of  linearly. Subsequently, these new values of  serve as new initial values to determine the angle of the phase gradient at further points, and so forth. The accuracy of this approximation depends on the distance between the sampling points. In this way  can be determined unambiguously in the whole plane if the assumptions made above are fulfilled and an initial value is given.
k,
d=I
冉冑 冊
1/2
共17兲
.
I
冉 冊 冉 冊 
−
y x
共14兲
The angle  is the value of interest, which shall be determined by means of the intensity distribution of the light field. Starting from Eq. (14), it is derived in detail in Appendix A that the partial derivatives of  in the x and y directions are given by
z
ⵜ2冑I
Consequently, for this case the direction of the twodimensional phase gradient can be determined in the 共x , y兲 plane only by considering the intensity distribution of the light if an initial value of  is known. The initial value that is required to determine the field of  cannot be chosen freely. Furthermore, only certain values guarantee that numerical integration of the partial derivatives of  results in a continuous field. A necessary condition for obtaining a continuous field is that the curl of the vector field, defined by the derivatives in the x and y directions, is equal to zero:
Eq. (11) can be expressed as
I

x y
共18兲
= 0.
Starting from this equation, it is possible to derive an algebraic expression that can be used to evaluate  directly for each point in the 共x , y兲 plane. For this to happen the derivatives that occur in Eq. (18) have to be applied to the right sides of Eqs. (15) and (16). When this is done, new terms with first-order derivatives of  will arise. These can be eliminated by resubstituting the expressions in Eqs. (15) and (16) once more. This long, but straightforward, calculation leads, after some transformations, to c0 = c1 cos  + c2 sin  + c3 cos  sin  + c4 cos2  + c5 sin2  , 共19兲 where the coefficients are given by c0 = − ⵜ2Tln d + ⵜTln I ⵜTln d +
c1 =
c2 =
冉冊 冉冊 冉
c3 = 2
c4 = 2
a ln共Id兲
d
x
a ln共Id兲
d
y
1 2I I x y
−
ln I ln d y
y
−
−
共 da 兲 x
共 da 兲 y
−
I y2
d
2
,
,
x
1 2I
a
,
ln I ln d y
冉冊
,
−
ln I ln d x
y
冊
,
Ervin Kolenovic´
c5 = 2
Vol. 22, No. 5 / May 2005 / J. Opt. Soc. Am. A
ln I ln d x
x
−
1 2I I x
2
.
共20兲
Multiplying both sides of Eq. (19) with any value or function may lead to a different, but equivalent, set of coefficients. With Eq. (19) it is finally possible not only to find the desired initial value for the direction of the phase gradient but also, by solving this equation, to determine  for each point in the 共x , y兲 plane. It can be shown that in general, analytical expressions for cos  and sin  can be derived from Eq. (19). To do so, either all cos  or all sin  expressions have to be eliminated, using the relation cos  = ± 共1 − sin2 兲1/2 or sin  = ± 共1 − cos2 兲1/2 . 共21兲 Then the resulting equation can be transformed into a fourth-order polynomial of the remaining trigonometric terms. Depending on which expressions were eliminated, the four roots of this quartic equation finally define the solutions of either cos  or sin . Quartic equations always have an analytical solution. However, the particular results found for the equations derived from Eq. (19) are far too long to be presented here, and their complexity makes them less useful in determining . To verify the correctness of Eq. (19), this equation has been solved numerically for the speckle field depicted in Fig. 1. For the numerical solution the values for the fields a and d were determined directly from the simulated intensity and phase maps, according to Eqs. (13), instead of evaluating them by the expressions given in Eqs. (17). This was necessary because it turned out that otherwise a and d could not be determined accurately enough in some regions of the speckle field, which means that the speckle field does not propagate strictly in only one direction. By using accurate values for a and d, one could examine the validity of Eq. (19) without inducing errors resulting from an invalid use of Eqs. (17).
903
The result from the numerical solution is shown in the right image of Fig. 3, and the left image depicts for comparison the field of  evaluated from the phase map of the speckle field. The data in both images show very good coincidence. Although, as mentioned above, Eq. (19) has in general four solutions for each point, surprisingly the original field could be recovered unambiguously. This was achieved by utilizing the whole map to determine  instead of considering only the solution for isolated points. As a first step in recovering the field, the real-valued solutions of  have been determined numerically for each point. Once this procedure was performed, it turned out that numerous points of the field have only one realvalued solution. By substituting the unique results for  into Eqs. (15) and (16), one could obtain a linear approximation for the surrounding points. Afterward, these approximated values were used for comparison with the precise, but ambiguous, results for  obtained by solving Eq. (19). If the values chosen were those closest to the approximated ones, the correct branch of the solutions could be found point by point. With this approach, errors that will occur by a simple integration of the partial derivatives of  can in general, be avoided. For the results in Fig. 3, a simple algorithm, which utilizes only Eq. (15) to approximate the surrounding values, has been used. This is the reason for the horizontal lines with mismatching data in the right image. With more sophisticated algorithms these artifacts can be eliminated. If the light propagates strictly into the z direction, then the crucial fields a and d can be calculated from the intensity distribution by means of Eqs. (17). In that case the direction of the two-dimensional phase gradient can be completely determined from noninterferometric intensity measurements. Considering this case, it can be shown that the transport-of-intensity equation8 appears as a subset of the basic expressions that were used in this section to determine . The transport-of-intensity equation can be directly obtained from Eq. (11) by substitution of Eq. (6), which is the condition that the light propagates
Fig. 3. Direction  of the phase gradient of the speckle field in Fig. 1. The left image is evaluated directly from the phase image, and the right one shows the directional field of  obtained by numerical solution of Eq. (19) with initially known fields a and d. The values of  are displayed from zero (black values) to 2 (brightest values).
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Ervin Kolenovic´
J. Opt. Soc. Am. A / Vol. 22, No. 5 / May 2005
only into the z direction.
3. CONCLUSIONS In this paper the correlation between intensity and phase has been investigated analytically and numerically. Based on the Helmholtz equation, it has been shown analytically that the absolute value, as well as the direction of the phase gradient, can be expressed by means of the three-dimensional intensity distribution of the light. The formulas that were derived have been verified by means of numerical investigations on simulated speckle fields. As presented in this work, for light that propagates strictly into one direction, the absolute value, as well as the direction of the phase gradient, can be evaluated from noninterferometric intensity measurements without further approximations. This can be used in future work to recover the phase of the light from noninterferometric intensity measurements. It has also been pointed out that the transport-of-phase and transport-of-intensity equations appear as subsets of the basic expressions that were used to determine the phase gradient. The first can be derived from an equation that was utilized to determine the absolute value of the phase gradient, while the latter appears as a part of the equation that was used to determine the direction.
APPENDIX A: PARTIAL DERIVATIVES OF THE DIRECTION OF THE PHASE GRADIENT In this section the explicit deduction of the partial derivatives of  in the x and y directions is presented. According to Eqs. (12) the direction of the phase gradient is in the 共x , y兲 coordinate system defined by the unity vector u =
冉 冊 cos  sin 
共A1兲
.
This vector always points to the direction of maximum phase change, and correspondingly the orthogonal unity vector u =
冉 冊 − sin 
共A2兲
cos 
points to the direction of constant phase. As depicted in Fig. 4, with these two vectors a 共 , 兲 coordinate system can be defined for each point of the 共x , y兲 plane. Depending on the direction of the phase gradient, this adapted coordinate system is rotated by the angle  with respect to the 共x , y兲 system. In the following the 共 , 兲 system will be used to simplify the subsequent calculations. Considering Eq. (14), by applying the partial derivatives on the expressions in parentheses, one can expand this equation to
ln d x
cos  +
ln d y
sin  +
冉
 y
cos  −
 x
冊
sin  =
a d
Fig. 4.
Vector relations between isophase curves.

a =
d
−
ln d
.
共A4兲
With this result assuming that a and d are known, the partial derivative of  in the direction of constant phase can be calculated for each point from the intensity distribution. As pointed out in Subsection 2.B, the fields a and d can also be evaluated from the intensity distribution if the light propagates strictly into the z direction. To determine the two-dimensional gradient of  completely, it is necessary to find the derivative of  in the direction of the phase gradient. For this step it is useful to consider the geometrical relations shown in Fig. 4. The two curves in this sketch depict the paths of equal phase for the values and + ⌬, respectively. For the following calculations the positions p1 to p4 are chosen close enough to each other to allow a linearization of the isophase curves and also to allow linear approximation of the phase changes between these points. After the geometrical relations between all points are defined, the desired derivative of  in the direction of the phase gradient will be determined precisely by letting the distances between these locations become infinitesimally small. The position p1 is an arbitrary location on the first isophase curve, and it marks the origin of the 共 , 兲 system that will be used to simplify subsequent considerations. Moving along the curve from p1 to a second location p2, the direction of the phase gradient changes by the angle ⌬. For a short distance ⌬, it can be described by
⌬ =
冉 冊 
⌬ ,
共A5兲
1
. 共A3兲
The terms on the left side can be identified as projections onto the and axes, so that this equation can be written in adapted coordinates as
where the partial derivative of  in the direction is determined by Eq. (A4). In the perpendicular direction, starting from p1 and moving in the direction pointed to by the phase gradient 共ⵜT兲1, the isophase + ⌬ is intersected at p4 at a distance
Ervin Kolenovic´
Vol. 22, No. 5 / May 2005 / J. Opt. Soc. Am. A
⌬1 =
⌬ 兩ⵜT兩1
共A6兲
.
Analogously, starting from p2 and moving in the direction pointed to by the phase gradient 共ⵜT兲2, the isophase + ⌬ will be intersected at location p3 at a distance ⌬2 =
⌬ 兩ⵜT兩2
For decreasing distance between p3 and p4, the directional vector v defined by these points becomes a tangent to the isophase in p4, which means that v is perpendicular to the phase gradient 共ⵜT兲4. The angle ⌬ between 共ⵜT兲4 and 共ⵜT兲1 is in linear approximation given by ⌬ =
冉 冊
兩ⵜT兩
⌬1 .
v ⬅ p3 − p4 =
冉冊 v v
共A9兲
;
v
⌬2 cos共⌬兲 ⌬ + ⌬2 sin共⌬兲
冊
,
冉 冊 ⌬1 0
共A11兲
.
With these position vectors, and by means of Eqs. (A5)–(A8), Eq. (A10) becomes

⌬ 兩ⵜ 兩 1 T 1
冦
兩ⵜT兩1−1
−
兩ⵜT兩2−1
= arctan ⌬
冋冉 冊 册 冋冉 冊 册
cos
⌬ + ⌬兩ⵜT兩2−1 sin

⌬
1

⌬
1
冧
兩ⵜT兩2 = 兩ⵜT兩1 +
⌬ .

⌬

⌬ + ⌬ sin 2
⌬
冥
.
冉 冊 
⌬ = 1,
冉 冊 
lim sin ⌬→0

⌬ = lim
⌬→0
⌬ .
共A15兲
If additionally the value ⌬ approaches zero, then the left side of Eq. (A14), and consequently also the right side, becomes very small. In that case the arcus tangent on the right side can be replaced by its argument. Then Eq. (A14) can be written as
ln兩ⵜT兩 
兩ⵜT兩−1 = lim ⌬→0
兩ⵜT兩 +
兩ⵜT兩
⌬ + ⌬

. 共A16兲
Finally, if the values of ⌬ and ⌬ are allowed to become zero, this equation shows the result for the derivative of  in the direction of the phase gradient:

=
ln兩ⵜT兩
共A17兲
.
With Eqs. (A4) and (A17), the gradient of  is given in 共 , 兲 coordinates. In 共x , y兲 coordinates these two equations can be written as
 x
cos  +
 y
sin  = −
ln兩ⵜT兩 x
sin  +
ln兩ⵜT兩 y
cos  , 共A18兲
.
共A12兲
The absolute value of the phase gradient in p2 can be expressed by
兩ⵜT兩1
冉 冊 冉 冊
⌬ − cos
共A14兲
⌬→0
p4 =
兩⌬T兩
lim cos
共A10兲
.
The position vectors p3 and p4 can be written in 共 , 兲 coordinates as
兩ⵜT兩⌬ +
共A8兲
冉 冊 − v
⌬ = arctan
冉 冊
If the value ⌬ approaches zero, then the trigonometric terms in the argument of the arcus tangent are given by
then, because of the orthogonality of 共ⵜT兲4 and v, the angle ⌬ can be determined from
冉
ln兩ⵜT兩
= arctan ⌬
1
The partial derivative of  in the direction is the value to be found. Let the components of the directional vector v in the 共 , 兲 system be denoted by
p3 =
冤
1+
⌬→0

 ⌬
共A7兲
.
905
共A13兲
By substitution of Eq. (A13) into Eq. (A12), the explicit dependence on the location can be omitted because all values are considered in the same position p1. With this substitution Eq. (A12) can be transformed to
−
 x
sin  +
 y
cos  =
a d
−
ln d x
cos  +
ln d y
sin  . 共A19兲
For the partial derivatives of  in the x and y directions, this defines a linear equation system. With use of the definition for d given in Eqs. (13), solving this system leads to the final result:
 x
=
ln d y
−
ln I y
cos2  +
ln I x
sin  cos  −
a d
sin  , 共A20兲
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Ervin Kolenovic´
J. Opt. Soc. Am. A / Vol. 22, No. 5 / May 2005
 y
=−
ln d x
+
ln I x
sin2  −
ln I y
sin  cos  +
a d
cos  .
8.
共A21兲
9.
On the basis of this result, it was discussed in Subsection 2.B how to determine .
10.
ACKNOWLEDGMENTS
11.
The author expresses his gratitude to W. Jüptner and W. Osten for inspiring discussions and for their continuous support on this paper. This work was funded by the German research community Deutsche Forschungsgemeinschaft–DFG under grant OS 111/14-2. Address correspondence
[email protected].
to
Ervin
Kolenovic´
at
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