Method and apparatus to determine spatial distribution of magnitude and phase of electro-magnetic fields especially optical fields

• US 3,950,103 A
• Filed: 06/21/1974
• Issued: 04/13/1976
• Est. Priority Date: 10/27/1972
• Status: Expired due to Term

• Current Assignee: University of Alberta
• Original Assignee: Canadian Patents and Development Limited
• Inventors: Schmidt-Weinmar, Heinz Gunter
• Primary Examiner(s): Smith, Alfred E.
• Assistant Examiner(s): Clark, Conrad J.
• Application Number:
  US05/481,881
• Time in Patent Office: 662 Days
• Field of Search: 356/106, 356/107, 356/108, 356/109, 356/113
• US Class Current: 356/450
• CPC Class Codes: G01J 9/02 by interferometric methods using interferometers for measuring optically the linear dimensions of objects G01B9/02

Abstract

A method and apparatus for numerical determination of the distribution of magnitude and phase of a time harmonic electromagnetic, particularly an optical, wave field by using the wavefront of a reference wave as the optical reference surface and by introducing additional information to the recording process such that the field becomes uniquely determined and this is realized by the recording of three interference fringe patterns which belong to variations of the reference field distribution. A reference field is chosen whose spatial distribution is similar to the unknown field whereby fringes are obtained the position of which varies slowly with the recording coordinates. The recorded data are numerically evaluated and this may be done conveniently in a computer. After these measurements are made the reference beam is shifted by an optical element e.g. a prism, gaseous wedge, electro-optical diffraction device placed in the reference optical path and a second set of measurements is made. In this way a reference field whose spatial distribution is not known precisely is introduced but the system is such that the reference spatial distribution information is used and then eliminated from the information finally obtained on the unknown wavefield.

Claims

1. A method of numerically determining the distribution of magnitude and phase of a time harmonic electromagnetic, particularly an optical, wave field comprising:

a. engendering a reference wave field such that its spatial distribution is similar to the field to be measured whereby fringes are obtained the position of which varies slowly with the recording coordinates;

b. using the reference wave as reference surface;

c. recording under far field conditions point by point by means of photodetector three interference fringe patterns which belong to the unknown field and variations of the said reference field distribution, the relationship of said variations as to magnitude and phase being such as to uniquely determine, for any recording point, the amplitude and phase distribution of the unknown electromagnetic wave field;

d. recording data in numeral form obtained from the photodetector;

e. placing an optical element in the said reference wave optical path such that the reference wave front is shifted in relation to the unknown wave front by a small known amount;

f. repeating the measurements as in (c) and (d); and

g. processing said data to obtain numeral values of magnitude and phase of the electromagnetic wave field being measured, such as to provide information in terms of a smooth spherical wavefront.

2. A method as in claim 1 wherein the optical elements is a prism.

3. A method as in claim 1 wherein the optical element is a prism having silvered surfaces providing two shifts of the reference wavefront beam.

4. A method as in claim 1 wherein the optical element is a gaseous wedge in the form of a wedge-shaped chamber inside a glass body which is filled alternately with two gases of different refractive indices to provide the necessary shifting.

5. A method as in claim 1 wherein the optical element is an electro-optical beam deflecting element formed of a rectangular parallelopiped block of optically homogeneous and isotropic material having two acoustical transducers mounted thereon such as to generate two standing acoustical waves in mutually perpendicular directions that act as a diffraction grating to cause deflection of the reference beam by diffraction.

6. A method of numerically determining the distribution of magnitude and phase of an unknown time harmonic electro-magnetic, particularly an optical wave field comprising:

a. engendering a reference electromagnetic wave field such that its spatial distribution is similar to the wave field to be measured, whereby on interference between the unknown and reference wave fields, fringes are obtained the position of which varies slowly with the recording coordinates;

b. using the wavefront of the reference wave as a reference of the field to be measured in particular the reference field being such that its spatial distribution is similar to the field to be measured, whereby the ratio of the complex-valued amplitude of object field and reference field depends weakly on the spatial coordinates and consequently the recording coordinates may be known only within limits large compared to the wavelength of the light;

c. recording point by point by means of a photodetector three or more interference fringe patterns which belong to the unknown field and variations of said reference field distributions, the relationship of said variations being such that for any recording point the complex-valued amplitude R_l of the lth variation, l = 1,2,3, . . . satisfies the relation

space="preserve" listing-type="equation">R.sub.l = (δ

.sub.l R)R

where R is some arbitrary complex number and where real and imaginary parts of the complex numbers δ

_l R = δ

_l R_r jδ

_l R_i are known and satisfy the condition

space="preserve" listing-type="equation">(δ

.sub.1 R.sub.r -δ

.sub.2 R.sub.r) (δ

.sub.2 R.sub.i -δ

.sub.3 R.sub.i) - (δ

.sub.2 R.sub.r -δ

.sub.3 R.sub.r) (δ

.sub.1 R.sub.i -δ

.sub.2 R.sub.i) ≠

0

d. solving by means of a computer the simultaneous system of equations ##EQU38## for α

_r and α

_i at every recording point with given data of J₁, J₂, J₃, . . . of the irradiance and with known values of |R|², δ

_l R_r, and δ

_l R_i, l = 1,2,3, . . . , and where Ε

_r, α

_i are real and imaginary part, respectively, of the ratio of the complex-valued amplitude of the field to be measured and the reference field at the recording point;

e. placing an optical element in the said reference wave optical path such that the reference wave front is shifted in relation to the unknown wave front by a small known amount;

f. repeating the steps of (c) and (d) such that the method is carried out twice with one object field of arbitrary spatial distribution but two variations of the reference field such that the complex-valued amplitude of the 1st reference field at the (k

     1)th recording point (R₁ (k

     1)) and the complex-valued amplitude of the second reference field at the kth recording point (R₂ (k)) satisfy the relation

space="preserve" listing-type="equation">R.sub.1 (k+1) = f(k) R.sub.2 (k),

where f(k) is a known complex number and whereby the complex-valued amplitude of the 1st reference field R₁ at the kth recording point can be derived (except some irrelevant constant (R₁¹) from the expression

space="preserve" listing-type="equation">R.sub.1 (k) = μ

(1) μ

(2) . . . μ

(k-1) f(1)f(2) . . . f(k-1) R.sub.1.sup.1

where the complex number μ

(k) is the ratio of the measurable quantities α

₁ (k) and α

₂ (k) obtained at the kth recording point with the said two variations of the reference field, ##EQU39## and g. processing said data to obtain numeral values of magnitude and phase of the electromagnetic wave field being measured, said data being sufficient to eliminate the reference wave information from the obtained information which can be related to a mathematically exact wavefront.

7. A method as in claim 6 wherein the optical element is a prism.

8. A method as in claim 6 wherein the optical element is a gaseous wedge in the form of a wedge-shaped chamber inside a glass body which is filled alternately with two gases of different refraction indices to provide the necessary shifting.

9. A method as in claim 6 wherein the optical element is an electro-optical beam deflecting element formed of a rectangular parallelopiped block of optically homogeneous and isotropic material having two acoustical transducers mounted thereon such as to generate two standing acoustical waves in mutually perpendicular directions that act as a diffraction grating to cause deflection of the reference beam by diffraction.