J. Eur. Opt. Society-Rapid Publ. 21, 1( 2025) 3
Fig
. 2. Measured imaged intensity distributions of the output beam( overlapped reference and signal beams) on the camera’ s image sensor as a function of the transverse dimensions x o and y o. The signal beam has varied degrees of horizontal tilt across it, yielding fringe spacings of K =( a) 387 lm,( b) 177 lm,( c) 117 lm, and( d) 87 lm.
and reference beams are overlapped / imaged on the camera, we then see the tilted signal wavefronts and flattened reference wavefronts form fringes with a fringe spacing K. Figure 2 shows such imaged fringe patterns for applied tilts yielding spatial pitches of K = 387lm inFigure 2a, 177lm in Figure 2b, 117 lm in Figure 2c, and 87 lm in Figure 2d. The significance of the aperture diameters and spatial pitch, together, can be understood by defining and characterizing the input, signal, reference, and output beams.
The electric field of the input beam ~ E i ðx i; y i Þ is defined in the input pupil plane, which is denoted as a violet dotted line at the input of the SRI in Figure 1b. It consists of an input beam amplitude profile with a maximum E 0 and radius x, spanning out to e �1 of the maximum, and an input beam phase profile / i( x i, y i). The electric field of the input beam can then be expressed as
E~ i ðx i; y i Þ ¼ E 0 e �ðx2 i þy2 i Þ = x2 e j / iðx i; y i Þ; ð1Þ
where x i and y i are coordinates along the horizontal and vertical dimensions, respectively.
The electric field of the signal beam ~ E s ðx f; y f Þ is defined in the focal plane of the signal arm, which is denoted as a blue dotted line within this arm in Figure 1b. It consists of a focused signal beam amplitude profile E s( x f, y f) and focused signal beam phase profile / s( x f, y f), such that the electric field of the signal beam is
E~ s ðx f; y f
Þ ¼ E s ðx f; y f 8 ><
F jk 0 f 1 >:
¼ ej2k 0f 1
Þe j / sðx f; y f Þ
E~ i ðx i; y i Þe j
9
2p ðf 1 = f 2 ÞK x i
>= >;
u ¼ x f: ð2aÞ k 0 f 1 v ¼ y f k 0 f 1
Here, x f and y f are coordinates along the horizontal and vertical dimensions, respectively, f 1 and f 2 are the focal lengths of the primary lens and secondary lens, respectively, k 0 = 2p / k 0 is the magnitude of the wavevector at a freespace wavelength k 0, andFfg is the Fourier transform operator with generalized transform variables u and v. The complex exponential inside the Fourier transform’ s argument is due to the aforementioned angling of the input beamsplitter, which establishes a horizontal phase shift across the transverse profile of the signal beam. Thus, we can apply this tilt at differing degrees to alter the linear phase shift across the signal beam and thereby vary the fringe spacing K in the output beam.
The electric field of the reference beam ~ E i ðx i; y i Þ is defined in the focal plane of the reference arm, coplanar with the pinhole aperture, as denoted by a red dotted line in Figure 1b. It consists of a focused reference beam amplitude profile E r( x f, y f) and focused reference beam phase profile / r( x f, y f), which give