4 J. Eur. Opt. Society-Rapid Publ. 21, 1( 2025)
¼ ej2k 0f 1
E~ r ðx f; y f
F E~ i ðx i; y jk 0 f i Þ
1
Þ ¼ E r ðx f; y f
u ¼ xf = ð k0 f 1 Þ v ¼ y f = ð k0 f 1 Þ
Þe j / rðx f; y f Þ
!
1 ðk 0 f 2 Þ px ð 2 f; y f Þ:
ð2bÞ
The rightmost factor in parentheses characterizes the pinhole aperture in the reference focal plane by way of its transmission coefficient p( x f, y f) and the multiplicative constant 1 /( k 0 f 2) 2, where the latter constant is included to give a normalized point-spread function.
The electric field of the output beam ~ E s ðx f; y f Þ is defined in the output pupil plane, coplanar with the camera’ s image sensor, as denoted by a black dotted line in Figure 1b. Itis formed as the superposition of the signal and reference beams’ electric fields with an amplitude profile E o( x o, y o) and phase profile / o( x o, y o). The electric field of this output beam can then be defined by
See Equation( 3) at the bottom of the page
where x o and y o are coordinates along the horizontal and vertical dimensions, respectively, denotes the convolution operation, P( x o /( k 0 f 2), y o /( k 0 f 2))/( k 0 f 2) 2 is the normalized point-spread function of the pinhole aperture, and K is the fringe spacing arising along the horizontal dimension( quantifying the degree of phase tilt applied across the signal beam).
Overall, the key parameters for the design of the SRI wavefront sensor arise within the first and second terms in the final expression of equation( 3), and manifest through the signal and reference beams, respectively. Namely, the
~ E o ðx o; y o Þ¼ ej2k 0 f 2 jk 0 f 2
2
6 4
¼ ej2k 0 f 2 jk 0 f 2
F ej2k 0 f 1
þF ej2k 0f 1
¼�
2
6F E~ s ðx f; y f Þ 4 u ¼ xo = ðk 0 f 2 Þ
8 >< n
jk 0 f 1
F E~ i ðx i; y i Þe j 2p >:
8 ><
F E~ i ðx i; y jk 0 f i Þ
>: 1
ð f1þf2 ej2k0 Þ
6 k 2 0 f ðk 0 f 1 Þ 2 E~ i ð�k 0 f 1
1f 2 ðf 1 = f 2 ÞK x i
v ¼ y o = ðk 0 f 2 Þ
o u ¼ xf = ðk 0 f 1 Þ
v ¼ y f = ðk 0 f 1 Þ
þ F E~ r ðx f; y f Þ
9 >=
>; tilt applied to the signal beam imparts the fringe spacing K on the output image, which then defines the resolution of spatial features( and the order of modes seen) in the image. At the same time, the aperturing applied to the reference beam flattens its wavefronts in the output pupil plane, which lessens distortion in the image.
2.2 Image processing
The optical design presented in the prior section establishes an intensity distribution on the camera’ s image sensor according to ~ E o ðx o; y o Þ ~ E o ðx o; y o Þ, where ~ E o ðx o; y o Þ is the electric field of the output beam across the horizontal x o and vertical y o dimensions, and * denotes the complex conjugate. We then process this image via Fourier fringe analysis [ 21 ] with four steps. In the first step, we apply a two-dimensional fast Fourier transform, F fft { }, to the imaged intensity distribution to give F fft f ~ E o ðx o; y o Þ ~ E o ðx o; y o Þ }.. This generates an image in reciprocal space with a large central peak at the origin, resulting from low-spatial- frequency( averaged) characteristics across the imaged intensity distribution, as well as negative and positive( side) peaks, displaced horizontally off the origin by 1 / K. The latter two peaks are due to the horizontal tilt applied to the signal beam and its resulting fringe( sinusoidal) pattern on the imaged intensity distribution. In the second step, we apply a circular reciprocal-space filter U RS to have it pass only the positive( side) peak. This yields the reciprocal-space distribution F fft f ~ E o ðx o; y o Þ ~ E o ðx o; y o Þ gU RS, where the filter U RS has a diameter equal to the displacement between the central and side peaks, 1 / K, with unity in its interior and zero elsewhere. Such filtering passes the full wavefront characteristics across the input beam while rejecting
u ¼ x o = ðk 0 f 2 Þ v ¼ y o = ðk 0 f 2 Þ
7 u ¼ xo = ðk 0 f 2 Þ 5 v ¼ y o = ðk 0 f 2 Þ
3 9
1
>= u ¼ xf = ðk 0 f 1 Þ ðk 0 f 2 Þ pðx 2 f; y f Þ
>; 7 v ¼ y f = ðk 0 f 1 Þ
u ¼ x o = ðk 0 f 2 Þ 5 v ¼ y o = ðk 0 f 2 Þ
2
Þ x o; ð�k 0 f 1 Þ y 2p j ð o e ðf 1 = f 2 ÞK �k 0f 1 Þ x o
4 k 0 f 2 þðk 0 f 1 Þ 2 E~ i ð�k 0 f 1 Þ x o
; ð�k 0 f 1 Þ y
o k 0 f 2 k 0 f 2 k 0 f 2 k 0 f 2
3
1 ðk 0 f 2 Þ P x o
; y #
o 2 k 0 f 2 k 0 f 2
¼� ej2k 0ðf 1 þf 2
Þ
E~ i � x o
; � y
o f 2 = f 1 f 2 = f 1 f 2 = f 1 e �j2p K xo
þ E~ i � x o
; � y
o f 2 = f 1 f 2 = f 1
1 ðk 0 f 2 Þ P x o
; y #
o ð3Þ 2 k 0 f 2 k 0 f 2