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Vectorial shearing cascaded Sagnac interferometric wavefront sensor

Pramod Panchal, Dinesh N Naik, and C. S. Narayanamurthy

Open Access Open Access

Abstract

We propose a highly stable and single-shot wavefront sensor using vectorial shearing interferometry. We design it using a pair of cascaded Sagnac interferometers that allow spatial frequency multiplexing in the interferogram in order to detect the vector gradients simultaneously and independently. The scheme avoids wavelength dispersion in sheared copies as compared to the vectorial shearing interferometer schemes based on diffractive optical elements. Capability to tune the amount of shear merely through the axial displacement of the imaging sensor, and with the added advantage of adjustable resolution controlled by the spatial frequency introduced, the near common-path geometry opens up its applications in optical testing even in a vibration sensitive environment.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

A wavefront sensor, with its capability to measure the aberrations in an optical wave front, found its immense use in optical shop testing, ophthalmology and in adaptive optics system. Wavefront being the locus of field with same phase and the direct phase at optical frequency being inaccessible due to limited response time of available detectors, it has to be encoded into intensity for any meaningful measurement. As a result, the intensity change brought forward by refraction, diffraction or interference, the fundamental aspects of light propagation paved the way for sensing the phase/phase gradients in the wavefront. The wavefront sensors that work on principle of intensity transport such as micro-lens (refraction) [1] and grating (diffraction) [2] seemed to be easily designed and handled as compared to the interferometric wavefront sensors where the phase gradients are detected from the resulting intensity redistribution.

Out of such available wavefront sensors (WFSs), in situation having strong turbulence and the associated high levels of scintillation, interferometric sensors such as the self-referencing interferometer (SRI) are considered to be good choices [3]. Shearing interferometers, with an inherent advantage that it requires no separate reference optical field, are a good choice for wave front sensing. However, the quantity that can be measured using such interferometers is not the phase of the optical field, but the phase gradient. Though the scalar interferometers like lateral, radial and rotational shear interferometers are proposed to measure the phase gradients, using either of them alone can give gradients along only single coordinate direction (x/y/radial/azimuthal). To achieve complete 2-D wavefront reconstruction from the gradient data, especially for wavefronts lacking symmetry in Cartesian/polar coordinates, gradients along both orthogonal coordinates (x and y or radial and azimuthal) have to be measured [4–6].

The scalar (1-D) interferometer is extensively used for optical testing of symmetric optical elements. Such interferometers cannot be used for adaptive optics corrections for telescopic imaging under strong atmospheric turbulence, or for microscopic imaging under scattering media. Optical testing of aspheric and asymmetric optical components also suggests to venture beyond the scalar interferometry. Vectorial Shearing Interferometer is an extension of the interferometer based on the 1-D lateral shearing; as it is able to displace the wavefront with a suitable amount in two orthogonal directions and measure the corresponding interference independently. The three wave lateral shearing interferometer [7–9] and its derivatives [10,11] that uses a predesigned diffractive optical element (DoE) [12] for generating 3 copies of the incoming wavefront have shown to encode the vector gradients in the interference pattern and reconstruct the wavefront through fringe analysis.

Though the DoE based schemes are achromatic, when quasi-monochromatic light is used, one could design and implement the scheme by making 3 copies of the wavefront using merely reflections instead of diffraction. For a robust sensing scheme, all the three wavefronts should ideally pass through the same path, encountering every element of the system, so that the system induced error does not reflect in the relative phase measurement. Moreover, the common-path geometry will ensure that no dynamically varying phase difference gets introduced between the interfering copies of wavefront. Keeping these points in mind, we design it using a pair of cascaded Sagnac interferometer that allows spatial frequency multiplexing in the interferogram in order to detect the vector gradients simultaneously and independently. By avoiding the use of any DoE or active Spatial Light Modulators, the proposed scheme can be a cost-effective approach for sensing dynamic wavefronts. By simultaneous extraction of vector gradients using Fourier transform method of fringe analysis followed by a two-dimensional numerical integration method, the complete wavefront W(x, y) gets sensed [13,14].

2. Principle

The conceptual diagram for common-path vectorial shearing wavefront sensor is shown in Fig. 1. The system is a combination of a pair of cascaded Sagnac interferometers and a 4-F telescopic system with two lenses of focal length f1 and f2 respectively. The core part of this system is the common path vectorial shearing interferometer. It is positioned between the lens L1 and its back focal plane. O is the object plane which is at Z1 distance from plane-A i.e. focal plane of lens L1. The combination of lens L1 and vectorial shearing interferometer will generate four copies of the Fourier transform of input wavefront at plane-F which can be equally separated along x and y direction by magnitude given by ${\hat{x}_s}$ and ${\hat{y}_s}$ respectively. The detailed alignment to achieve it will be explained in experiment section. Plane-B is the back focal plane of the lens L2 where all four wavefront copies will completely overlap. Plane-I is the shearing plane at which wavefront can be sheared long two perpendicular directions simultaneously. Plane-I is at Z2 distance from back focal plane of lens L2.

 figure: Fig. 1.

Fig. 1. Conceptual diagram for common-path vectorial shearing wavefront sensor

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Let ${u_O}({x^{\prime},y^{\prime}} )$ be the field at object plane-O, then field at plane-A will be given as [15]:

$${u_A}(x^{\prime},y^{\prime}) = F_{({{f_1}} )}^{ - 1}\left\{ {{F_{({{f_1}} )}}[{\mathop u\nolimits_o (x^{\prime},y^{\prime})} ]\times exp\left( { - i\frac{{2\pi {z_1}}}{\lambda }\sqrt {1 - \frac{{{{\hat{x}}^2}}}{{f_1^2}} - \frac{{{{\hat{y}}^2}}}{{f_1^2}}} } \right)} \right\}$$
Where ${F_{({{f_1}} )}}$ and $F_{({{f_1}} )}^{ - 1}$ represents the Fourier and inverse Fourier transform due to lens L1. Field at plane F will be Fourier transform of field at plane-A and can be written as
$${U_F}\left( {\frac{{\hat{x}}}{{\lambda {f_1}}},\frac{{\hat{y}}}{{\lambda {f_1}}}} \right) = {F_{({{f_1}} )}}[{{u_O}(x^{\prime},y^{\prime})} ]\times exp\left( { - i\frac{{2\pi {z_1}}}{\lambda }\sqrt {1 - \frac{{{{\hat{x}}^2}}}{{f_1^2}} - \frac{{{{\hat{y}}^2}}}{{f_1^2}}} } \right)$$
The four copies of the Fourier transform of the wavefront will be generated in plane-F using cascaded Sagnac interferometer and lens L2. One of the shifted Fourier transform at plane F can be written as
$$\begin{aligned} {U_F}\left( {\frac{{\hat{x} - {{\hat{x}}_{s,n}}}}{{\lambda {f_1}}},\frac{{\hat{y} - {{\hat{y}}_{s,n}}}}{{\lambda {f_1}}}} \right) &= {U_O}\left( {\frac{{\hat{x} - {{\hat{x}}_{s,n}}}}{{\lambda {f_1}}},\frac{{\hat{y} - {{\hat{y}}_{s,n}}}}{{\lambda {f_1}}}} \right) \times \\ &exp\left( { - i\frac{{2\pi {z_1}}}{\lambda }\sqrt {1 - \frac{{{{({\hat{x} - {{\hat{x}}_{s,n}}} )}^2}}}{{f_1^2}} - \frac{{{{({\hat{y} - {{\hat{y}}_{s,n}}} )}^2}}}{{f_1^2}}} } \right) \end{aligned}$$
where, ${\hat{x}_{s,n}}$ and ${\hat{y}_{s,n}}$ is the separation along $\hat{x}$ and $\hat{y}$ direction respectively. Where, n represents the one of the copy of input field generated by cascaded Sagnac interferometer. Since we will select only three copies, so n can have values of 1,2 and 3. Field at plane B will be another Fourier transform of the field at input focal plane of lens L2 (having focal length of f2)
$${u_{B,n}}({x,y} )= {F_{({{f_2}} )}}\left\{ {{U_F}\left( {\frac{{\hat{x} - {{\hat{x}}_{s,n}}}}{{\lambda {f_1}}},\frac{{\hat{y} - {{\hat{y}}_{s,n}}}}{{\lambda {f_1}}}} \right)} \right\}$$
Now, field at shearing plane I, can written as
$${u_{I,n}}(x,y) = F_{({{f_2}} )}^{ - 1}\left\{ {{F_{({{f_2}} )}}[{{u_{B,n}}({x,y} )} ]\times \exp \left( {i\frac{{2\pi {z_2}}}{\lambda }\sqrt {1 - \frac{{{{\hat{x}}^2}}}{{f_2^2}} - \frac{{{{\hat{y}}^2}}}{{f_2^2}}} } \right)} \right\}$$
Substituting the value of ${u_{B,n}}({x,y} )$ from Eq. (4) in above Eq. (5)
$${u_{I,n}}({x,y} )= F_{({{f_2}} )}^{ - 1}\left[ {{F_{({{f_2}} )}}\left[ {{F_{({{f_2}} )}}\left\{ {{U_F}\left( {\frac{{\hat{x} - {{\hat{x}}_{s,n}}}}{{\lambda {f_1}}},\frac{{\hat{y} - {{\hat{y}}_{s,n}}}}{{\lambda {f_1}}}} \right)} \right\}} \right] \times \exp \left( {i\frac{{2\pi {z_2}}}{\lambda }\sqrt {1 - \frac{{{{\hat{x}}^2}}}{{f_2^2}} - \frac{{{{\hat{y}}^2}}}{{f_2^2}}} } \right)} \right]$$
Substituting the value of ${U_F}({{{({\hat{x} - {{\hat{x}}_{s,n}}} )} \mathord{\left/ {\vphantom {{({\hat{x} - {{\hat{x}}_{s,n}}} )} {\lambda {f_1},}}} \right.} {\lambda {f_1},}}{{({\hat{y} - {{\hat{y}}_{s,n}}} )} \mathord{\left/ {\vphantom {{({\hat{y} - {{\hat{y}}_{s,n}}} )} {\lambda {f_1}}}} \right.} {\lambda {f_1}}}} )$ from Eq. (3) in Eq. (6)
$${u_{I,n}}({x,y} )= F_{({{f_2}} )}^{ - 1}\left[ \begin{array}{l} {U_O}\left( {\frac{{ - ({\hat{x} - {{\hat{x}}_{s,n}}} )}}{{\lambda {f_1}}},\frac{{ - ({\hat{y} - {{\hat{y}}_{s,n}}} )}}{{\lambda {f_1}}}} \right) \times \exp \left( { - i\frac{{2\pi {z_1}}}{\lambda }\sqrt {1 - \frac{{{{({\hat{x} - {{\hat{x}}_{s,n}}} )}^2}}}{{f_1^2}} - \frac{{{{({\hat{y} - {{\hat{y}}_{s,n}}} )}^2}}}{{f_1^2}}} } \right)\\ \times \exp \left( {i\frac{{2\pi {z_2}}}{\lambda }\sqrt {1 - \frac{{{{\hat{x}}^2}}}{{f_2^2}} - \frac{{{{\hat{y}}^2}}}{{f_2^2}}} } \right) \end{array} \right]$$
Now, substituting ${z_1} = {{{z_2}} \mathord{\left/ {\vphantom {{{z_2}} {{m^2}}}} \right.} {{m^2}}}, \,m = {{{f_2}} \mathord{\left/ {\vphantom {{{f_2}} {{f_1}}}} \right.} {{f_1}}}, \,{z_1} = {z_2} \times ({{{f_1^2} \mathord{\left/ {\vphantom {{f_1^2} {f_2^2}}} \right.} {f_2^2}}} )$ where m is lateral magnification and m2 is longitudinal magnification. After doing binomial expansion and rearranging the Eq. (7), field of one of the wavefront at plane-I can be written as:
$$\begin{aligned} {u_{I,n}}({x,y} )&= exp\left( {i\frac{{2\pi {z_2}}}{\lambda }\left( {1 - \frac{{f_1^2}}{{{f_2}^2}} + \frac{{\hat{x}_{s,n}^2 + \hat{y}_{s,n}^2}}{{2{f_2}^2}}} \right)} \right)exp\left( { - i\frac{{2\pi }}{{\lambda {f_2}}}({x{{\hat{x}}_{s,n}} + y{{\hat{y}}_{s,n}}} )} \right)\\ &\times \,{u_O}\left( {\frac{{ - 1}}{m}\left( {x - \frac{{{{\hat{x}}_{s,n}}{z_2}}}{{{f_2}}}} \right),\frac{{ - 1}}{m}\left( {y - \frac{{{{\hat{y}}_{s,n}}{z_2}}}{{{f_2}}}} \right)} \right) \end{aligned}$$
The field at plane-I can be divided into three parts. The first shows a constant phase depending on the focal lengths of lenses L1 and L2 and image plane or shearing plane distance Z2 from plane-B (the back focal plane of lens L2). Second part shows the presence of a linear phase which is a function of focal length of lens L2, the separation introduced at the output of the cascaded Sagnac interferometer i.e. ${\hat{x}_{s,n}}$ and ${\hat{y}_{s,n}}$ respectively. Part three shows the complex field of wavefront at plane O which is inverted, scaled by factor of $m = {{{f_2}} \mathord{\left/ {\vphantom {{{f_2}} {{f_1}}}} \right.} {{f_1}}}$ and shifted by ${{{{\hat{x}}_{s,n}}{z_2}} \mathord{\left/ {\vphantom {{{{\hat{x}}_{s,n}}{z_2}} {{f_2}}}} \right.} {{f_2}}}$ along x-direction and by ${{{{\hat{y}}_{s,n}}{z_2}} \mathord{\left/ {\vphantom {{{{\hat{y}}_{s,n}}{z_2}} {{f_2}}}} \right.} {{f_2}}}$ along y-direction. Out of four copies generated, the three orthogonally sheared copies of the wavefront alone can be selected by spatially filtering them at plane-F. These three copies interfere at plane-I as shown in Eq. (9) and the resulting intensity is recorded by a CCD.
$${I_I}({x,y} )= {|{{u_{I,1}}({x,y} )+ {u_{I,2}}({x,y} )+ {u_{I,3}}({x,y} )} |^2}$$

3. Experimental procedure

The experimental set-up for the implementation of the vectorial shearing is shown in Fig. 1. It consists of telecentric system and cascaded Saganc interferometer. The two cascaded Sagnac interferometers are shown in Fig. 2. First Sagnac interferometer shown in Fig. 2(a) consist of mirrors M1, M2 and beam splitter BS1. It generates two copies (1-red and 2-green) of input wavefront (blue) separated symmetrically along y-axis such that such separation between two copies is $2{\hat{x}_s}$ with ${\hat{x}_{s,1}} = - {\hat{x}_s}$ and ${\hat{x}_{s,2}} = {\hat{x}_s}$. The output of Sagnac interferometer-1 is given to the Sagnac interferometer-2 shown in Fig. 2(b). Sagnac interferometer-2 consists of mirrors M3, M4 and beam splitter BS2. It also generates two copies for every single wavefront at its input, but separate them symmetrically along x-axis. Thus the output of Sagnac interferometer-2 contains four copies i.e. 1’, 2’, 3’ and 4’. The separation along y-direction is $2{\hat{y}_s}$. This separation of wavefront along x and y direction can be controlled by shifting one of the mirror in Sagnac interferometer-1 and equal tilting of both the mirrors in Sagnac interferometer-2 as represented in Fig. 2. By doing this we achieve co-propagating but spatially separated copies of wavefront.

 figure: Fig. 2.

Fig. 2. Generation of four copies of the wavefront using two Sagnac interferometer in kept in series.

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By introducing the lens L1 (focal length f1 = 500 mm) at the input of the cascaded Sagnac interferometers, four copies represented as 1’, 2’, 3’ and 4’ becomes Fourier transform ${U_F}\left( {\frac{{\hat{x} - {{\hat{x}}_{s,n}}}}{{\lambda {f_1}}},\frac{{\hat{y} - {{\hat{y}}_{s,n}}}}{{\lambda {f_1}}}} \right)$ related to the input wavefront ${u_O}({x^{\prime},y^{\prime}} )$ as represented in Eq. (3). The lens L2 (focal length f2 = 200 mm), introduced at the output of the cascaded Sagnac interferometer performs another Fourier transform in order to obtain ${u_{I,n}}({x,y} )$ at plane I. We record the interferogram of laterally sheared beams, described by Eq. (9), on CCD (PCO pixelfly, 1392 × 1040, with pixel size of 6.4um) kept at plane I, located at a distance z2 = 18 mm from plane-B, the back focal plane of lens L2. Being nearly common-path geometry allows us to cancel any system induced errors and surrounding vibrations. .

Experimental set-up to validate vectorial shearing cascaded Sagnac interferometric wavefront sensor is shown in Fig. 3. In this set-up, a He-Ne laser (633 nm) is used as a source whose output is spatially filtered and then collimated with lens L having focal length f of 100 mm. This collimated beam is fed as input to the vectorial shearing wavefront sensor followed by CCD. As the position of the collimating lens L changes (f + dr), the shape of the beam changes. Depending upon the direction in which the lens is moved the beam output may be converging or diverging. The collimating lens is moved in direction towards the VSWFS with step size of 0.5 mm. Corresponding to each position of lens the interferogram is recorded with 8-bit CCD as shown in Fig. 4(a).

 figure: Fig. 3.

Fig. 3. Wavefront Sensing using vectorial shearing interferometer.

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 figure: Fig. 4.

Fig. 4. (a) Vectorial sheared fringe pattern, (b) Fourier transform of fringe pattern.

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4. Results and analysis

The recorded interferogram is shown in Fig. 4(a) when the collimating lens is shifted by an amount of dr = 3 mm from the initial position. The selected region of interferogram for processing is marked with white color square (having each side of 2.6 mm). The spatial carrier frequency multiplexed interfered data corresponds to the superposition of three sheared wavefronts. The phase gradients can be obtained using Fourier transform method of fringe analysis [16,17]. Figure 4(b) shows the 2D Fourier spectrum of the recorded interferogram with spatial frequencies ${f_x}$ and ${f_y}$ represented as ${{2{{\hat{x}}_s}} \mathord{\left/ {\vphantom {{2{{\hat{x}}_s}} {\lambda {f_2}}}} \right.} {\lambda {f_2}}}$ and ${{2{{\hat{y}}_s}} \mathord{\left/ {\vphantom {{2{{\hat{y}}_s}} {\lambda {f_2}}}} \right.} {\lambda {f_2}}}$. There are seven spots in Fig. 4(b) in which the center one is the zeroth order term representing the dc-part in the recorded interferogram. The upper three are resulting from the mutual superposition of sheared wavefronts. They get separated due to the spatial carrier frequency introduced. Their conjugates appear at the lower side. Out of the upper three spots, one along x-direction and one along y-direction is selected (marked with yellow circle) and filtered out. The spatial filter having diameter of 400um is used. After doing inverse Fourier transform of the selected spot, phase of the complex field can be calculated. However, precise amount of the spatial carrier frequency introduced should be known. In addition, it should be noted here that during the alignment of the cascaded Sagnac interferometers and the accompanying lenses, there can be system induced aberrations which should be cancelled. Therefore, the phase of the complex field obtained for a particular case is always compared to that obtained for the initial condition of a well collimated beam; thereby automatically cancelling the spatial carrier frequency and the system induced aberrations.

Further, proper phase unwrapping algorithm is performed to get the phase gradient information without 2π jumps. Figure 5(a) and 5(b) shows the calculated phase gradient information along x and y directions respectively when the collimating lens is shifted by an amount of 3 mm from the initial position. We use two-dimensional trapezoidal numerical integration method to integrate these x and y gradient that will reconstruct the complete wavefront W(x,y). Figure 5(c) shows the reconstructed wavefront from obtained phase gradients. The same process is repeated for recorded interferograms at different positions of collimating lens L (dr = 0.5 mm, 1.0 mm, 1.5 mm, 2.0 mm, and 2.5 mm) from the initial position. Then lens L is moved back to its intial collimation position with same step size (dr = 2.0 mm, 1.5 mm, 1.0 mm, 0.5 mm and final collimation state).

 figure: Fig. 5.

Fig. 5. (a) Gradient along X-direction, (b) Gradient along Y-direction (c) Reconstructed Wavefront.

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Using the parameters used for the experiment, a numerical simulation was performed using MATLAB software for wavefront reconstruction by using phase gradients obtained by orthogonal shearing of wavefront with different position of collimating lens as done in experiment. Figure 6 (a)–(f) shows the experimental (Red and Green dashed line) and simulation (Blue line) results. Red dashed line represents the wavefront reconstructed when lens moved forward along z-direction with step size of 0.5 mm. Green dashed line represents the wavefront when lens moved in backward direction. The matching results confirm the accurate and stable wavefront reconstruction achieved using the scheme. From the results shown in Fig. 6, a deviation in wavefront shows a gradually increasing/decreasing tilt. This can be attributed to the in-plane mechanical movement of lens mount when the collimating lens is shifted in z-direction.

 figure: Fig. 6.

Fig. 6. (a), (b), (c), (d), (e) and (f) shows the reconstructed wavefront at different positions of collimating lens along z-direction with step size of 0.5 mm. Red and green colored dashed lines show experimental results and blue colored line shows simulation results. Figure (g), (h), (i), (j) and (l) shows the wavefront deviations when collimating lens moves in forward direction. Figure (m), (n), (o), (p) and (q) shows the wavefront deviations when collimating lens moves in backward direction.

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Experimental results of calculated wavefronts at different positions of collimating lens are compared with MATLAB simulated results and the error images are shown in Fig. 6 (g)–(q). The color bar in images shows the value in term of wavelength (λ). For each wavefront Peak to Valley (PV) and root mean square (RMS) values are calculated and these values are shown in Table 1.

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Table 1. Shows PV and RMS values of calculated wavefront at different position of collimating lens L.

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The laser beam was collimated when lens L positioned at distance f = 100 mm from the pin hole of spatial filter assembly (dr = 0). This condition was considered as reference for above experiment. When lens L is moved, the change in wavefront should match with simulated wavefront. But as Table 1 shows PV and RMS values are increasing as the lens L is moved in forward direction. It is also observed that when lens L is moved back towards collimation position, PV and RMS values are not matching with that of forward direction at same position. This error may be result of inline mechanical moment of lens which might introduce tip and tilt to wavefront. Even when lens L reaches the initial collimation position, there is error present in calculated wavefront. Thus these types of wavefront errors can also be detected with Vectorial sheared cascaded Sagnac wavefront sensor.

5. Conclusion

We proposed and experimentally demonstrated Vectorial shearing interferometry using common-path Sagnac interferometers and telecentric lens system. While the amount of spatial carrier frequency for multiplexing the shearing interferogram can be fixed by introducing controlled mirror alignments of cascaded Sagnac interferometer, the lateral shear can still be varied by moving the detector plane alone along z. Wavefront gradient along two perpendicular directions computed simultaneously using that wavefront is determined. By changing the position of collimation lens we showed that mechanical errors inducing wavefront deviations of the order of fraction of wavelength can be detected using our proposed scheme. The near common-path geometry makes it stable in vibration sensitive environment allowing one to pick only the wavefront deviations of the input light.

References

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10. S. Velghe, J. Primot, N. Guerineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30(3), 245–247 (2005). [CrossRef]  

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12. J. C. Chanteloup, F. Druon, M. Nantel, A. Maksimchuk, and G. Mourou, “Single-shot wave-front measurements of high-intensity ultrashort laser pulses with a three-wave interferometer,” Opt. Lett. 23(8), 621–623 (1998). [CrossRef]  

13. L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19(10), 2100–2111 (2002). [CrossRef]  

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Figures (6)
Fig. 1.
Fig. 1. Conceptual diagram for common-path vectorial shearing wavefront sensor

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Fig. 2.
Fig. 2. Generation of four copies of the wavefront using two Sagnac interferometer in kept in series.

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Fig. 3.
Fig. 3. Wavefront Sensing using vectorial shearing interferometer.

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Fig. 4.
Fig. 4. (a) Vectorial sheared fringe pattern, (b) Fourier transform of fringe pattern.

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Fig. 5.
Fig. 5. (a) Gradient along X-direction, (b) Gradient along Y-direction (c) Reconstructed Wavefront.

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Fig. 6.
Fig. 6. (a), (b), (c), (d), (e) and (f) shows the reconstructed wavefront at different positions of collimating lens along z-direction with step size of 0.5 mm. Red and green colored dashed lines show experimental results and blue colored line shows simulation results. Figure (g), (h), (i), (j) and (l) shows the wavefront deviations when collimating lens moves in forward direction. Figure (m), (n), (o), (p) and (q) shows the wavefront deviations when collimating lens moves in backward direction.

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Tables (1)

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Table 1. Shows PV and RMS values of calculated wavefront at different position of collimating lens L.

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Equations (9)

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u A ( x ′ , y ′ ) = F ( f 1 ) − 1 { F ( f 1 ) [ u o ⁡ ( x ′ , y ′ ) ] × e x p ( − i 2 π z 1 λ 1 − x ^ 2 f 1 2 − y ^ 2 f 1 2 ) }
U F ( x ^ λ f 1 , y ^ λ f 1 ) = F ( f 1 ) [ u O ( x ′ , y ′ ) ] × e x p ( − i 2 π z 1 λ 1 − x ^ 2 f 1 2 − y ^ 2 f 1 2 )
U F ( x ^ − x ^ s , n λ f 1 , y ^ − y ^ s , n λ f 1 ) = U O ( x ^ − x ^ s , n λ f 1 , y ^ − y ^ s , n λ f 1 ) × e x p ( − i 2 π z 1 λ 1 − ( x ^ − x ^ s , n ) 2 f 1 2 − ( y ^ − y ^ s , n ) 2 f 1 2 )
u B , n ( x , y ) = F ( f 2 ) { U F ( x ^ − x ^ s , n λ f 1 , y ^ − y ^ s , n λ f 1 ) }
u I , n ( x , y ) = F ( f 2 ) − 1 { F ( f 2 ) [ u B , n ( x , y ) ] × exp ⁡ ( i 2 π z 2 λ 1 − x ^ 2 f 2 2 − y ^ 2 f 2 2 ) }
u I , n ( x , y ) = F ( f 2 ) − 1 [ F ( f 2 ) [ F ( f 2 ) { U F ( x ^ − x ^ s , n λ f 1 , y ^ − y ^ s , n λ f 1 ) } ] × exp ⁡ ( i 2 π z 2 λ 1 − x ^ 2 f 2 2 − y ^ 2 f 2 2 ) ]
u I , n ( x , y ) = F ( f 2 ) − 1 [ U O ( − ( x ^ − x ^ s , n ) λ f 1 , − ( y ^ − y ^ s , n ) λ f 1 ) × exp ⁡ ( − i 2 π z 1 λ 1 − ( x ^ − x ^ s , n ) 2 f 1 2 − ( y ^ − y ^ s , n ) 2 f 1 2 ) × exp ⁡ ( i 2 π z 2 λ 1 − x ^ 2 f 2 2 − y ^ 2 f 2 2 ) ]
u I , n ( x , y ) = e x p ( i 2 π z 2 λ ( 1 − f 1 2 f 2 2 + x ^ s , n 2 + y ^ s , n 2 2 f 2 2 ) ) e x p ( − i 2 π λ f 2 ( x x ^ s , n + y y ^ s , n ) ) × u O ( − 1 m ( x − x ^ s , n z 2 f 2 ) , − 1 m ( y − y ^ s , n z 2 f 2 ) )
I I ( x , y ) = | u I , 1 ( x , y ) + u I , 2 ( x , y ) + u I , 3 ( x , y ) | 2
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