This research was supported by Generalitat Valenciana (PROMETEO 2016-079), Universitat Jaume I (UJI) (UJIB2016-19); and Ministerio de Econom&#237;a y Competitividad (MINECO) (FIS2016-75618-R). The authors are very grateful to the SCIC of the Universitat Jaume I for the use of the femtosecond laser.

1. Encoding the Complex Field into a Single Phase Element
<ol>
	<li>From the technical specifications of the SLM, find its spatial resolution (for instance 1920 pixels x 1800 pixels).</li>
	<li>Define and generate the desired amplitude A(x,y) and phase &#966;(x,y) patterns as digital images.
	<ol>
		<li>Set the spatial resolution of abovementioned digital images equal to that of the SLM display.</li>
		<li>Set abovementioned digital images in gray level format.</li>
		<li>Set the minimum and maximum values of the amplitude and phase images from 0 to 255, and from -&#960;/2 to &#960;/2, respectively.</li>
		<li>Set Amax = 2 in equations 2 and 3, and computer-generate the phase patterns &#977;(x,y) and &#952;(x,y) from them.</li>
	</ol>
	</li>
	<li>Computer generate the checkerboard patterns M1(x,y) and M2(x,y).
	<ol>
		<li>Set the spatial resolution of these checkerboard patterns equal to that of the SLM display.</li>
		<li>To reduce the effect of pixel crosstalk, generate other pairs of checkerboard patterns M1(x,y) and M2(x,y) constructed with different pixel cells having an increased number of pixels (for instance: 2x2, 3x3, and 4x4 pixel cells, etc.). 
		CAUTION: When increasing the pixel cell, the total number of pixels of checkerboard patterns must be kept unchanged and equal to the spatial resolution of the SLM. Ensure that final number of pixels of all checkerboard patterns remains the same after modifying their pixel cells.</li>
	</ol>
	</li>
	<li>Computer generate the single phase element &#945;(x,y) from equation 5. 
	NOTE: See supplemental material named &#8220;MATLAB_code_1.m&#8221; for related tasks on step 1 of this protocol.</li>
</ol>
2. Reconstruction of the Complex Field
<ol>
	<li>Use a collimated, linear polarized, and spatially coherent laser beam as a light source.</li>
	<li>Use a phase-only SLM with at least 2&#960; phase range.</li>
	<li>When necessary, use a proper beam expander to adjust the size of the beam to the size of the SLM display.</li>
	<li>When necessary, use an optical polarizer to set laser beam polarization to the horizontal direction. This is usually important for the proper operation of phase-only SLMs, which are typically designed to modulate the spatial phase of the electromagnetic field that oscillate in the horizontal direction, keeping unchanged its vertical components.</li>
	<li>In order to send a phase pattern to the SLM, follow standard communication protocols given by the SLM&#8217;s manufacturer to connect and control the SLM with the computer. 
	NOTE: Common protocol for this purpose includes the use of a calibration curve to transform the values in radians (due to mathematical operations with angles) into gray level ones, which the electronic control unit of the SLM will finally convert into voltage levels. Additionally, as the SLM is connected to computer as an external device with its own screen, an extension of the computer screen is usually necessary, as well as a proper program to send the corresponding gray level images to this extra screen. An example of these codes is also included as supplemental material (please, see MATLAB_code_2.m).</li>
	<li>Implement an image optical system and put the display of the SLM in the input plane of this system.
	<ol>
		<li>Use a refractive lens of a focal length f to construct a 2f x 2f optical image system (a 4f optical system is also valid for this task). In accordance with the expected output size of the complex field, beam width, wavelength of light, and the available physical space, employ lens/lenses with suited technical specifications (e.g., coating, size, focal length, etc.).</li>
		<li>To find the position of the output plane of the imaging system, send the phase pattern &#945;(x,y) to the SLM and visually look for the recorded image (depending on the position of the camera) with the best spatial resolution. 
		CAUTION: In the case of low-size pixel cells (for instance, 1x1 pixel cells) and SLM displays with pixel widths of a few microns (for instance, 8 &#181;m), only beam propagation can produce interference between encoded uniform waves, getting a reconstructed images without including the circular iris in the imaging system. Use low-size pixel cells to locate the position of the output plane.</li>
		<li>Place a circular iris of variable diameter at the Fourier plane of the optical system and align its center with that of the laser beam focus.</li>
		<li>To adjust the size of the circular iris at the Fourier plane, send the phase pattern &#945;(x,y) and visually look for the recorded image (depending on the diameter of the circular iris) with best spatial resolution. 
		CAUTION: In the case of long-size pixel cells (for instance, 4x4 pixel cells), the interference between encoded uniform waves is basically carried out with the spatial filter. Use long-size pixel cell to adjust the size of the circular iris. In this protocol, the terms low-size and long-size are referred to the number of pixels contained within a pixel cell. However, the abovementioned interference depends also on the pixel width. Employ SLMs with pixel widths equal or less than 8 &#181;m.</li>
	</ol>
	</li>
	<li>Send the gray level image corresponding to the phase element &#945;(x,y) to the SLM.
	<ol>
		<li>To minimize the crosstalk effect, look for the best pixel cell size which allow achieving the recorded image with the higher spatial resolution.</li>
	</ol>
	</li>
</ol>
3. Measure the Reconstructed Complex Field
<ol>
	<li>Implement the polarization-based phase shifting technique8.
	<ol>
		<li>Place and align the rotation angle of the first optical polarizer, located just before the SLM (see Figure 2). To set the rotation angle of the first polarizer, visually look for the maximum and minimal light transmittance in the CCD camera (placed at the output plane of the imaging system), depending on the rotation of the polarizer. Write down the two corresponding angles of the polarizer. Fix the final angle of the polarizer to that between the two previous-recorded angles.</li>
		<li>Place and align the rotation angle of the second optical polarizer, located after the Fourier plane of the imaging system (see Figure 2). To set the rotation angle of the second polarizer, visually look for the sharpest and most blurred images in the CCD camera (placed at the output plane of the imaging system) after sending the phase pattern &#945;(x,y) to the SLM. Write down the two corresponding angles of the polarizer. Fix the final angle of the second polarizer to that between the previous-recorded angles.</li>
	</ol>
	</li>
	<li>Record the interferograms.
	<ol>
		<li>Keep the CCD camera at the output plane of the imaging system.</li>
		<li>To record the first interferogram, add a matrix of 0 radians to the phase element &#945;(x,y) and send it to the SLM. Record corresponding image I1(x,y) with the CCD.</li>
		<li>To record the second interferogram, add a matrix of &#960;/2 radians to the phase element &#945;(x,y) and send it to the SLM. Record corresponding image I2(x,y) with the CCD camera.</li>
		<li>To record the third interferogram, add a matrix of &#960; radians to the phase element &#945;(x,y) and send it to the SLM. Record corresponding image I3(x,y) with the CCD camera.</li>
		<li>To record the fourth and last interferogram, add a matrix of 3&#960;/2 radians to the phase element &#945;(x,y) and send it to the SLM. Record corresponding image I4(x,y) with the CCD camera.</li>
	</ol>
	</li>
	<li>Reconstruct the complex field. 
	NOTE: See supplemental material named &#8220;MATLAB_code_3.m&#8221; for related tasks on this point of the protocol.
	<ol>
		<li>Retrieve the amplitude of the complex field Aretrieved(x,y) by using the expression 
		<img alt="Equation 8" src="/files/ftp_upload/59158/59158eq8.jpg" />&#160;(8)</li>
		<li>Retrieve the phase of the complex field &#966;retrieved(x,y) by using the expression 
		<img alt="Equation 9" src="/files/ftp_upload/59158/59158eq9.jpg" />&#160;(9)</li>
	</ol>
	</li>
</ol>

<ol>
	<li>Hsueh, C. K., Sawchuk, A. A. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Computer-generated+double-phase+holograms.">Computer-generated double-phase holograms.</a> Applied Optics. 17 (24), 3874-3883 (1978).</li><li>Arriz&#243;n, V. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Complex+modulation+with+a+twisted-nematic+liquid-crystal+spatial+light+modulator:+double-pixel+approach.">Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach.</a> Optics Letters. 28 (15), 1359-1361 (2003).</li><li>Arriz&#243;n, V., Ruiz, U., Carrada, R., Gonz&#225;lez, L. A. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Pixelated+phase+computer+holograms+for+the+accurate+encoding+of+scalar+complex+fields.">Pixelated phase computer holograms for the accurate encoding of scalar complex fields.</a> Journal of the Optical Society of America A. 24 (11), (2007).</li><li>Shibukawa, A., Okamoto, A., Takabayashi, M., Tomita, A. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Spatial+cross+modulation+method+using+a+random+diffuser+and+phase-only+spatial+light+modulator+for+constructing+arbitrary+complex+fields.">Spatial cross modulation method using a random diffuser and phase-only spatial light modulator for constructing arbitrary complex fields.</a> Optics Express. 22 (4), 3968-3982 (2014).</li><li>Mart&#237;nez-Fuentes, J. L., Moreno, I. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Random+technique+to+encode+complex+valued+holograms+with+on+axis+reconstruction+onto+phase-only+displays.">Random technique to encode complex valued holograms with on axis reconstruction onto phase-only displays.</a> Optics Express. 26 (5), 5875-5893 (2018).</li><li>Clark, T. W., Offer, R. F., Franke-Arnold, S., Arnold, A. S., Radwell, N. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Comparison+of+beam+generation+techniques+using+a+phase+only+spatial+light+modulator.">Comparison of beam generation techniques using a phase only spatial light modulator.</a> Optics Express. 24 (6), 6249-6264 (2016).</li><li>Mendoza-Yero, O., M&#237;nguez-Vega, G., Lancis, J. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Encoding+complex+fields+by+using+a+phase-only+optical+element.">Encoding complex fields by using a phase-only optical element.</a> Optics Letters. 39 (7), 1740-1743 (2014).</li><li>Yamaguchi, I., Zhang, T. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Phase-shifting+digital+holography.">Phase-shifting digital holography.</a> Optics Letters. 22 (16), 1268-1270 (1997).</li><li>Shao, Y., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Addressable+multiregional+and+multifocal+multiphoton+microscopy+based+on+a+spatial+light+modulator.">Addressable multiregional and multifocal multiphoton microscopy based on a spatial light modulator.</a> Journal of Biomedical Optics. 17 (3), 030505 (2012).</li><li>Mendoza-Yero, O., Carbonell-Leal, M., M&#237;nguez-Vega, G., Lancis, J. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Generation+of+multifocal+irradiance+patterns+by+using+complex+Fresnel+holograms.">Generation of multifocal irradiance patterns by using complex Fresnel holograms.</a> Optics Letters. 43 (5), 1167-1170 (2018).</li><li>Kuang, Z., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Diffractive+Multi-beam+Ultra-fast+Laser+Micro-processing+Using+a+Spatial+Light+Modulator+(Invited+Paper).">Diffractive Multi-beam Ultra-fast Laser Micro-processing Using a Spatial Light Modulator (Invited Paper).</a> Chinese Journal of Lasers. 36 (12), 3093-3115 (2009).</li><li>Kuang, Z., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=High+throughput+diffractive+multi-beam+femtosecond+laser+processing+using+a+spatial+light+modulator.">High throughput diffractive multi-beam femtosecond laser processing using a spatial light modulator.</a> Applied Surface Science. 255, 2284-2289 (2008).</li></ol>

The authors have nothing to disclose.

In this protocol, practical parameters as the pixel width of the phase-only SLM or the number of pixels contained within pixel cells of a computer-generated pattern are key points to successfully implement the encoding method. In steps 1.2, 1.3, and 1.4 of the protocol, the shorter the pixel width, the better the spatial resolution of the retrieved amplitude and phase patterns. In addition, as the codification into the SLM of abrupt pixel-to-pixel phase modulations can originate unexpected phase responses (pixel crosstalk), the construction of checkerboard patterns (as described in step 1.3) should be linked to the increment of the number of pixels within pixel cells. The main reason to do that is to mitigate the effects of pixel crosstalk on the retrieved amplitude and phase patterns. However, when increasing the number of pixels within the pixel cells, the spatial resolution of the recorded complex field patterns Aretrieved(x,y)&#160;and &#966;retrieved(x,y) is decreased. Hence, having high spatial resolution SLMs with low pixel widths allows reducing possible crosstalk effects, without losing significant spatial resolution in the retrieved amplitude and phase patterns.
Furthermore, in step 1.2.3 of the protocol, the phase of the complex field is defined from - &#960;/2 to &#960;/2. The main reason for setting such phase range is to generate a phase element &#945;(x,y) ranging from -&#960; to &#960;, which can be implemented into a SLM with 2&#960; of phase range. However, if the phase range of the available SLM is greater than 2&#960;, the phase of the complex field could be defined within an extended range (for instance: for &#966;(x,y) ranging from -&#960; to &#960;, the phase element &#945;(x,y) may range from -3&#960;/2 to 3&#960;/2, and consequently the phase range of the SLM must be, at least, 3&#960;).
The characteristics of the laser beam could also influence the results of the encoding method. Pay special attention to the steps 2.1-2.4, setting the right polarization direction, collimation, and transversal size of the laser beam before following the remaining steps of the protocol. Furthermore, as phase-only SLMs are basically diffractive-dependent optical devices based on the interference phenomenon, it is necessary to use laser beams with high/good spatial coherence.
On the other hand, instead of quasi-monochromatic, ultrashort pulsed illumination also allows obtaining good results. In this case, the different spectral components of the pulse are phase modulated (in a very similar manner) just with the single phase element &#945;(x,y). Here, to show the effect of a broadband light source on the encoding method, we repeat all steps of the protocol, but this time for pulsed radiation (an ultrashort pulse of about 12 fs FWHM, centered at 800 nm, spectral bandwidth of 100 nm FWHM, emitted by a mode-locked Ti:Sapphire laser from the femtolaser, at a 75 MHz repetition rate). The results are shown in Figure 4. Note that, due to the mix of the different spectral components of the pulse, the retrieved patterns are very close to the expected ones.
<img alt="Figure 4" class="xfigimg" src="/files/ftp_upload/59158/59158fig4.jpg" /> 
Figure 4: Representative experimental results under ultrashort pulsed illumination. (A) User-defined amplitude pattern of the complex field. (B) User-defined phase pattern of the complex field. (C) Interferograms associated with the polarization-based phase shifting technique developed in four steps and obtained after adding 0, &#960;/2, &#960;, and 3&#960;/2 to the phase element &#945;(x,y). (D) Retrieved experimental amplitude pattern. (E) Retrieved experimental phase pattern. <a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/59158/59158fig4large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
Laser beams are intrinsically complex fields, so in most potential applications one should be able to modify their amplitude and phase, simultaneously. The present method allows to do that by means of a single phase element (implemented or not into a phase-only SLM). We believe that, in a near future, this method could be employed, for instance in the illumination path of microscopes9,10 for simultaneous linear and non-linear excitation of different zones of biological samples, or in parallel micro-processing11,12 of materials. In both applications the role of amplitude modulation is apparent, meanwhile phase modulation can be utilized, at the same time, for compensation of optical aberrations at the sample/processing plane. Finally, it should be mentioned that the encoding method described with the present protocol is not limited to the utilization of SLMs. Fixed phase elements &#945;(x,y) constructed with other techniques (for instance: photolithographic techniques) can be a different, but equally valid option to implement this protocol.

Almost all laser applications are in close relation with the management of the optical wavefront of light. In the paraxial approximation, the complex field associated with the laser radiation can be described by two terms, the amplitude and the phase. Having control over these two terms is necessary to modify both the temporal and the spatial structure of laser beams at will. In general, the amplitude and the phase of a laser beam can be properly changed by several methods including the use of optical components that range from single bulk lenses, beam splitters and mirrors to most complex devices like deformable mirrors or spatial light modulators. Here, we show a method for encoding and reconstructing the complex field of coherent laser beams, which is based on dual-phase hologram theory1, and the utilization of a common-path interferometer.
Nowadays, there exists a wide variety of methods to encode the complex fields of laser beams2,3,4,5. In this context, some well-established methods to produce phase and amplitude modulation rely on the use of digital holograms6. A common point in all these methods is the necessity of generating a spatial offset to separate the desired output beam from the zeroth-order coming from the reflection of light at the SLM display. These methods are basically off-axis (usually applying for the first diffraction order of the grating), employing phase grating not only to encode the phase, but also to introduce necessary amplitude modulation. In particular, amplitude modulation is performed by spatially lowering the grating height, which clearly degrades the diffraction efficiency. The hologram reconstruction process mostly gets an approximate, but not exact, reconstruction of the amplitude and phase of the desired complex field. Discrepancies between theory and experiment seem to appear from an inaccurate encoding of the amplitude information as well as other experimental issues happening during the spatial filtering of the first diffraction order or due to SLM pixilation effects. In addition, the intensity profile of the input beam can introduce restrictions on the output power.
In contrast, with the introduced method7, all light management is carried out on-axis, which is very convenient from an experimental point of view. Additionally, it takes advantage of considering, in the paraxial approximation, the complex field associated with laser beams as a sum of two uniform waves. The amplitude information is synthetized by the interference of these uniform waves. In practice, such interference is carried out by spatial filtering of light frequencies at the Fourier plane of a given imaging system. Previously, the phase patterns associated with the uniform waves are spatially multiplexed and encoded into a phase-only SLM (placed at the entrance plane of this imaging system). Hence, the whole optical setup can be regarded as a common-path interferometer (very robust against mechanical vibrations, temperature changes, or optical misalignments). Please, note that the abovementioned interference process can be alternatively accomplished by using other optical layouts: with a couple of phase-only SLMs properly placed within a typical two-arm interferometer, or by time sequentially encoding the two phase patterns into the SLM (previous introduction of a reference mirror in the optical setup). In both cases, there is no necessity of spatial filtering, and consequently no loss of spatial resolution, at the expense of increasing the complexity of the optical system, as well as the alignment process. Here, it should be also emphasized that by using this encoding method, the full spectrum of the desired complex field can be exactly retrieved at the Fourier plane, after filtering all diffraction orders but the zeroth one.
On the other hand, the efficiency of the method depends on several factors: the manufacturer&#39;s specifications of the SLM (e.g., fill factor, reflectivity, or diffraction efficiency), the size of the encoded pattern, and the way at which the light impinges onto the SLM (reflection with a small hitting angle, or normal incidence by using a beam splitter). At this point, under proper experimental conditions, the measured total light efficiency can be more than 30%. However, note that that the total light efficiency just due to the use of the SLM can be less than 50%. The lack of random or diffuser elements within the optical setup allows the retrieving of amplitude and phase patterns without coherent noise (speckle). Other significant aspects to point out are the utilization of a direct codification algorithm rather than iterative procedures and its ability to perform arbitrary and independent amplitude and phase modulation at the frequency refresh time of the SLM (up to hundreds of hertz according to the current technology).
In principle, the method7 is intended to be used with input plane waves, but it is not limited to that. For instance, if a Gaussian beam is hitting the SLM, it is possible to modify its irradiance shape at the output of the system by encoding a suited amplitude pattern into the SLM. However, as the intensity of the output beam cannot exceed that of the input beam at any transversal position (x,y), the shaping of the amplitude is performed by intensity losses originated by a partially destructive interference process.
The theory underlining the encoding method7 is as follows. Any complex field represented in the form U(x,y)= A(x,y)ei&#966;(x,y) can be also rewritten as:
<img alt="Equation 1" src="/files/ftp_upload/59158/59158eq1.jpg" />&#160;(1)
where
<img alt="Equation 2" src="/files/ftp_upload/59158/59158eq2.jpg" />&#160;(2)
<img alt="Equation 3" src="/files/ftp_upload/59158/59158eq3.jpg" />&#160;(3)
In equations 1-3, the amplitude and phase of the two-dimensional complex field U(x,y)is given by A(x,y) and &#966;(x,y), respectively. Note that, terms Amax (maximum of A(x,y)) and B = Amax/2 do not depend on the transversal coordinates (x,y). From the theory, if we set Amax=2, then B =1. Hence, the complex field U(x,y) can be obtained, in a simple manner, from the coherent sum of uniform waves Bei&#977;(x,y) and Bei&#952;(x,y). In practice, this is accomplished with a common-path interferometer made up of a single phase element &#945;(x,y), placed at the input plane of an imaging system. The single phase element is constructed by spatial multiplexing of the phase terms &#977;(x,y)
and &#952;(x,y) with the help of two-dimensional binary gratings (checkerboard patterns) M1(x,y) and M2(x,y) as follows
<img alt="Equation 4" src="/files/ftp_upload/59158/59158eq4.jpg" />&#160;(4)
hence,
<img alt="Equation 5" src="/files/ftp_upload/59158/59158eq5.jpg" />&#160;(5)
These binary patterns fulfill the condition M1(x,y) + M2(x,y) = 1. Note that, the interference of uniform waves cannot happen if we do not mix the information contained in the phase element&#945;(x,y). In the present method, this is carried out by using a spatial filter able to block all diffraction orders but the zeroth one. In this way, after the filtering process at the Fourier plane, the spectrum H(u,v)= F{ei&#945;(x,y)} of the encoded phase function is related to the spectrum of the complex field F{U(x,y)} by the expression
<img alt="Equation 6" src="/files/ftp_upload/59158/59158eq6.jpg" />&#160;(6)
In Eq. (6), (u,v) denote coordinates in the frequency domain, P(u,v) holds for the spatial filter, whereas the Fourier transform of a given function &#920;(x,y) is represented in the form F{&#920;(x,y)}. From Eq. (6), it follows that, at the output plane of the imaging system, the retrieved complex field URET(x,y), (without considering constant factors), is given by the convolution of the magnified and spatially reversed complex field U(x,y) with the Fourier transform of the filter mask. That is:
<img alt="Equation 7" src="/files/ftp_upload/59158/59158eq7.jpg" />&#160;(7)
In Eq. (7), the convolution operation is denoted by the symbol <img alt="Equation 10" src="/files/ftp_upload/59052/59052eq10.jpg" />, and the term Mag represents the magnification of the imaging system. Hence, the amplitude and phase of U(x,y) is fully retrieved at the output plane, except for some loss of spatial resolution due to the convolution operation.

<table><tbody><tr><td>Achromatic Doublet</td><td>THORLABS</td><td>AC254-100-B-ML</td><td>Lens Diameter 25.4 mm, focal length 100 mm</td></tr><tr><td>Achromatic Galilean Beam Expander</td><td>THORLABS</td><td>GBE05-A</td><td>AR Coated: 400 - 650 nm</td></tr><tr><td>Basler camera</td><td>BASLER</td><td>avA1600-50gm GigE camera</td><td>sensor size 8.8 mm x 6.6 mm, pizel size 5.5 microns</td></tr><tr><td>Mounted Zero-Aperture Iris</td><td>THORLABS</td><td>ID12Z/M</td><td>Max Aperture 12 mm</td></tr><tr><td>Pellicle Beamsplitter</td><td>THORLABS</td><td>CM1-BP145B2</td><td>45:55 (R:T), Coating: 700 - 900 nm</td></tr><tr><td>PLUTO Spatial Light Modulator</td><td>HOLOEYE Photonics AG</td><td>NIR-II</td><td>Phase Only Spatial Light Modulator (Optimized for 700 -1000 nm)</td></tr><tr><td>Two thin film laser polarizers</td><td>EKSMA OPTICS</td><td>420-0526M</td><td>material BK7, diameter 50 mm, wavelength 780-820 nm</td></tr></tbody></table>

shaping the amplitude and phase of laser beams by using a phase-only spatial light modulator

The aim of this article is to visually demonstrate the utilization of an interferometric method for encoding complex fields associated with coherent laser radiation. The method is based on the coherent sum of two uniform waves, previously encoded into a phase-only spatial light modulator (SLM) by spatial multiplexing of their phases. Here, the interference process is carried out by spatial filtering of light frequencies at the Fourier plane of certain imaging system. The correct implementation of this method allows arbitrary phase and amplitude information to be retrieved at the output of the optical system.
It is an on-axis, rather than off-axis encoding technique, with a direct processing algorithm (not an iterative loop), and free from coherent noise (speckle). The complex field can be exactly retrieved at the output of the optical system, except for some loss of resolution due to the frequency filtering process. The main limitation of the method might come from the inability to operate at frequency rates higher than the refresh rate of the SLM. Applications include, but are not limited to, linear and non-linear microscopy, beam shaping, or laser micro-processing of material surfaces.

The spatial resolution of the employed phase-only SLM is 1920 pixels x 1080 pixels, with a pixel pitch of 8 &#181;m. The selected amplitude A(x,y) and phase &#966;(x,y) of the complex field are defined by two different gray level images corresponding to the well-known Lenna&#8217;s picture (amplitude pattern) and a young girl sticking out her tongue (phase pattern), respectively. In general, for both, the generation of necessary patterns, and the control of the SLM, Matlab codes are utilized. The spatial resolution of these images is set to be 1920 pixels x 1080 pixels. Then, equations 2 and 3 are used to determine the phase patterns &#977;(x,y) and &#952;(x,y) for Amax = 2. Note that, the numerical value given to Amax guaranties that term B = 1 and consequently, the complex field U(x,y) &#160;described by Eq. (1) can be understood as the sum of two uniform waves in the simplest form U(x,y) = ei&#977;(x,y) + ei&#952;(x,y). Now, different pairs of binary checkerboard patterns M1(x,y) and M2(x,y) (for increased pixel cell sizes), but equal spatial resolution (1920 pixels x 1080 pixels), are computer generated. Particularly, checkerboard patterns made up of 1x1, 2x2, 3x3 and 4x4 pixel cells are digitally constructed by using a programed Matlab function. All abovementioned patterns A(x,y), &#966;(x,y), &#977;(x,y), &#952;(x,y), M1(x,y), and M2(x,y) are shown in parts A, B, C, D, E, and F of Figure 1, respectively. In parts E and F, and just to get a better visualization of the structure of checkerboard patterns, the constituent pixel cells are of 240 pixels x 240 pixels. From Eq. 5, a set of phase elements &#945;(x,y) for each pair of previously-designed checkerboard patterns are digitally constructed.
<img alt="Figure 1" class="xfigimg" src="/files/ftp_upload/59158/59158fig1.jpg" /> 
Figure 1: Computer generated patterns associated with the introduced encoding method. (A) User-defined amplitude pattern of the complex field. (B) User-defined phase pattern of the complex field. (C) Phase pattern corresponding to the first uniform wave in equation 1. (D) Phase pattern corresponding to the second uniform wave in equation 1. (E) First checkerboard pattern following the sampling process described with equation 4. (F) Second checkerboard pattern following the sampling process described with equation 4. <a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/59158/59158fig1large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
At this point, the expected complex field U(x,y) can be experimentally retrieved at the output plane of an imaging system, once the phase element &#945;(x,y) is sent to the phase-only SLM and the interference between encoded uniform waves takes place. To perform this interference, a spatial filter (for instance, a circular iris) is adjusted in size to block all frequencies but the zeroth one at the Fourier plane of the imaging system (Figure 2).
<img alt="Figure 2" class="xfigimg" src="/files/ftp_upload/59158/59158fig2.jpg" /> 
Figure 2: Optical setup used to accomplish the encoding method. Imaging system composed of a spatial light modulator (SLM), beam splitter (BS), and single refractive lens (L) of focal length 200 mm. In the Fourier plane is included a hard iris, which is employed as a spatial filter (SF) to block all frequencies but the zero one. In addition, at the output plane of the imaging system is placed a camera (CCD) to record amplitude patterns, and interferograms. Only to measure the generated complex field by means of polarization-based phase shifting technique, a couple of optical polarizers (P) are properly located within the optical setup. <a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/59158/59158fig2large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
As a light source, a Ti: Sapphire laser oscillator (working out of the mode locked condition to emit a quasi-monochromatic laser radiation of about 10 nm intensity full width at half maximum (FWHM) and centered at 800 nm) is employed. In addition, to fill almost all the active area of the SLM display (8.64 cm x 15.36 cm) with the laser beam, a commercial 5x telescope beam expander is used. The laser beam is sent (in normal incident) to the display of the SLM by means of a pellicle beam splitter. A refractive lens of focal length 100 mm is placed 200 mm after the SLM and aligned with respect to the optical axis of the laser beam reflected back from the SLM. To locate the position of the output plane of the imaging system, the image of A(x,y) recorded the CCD camera was found. This is done once the phase element &#945;(x,y) (formed with 1x1 pixel cells) is sent to the SLM. Then, a circular iris is placed at the Fourier plane of the optical system, and aligned with respect to the focus of the laser beam. In addition, to adjust the size of the circular iris, its diameter is varied until better image reconstruction is achieved by visual inspection in the CCD camera. To this purpose, the phase element &#945;(x,y) (digitally constructed with 4x4 pixel cells) was previously sent to the SLM. To minimize the effect of pixel crosstalk, the best phase element &#945;(x,y) (depending on the pixel cell size) that allows achieving the image with higher spatial resolution in the CCD is found.
In order to corroborate that the desired complex field is reconstructed at the output plane of the imaging system, the already-mentioned polarization-based phase shifting technique is used to measure its amplitude and phase. To do that, a couple of polarizers p (one placed before the SLM, and another after the output plane of the imaging system) are properly aligned within the optical setup (see Figure 2), following the procedure described in steps 3.1.1&#160;and 3.1.2&#160;of the protocol. Then, the interferograms associated with the four-step phase shifting technique I1(x,y), I2(x,y), I3(x,y), and I4(x,y) are recorded with the CCD camera (already placed at the output plane of the imaging system). Here, it should be recalled that these four interferograms are recorded with the camera after the addition of 0, &#960;/2, &#960;, and 3&#960;/2 to the phase element &#945;(x,y) (see steps 3.2.2 - 3.2.5 of the protocol for details). Finally, using equations 8 and 9, the amplitude and phase of the reconstructed complex field can be retrieved. For this experiment, the results are shown in Figure 3.
<img alt="Figure 3" class="xfigimg" src="/files/ftp_upload/59158/59158fig3.jpg" /> 
Figure 3: Representative experimental results under quasi-monochromatic illumination. (A) User-defined amplitude pattern of the complex field. (B) User-defined phase pattern of the complex field. (C) Interferograms associated with the polarization-based phase shifting technique developed in four steps and obtained after adding 0, &#960;/2, &#960;, and 3&#960;/2 to the phase element &#945;(x,y). (D) Retrieved experimental amplitude pattern. (E) Retrieved experimental phase pattern. <a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/59158/59158fig3large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
Supplemental Materials.&#160;<a href="https://www-jove-com.remotexs.ntu.edu.sg/files/ftp_upload/59158/Supplemental_Materials.zip" target="_blank">Please click here to download the files.</a>

Watch this Scientific Journal Video about Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator at JoVE.com

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator

We show how to encode the complex field of laser beams by using a single phase element. A common-path interferometer is employed to mix the phase information displayed into a phase-only spatial light modulator to finally retrieve the desired complex field pattern at the output of an optical imaging system.

The aim of this article is to visually demonstrate the utilization of an interferometric method for encoding complex fields associated with coherent laser ...

shaping-amplitude-phase-laser-beams-using-phase-only-spatial-light

Nanyang Technological University

Research

JoVE Journal

Engineering

9.7K Views.  Universitat Jaume I. We show how to encode the complex field of laser beams by using a single phase element. A common-path interferometer is employed to mix the phase information displayed into a phase-only spatial light modulator to finally retrieve the desired complex field pattern at the output of an optical imaging system.

Video: Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator

Quasi-light Storage for Optical Data Packets

Today's telecommunication is based on optical packets which transmit the information in optical fiber networks around the world. Currently, the processing of the signals is done in the electrical domain. Direct storage in the optical domain would avoid the transfer of the packets to the electrical and back to the optical domain in every network node and, therefore, increase the speed and possibly reduce the energy consumption of telecommunications. However, light consists of photons which propagate with the speed of light in vacuum. Thus, the storage of light is a big challenge. There exist some methods to slow down the speed of the light, or to store it in excitations of a medium. However, these methods cannot be used for the storage of optical data packets used in telecommunications networks. Here we show how the time-frequency-coherence, which holds for every signal and therefore for optical packets as well, can be exploited to build an optical memory. We will review the background and show in detail and through examples, how a frequency comb can be used for the copying of an optical packet which enters the memory. One of these time domain copies is then extracted from the memory by a time domain switch. We will show this method for intensity as well as for phase modulated signals.

The article describes a procedure to store optical data packets with an arbitrary modulation, wavelength, and data rate. These packets are the basis of modern telecommunications.

Today's telecommunication is based on optical packets which transmit the information in optical fiber networks around the world. Currently, the processing ...

Construction of a High Resolution Microscope with Conventional and Holographic Optical Trapping Capabilities

High resolution microscope systems with optical traps allow for precise manipulation of various refractive objects, such as dielectric beads 1 or cellular organelles 2,3, as well as for high spatial and temporal resolution readout of their position relative to the center of the trap. The system described herein has one such "traditional" trap operating at 980 nm. It additionally provides a second optical trapping system that uses a commercially available holographic package to simultaneously create and manipulate complex trapping patterns in the field of view of the microscope 4,5 at a wavelength of 1,064 nm. The combination of the two systems allows for the manipulation of multiple refractive objects at the same time while simultaneously conducting high speed and high resolution measurements of motion and force production at nanometer and piconewton scale.

The system described herein employs a traditional optical trap as well as an independent holographic optical trapping line, capable of creating and manipulating multiple traps. This allows for the creation of complex geometric arrangements of refractive particles while also permitting simultaneous high-speed, high-resolution measurements of the activity of biological enzymes.

High resolution microscope systems with optical traps allow for precise manipulation of various refractive objects, such as dielectric beads 1 or cellular ...

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Thermal noise in high-reflectivity mirrors is a major impediment for several types of high-precision interferometric experiments that aim to reach the standard quantum limit or to cool mechanical systems to their quantum ground state. This is for example the case of future gravitational wave observatories, whose sensitivity to gravitational wave signals is expected to be limited in the most sensitive frequency band, by atomic vibration of their mirror masses. One promising approach being pursued to overcome this limitation is to employ higher-order Laguerre-Gauss (LG) optical beams in place of the conventionally used fundamental mode. Owing to their more homogeneous light intensity distribution these beams average more effectively over the thermally driven fluctuations of the mirror surface, which in turn reduces the uncertainty in the mirror position sensed by the laser light.
We demonstrate a promising method to generate higher-order LG beams by shaping a fundamental Gaussian beam with the help of diffractive optical elements. We show that with conventional sensing and control techniques that are known for stabilizing fundamental laser beams, higher-order LG modes can be purified and stabilized just as well at a comparably high level. A set of diagnostic tools allows us to control and tailor the properties of generated LG beams. This enabled us to produce an LG beam with the highest purity reported to date. The demonstrated compatibility of higher-order LG modes with standard interferometry techniques and with the use of standard spherical optics makes them an ideal candidate for application in a future generation of high-precision interferometry.

Large laser-interferometers are being constructed to create a new type of astronomy based on gravitational waves. Their sensitivities, as for many other high-precision experiments, are approaching fundamental noise limits such as the atomic vibration of their components. We are pioneering technologies to overcome these limits using novel laser beam shapes.

Thermal noise in high-reflectivity mirrors is a major impediment for several  types of high-precision interferometric experiments that aim to reach the  ...

Automation of Mode Locking in a Nonlinear Polarization Rotation Fiber Laser through Output Polarization Measurements

When a laser is mode-locked, it emits a train of ultra-short pulses at a repetition rate determined by the laser cavity length. This article outlines a new and inexpensive procedure to force mode locking in a pre-adjusted nonlinear polarization rotation fiber laser. This procedure is based on the detection of a sudden change in the output polarization state when mode locking occurs. This change is used to command the alignment of the intra-cavity polarization controller in order to find mode-locking conditions. More specifically, the value of the first Stokes parameter varies when the angle of the polarization controller is swept and, moreover, it undergoes an abrupt variation when the laser enters the mode-locked state. Monitoring this abrupt variation provides a practical easy-to-detect signal that can be used to command the alignment of the polarization controller and drive the laser towards mode locking. This monitoring is achieved by feeding a small portion of the signal to a polarization analyzer measuring the first Stokes parameter. A sudden change in the read out of this parameter from the analyzer will occur when the laser enters the mode-locked state. At this moment, the required angle of the polarization controller is kept fixed. The alignment is completed. This procedure provides an alternate way to existing automating procedures that use equipment such as an optical spectrum analyzer, an RF spectrum analyzer, a photodiode connected to an electronic pulse-counter or a nonlinear detecting scheme based on two-photon absorption or second harmonic generation. It is suitable for lasers mode locked by nonlinear polarization rotation. It is relatively easy to implement, it requires inexpensive means, especially at a wavelength of 1550 nm, and it lowers the production and operation costs incurred in comparison to the above-mentioned techniques.

A protocol to detect and automate mode locking in a pre-adjusted nonlinear polarization rotation fiber laser is presented. The detection of a sudden change in the output polarization state when mode locking occurs is used to command the alignment of an intra-cavity polarization controller in order to find mode-locking conditions.

When a laser is mode-locked, it emits a train of ultra-short pulses at a repetition rate determined by the laser cavity length. This article outlines a ...

Free-form Light Actuators &#8212; Fabrication and Control of Actuation in Microscopic Scale

Liquid crystalline elastomers (LCEs) are smart materials capable of reversible shape-change in response to external stimuli, and have attracted researchers&#39; attention in many fields. Most of the studies focused on macroscopic LCE structures (films, fibers) and their miniaturization is still in its infancy. Recently developed lithography techniques, e.g., mask exposure and replica molding, only allow for creating 2D structures on LCE thin films. Direct laser writing (DLW) opens access to truly 3D fabrication in the microscopic scale. However, controlling the actuation topology and dynamics at the same length scale remains a challenge.
In this paper we report on a method to control the liquid crystal (LC) molecular alignment in the LCE microstructures of arbitrary three-dimensional shape. This was made possible by a combination of direct laser writing for both the LCE structures as well as for micrograting patterns inducing local LC alignment. Several types of grating patterns were used to introduce different LC alignments, which can be subsequently patterned into the LCE structures. This protocol allows one to obtain LCE microstructures with engineered alignments able to perform multiple opto-mechanical actuation, thus being capable of multiple functionalities. Applications can be foreseen in the fields of tunable photonics, micro-robotics, lab-on-chip technology and others.

Here, we fabricate 3D polymeric micro/nano structures in which both the shape and the molecular alignment can be engineered with nanometer scale accuracy by the use of direct laser writing. Light induced deformation of several types of liquid crystalline elastomer microstructures can be controlled in the microscopic scale.

Liquid crystalline elastomers (LCEs) are smart materials capable of reversible shape-change in response to external stimuli, and have attracted ...

Direct Imaging of Laser-driven Ultrafast Molecular Rotation

We present a method for visualizing laser-induced, ultrafast molecular rotational wave packet dynamics. We have developed a new 2-dimensional Coulomb explosion imaging setup in which a hitherto-impractical camera angle is realized. In our imaging technique, diatomic molecules are irradiated with a circularly polarized strong laser pulse. The ejected atomic ions are accelerated perpendicularly to the laser propagation. The ions lying in the laser polarization plane are selected through the use of a mechanical slit and imaged with a high-throughput, 2-dimensional detector installed parallel to the polarization plane. Because a circularly polarized (isotropic) Coulomb exploding pulse is used, the observed angular distribution of the ejected ions directly corresponds to the squared rotational wave function at the time of the pulse irradiation. To create a real-time movie of molecular rotation, the present imaging technique is combined with a femtosecond pump-probe optical setup in which the pump pulses create unidirectionally rotating molecular ensembles. Due to the high image throughput of our detection system, the pump-probe experimental condition can be easily optimized by monitoring a real-time snapshot. As a result, the quality of the observed movie is sufficiently high for visualizing the detailed wave nature of motion. We also note that the present technique can be implemented in existing standard ion imaging setups, offering a new camera angle or viewpoint for the molecular systems without the need for extensive modification.

We present a protocol for creating a real-time movie of a molecular rotational wave packet using a high-resolution Coulomb explosion imaging setup.

We present a method for visualizing laser-induced, ultrafast molecular rotational wave packet dynamics. We have developed a new 2-dimensional Coulomb ...

Chemistry

High-speed Continuous-wave Stimulated Brillouin Scattering Spectrometer for Material Analysis

Recent years have witnessed a significant increase in the use of spontaneous Brillouin spectrometers for non-contact analysis of soft matter, such as aqueous solutions and biomaterials, with fast acquisition times. Here, we discuss the assembly and operation of a Brillouin spectrometer that uses stimulated Brillouin scattering (SBS) to measure stimulated Brillouin gain (SBG) spectra of water and lipid emulsion-based tissue-like samples in transmission mode with &lt;10 MHz spectral-resolution and &lt;35 MHz Brillouin-shift measurement precision at &lt;100 ms. The spectrometer consists of two nearly counter-propagating continuous-wave (CW) narrow-linewidth lasers at 780 nm whose frequency detuning is scanned through the material Brillouin shift. By using an ultra-narrowband hot rubidium-85 vapor notch filter and a phase-sensitive detector, the signal-to-noise-ratio of the SBG signal is significantly enhanced compared to that obtained with existing CW-SBS spectrometers. This improvement enables measurement of SBG spectra with up to 100-fold faster acquisition times, thereby facilitating high spectral-resolution and high-precision Brillouin analysis of soft materials at high speed.

We describe the construction of a rapid continuous-wave-stimulated-Brillouin-scattering (CW-SBS) spectrometer. The spectrometer employs single-frequency diode-lasers and an atomic vapor notch-filter to acquire transmission spectra of turbid/non-turbid samples with high spectral-resolution at speeds up to 100-fold faster than those of existing CW-SBS spectrometers. This improvement enables high-speed Brillouin material analysis.

Recent years have witnessed a significant increase in the use of spontaneous Brillouin spectrometers for non-contact analysis of soft matter, such as ...

Low-cost Custom Fabrication and Mode-locked Operation of an All-normal-dispersion Femtosecond Fiber Laser for Multiphoton Microscopy

A protocol is presented to build a custom low-cost yet high-performance femtosecond (fs) fiber laser. This all-normal-dispersion (ANDi) ytterbium-doped fiber laser is built completely using commercially available parts, including $8,000 in fiber optic and pump laser components, plus $4,800 in standard optical components and extra-cavity accessories. Researchers new to fiber optic device fabrication may also consider investing in basic fiber splicing and laser pulse characterization equipment (~$63,000). Important for optimal laser operation, methods to verify true versus apparent (partial or noise-like) mode-locked performance are presented. This system achieves 70 fs pulse duration with a center wavelength of approximately 1,070 nm and a pulse repetition rate of 31 MHz. This fiber laser exhibits the peak performance that may be obtained for an easily assembled fiber laser system, which makes this design ideal for research laboratories aiming to develop compact and portable fs laser technologies that enable new implementations of clinical multiphoton microscopy and fs surgery.

A method is presented to build a custom low-cost, mode-locked femtosecond fiber laser for potential applications in multiphoton microscopy, endoscopy, and photomedicine. This laser is built using commercially available parts and basic splicing techniques.

A protocol is presented to build a custom low-cost yet high-performance femtosecond (fs) fiber laser. This all-normal-dispersion (ANDi) ytterbium-doped ...

Characterization of SiN Integrated Optical Phased Arrays on a Wafer-Scale Test Station

Optical phased arrays (OPAs) can produce low-divergence laser beams and can be used to control the emission angle electronically without the need for moving mechanical parts. This technology is particularly useful for beam steering applications. Here, we focus on OPAs integrated into SiN photonic circuits for a wavelength in the near infrared. A characterization method of such circuits is presented, which allows the output beam of integrated OPAs to be shaped and steered. Furthermore, using a wafer-scale characterization setup, several devices can easily be tested across multiple dies on a wafer. In this way, fabrication variations can be studied, and high-performance devices identified. Typical images of OPA beams are shown, including beams emitted from OPAs with and without a uniform waveguide length, and with varying numbers of channels. In addition, the evolution of output beams during the phase optimization process and beam steering in two dimensions is presented. Finally, a study of the variation in the beam divergence of identical devices is performed with respect to their position on the wafer.

Here, we describe the operation of a SiN integrated photonic circuit containing optical phased arrays. The circuits are used to emit low divergence laser beams in the near infrared and steer them in two dimensions.

Optical phased arrays (OPAs) can produce low-divergence laser beams and can be used to control the emission angle electronically without the need for ...

Transmission of Multiple Signals through an Optical Fiber Using Wavefront Shaping

The transmission of multiple independent optical signals through a multimode fiber is accomplished using wavefront shaping in order to compensate for the light distortion during the propagation within the fiber. Our methodology is based on digital optical phase conjugation employing only a single spatial light modulator, where the optical wavefront is individually modulated at different regions of the modulator, one region per light signal. Digital optical phase conjugation approaches are considered to be faster than other wavefront shaping approaches, where (for example) a complete determination of the wave propagation behavior of the fiber is performed. In contrast, the presented approach is time-efficient since it only requires one calibration per light signal. The proposed method is potentially appropriate for spatial division multiplexing in communications engineering. Further application fields are endoscopic light delivery in biophotonics, especially in optogenetics, where single cells in biological tissue have to be selectively illuminated with high spatial and temporal resolution.

We demonstrate the transmission of multiple independent signals through a multimode fiber using wavefront shaping employing a single spatial light modulator. By modulating the wavefront for each signal individually, spatially separated foci are transmitted. Potential applications are multiplexed data transfer in communications engineering and endoscopic light delivery in biophotonics.

The transmission of multiple independent optical signals through a multimode fiber is accomplished using wavefront shaping in order to compensate for the ...