Beam control key lab of Chinese academy of Sciences, the Institute of optics and electronics the Chinese Academy of Sciences, Chengdu, 610209, China

where , is the central position of the jth-Gaussian beam, ω0 = 0.0025 m is the waist width of each Gaussian beams, is the hard aperture and is expressed by

Planchon, T. A. et al. Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination. Nat. methods 8, 417–423 (2011).

Comparison of the amplitude distribution between coherently combined beams and zero-order BG beams, (a) coherently combined beams (b) zeroth-order BG beams.

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The comparison of the intensity profiles between combined BG beam and second-order BG beam is shown in Fig. 4 when the propagation distance is 250 m. It can be seen from Fig. 4 that the evolution of the intensity at the center area for the combined beam is similar to that of the BG beam. For example, the second-order ring near the main ring appears. However, there are many petals outside the rings of combined beam which is different from the BG beam. When increasing propagation distance, such as z= 500 m we can see from Fig. 5 that the coherently combined beam has the same multi-ring shape as that of the BG beam. However, the petals outside the rings of the combined beam still exist. Further increasing the propagation distance (z= 750 m), the intensity profiles of combined beam is very similar to those of the BG beam at the central area. Meanwhile, petals outside the rings of the combined beam still remain, as shown in Fig. 6.

Comparison of the intensity between coherently combined beams and second-order BG beams where z = 500 m, (a) coherently combined beams (b) second-order BG beams.

Zhou, P. et al. Coherent beam combination of three two-tone fiber amplifiers using stochastic parallel gradient descent algorithm. Opt. Lett. 34, 2939–2941 (2009).

Lightroom's Lens Corrections tool is great. Use the Basic tab for simple correction of chromatic aberration (the program should be able to recognise your lens from its database) or go to the Manual option for a finer degree of control over vertical and horizontal distortion, scale, and other parameters.

Ishaaya, A., Davidson, N. & Friesem, A. Passive Laser Beam Combining With Intracavity Interferometric Combiners. IEEE J. Sel. Top. Quant, 15, 301–311 (2009).

Lhermite, J. et al. Coherent combining of 49 laser beams from a multiple core optical fiber by a spatial light modulator. Opt. Express, 18, 4783–4789 (2010).

Comparison of the intensity evolution between coherently combined beams and zero-order BG beams where z = 250 m, (a) coherently combined beams (b) zero-order BG beams.

Milne, G., Jeffries, G. D. & Chiu, D. T. Tunable generation of Bessel beams with a fluidic axicon. Appl. Phys. Lett. 92, 261101 (2008).

A seed laser is split into 16 channels by the splitter. Each beam which phase is controlled by the phase modulators (PM) is amplified by multiple cascaded amplifiers. Then these beams pass through the collimators (CO) array and the beam combiner to reform the arrays as shown in Fig. 1a. The combined arrays are split into two beams by the mirror with high reflectivity. Most of the laser energy is reflected by the mirror. The residual transmitted laser which phase is modulated by the SLM is focused by a lens and coupled into the photoelectric sensor with a pinhole to offer the feedback signal to the phase control system. The optical phase in each channel is decoupled by the phase control system and the control signal is added to the phase modulator, so that the optical phase in each channel is the same when the phase control system is in a closed loop32,33. Because the phase of the residual transmitted laser is modulated by the discontinuous piston phase as shown in Fig. 1c, the reflected laser has the conjugate phase of SLM.

Gaussianbeamcalculator

where Jn represents the nth-order Bessel function of the first kind, (r0, θ0) is a pair of polar coordinates, a is a scale factor, w0 is the waist width of Gaussian part. Because a high order Bessel-beam () is a hollow beam with many rings, a Gaussian annular aperture is introduced to restrict the beam where b is the radius of the annular.

Prime lenses tend to reveal less distortion, and you can pick up some bargain 50mm and 85mm examples online – they tend to have constant wide apertures too, making them great for people shots. It's also got a lot easier to fix lens distortion in software.

Again, this is easily spotted when using a more extreme wide-angle lens – you'll notice straight lines at the edges of the picture 'bowing out,' like the shape of a barrel. It's obvious on a fisheye lens, too. Indeed barrel distortion helps to give fisheye lenses their distinctive look.

Gaussianbeampdf

To confirm the validity of the method, the comparison of the intensity profiles between the combined beam and the BG beam at different propagation distance is simulated by the see-light software developed by National University of Defense Technology and Institute of software Chinese academy of Sciences where the wavelength is 550 nm. The see-light software is based on the angular spectrum approach and two-dimensional fast Fourier transform. Simulation block diagram for the generation and propagation of combined second-order BG beam is shown in Fig. 3. Namely, 16-Gaussian beams are arranged as in Fig. 1a. The phase of each Gaussian beam is modulated by different piston phase where .

This effect, were lines do the opposite and bend inwards, is a common problem with longer telephoto lenses. Sometimes you have to look hard to spot this effect, but it's pretty obvious if you are shooting a rectangular shape, such as a window frame.

The project was supported by National Natural Science Foundation of China (No.11374264) and beam control key lab foundation of Chinese academy of Sciences (2014LBC004).

A Bessel beam which is the exact solution of the Helmholtz equation has infinite energy and cannot be generated in practice. Research shows that BG beam with finite energy has some properties as that of Bessel beam9,10. In this paper, the expression for the BG beam at the initial plane (z = 0) is given as

Here R= 1.1 ω0 is the radius of the hard aperture. The vortex phase is composed approximately of piston phases, namely , see Fig. 1c. In practice, the different piston phase of each Gaussian beam can be modulated by the phase locking technology.

Comparison of the intensity evolution between coherently combined beams and zero-order BG beams where z = 500 m, (a) coherently combined beams (b) zero-order BG beams.

Comparison of the intensity evolution between coherently combined beams and zero-order BG beams where z = 750 m, (a) coherently combined beams (b) zero-order BG beams.

Gaussianbeamintensity formula

Comparison of the intensity evolution between coherently combined beams and second-order BG beams where z = 250 m, (a) coherently combined beams (b) second-order BG beams.

As mentioned, you shouldn't get OCD about lens distortion and try to eradicate it altogether. It's a fact of life, but having a good range of lenses (not just the cheap 18-5mm kit lens that came with your camera) will mean it's less of a headache.

Xiao, Y. et al. 1-kilowatt CW all-fiber laser oscillator pumped with wavelength-beam-combined diode stacks. Opt. Express, 20, 3296–3301 (2012).

Hwang, C., Kim, K. & Lee, B. Bessel-like beam generation by superposing multiple Airy beams. Opt. Express, 19, 7356–7364 (2011).

Here is the intensity distribution. We can see that the variational trend of the second moment for the combined beam is the same as that of the BG beam. However, the difference is growing with the increase of propagation distance.

Eyyuboglu, H. T. et al. Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere. Appl. Phys. B., 93, 605–611 (2008).

Secondly, we can also generate the zeroth-order BG beam by the coherently combined beams. The amplitude distribution of the zeroth-order BG beam and combined Gaussian beams are shown in Fig. 2 where the BG beam is also expressed in Eq.(1) with n= 0, a= 200 m−1, w0 = 0.015 m and b= 0. The combined beams are composed of 9 Gaussian beams and are expressed as

The simulation software is developed by National University of Defense Technology, China and Institute of software Chinese academy of Sciences. The principle of the software is based on the angular spectrum theory and fast Fourier transform. Because fiber lasers have lots of advantages over other kinds of lasers, such as high conversion efficiency, convenient heat management and excellent beam quality32,33, to confirm the validity of the technique proposed in this paper, an experimental setup for the generation of 2-order BG beams by fiber lasers is depicted in Fig. 12.

Gaussianbeamdivergence

Litvin, I. A., McLaren, M. G. & Forbes, A. A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles. Opt. Commun. 282, 1078–1082 (2009).

How to cite this article: Chu, X. et al. Generating a Bessel-Gaussian beam for the application in optical engineering. Sci. Rep. 5, 18665; doi: 10.1038/srep18665 (2015).

Chu, X., Sun, Q., Wang, J. et al. Generating a Bessel-Gaussian beam for the application in optical engineering. Sci Rep 5, 18665 (2016). https://doi.org/10.1038/srep18665

Apart from distortion to the shape of objects and lines and edges, lens distortion can cause fringing. Use a cheap and nasty lens and zoom in on some edges and you can often spot chromatic aberration, or colour fringing. It tends to be less of a problem on better quality lenses, and is also harder to spot at more conservative focal lengths.

X.C. and X.X. wrote the main manuscript text and prepared Figures 1,2,3, Q.S. and W.X. prepared Figures 4,5,6,7,8,9. J.W. and P.L. simulated and compared the propagation. All authors reviewed the manuscript.

Because the multi-ring-shaped beam has formed when the propagation distance is larger than 200 m, the energy efficiency after 200 m has been plotted. From Fig. 7 we can see that the energy efficiency is very lower when the propagation distance is shorter. With the increase of propagation distance, the energy efficiency gradually tends to a constant value. However, the energy efficiency of the combined beam is much less than that of the BG beam.

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Gaussianbeamsoftware

Tsampoula, X. et al. Femtosecond cellular transfection using a nondiffracting light beam. Appl. Phys. Lett. 91, 053902–053904 (2007).

Li, J. et al. High-power coherent beam combining of two photonic crystal fiber lasers. IEEE Photonic Tech. L. 20, 888–890 (2008).

Gong, L. et al. Generation of nondiffracting Bessel beam using digital micromirror device. Appl. Optics, 52, 4566–4575 (2013).

Since Bessel beam is the important member of the family of non-diffracting beams, many methods have been proposed to generate this beam14,15,16,17,18,19,20,21,22,23. However, because of beam spreading and the loss of energy during propagation, the energy of Bessel beam generated by those methods for long distance propagation is still a problem, such as in quantum and optical communication. In optical engineering, phase locking and coherent combining technology can be used to generate laser source of high brightness by several laser sources25,26,27,28,29,30,31,32. In the past few years, coherent beam combining have multiplied their output power reaching kilowatts24,33. In this paper, we investigate the validity for generating Bessel-Gaussian (BG) beam by coherent combining technology. Meanwhile, we suggest an idea to form a vortex phase by using discontinuous piston phase which can be used in the generation of many other vortex beams23.

If you want to up your photography game, join award-winning photographer, author and teacher Michael Freeman on his online Photography Foundation course. The course covers all aspects of composition, and you'll receive a thorough grounding for pursuing photography seriously.

Science and technology on integrated information system laboratory, Institute of software Chinese Academy of Sciences, Beijing, 100190, China

At first, we compose a second-order BG beam (n = 2) by using 16-Gaussian beams. The intensity distribution and phase map of the combined beams and the second-order BG beam is given in Fig. 1 where a = 200m−1, w0 = 0.0025 m and b = 0.015 m. To compose the second-order BG beam, as shown in Fig. 1a, the 16-Gaussian beams are given as

Gaussianbeamradius

Comparison of the intensity between coherently combined beams and second-order BG beams where z = 750 m, (a) coherently combined beams (b) second-order BG beams.

Mazilu, M., Stevenson, D., Gunn-Moore, F. & Dholakia, K. Light beats the spread: “non-diffracting” beams. Laser Photonics Rev. 4, 529–547 (2010).

Bessel beam is the important member of the family of non-diffracting beams and has many novel properties which can be used in many areas. However, the source of Bessel beam generated by the existing methods can be used only in a short distance due to its low power. In this paper, based on the coherent combining technology, we have proposed a method which can be used to generate a high-power Bessel beam. Even more, we give an innovative idea to form vortex phase by using discontinuous piston phase. To confirm the validity of this method, the intensity evolution of the combined beam and the Bessel-Gaussian beam at different propagation distance have been studied and compared. Meanwhile, the experimental realization has been discussed from the existing experimental result related to the coherent combining technology.

Fahrbach, F. O. et al. A. Self-reconstructing sectioned Bessel beams offer submicron optical sectioning for large fields of view in light-sheet microscopy. Opt. express, 21, 11425–11440 (2013).

A prime lens with a similar focal length, e.g. 24mm, will reveal less distortion as it's easier for makers to compensate for distortion when they only have one focal length to worry about. This relative lack of distortion is one of the reason that prime 85mm lenses are considered the 'perfect' portrait lens, for example.

Experimental setup for the generation of a BG beam by the coherent combining technology where PM is phase modulators array and CO is collimators array.

Zhou, P. et al. Coherent Beam Combining of Fiber Amplifiers Using Stochastic Parallel Gradient Descent Algorithm and Its Application. IEEE J. Sel. Top. Quant, 15, 248–256 (2009).

To compare qualitatively the evolution of the intensity for the two beams, the variation of the second moment of intensity is shown in Fig. 11 where the second moment of intensity is defined as

Nelson, W., Palastro, J. P., Davis, C. C. & Sprangle, P. Propagation of Bessel and Airy beams through atmospheric turbulence. J. Opt. Soc. Am. A. 31, 603–609 (2014).

Gaussianbeamformula

One of the easiest ways to spot lens distortion is to attach a wide-angle lens, or shoot at the wide end of a zoom lens. The wider angled the lens, the more curved that straight lines will appear, as you can see with the distorted edges in the image below, taken with an otherwise good-quality, 24-70mm f/2.8 zoom.

In recent years, many new interesting properties of non-diffracting beam were intensively investigated. The particular attention was focused on the applications in both the fundamental and applied physics1,2,3,4. For example, the self-healing property of Airy beams leads to a new feature in optical micromanipulation5, a holographically shaped and scanned Bessel beam not only reduces scattering artefacts6, but also simultaneously increases the quality of the image and penetration in dense media7,8. The interesting properties and successful application of non-diffracting beam in micro-imaging have attracted many attentions in other fields. In free-space, quantum communication or between quantum processing systems, Bessel beam offers advantages to overcome the losses of quantum entanglement between photon pairs9,10. In the applications in atmosphere, such as in optical communication, non-diffracting beam exhibits more resilience against perturbations11,12,13.

Just be careful, however, as it's easy to widen or lengthen people at the same time as you are straightening the edges of buildings behind them!

Gaussianbeamprofile

Chattrapiban, N. et al. Generation of nondiffracting Bessel beams by use of a spatial light modulator. Opt. Lett. 28, 2183–2185 (2003).

Existing results show that the fiber pigtailed phase modulator and acoustics-optics frequency shifter make possible the active phase control with MHz-level control bandwidth which could be used for phase locking in the experiment32,33. Many experiments show that this method can obtain a satisfied result in beam combining with high precise phase control. Since not only the distribution of the intensity given in Fig. 1a, but also the discontinuous piston phase as shown in Fig. 1a can be generated, these results confirm the validity of the method.

Baumgartl, J., Mazilu, M. & Dholakia, K. Optically mediated particle clearing using Airy wavepackets. Nat. Photon. 2, 675–678 (2008).

Zhu, K. C. et al. Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere. Opt. Express, 16, 21315–21320 (2008).

Cizmar, T. et al. Generation of multiple Bessel beams for a biophotonics workstation. Opt. Express, 16, 14024–14035 (2008).

Carbajal-Dominguez, A., Bernal, J., Martin-Ruiz, A. & Niconoff, G. Generation of J0 Bessel beams with controlled spatial coherence features. Opt. Express, 18, 8400–8405 (2010).

The evolution of the combined zeroth-order BG beam and the corresponding BG beam also are simulated by the see-light software where the wavelength is 550 nm. The intensity distributions with different propagation distance are shown in Figs 8, 9, 10. It can be seen that the main spots of the combined zeroth-order BG beam is very similar to that of the BG beam during propagation. For example, with the increase of propagation distance, the central spots of the two types of the beams spread with the same speed and the second ring come together with the central spots. Meanwhile, the intensity uniformity of the combined beam becomes better during propagation.

Zhou, P. et al. Coherent beam combination of two-dimensional high power fiber amplifier array using stochastic parallel gradient descent algorithm. Appl. Phys. Lett. 94, 231106 (2009).

where is the hard aperture with radius 0.012 m located at the center, , m, m is the central position of the jth-Gaussian beam, ω1 = 0.006 m is the waist width of the Gaussian beam located at the position of the first ring, is the hard aperture with radius 0.007 m located at (cxj, cyj), A0 = 0.1 and express the maximum amplitude and the phase of the 8-Gaussian beams located at the location of the first ring, respectively.

Comparison of the intensity distribution and phase map between coherently combined beams and second-order BG beams (a) intensity distribution of coherently combined beams (b) intensity distribution of second-order BG beams (c)phase map of coherently combined beams (d) phase map of second-order BG beams.

Vasilyeu, R., Dudley, A., Khilo, N. & Forbes, A. Generating superpositions of higher-order Bessel beams. Opt. Express, 17, 23389–23395 (2009).

Kaminer, I., Bekenstein, R., Nemirovsky, J. & Segev, M. Nondiffracting Accelerating Wave Packets of Maxwell’s Equations. Phys. Rev. Lett. 108, 163901 (2012).

The evolution of the intensity for coherently combined beams shows that the petals outside the rings always exist. Comparing with the BG beams, the petals will cause the loss of the energy. The variation of the loss of the energy with propagation distance is shown in Fig. 7. It should be pointed out that the energy efficiency of the combined beams is defined as the ratio between the energy of the first ring and the total energy. Because our purpose is to generate BG beam, the energy efficiency between the BG beams and the combined beams is compared.

You may think that once you have bought a decent lens for your SLR, or a compact camera with a decent lens built-in, that you can just forget about it.

Unfortunately, there is no such thing as a perfect lens, just as it's hard to find a perfect person, and there will always be some distortion – you just notice it a lot more with cheaper lenses. It's not necessarily about spending more, however. Certain types of lenses, such as wide-angles and zooms, will always be prone to some distortion.