Increasing the numerical aperture of an optical system may increase the resolving power, but it’s not a foolproof way to improve performance.

Key Takeaways: The paraxial approximation simplifies optical calculations by assuming small angular deviations, useful for systems like lasers and optical fibers. It’s accurate for angles under 10 degrees, where errors are less than 0.5%, but fails at larger angles, requiring more precise methods.  For waveguides with small effective mode areas or strong focusing systems, the approximation becomes unreliable.  In such cases, more advanced techniques are necessary, including distinguishing between meridional and sagittal rays or using beam propagation methods for precise results. Understanding the Paraxial Approximation The paraxial approximation (approximation at small angles) is a useful mathematical trick for doing ray tracing and simplifying optical calculations. It assumes that angular deviations in propagation from a beam axis are minimal, and is a key part of the optical designer’s toolbox, especially when it comes to laser physics and fiber optics. But it’s important to check and double-check to be sure that this approximation is valid for your optical system.  While this approximation works well for many small angles, it becomes more and more problematic as ray angles increase. If you utilize the paraxial approximation where it doesn’t apply you may end up with a system that performs very differently in practice than it did on paper.  The paraxial approximation leads to minimal errors in many applications, but there are other situations in which the error ends up being very large. This article is the sixth part of a multi-part series on common mistakes in the optics industry. Each article will look at a topic that is often confused, and provide a brief, easy-to-understand explanation of the key foundational concepts involved.  What is the Paraxial Approximation? In geometrical optics we consider that light is propagated in the form of geometric rays. When we do this,  the paraxial approximation allows us to assume that this propagation doesn’t deviate much from the propagation axis. More specifically, we can assume that the tangent and sine of all angles are the same as the actual angles, in radians. We could write this: tan(u) = u   and sin(u) =U This is illustrated below for an optical system in which the index of refraction goes from n to n’ at the y-axis. In the system below the paraxial approximation allows us to assume that sin(i) = I, sin(I’)=i, sin(u)= u, and sin (u’) = u’.  The paraxial approximation allows us to perform simple ray tracing for small angles. When Does the Paraxial Approximation Not Apply? The paraxial approximation is a suitable choice in many situations in which one might be working with small angles; with light rays close to the optical axis. In general, it can be considered a good approximation (deviating by no more than 0.5%) if the angle is under 10 degrees.  But what happens when the angle gets larger? Then the paraxial approximation is no longer valid, and you’ll need to do more explicit calculations. When you’re working with large angles you may need to distinguish between those angles that lie on a plane with the optical axis—- we call these meridional rays— and those that are on another plane, sagittal rays.  One situation in which the paraxial approximation is often used is in the analysis of propagation modes of waveguides for optical fibers. When divergence in the beams exiting a given wavelength is small and the effective mode area is large, this approximation leads to simple, straightforward calculations and an accurate estimate of how the light will propagate. The paraxial approximation shouldn’t be used with large beam divergence or with a small effective mode area.   Another area in which you can’t use the paraxial approximation is when working with extra strong focusing. In this situation it isn’t only the large angle that complicates the issue; polarity also cannot be ignored. For optical systems with very strong focusing, you will want to use beam proportions methods that involve a two-dimensional array of complex field amplitudes rather than the paraxial approximation.  Paraxial approximation is key to ray tracing in many fields, but it’s important to be aware of when it works— and when it doesn’t] Ray Tracing and Optical Design at Avantier When you work with optical designers and engineers at Avantier, you know you’re working with world-class experts. Our optical engineers can walk you through important design considerations for your application, and help you determine whether or not the paraxial approximation applies in your case. More than fifty years working on optical systems like yours have given our teams the know-how to avoid optical pitfalls and design effective, high-quality optical systems that provide you with the ultimate performance you need. Contact us today to set up your first consultation or place a custom order.  Related Content

KeyTakeaways: Optical medical imaging is transforming diagnostics, surgery, and research. Diffusive optical imaging (DOI) uses NIR spectroscopy to study soft tissue, while ballistic optical imaging detects unscattered photons for high-resolution imaging, offering better optical quality than MRI or ultrasound.  Photoacoustic imaging combines optical and ultrasonic methods for deep, detailed scans. Positron emission tomography (PET) tracks radioactive tracers for real-time molecular insights, though radiation precautions are needed.  Avantier offers custom optical solutions for advanced medical applications. Optical Medical Imaging Whether your focus is medical diagnostics, surgery, clinical care, or biomedical research, optical medical imaging can play a transformational role in your practice. Recent advances in optical research are pushing the bar on medical imagery, and today’s practitioners have tools at their fingertips that could only have been dreamed of a few decades ago.  Here we’ll look at two basic types of optical medical imagery that are both seeing transformative advances: diffusive optical imaging and ballistic optical imaging. We’ll also look briefly at nuclear imaging and a special hybrid type of medical imaging: photoacoustic imaging.  Diffusive Optical Imaging Diffusive optical imaging (DOI) involves using NIR spectroscopy or fluorescence-based imaging to elucidate the optical properties of biological tissue. Most useful for soft tissue, a strongly diffusive media, this type of imaging can be used to determine the real-time concentration of key chromophores in the tissue through measurement of absorption and scattering coefficients. For instance, DOI can be used to monitor changes in oxygenated or deoxygenated hemoglobin, or the redox states of cytochromes.  One possible setup for DOI involves time-resolved excitation and detection schemes. A multiple picosecond diode laser sends pulses to the tissue to be imaged. As the laser pulses exit the tissue, temporal profiles are recorded using time-correlated single photon counting (TSPC). The shape of the pulses, analyzed with physical models of the absorption and scattering properties of the tissue, is used to generate images.  When diffusive optical imaging is used to create 3D images it is called diffusive optical tomography (DOT).  Breast cancer imaging, stroke detection, muscle functional studies, and brain functional imaging are just a few of the fields in which DOI and DOT are applied on a regular basis.  A mammography captured by diffusive optical imaging, a noninvasive form of optical medical imaging. Ballistic Optical Imaging Ballistic optical imaging produces high-resolution images of biochemical tissue using techniques like optical coherence tomography (OCT) and ballistic scanners featuring ultrafast time gates. Unlike diffusive optical imaging, this type of imaging relies on detecting photons that travel through the tissue in a straight line without being scattered or ‘diffused’. Although ballistic optical imaging technology has a limited working depth, the images it produces can have better optical  qualities than images produced by alternate imaging technology like magnetic resonance imaging (MRI), ultrasound, or X-ray.  When the procedure is used for deeper tissue, image processing techniques are used to reconstruct higher-quality images from multiple raw ballistic images.  Photoacoustic Medical Imaging Photoacoustic imaging (also known as optoacoustic imaging) is a revolutionary new type of medical imaging that combines high contrast optical imaging with deep penetrating ultrasonic imaging. The result is high resolution images that provide both structural and functional information.  Non-invasive and non-ionizing, the method produces diverse endogenous and exogenous contrast and good imaging depth. Types of photoacoustic medial imagery include photoacoustic tomography (PAT) and photoacoustic microscopy (PAM). Different configurations for photoacoustic medical imagery enable different penetrations and resolutions. Positron Emission Tomography  Positron emission tomography (PET) works in a way fundamentally different from the other imaging modalities we’ve looked at here. A type of nuclear imaging, PET is based on tracking a radio tracer, an injectable radioactive chemical that is absorbed at different rates in different tissues. High absorption of the radio tracer signals diseased or cancerous cells, allowing medical professionals to diagnose issues that might be hard to identify with other imaging methods.  Unlike computer tomography (CT) and magnetic resonance imaging (MRI), a PET scan is a imaging procedure that focuses on real-time molecular activity and processes. It may be performed at the same time as other imaging techniques for high resolution 3D images.  While the method forms an important part of medical diagnostics, the procedures are time consuming, and precautions around the use of radioactive materials used have limited the ease of use of this method.  Optical Medical Imaging at Avantier At Avantier, we produce high performance custom optics for cutting-edge medical research, biomedical engineers, and clinical applications. Whether you know exactly what optic you require or need assistance to design an imaging system for your medical application, we can work with you.  Our experienced team of optical designers and engineers is available to put their expertise to work for your application, and we are proud of our track record of providing custom optical systems to satisfied clients worldwide. Contact us today to set up an initial consult or inquire about the custom order process.  Related Content

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