Assuming that the thickness of the lens is small compared to the radius of curvature, the thin lens approximation can be used to determine the appropriate focal length for the asphere. Assuming a divergence angle of 30° (FWHM) and desired beam diameter of 3 mm:

Estimated inclination coefficients for evolution of light scattering in the lens as a function of time and their dependence on temperature.

Using the laser diode and aspheric lens chosen above, we can use an anamorphic prism pair to convert our collimated, elliptical beam into a circular beam.

The specifications for the P1-630A-FC-2, 630 nm, FC/PC single mode patch cable indicate that the mode field diameter (MFD) is 4.3 μm. This specification should be matched to the diffraction-limited spot size given by the following equation:

Whereas earlier we considered only the larger divergence angle, we now look at the smaller beam divergence of 8°. From this, and using the effective focal length of the A390-B aspheric lens chosen in Example 1, we can determine the length of the semi-minor axis of the elliptical beam after collimation:

For a certain tissue, the original concentration of native state molecules is constant. Then, Eq. (7) can be simplified by substituting A and [N] with a rate constant, P (mol·l−1·s−1) dΩdt=P·e−EaRTor Eq. (7)lndΩdt=ln P−EaR·1T.

Molded glass aspheres are manufactured from a variety of optical glasses to yield the indicated performance. The molding process will cause the properties of the glass (e.g., Abbe number) to deviate slightly from those given by glass manufacturers. Specific material properties for each lens can be found by clicking on the Info Icon  in the tables below and selecting the Glass tab.

A selection of the lenses sold on this page are designed for collimating laser diodes. As seen in the tables below, a compatible laser window thickness is listed for these lenses. In these instances, the numerical aperture (NA), working distance (WD), and wavefront error of these lenses are defined based on the presence of a laser window of the indicated thickness (not included).

To determine the dependence of denaturation rate on heat load in vivo, it would be necessary to measure denaturation rate as heat-induced rate of back scattering. In vitro determined dependence of denaturation rate on temperature can be used to estimate the heat load-induced lens temperature. Consequently, the relationship between temperature in the lens and in vivo heat load exposure of the eye can be determined. Then, the critical temperature in the lens can be estimated by in vivo exposures at incrementing heat load exposure of the eye with subsequent postexposure measurements of permanent light scattering in the lens (Fig. 6).

(a) The estimated confidence interval for predicted temperature as a function of sample size of additional measurements, adopting the outcome of the estimates of inclination coefficients for increase of light scattering as a function of time and temperature. (b) Continuous line is the regression line for predicted temperature as a function of inclination coefficient. Dashed lines are CI(0.95) for the regression line. Vertical line corresponds to an example of an estimated inclination coefficient averaged from 20 additional measurements. Horizontal lines indicate the associated confidence limits for predicted temperature.

The Arrhenius equation models the denaturation rate as a function of temperature. On the assumption that light scattering in the lens depends on denaturation of lens proteins, the Arrhenius equation can be applied to experimentally establish the relationship between temperature at exposure and rate of light scattering increase (Appendix A). The long-term goal is to establish data on in vivo lens protein denaturation after heat load that allows rational criteria for safe exposure to infrared radiation.

Averaged intensity of light scattering as a function of exposure time from 30 to 300 s in lenses exposed to the temperatures in the interval 37 to 46°C.

Schematic of temperature-controlled lens cuvette. (a) Water bath and associated pump driving temperature-controlled water through a heater, indirectly regulating the temperature in the lens cuvette filled with BSS and (b) water channel bypassing the cuvette. Arrows indicate in and out flow of circulating water.

The finding that the inclination coefficient for light scattering increase increased with temperature (Fig. 3 and Table 1) agrees with the Arrhenius equation (Appendix A). Therefore, for the selected time and temperature range, it is possible to estimate temperature in the lens with light scattering increase rate.

The light scattering measurement was recorded for 5 min during exposure. At the beginning of exposure, the heat transfer to the lens still occurred. Therefore, the intensity of forward light scattering in the time window 30 to 300 s was selected for fitting light scattering as a function of time with regression.

At temperature saturation, heat application is in balance with heat loss through heat diffusion and convection. Thus, temperature at saturation, T, is independent of the exposure time, and the denaturation rate, dΩ/dt, is directly proportional to the exposure time, t. Therefore, Eq. (7) can be rewritten as follows: Ω=t·P·e−EaRTor Eq. (8)ln(Ω)=ln(t)+ln(P)−EaR·1T.

The aspheric surfaces of these lenses may be described using a polynomial expansion in Y, the radial distance from the optical axis. The surface profile or sagitta (often abbreviated as sag) is denoted by z, and is given by the following expression:

The natural logarithm of the inclination coefficient for light scattering increase as a function of the inverse of the absolute temperature. Open circles are estimates of the natural logarithm of the inclination coefficient for each lens. Continuous line is the regression line. Dashed lines are CI(0.95) for the regression line. Dotted lines are CI(0.95) for one additional measurement of inclination coefficient for prediction of the inverse temperature.

Since the output of a laser diode is highly divergent, collimating optics are necessary. Aspheric lenses do not introduce spherical aberration and therefore are commonly chosen when the collimated laser beam is to be between one and five millimeters. A simple example will illustrate the key specifications to consider when choosing the correct lens for a given application. The second example below is an extension of the procedure, which will show how to circularize an elliptical beam.

Arrhenius demonstrated that the reaction rate constant, k(T) (s−1), varies exponentially with the inverse of the molecular enthalpy, the general gas constant, R·(8.31  J·mol−1·K−1), multiplied with the temperature, T (K−1), and a proportionality constant, the activation energy, Ea (J·mol−1), determined by a proportionality constant, A (s−1), the pre-exponential factor Eq. (5)k(T)=Ae−EaRT.

Clinically, a cataract infers permanent light scattering in the lens that decreases vision perception. To predict permanent light scattering in the lens induced by thermal exposure, experimentally or by modeling, it is important to know the critical temperature for permanent light scattering in the lens. The critical temperature for permanent light scattering in the lens is currently unknown because of the difficulties in measuring the in vivo temperature during thermal exposure.

The activation energy calculated for temperature-induced aggregation of protein in whole lens, 8.0±2.0×101  kJ·mol−1, based on the outcome depicted in Fig. 4, is in the range of what has previously been reported for temperature-induced conformational change of γ-crystalline tryptophan,13 temperature-induced α-crystalline aggregation,14 and chemically induced aggregation of γF-crystallin.15

The relative intensity of scattered light, Is, can be considered directly proportional to the concentration of denatured stated proteins, Ω (mol·l−1), with a proportionality constant, l (mol−1·l) Eq. (2)Is=l·Ω.

The purpose of this study was to explore the feasibility for experimental indirect estimation of lens temperature based on light scattering measurement in the lens.

Refer to the diagram to the right for α1 and α2 definitions. Our 780 nm laser will experience slightly less magnification than a 670 nm beam passing through the prisms at these angles. Some trial and error may be required to achieve the exact desired magnification. In general:

Averaging a very large sample of additional measurements, the estimated confidence limits for the estimated temperature approximate ±0.9°C.

All of the molded glass lenses featured on this page are available with an antireflection coating for either the 600 - 1050 nm or 650 - 1050 nm range deposited on both sides. Other AR coating options are listed in the Aspheric Lens Selection Guide table at right.

The currently presented method for measurement of lens temperature during heat load is to our knowledge the only available method that allows temperature measurement without disturbance from the measurement sensor.

where R is the radius of curvature, k is the conic constant, and the An are the nth order aspheric coefficients. The sign of R is determined by whether the center of curvature for the lens surface is located to the right or left of the lens' vertex; a positive R indicates that the center of curvature is located to the right of the vertex, while a negative R indicates that the center of curvature is located to the left of the vertex. For example, the radius of curvature for the left surface of a biconvex lens would be specified as positive, while the radius of curvature for its right surface would be specified as negative.

The exposure setup consisted of a cuvette that has an inner water channel bypassing a central well (Fig. 2). A pump (MD-10, IWAKI Co., Japan) drives water from a temperature-controlled water bath (VWB 12, VWR, Germany) in a closed loop through a heater, indirectly regulating the temperature in the lens cuvette filled with a sterile balanced salt solution (BSS: BSS Sterile Irrigating Solution, Alcon).

To measure the in vivo temperature in the lens during thermal exposure is challenging. A thermocouple probe damages the lens and decreases the heat load due to heat conduction in the probe.5–7 Fiber optic sensors are based on fluorescence excited in the near-infrared waveband and are therefore unsuitable for temperature measurement due to infrared radiation heat load.

The absolute resolution limit of the currently presented method is around ±1°C even if a very large sample size is averaged [Fig. 5(a)]. The resolution limit is determined by the variability in scattering-time response as a function of temperature among individuals (Fig. 4). However, with 20 additional measurements of inclination coefficients, the resolution is on the order of ±2°C. For experiments aiming at estimating lens sensitivity to heat load, ±2°C is sufficient.

Experimentally, the difference of light scattering between temperature-induced intensity of light scattering and base line intensity of light scattering, ΔIs, is a measure of the temperature-induced concentration of denatured state proteins ΔIs=l·Ωor Eq. (3)Ω=ΔIsl.

The PS883-B mounted prism pair provides a magnification of 4.0 for a 950 nm wavelength beam. Because shorter wavelengths undergo greater magnification when passing through the prism pair, we can expect our 780 nm beam to be magnified by slightly more than 4.0X. Thus, the beam will still maintain a small degree of ellipticity.

When choosing a collimation lens, it is essential to know the divergence angle of the source being used and the desired output diameter. The specifications for the L780P010 laser diode indicate that the typical parallel and perpendicular FWHM beam divergences are 8° and 30°, respectively. Therefore, as the light diverges, an elliptical beam will result. To collect as much light as possible during the collimation process, consider the larger of these two divergence angles in any calculations (i.e., in this case, use 30°). If you wish to convert your elliptical beam into a round one, we suggest using an anamorphic prism pair, which magnifies one axis of your beam; for details, see Example 2 below.

Thermal cataracts were first identified in the late 1800s.1 Both clinical observations of cataracts in glassblowers and steel workers and experimental studies have indicated an association between thermal exposure and cataract formation.2–4

Per Söderberg is a professor and chair of ophthalmology at Uppsala University. He focuses on the interaction of optical radiation and the eye, cataract measurement, contrast sensitivity, virtual reality cataract surgery, corneal endothelium biometry, and digital visual acuity charts.

Aspheric lenses focus or collimate light without introducing spherical aberration into the transmitted wavefront. For monochromatic sources, spherical aberration often prevents a single spherical lens from achieving diffraction-limited performance when focusing or collimating light. Aspheric lenses are designed to mitigate the impacts of spherical aberration and are often the best single element solution for many applications including collimating the output of a fiber or laser diode, coupling light into a fiber, spatial filtering, or imaging light onto a detector.

These lenses can be purchased unmounted or premounted in nonmagnetic 303 stainless steel lens cells that are engraved with the Item # for easy identification. All mounted aspheres have a metric thread that make them easy to integrate into an optical setup or OEM application; they can also be readily used with our SM1-threaded (1.035"-40) lens tubes by using our aspheric lens adapters. When combined with our microscope objective adapter extension tube, mounted aspheres can be used as a drop-in replacement for multi-element microscope objectives.

A good rule of thumb is to pick a lens with an NA twice that of the laser diode NA. For example, either the A390-B or the A390TM-B could be used as these lenses each have an NA of 0.53, which is more than twice the approximate NA of our laser diode (0.26). These lenses each have a focal length of 4.6 mm, resulting in an approximate major beam diameter of 2.5 mm. In general, using a collimating lens with a short focal length will result in a small collimated beam diameter and a large beam divergence, while a lens with a large focal length will result in a large collimated beam diameter and a small divergence.

Up to this point, we have been using the full-width at half maximum (FWHM) beam diameter to characterize the beam. However, a better practice is to use the 1/e2 beam diameter. For a Gaussian beam profile, the 1/e2 diameter is almost equal to 1.7X the FWHM diameter. The 1/e2 beam diameter therefore captures more of the laser diode's output light (for greater power delivery) and minimizes far-field diffraction (by clipping less of the incident light).

To allow for a short time constant for temperature equilibrium, a small mass should be heated. Therefore, the rat lens was selected for this experiment.

The minor beam diameter is double the semi-minor axis, or 0.64 mm. In order to magnify the minor diameter to be equal to the major diameter of 2.5 mm, we will need an anamorphic prism pair that yields a magnification of 3.9. Thorlabs offers both mounted and unmounted prism pairs. Mounted prism pairs provide the benefit of a stable housing to preserve alignment, while unmounted prism pairs can be positioned at any angle to achieve the exact desired magnification.

To allow temperature measurement based on denaturation, the time constant for heat transfer must be small in relation to the time constant for denaturation. An estimate assuming Newton’s law of heating indicated that for the current experiment this requirement holds.

The intensity of light scattering, ΔIs, recorded at exposure times between 30 and 300 s was fit to a linear model [Eq. (1)], considering an initial light scattering, k0 (tEDC), and an inclination coefficient, k1 (tEDC/s) Eq. (1)ΔIs=k0+k1·t.

An in vitro study of cold cataracts indicated that temperature-induced light scattering varies little in the range from 16 to 37°C.10 In a pre-experiment, we found that the inclination coefficient for light scattering increases as a function of time up to a temperature of 48°C and approximates an asymptote toward higher temperatures. A study of lenses from multiple species showed that at temperatures between 55°C and 65°C most of the soluble proteins are lost.11 In the present investigation, we aimed at temperatures just above the threshold for denaturation. Therefore, the minimum exposure temperature was set to 37°C and the maximum was set to 46°C.

Here, f is the focal length of the lens, λ is the wavelength of the input light, and D is the diameter of collimated beam incident on the lens. Solving for the desired focal length of the collimating lens yields

Considering the sample size in the current experiment, the confidence interval for the inclination coefficient as a function of temperature (Table 1) reflects a substantial variation of scattering-time response among lenses from different animals. The temperature in the cuvette was controlled to within ±0.1°C. Therefore, the observed variation in scattering-time response most probably reflects variability in temperature sensitivity in the lens among individuals. This is also revealed by the individual estimates of the natural logarithm of the inclination coefficient, plotted in Fig. 4. This may implicate a substantial variability in sensitivity to sudden heat load in the lens among individuals, which has to be considered when setting safety guidelines for human exposure. Considering the Arrhenius equation, denaturation rate is directly dependent on temperature. Therefore, it is possible that a very small heat load over a long time may accelerate cataract formation.12

Temperature was measured with manufacturer calibrated thermocouples (HYP0, OMEGA) connected to an integrated analogue–digital converter (TC-08, OMEGA).

To judge the practical validity of the described strategy, it is necessary to estimate the precision of predicted temperature as a function of sample size. Adopting the outcome of the current estimates of inclination coefficients for an increase of light scattering as a function of time and temperature, the confidence interval for predicted temperature based on averages of additional measurements of inclination coefficients can be estimated.9 It was found that averaging 20 additional measurements of inclination coefficients, the confidence limits around the estimated temperature correspond to ±1.9°C (Fig. 5).

The intensity of forward light scattering in the lens was measured on a dark field source (Fig. 1). After reflection with mirrors, the light from the dark field source hits the cuvette cavity in a narrow circular beam from below at a 45-deg angle. The cuvette is imaged on a photodiode, which generates a current. Current readings are converted according to a calibration curve to the concentration of a known light scattering standard concentration, a commercial lipid emulsion of Diazepam (Stesolid Novum, Actavis AB, Sweden). To make measurements normally distributed, the concentrations of Diazepam are log transformed, as transformed equivalent diazepam concentration (tEDC).8

Aspheric lenses are commonly chosen to couple incident light with a diameter of 1 - 5 mm into a single mode fiber. A simple example will illustrate the key specifications to consider when trying to choose the correct lens.

The table below contains all molded visible and near-IR aspheric lenses offered by Thorlabs. For our selection of IR molded aspheres, click here. The Item # listed is that of the unmounted, uncoated lens. An "X" in any of the five AR Coating Columns indicates the lens is available with that coating (note that the V coating availability is indicated with the AR coating wavelength). The table to the right defines each letter and lists the specified AR coating range. Clicking on the X takes you to the landing page where that lens (mounted or unmounted) can be purchased.

Six-week-old albino Sprague–Dawley female rats were kept and treated according to the ARVO Statement for the Use of Animals in Ophthalmic and Vision Research. Ethical approval was obtained by Uppsala Djurförsöksetiska Nämnd (C29/16).

The natural logarithm of the inclination coefficient for light scattering increase, as a function of the inverse of the absolute temperature, was fit to a linear model [Appendix Eq. (9), Fig. 4]. The proportionality constant Ea/R (Ea=the activation energy, R=the general gas constant) and the intercept ln(l·P) were estimated as CI(0.95), 9.6±2.4×103  K and 22.8±7.7. Since R is known, Ea was estimated as a CI(0.95), 8.0±2.0×101  kJ mol−1.

The temperature-controlled cuvette filled with BSS was preheated by the circulating water to the planned exposure temperature. The temperature was confirmed by the thermocouple.

The current study intended to explore the possibility of using denaturation as an indirect method to experimentally measure lens temperature due to heat load, applying the Arrhenius equation. Denaturation was measured as light scattering.

A calibration curve for dependence of rate of light scattering increase (here, inclination coefficient) on temperature can be established experimentally. The confidence interval for predicted temperature can be estimated from an average of inclination coefficients.9 The precision depends on the residual variance, the sample size constituting the basis of the average, and the number of additional measurements of inclination coefficients averaged. The lowest precision is associated with one additional measurement of inclination coefficient in a future experiment for temperature probing (Fig. 4).

Due to the rotational symmetry of the lens surface, only even powers of Y are contained in the polynomial expansion above. The target values of the aspheric coefficients for each product can be found by clicking either on the blue Info Icons in the tables below () or on the red documents icon () next to each lens sold below.

If an unmounted aspheric lens is being used to collimate the light from a point source or laser diode, the side with the greater radius of curvature (i.e., the flatter surface) should face the point source or laser diode. To collimate light using one of our mounted aspheric lenses, orient the housing so that the externally threaded end of the mount faces the source.

Such measurements of permanent light scattering should be done at postexposure time intervals long enough to exclude immediate reversible light scattering. Knowledge about experimentally determined critical temperature in lenses from warm-blooded animals can be used for extrapolation to human lens.

Alternatively, we can use the PS871-B unmounted prism pair to achieve the precise magnification of the minor diameter necessary to produce a circular beam. Using the data available here, we see that the PS871-B achieves a magnification of 4.0 when the prisms are positioned at the following angles for a 670 nm wavelength beam:

The rate of production of denatured proteins, r (mol·l−1·s−1), is proportional to the concentration of nature state protein, [N] (mol·l−1), with a proportionality constant, k(T) (s−1), that varies with temperature, T (K) r=k(T)[N]or Eq. (4)dΩdt=k(T)[N].

The pre-exponential factor expresses the maximum rate at infinitely high temperature and reflects the increase in entropy associated with the loss of order due to the reaction.

Thorlabs offers a large selection of mounted and unmounted aspheric lenses to choose from. The aspheric lens with a focal length that is closest to 16 mm has a focal length of 15.29 mm (Item # 354260-B or A260-B). This lens also has a clear aperture that is larger than the collimated beam diameter. Therefore, this option is the best choice given the initial parameters (i.e., a P1-630A-FC-2 single mode fiber and a collimated beam diameter of 3 mm). Remember, for optimal coupling, the spot size of the focused beam must be less than the MFD of the single mode fiber. As a result, if an aspheric lens is not available that provides an exact match, then choose one with a focal length that is shorter than the calculation above yields. Alternatively, if the clear aperture of the aspheric lens is large enough, the beam can be expanded before the aspheric lens, which has the result of reducing the spot size of the focus beam.

The current study was supported by Carmen och Bertil Regnérs fond för forskning, Gun och Bertil Stohnes Stiftelse, Karin Sandqvists Stiftelse, Svenska Läkaresällskapet Resebidrag, Konung Gustav V:s och Drottning Victorias Frimurarstiftelse, Uppsala Läns Landsting’s Research grants (ALF), Ögonfonden, Stiftelsen Sigurd och Elsa Goljes Minne.

The animal was sacrificed. Then one lens was extracted and put into the preheated BSS in the cuvette; intensity of forward light scattering in the lens was measured for 5 min, as described elsewhere. This assured a fast increase of lens temperature.

With this information known, it is now time to choose the appropriate collimating lens. Thorlabs offers a large selection of aspheric lenses. For this application, the ideal lens is a molded glass aspheric lens with focal length near 5.6 mm and our -B antireflection coating, which covers 780 nm. The C171TMD-B (mounted) or 354171-B (unmounted) aspheric lenses have a focal length of 6.20 mm, which will result in a collimated beam diameter (major axis) of 3.3 mm. Next, check to see if the numerical aperture (NA) of the diode is smaller than the NA of the lens: