Ultrastable lasers are critical for advanced research and applications requiring high-spectral purity, such as optical atomic clocks1, photonic microwave generation2,3, communications4, and very long baseline interferometry5. The combination of Fabry–Perot (FP) cavities as frequency references and the Pound–Drever–Hall (PDH)6 locking technique represents the “golden rule” for laser frequency stabilization, leading to state-of-the-art frequency stability on the level of 10−16 or better7,8,9,10,11. To support such quiet operation, a sub-femtometer level of cavity length fluctuation is required. This high demand typically necessitates that the FP cavity should be operated in a laboratory environment with a complex isolating system comprising of the maintained vacuum, temperature stabilization (either at room temperature or cryogenics), and vibration isolation, leading to relatively large size, weight, and the power consumption (SWaP).

The system of f-numbers for specifying relative apertures evolved in the late nineteenth century, in competition with several other systems of aperture notation.

Camera lenses often include an adjustable diaphragm, which changes the size of the aperture stop and thus the entrance pupil size. This allows the user to vary the f-number as needed. The entrance pupil diameter is not necessarily equal to the aperture stop diameter, because of the magnifying effect of lens elements in front of the aperture.

Beck and Andrews in 1902 talk about the Royal Photographic Society standard of f/4, f/5.6, f/8, f/11.3, etc.[30] The R.P.S. had changed their name and moved off of the U.S. system some time between 1895 and 1902.

Light falloff is also sensitive to f-stop. Many wide-angle lenses will show a significant light falloff (vignetting) at the edges for large apertures.

Moreover, the photonic resonator enables a relatively simple and robust approach to highly coherent photonics-based millimeter-wave generation. Figure 5 presents a comparison of the phase noise in the avenue of photonic heterodyne approaches. The green line shows the results in ref. 22, where two semiconductor lasers are injection-locked to separated WGMRs, and the blue line presents the self-injection-locked based on Si photonic integrated circuits45 (Si PICs). The phase noise based on locking lasers to a common photonic resonator, as shown in Fig. 4b, exhibits up to 80 dB improvement for Fouirer frequencies from 10 Hz to 10 kHz. Since the phase noise presented in Fig. 4b is limited by the inloop error of the locking, it can be improved by using a laser with lower free-running noise. We have achieved −110 dBc/Hz at 10 kHz (10 dB improvement over Fig. 4b result) by using another RIO PLANEX as laser 2, though the frequency distinct is 4 GHz from laser 1. The low noise photonic millimeter-wave synthesis may find use for an array of sensing and communications applications. For example, low-noise oscillators operating directly at high frequencies can directly enhance the performance of future radar instruments, offering improvements to resolution46, reconfigurability47,48, and sensitivity22,49. Phase noise level, in particular, impacts the achievable dynamic range of pulse-compressed (such as modulated cw) radar measurements, and as a result, the low transmitted noise level will be key to suppressing clutter in future Earth-observing scientific radar instruments50,51.

The word stop is sometimes confusing due to its multiple meanings. A stop can be a physical object: an opaque part of an optical system that blocks certain rays. The aperture stop is the aperture setting that limits the brightness of the image by restricting the input pupil size, while a field stop is a stop intended to cut out light that would be outside the desired field of view and might cause flare or other problems if not stopped.

Kittlaus, E. A. et al. A low-noise photonic heterodyne synthesizer and its application to millimeter-wave radar. Nat. Commun. 12, 4397 (2021).

Heinert, D., Gurkovsky, A. G., Nawrodt, R., Vyatchanin, S. P. & Yamamoto, K. Thermorefractive noise of finite-sized cylindrical test masses. Phys. Rev. D 84, 062001 (2011).

while shutter speeds in reciprocal seconds have a few conventional differences in their numbers (1⁄15, 1⁄30, and 1⁄60 second instead of 1⁄16, 1⁄32, and 1⁄64).

Photojournalists have a saying, "f/8 and be there", meaning that being on the scene is more important than worrying about technical details. Practically, f/8 (in 35 mm and larger formats) allows adequate depth of field and sufficient lens speed for a decent base exposure in most daylight situations.[16]

Liu, Y. et al. Thermal-noise-limited, compact optical reference cavity operated without a vacuum enclosure. Preprint at arXiv https://doi.org/10.48550/arXiv.2307.04758 (2023).

Kelleher, M. L. et al. Compact, portable, thermal-noise-limited optical cavity with low acceleration sensitivity. Opt. Express 31, 11954–11965 (2023).

Robinson, J. M. et al. Crystalline optical cavity at 4 k with thermal-noise-limited instability and ultralow drift. Optica 6, 240–243 (2019).

Zhang, W., Kittlaus, E., Savchenkov, A. et al. Monolithic optical resonator for ultrastable laser and photonic millimeter-wave synthesis. Commun Phys 7, 177 (2024). https://doi.org/10.1038/s42005-024-01660-3

Chalermsongsak, T. et al. Coherent cancellation of photothermal noise in gaas/al0.92ga0.08as bragg mirrors. Metrologia 53, 860 (2016).

Cavity modes inlaser

Kedar, D., Yao, Z., Ryger, I., Hall, J. L. & Ye, J. Synthetic fm triplet for am-free precision laser stabilization and spectroscopy. Optica 11, 58–63 (2024).

The angle of transverse oscillation is called the "polarization angle" (see Figure 1). By the way, when we say that a certain light source is "unpolarized" we ...

At the same time, there were a number of aperture numbering systems designed with the goal of making exposure times vary in direct or inverse proportion with the aperture, rather than with the square of the f-number or inverse square of the apertal ratio or intensity ratio. But these systems all involved some arbitrary constant, as opposed to the simple ratio of focal length and diameter.

An example of the use of f-numbers in photography is the sunny 16 rule: an approximately correct exposure will be obtained on a sunny day by using an aperture of f/16 and the shutter speed closest to the reciprocal of the ISO speed of the film; for example, using ISO 200 film, an aperture of f/16 and a shutter speed of 1⁄200 second. The f-number may then be adjusted downwards for situations with lower light. Selecting a lower f-number is "opening up" the lens. Selecting a higher f-number is "closing" or "stopping down" the lens.

Although he did not yet have access to Ernst Abbe's theory of stops and pupils,[23] which was made widely available by Siegfried Czapski in 1893,[24] Dallmeyer knew that his working aperture was not the same as the physical diameter of the aperture stop:

Savchenkov, A. A., Matsko, A. B., Ilchenko, V. S., Yu, N. & Maleki, L. Whispering-gallery-mode resonators as frequency references. II. Stabilization. J. Opt. Soc. Am. B 24, 2988–2997 (2007).

Beasley, P. D. The influence of transmitter phase noise on FMCW radar performance. In 2006 European Microwave Conference, 1810–1813 (IEEE, 2006).

Photographers sometimes express other exposure ratios in terms of 'stops'. Ignoring the f-number markings, the f-stops make a logarithmic scale of exposure intensity. Given this interpretation, one can then think of taking a half-step along this scale, to make an exposure difference of a "half stop".

The idea of the study was conceived by W.Z. and A.M. W.Z. designed the monolithic resonator built under the supervision of A.M. and S.P. W.Z. and E.K. constructed the measurement apparatuses and conducted the experiments with input from L.Y., A.S., and V.I. All authors contributed to the project implementation and to the writing of this article.

An H-stop (for hole, by convention written with capital letter H) is an f-number equivalent for effective exposure based on the area covered by the holes in the diffusion discs or sieve aperture found in Rodenstock Imagon lenses.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Resonator laserapp

Beyond applications in spectrally pure stable lasers, the monolithic resonator permits a simple approach to low-noise photonic millimeter-wave synthesis22. As a comparison, the conventional electronic oscillators degrade the phase noise at higher frequencies generated through multiplier chains, regardless of 10–100 W power consumption. The frequency combs either based on the modelocked laser2 or microresonator29 allow low noise optical-to-microwave synthesis in lab environment, however the complexity of the system constrains their fieldable applications. In this article, we separately lock two lasers to distinct modes of the photonic resonator and down-mix their heterodyne-beatnote from optical to microwave frequencies by using a fast photodiode, translating the superb low-phase noise of the resonator to the radiofrequency domain. In this photonic synthesis approach, the phase noise of the heterodyne signal is independent of the output frequency. We demonstrate 4 and 96 GHz signals with single sideband phase noise of −100 dBc/Hz at 10 kHz, representing a simple approach for low-noise millimeter-wave synthesis via optical-to-microwave conversion. Through these experiments, the maximum frequency was limited only by the photodiode and characterization apparatus available; looking forward, the same performance level can be translated to sub-millimeter wave or THz frequencies by using higher frequency photomixers30,31, offering new capabilities for coherent THz sensing and high-frequency radars. Altogether, these results establish an alternative promising approach to achieve high-performance reference cavities and photonic mm-wave synthesis while permitting ambient environment operation in a robust, compact system.

Photodiode in an enclosure with built-in filters for the Photoelectric Effect experiment.

Communications Physics thanks Gregory Moille and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

On modern cameras, especially when aperture is set on the camera body, f-number is often divided more finely than steps of one stop. Steps of one-third stop (1⁄3 EV) are the most common, since this matches the ISO system of film speeds. Half-stop steps are used on some cameras. Usually the full stops are marked, and the intermediate positions click but are not marked. As an example, the aperture that is one-third stop smaller than f/2.8 is f/3.2, two-thirds smaller is f/3.5, and one whole stop smaller is f/4. The next few f-stops in this sequence are:

Computing the f-number of the human eye involves computing the physical aperture and focal length of the eye. Typically, the pupil can dilate to be as large as 6–7 mm in darkness, which translates into the maximal physical aperture. Some individuals' pupils can dilate to over 9 mm wide.

By 1920, the term f-number appeared in books both as F number and f/number. In modern publications, the forms f-number and f number are more common, though the earlier forms, as well as F-number are still found in a few books; not uncommonly, the initial lower-case f in f-number or f/number is set in a hooked italic form: ƒ.[31]

Hewlett-Packard. Phase noise characterization of microwave oscillators, frequency discriminator method. HP Product Note 11729C-2 (1985).

In every lens there is, corresponding to a given apertal ratio (that is, the ratio of the diameter of the stop to the focal length), a certain distance of a near object from it, between which and infinity all objects are in equally good focus. For instance, in a single view lens of 6-inch focus, with a 1⁄4 in. stop (apertal ratio one-twenty-fourth), all objects situated at distances lying between 20 feet from the lens and an infinite distance from it (a fixed star, for instance) are in equally good focus. Twenty feet is therefore called the 'focal range' of the lens when this stop is used. The focal range is consequently the distance of the nearest object, which will be in good focus when the ground glass is adjusted for an extremely distant object. In the same lens, the focal range will depend upon the size of the diaphragm used, while in different lenses having the same apertal ratio the focal ranges will be greater as the focal length of the lens is increased. The terms 'apertal ratio' and 'focal range' have not come into general use, but it is very desirable that they should, in order to prevent ambiguity and circumlocution when treating of the properties of photographic lenses.[21]

Sartorius, B. et al. Continuous wave terahertz systems exploiting 1.5 μm telecom technologies. Opt. Express 17, 15001–15007 (2009).

Notations for f-numbers were also quite variable in the early part of the twentieth century. They were sometimes written with a capital F,[32] sometimes with a dot (period) instead of a slash,[33] and sometimes set as a vertical fraction.[34]

Hall, J. & Zhu, M. In Laser Manipulation of Atoms and Ions, Proceedings Internat. School of Physics Enrico Fermi,Vol. Course CXVIII, 671–702 (North Holland-Elsevier, Amsterdam, 1992).

The camera equation, or G#, is the ratio of the radiance reaching the camera sensor to the irradiance on the focal plane of the camera lens:[19]

Häfner, S. et al. 8 × 10−17 fractional laser frequency instability with a long room-temperature cavity. Opt. Lett. 40, 2112–2115 (2015).

Nov 2, 2005 — Creating a sphere or semi spherical shape Could any one tell me how to make a sphere or semi-spherical shape. I know this is probably a...

Depth of field increases with f-number, as illustrated in the image here. This means that photographs taken with a low f-number (large aperture) will tend to have subjects at one distance in focus, with the rest of the image (nearer and farther elements) out of focus. This is frequently used for nature photography and portraiture because background blur (the aesthetic quality known as 'bokeh') can be aesthetically pleasing and puts the viewer's focus on the main subject in the foreground. The depth of field of an image produced at a given f-number is dependent on other parameters as well, including the focal length, the subject distance, and the format of the film or sensor used to capture the image. Depth of field can be described as depending on just angle of view, subject distance, and entrance pupil diameter (as in von Rohr's method). As a result, smaller formats will have a deeper field than larger formats at the same f-number for the same distance of focus and same angle of view since a smaller format requires a shorter focal length (wider angle lens) to produce the same angle of view, and depth of field increases with shorter focal lengths. Therefore, reduced–depth-of-field effects will require smaller f-numbers (and thus potentially more difficult or complex optics) when using small-format cameras than when using larger-format cameras.

This point is further emphasized by Czapski in 1893.[24] According to an English review of his book, in 1894, "The necessity of clearly distinguishing between effective aperture and diameter of physical stop is strongly insisted upon."[25]

For direct phase noise characterization of the 96 GHz signal, we implemented a hybrid microwave-photonic frequency discriminator using a fiber optic delay line (see Methods)22,34,35. The measured phase noise of the output signal at a carrier frequency of 96 GHz is plotted in Fig. 4b, revealing phase noise of below −60 dBc/Hz at 100 Hz offset and below −100 dBc/Hz at 10 kHz offset. These close-in phase noise results are around 100 × lower than our prior results using lasers injection-locked to two separate microresonator cavities22, resulting from both the superb stability of the monolithic photonic resonator and the common-mode cancellation of technical noise achieved by locking both lasers to a single resonator. The inloop error from laser 1 limits the phase noise from DC to 1 kHz, and the servo bump due to feedback to PZT of laser 2 dominates from 1 to 10 kHz. and the combined servo bump from the two laser locking by current modulation is seen above >10 kHz, in which the 500 kHz corresponds to laser 1 and 300 kHz is from laser 2. Note the spikes around 10–20 kHz from laser 2 elevates the measured noise level, and could be mitigated by using a laser without this intrinsic noise. The output phase noise of the resonator-based source was corroborated by direct measurements with frequency detuning between the two locked lasers reduced to 4 GHz (gray), despite negligible difference around the servo bumps. In principle, this signal generation can be tuned to any multiple of the cavity FSR, provided a suitable laser is available. As long as the phase noise of the locked laser does not change, then the beat-note phase noise is essentially independent of output frequency. As a result, the same performance could be extended to the sub-mm wave or THz regime by replacing the commercial photodiodes used with a THz photomixer30,31.

Drever, R. W. P. et al. Laser phase and frequency stabilization using an optical resonator. Appl. Phys. B 31, 97–105 (1983).

In photography this means that as one focuses closer, the lens's effective aperture becomes smaller, making the exposure darker. The working f-number is often described in photography as the f-number corrected for lens extensions by a bellows factor. This is of particular importance in macro photography.

In practice the maximum aperture of a lens is often not an integral power of √2 (i.e., √2 to the power of a whole number), in which case it is usually a half or third stop above or below an integral power of √2.

To illustrate the high spectral purity enabled by our approach in the context of state-of-the-art monolithic, vacuum-free reference resonators, Table 1 presents the comparison of the integrated laser linewidth33 <100 Hz and measured Allan deviation for recent systems. A laser locked to a mm-scale fused silica microrod25 achieves a thermal-noise-limited laser linewidth of 62 Hz. By increasing the mode volume, a laser locked to an on-chip coil19 can narrow the linewidth to 36 Hz. Self-injection locking a laser to the WGMR based on crystalline MgF221 achieves similar results. A prior design for the photonic resonator with 20,000 finesse26, due to the material loss of the fused silica, demonstrates the linewidth of 25 Hz. In this work, there are three improvements summarized as follows to upgrade the frequency stability presented in26. In a general aspect, the resonator has a larger ROC on the facet (1 m vs 0.5 m), leading to thermal-noise-limit laser linewidth (15 Hz vs 21 Hz). For the improvement in frequency domain measurement, the higher finesse (170,000 vs 20,000) due to lower loss is critical. The enhanced signal-to-noise ratio of PDH locking allows a tight locking up to 300 Hz (red line in Fig. 2b). The improvement in the time domain comes from the enhanced thermal isolation, leading to a more deterministic linear drift. Though the FFS with linear drift (green line in Fig. 3) is approximately a factor of 2 improvements compared with the results in26, the FFS with linear drift removed can achieve <10−13, approaching the thermal noise floor (gray line in Fig. 3). Both measurements in frequency and time domains achieve the best performance so far among monolithic optical references.

N w ≈ 1 2 N A i ≈ ( 1 + | m | P ) N , {\displaystyle N_{w}\approx {1 \over 2\mathrm {NA} _{i}}\approx \left(1+{\frac {|m|}{P}}\right)N\,,}

Most twentieth-century cameras had a continuously variable aperture, using an iris diaphragm, with each full stop marked. Click-stopped aperture came into common use in the 1960s; the aperture scale usually had a click stop at every whole and half stop.

Modern electronically controlled interchangeable lenses, such as those used for SLR cameras, have f-stops specified internally in 1⁄8-stop increments, so the cameras' 1⁄3-stop settings are approximated by the nearest 1⁄8-stop setting in the lens.[citation needed]

Most modern lenses use a standard f-stop scale, which is an approximately geometric sequence of numbers that corresponds to the sequence of the powers of the square root of 2: f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32, f/45, f/64, f/90, f/128, etc. Each element in the sequence is one stop lower than the element to its left, and one stop higher than the element to its right. The values of the ratios are rounded off to these particular conventional numbers, to make them easier to remember and write down. The sequence above is obtained by approximating the following exact geometric sequence:

We implemented a dual stabilized laser system as diagrammed in Fig. 4a. Two CW lasers (RIO PLANEX labeled as laser 1 and Velocity TLB 6700 as laser 2) are operating around two distinct cavity modes, 193.4 THz (f1) and 193.496 THz (f2), respectively. Each laser has independent phase modulation (9.8 MHz for Laser 1 and 60.2 MHz for Laser 2) provided by the corresponding fiber-based EOM. In-line isolators are placed in the system to eliminate back-reflection and unexpected interference. The two lasers are combined by a fiber coupler (50/50 coupling ratio) and launched into the cavity. Approximately 2 μW optical power from each laser is received by the photodetector for PDH error signal generation, which is demodulated at the corresponding frequency. The laser 1 is frequency-stabilized by controlling the current, and laser 2 is by current and piezoelectric transducer (PZT). The majority part of the optical power from each laser, approximately 10 mW, is combined by a fiber coupler and incident on a fast photodiode (Finisar XPDV4120R, bandwidth > 90 GHz), leading to an output power of 150 μW at f = 96 GHz, corresponding to 24 free-spectral range (FSR) of the resonator. This is then boosted to 11 dBm (12.5 mW) using a W-band amplifier (Eravant BP-7531142515-1010-E1) for signal analysis.

Kryhin, S., Hall, E. D. & Sudhir, V. Thermorefringent noise in crystalline optical materials. Phys. Rev. D 107, 022001 (2023).

There are several aspects that would likely improve the performance of the photonic resonator. (1) Finesse. Since the loss of the resonator is measured at ~10 ppm, the finesse can be further increased by >250,000 by reducing the transmission while keeping sufficient cavity contrast for PDH locking with low residual locking noise. To intrinsically reduce the loss of the resonator, we are investigating the material properties of fused silica and the manufacturing process. (2) Thermorefractive noise floor. The ROC of the surface can be readily increased up to 10 m, which enlarges the mode size to lower the thermorefractive noise floor. Using crystalline materials36 of which the thermorefractive index is lower than fused silica would be a possible solution, though other technical factors (e.g., thermal expansion and vibration sensitivity) should be considered. With these two practical improvements, the photonic resonator can support the stabilized laser with sub-10 Hz integrated linewidth, a performance typically requiring vacuum-gap FP cavities with compact size37. To reach similar frequency stability, a recent work38 presents a 20 mm FP cavity, which is assembled in a vacuum and thus does not require a continuous running vacuum pump during the operation. It is worth mentioning that the coherent cancellation of the thermal noise39,40,41 might be a path to reduce the TR noise of the photonic resonator. (3) The residual amplitude modulation (RAM)42 and frequency drift compensation. The laser frequency stabilization based on the monolithic resonator can afford up to 300 ppm (FFS ≈ 4 × 10−14), which is comparable with the resonator thermal noise floor. Though the free-running RAM is highly up to the PDH setup and the temperature control of the EOM, such an amount of RAM contribution would need a relatively long averaging time to dominate the overall stability performance. For further development of this resonator with more deterministic linear drift or with drift cancellation36,43 by which the FFS can be maintained on 10−14 level, the RAM stabilization42,44would be necessary.

The adopted value for vacuum wavelength of an uncalibrated helium-neon laser is 632.990 8 nm, and the relative standard uncertainty is 1.5 × 10−6. This ...

Liu, J. et al. Photonic microwave generation in the x- and k-band using integrated soliton microcombs. Nat. Photonics 14, 486–491 (2020).

Even though the principles of focal ratio are always the same, the application to which the principle is put can differ. In photography the focal ratio varies the focal-plane illuminance (or optical power per unit area in the image) and is used to control variables such as depth of field. When using an optical telescope in astronomy, there is no depth of field issue, and the brightness of stellar point sources in terms of total optical power (not divided by area) is a function of absolute aperture area only, independent of focal length. The focal length controls the field of view of the instrument and the scale of the image that is presented at the focal plane to an eyepiece, film plate, or CCD.

For example, the SOAR 4-meter telescope has a small field of view (about f/16) which is useful for stellar studies. The LSST 8.4 m telescope, which will cover the entire sky every three days, has a very large field of view. Its short 10.3 m focal length (f/1.2) is made possible by an error correction system which includes secondary and tertiary mirrors, a three element refractive system and active mounting and optics.[18]

It must be observed, however, that in order to find the real intensity ratio, the diameter of the actual working aperture must be ascertained. This is easily accomplished in the case of single lenses, or for double combination lenses used with the full opening, these merely requiring the application of a pair of compasses or rule; but when double or triple-combination lenses are used, with stops inserted between the combinations, it is somewhat more troublesome; for it is obvious that in this case the diameter of the stop employed is not the measure of the actual pencil of light transmitted by the front combination. To ascertain this, focus for a distant object, remove the focusing screen and replace it by the collodion slide, having previously inserted a piece of cardboard in place of the prepared plate. Make a small round hole in the centre of the cardboard with a piercer, and now remove to a darkened room; apply a candle close to the hole, and observe the illuminated patch visible upon the front combination; the diameter of this circle, carefully measured, is the actual working aperture of the lens in question for the particular stop employed.[22]

( 2 ) 0 3 ,   ( 2 ) 1 3 ,   ( 2 ) 2 3 ,   ( 2 ) 3 3 ,   ( 2 ) 4 3 ,   … {\displaystyle ({\sqrt {2}})^{\frac {0}{3}},\ ({\sqrt {2}})^{\frac {1}{3}},\ ({\sqrt {2}})^{\frac {2}{3}},\ ({\sqrt {2}})^{\frac {3}{3}},\ ({\sqrt {2}})^{\frac {4}{3}},\ \ldots }

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As an example application of the superb laser noise enabled by the photonic resonator, we next use it to construct a millimeter-wave oscillator based on the heterodyning of two lasers locked to distinct cavity modes. Typically, heterodyne down-mixing of off-the-shelf lasers will result in an output RF signal with relatively high phase noise given by the quadrature sum of the phase noises of the two independent lasers. The ultra-stable photonic resonator offers two benefits. First, locking a laser to the resonator greatly reduces phase noise and improves long-term frequency stability via active feedback. Second, two independent lasers can be separately locked to disparate modes in the same resonator, permitting common-mode rejection of technical noise related to the resonator environment.

where N is the uncorrected f-number, NAi is the image-space numerical aperture of the lens, | m | {\displaystyle |m|} is the absolute value of the lens's magnification for an object a particular distance away, and P is the pupil magnification. Since the pupil magnification is seldom known it is often assumed to be 1, which is the correct value for all symmetric lenses.

Cooper, K. B. et al. G-band radar for humidity and cloud remote sensing. IEEE Trans. Geosci. Remote Sens. 59, 1106–1117 (2021).

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

ZWB2+ QB39 UV Pass Filter Set ... This is a very versatile and economic pair of Chinese made filters, which when used together pass just UV, so can be used for ...

The 1961 ASA standard PH2.12-1961 American Standard General-Purpose Photographic Exposure Meters (Photoelectric Type) specifies that "The symbol for relative apertures shall be ƒ/ or ƒ: followed by the effective ƒ-number." They show the hooked italic 'ƒ' not only in the symbol, but also in the term f-number, which today is more commonly set in an ordinary non-italic face.

Opticalresonatorinlaserpdf

The f-number accurately describes the light-gathering ability of a lens only for objects an infinite distance away.[20] This limitation is typically ignored in photography, where f-number is often used regardless of the distance to the object. In optical design, an alternative is often needed for systems where the object is not far from the lens. In these cases the working f-number is used. The working f-number Nw is given by:[20]

Optical resonators are indispensable tools in optical metrology that usually benefit from an evacuated and highly-isolated environment to achieve peak performance. Even in the more sophisticated design of Fabry-Perot (FP) cavities, the material choice limits the achievable quality factors. For this reason, monolithic resonators are emerging as promising alternative to traditional designs, but their design is still at preliminary stage and far from being optimized. Here, we demonstrate a monolithic FP resonator with 4.5 cm3 volume and 2 × 105 finesse. In the ambient environment, we achieve 18 Hz integrated laser linewidth and 7 × 10−14 frequency stability measured from 0.08 s to 0.3 s averaging time, the highest spectral purity and stability demonstrated to date in the context of monolithic reference resonators. By locking two separate lasers to distinct modes of the same resonator, a 96 GHz microwave signals is generated with phase noise -100 dBc/Hz at 10 kHz frequency offset, achieving orders of magnitude improvement in the approach of photonic heterodyne synthesis. The compact monolithic FP resonator is promising for applications in spectrally-pure, high-frequency microwave photonic references as well as optical clocks and other metrological devices. ©2024. All rights reserved.

In astronomy, the f-number is commonly referred to as the focal ratio (or f-ratio) notated as N {\displaystyle N} . It is still defined as the focal length f {\displaystyle f} of an objective divided by its diameter D {\displaystyle D} or by the diameter of an aperture stop in the system:

Kedar, D. et al. Frequency stability of cryogenic silicon cavities with semiconductor crystalline coatings. Optica 10, 464–470 (2023).

… 16 / 13 ∘ ,   20 / 14 ∘ ,   25 / 15 ∘ ,   32 / 16 ∘ ,   40 / 17 ∘ ,   50 / 18 ∘ ,   64 / 19 ∘ ,   80 / 20 ∘ ,   100 / 21 ∘ ,   125 / 22 ∘ ,   … {\displaystyle \ldots 16/13^{\circ },\ 20/14^{\circ },\ 25/15^{\circ },\ 32/16^{\circ },\ 40/17^{\circ },\ 50/18^{\circ },\ 64/19^{\circ },\ 80/20^{\circ },\ 100/21^{\circ },\ 125/22^{\circ },\ \ldots }

Zhang, W. et al. Ultrastable silicon cavity in a continuously operating closed-cycle cryostat at 4 k. Phys. Rev. Lett. 119, 243601 (2017).

( 2 ) 0 2 ,   ( 2 ) 1 2 ,   ( 2 ) 2 2 ,   ( 2 ) 3 2 ,   ( 2 ) 4 2 ,   … {\displaystyle ({\sqrt {2}})^{\frac {0}{2}},\ ({\sqrt {2}})^{\frac {1}{2}},\ ({\sqrt {2}})^{\frac {2}{2}},\ ({\sqrt {2}})^{\frac {3}{2}},\ ({\sqrt {2}})^{\frac {4}{2}},\ \ldots }

Alnis, J. et al. Thermal-noise-limited crystalline whispering-gallery-mode resonator for laser stabilization. Phys. Rev. A 84, 011804 (2011).

What is alasercavity

The rapidity of a lens depends upon the relation or ratio of the aperture to the equivalent focus. To ascertain this, divide the equivalent focus by the diameter of the actual working aperture of the lens in question; and note down the quotient as the denominator with 1, or unity, for the numerator. Thus to find the ratio of a lens of 2 inches diameter and 6 inches focus, divide the focus by the aperture, or 6 divided by 2 equals 3; i.e., 1⁄3 is the intensity ratio.[22]

To evaluate the utility of the stable photonic resonator to millimeter-wave signal generation, we implemented a dual stabilized laser system as diagrammed in Fig. 4a. Two CW lasers are operating around two distinct cavity modes, and the corresponding beatnote signal at f = 96 GHz is detected (see Methods). The inset of Fig. 4a illustrates the expected phase noise of the millimeter-wave signal based on the dual stabilized laser system. One laser locked to one resonator mode should experience a nearly identical (coherent) thermal fluctuation process (dominated by TR noise) as the other laser on a distant mode. The incoherence between the two lasers is contributed by the difference in beam size, which is negligible in 100 GHz space. As a result, the phase noise of the heterodyned signal shows a common noise rejection relative to the TR noise and is limited by the incoherent contribution, i.e., the residual noise of PDH locking for each laser. The noise rejection, either in amplitude or ineffective Fourier frequency where TR noise is equal to measured phase noise, is determined by the free-running noise of each laser and the capability of the frequency locking.

Matsko, A. B., Savchenkov, A. A., Yu, N. & Maleki, L. Whispering-gallery-mode resonators as frequency references. I. Fundamental limitations. J. Opt. Soc. Am. B 24, 1324–1335 (2007).

f / 1 = f ( 2 ) 0 ,   f / 1.4 = f ( 2 ) 1 ,   f / 2 = f ( 2 ) 2 ,   f / 2.8 = f ( 2 ) 3 ,   … {\displaystyle f/1={\frac {f}{({\sqrt {2}})^{0}}},\ f/1.4={\frac {f}{({\sqrt {2}})^{1}}},\ f/2={\frac {f}{({\sqrt {2}})^{2}}},\ f/2.8={\frac {f}{({\sqrt {2}})^{3}}},\ \ldots } In the same way as one f-stop corresponds to a factor of two in light intensity, shutter speeds are arranged so that each setting differs in duration by a factor of approximately two from its neighbour. Opening up a lens by one stop allows twice as much light to fall on the film in a given period of time. Therefore, to have the same exposure at this larger aperture as at the previous aperture, the shutter would be opened for half as long (i.e., twice the speed). The film will respond equally to these equal amounts of light, since it has the property of reciprocity. This is less true for extremely long or short exposures, where there is reciprocity failure. Aperture, shutter speed, and film sensitivity are linked: for constant scene brightness, doubling the aperture area (one stop), halving the shutter speed (doubling the time open), or using a film twice as sensitive, has the same effect on the exposed image. For all practical purposes extreme accuracy is not required (mechanical shutter speeds were notoriously inaccurate as wear and lubrication varied, with no effect on exposure). It is not significant that aperture areas and shutter speeds do not vary by a factor of precisely two.

Lasercavity length formula

A growing research activity seeking to demonstrate high-stability lasers with minimization of SWaP arises from potential fieldable applications, such as geodesy12, space-based operation13,14, or transportable optical clocks15, where system robustness, as well as integration, are primary considerations. Besides efforts centered around the sophisticated design of the FP cavity and its support structures16,17, monolithic reference resonators provide an alternative solution. In this case, the optical field is mostly confined to the host material of the solid-state resonator, so the physical parameters of this medium are directly related to the achievable performance. For instance, material attenuation limits the quality factor of a resonator and, thus, the phase noise and stability of the device. The environmental sensitivity of the material readily imprints on the environmental sensitivity of the resonator. The nonlinearity of the material restricts the maximum intracavity power and hence limits the achievable signal-to-noise ratio of the system, restricting both short-term stability as well as the spectral purity of an oscillator. With careful design, several types of monolithic resonators to date have been successfully developed to reach 100 Hz linewidth level or less, including chip-scale waveguides18,19, whispering-gallery-mode resonators (WGMR)20,21,22,23,24, and microrods25. In contrast to microfabricated devices, monolithic photonic resonators based a bulk-optic and cylindrical cavity have demonstrated the narrowest linewidth among dielectric reference cavities26, thanks to the combination large mode volume, design flexibility, and narrow linewidth of the resonator modes.

( 2 ) 0 ,   ( 2 ) 1 ,   ( 2 ) 2 ,   ( 2 ) 3 ,   ( 2 ) 4 ,   … {\displaystyle ({\sqrt {2}})^{0},\ ({\sqrt {2}})^{1},\ ({\sqrt {2}})^{2},\ ({\sqrt {2}})^{3},\ ({\sqrt {2}})^{4},\ \ldots }

Zhang, W., Baynes, F., Diddams, S. A. & Papp, S. B. Microrod optical frequency reference in the ambient environment. Phys. Rev. Appl. 12, 024010 (2019).

Xiang, C. et al. 3d integration enables ultralow-noise isolator-free lasers in silicon photonics. Nature 620, 78–85 (2023).

f / 4.5 ,   f / 5 ,   f / 5.6 ,   f / 6.3 ,   f / 7.1 ,   f / 8 ,   … {\displaystyle f/4.5,\ f/5,\ f/5.6,\ f/6.3,\ f/7.1,\ f/8,\ \ldots }

Beyond focus, image sharpness is related to f-number through two different optical effects: aberration, due to imperfect lens design, and diffraction which is due to the wave nature of light.[15] The blur-optimal f-stop varies with the lens design. For modern standard lenses having 6 or 7 elements, the sharpest image is often obtained around f/5.6–f/8, while for older standard lenses having only 4 elements (Tessar formula) stopping to f/11 will give the sharpest image.[citation needed] The larger number of elements in modern lenses allow the designer to compensate for aberrations, allowing the lens to give better pictures at lower f-numbers. At small apertures, depth of field and aberrations are improved, but diffraction creates more spreading of the light, causing blur.

For example, the Uniform System (U.S.) of apertures was adopted as a standard by the Photographic Society of Great Britain in the 1880s. Bothamley in 1891 said "The stops of all the best makers are now arranged according to this system."[27] U.S. 16 is the same aperture as f/16, but apertures that are larger or smaller by a full stop use doubling or halving of the U.S. number, for example f/11 is U.S. 8 and f/8 is U.S. 4. The exposure time required is directly proportional to the U.S. number. Eastman Kodak used U.S. stops on many of their cameras at least in the 1920s.

Blumenthal, D. J. et al. Frequency-stabilized links for coherent wdm fiber interconnects in the datacenter. J. Lightwave Technol. 38, 3376–3386 (2020).

With 8% loss per air-glass surface on lenses without coating, multicoating of lenses is the key in lens design to decrease transmittance losses of lenses. Some reviews of lenses do measure the T-stop or transmission rate in their benchmarks.[11][12] T-stops are sometimes used instead of f-numbers to more accurately determine exposure, particularly when using external light meters.[13] Lens transmittances of 60%–95% are typical.[14] T-stops are often used in cinematography, where many images are seen in rapid succession and even small changes in exposure will be noticeable. Cinema camera lenses are typically calibrated in T-stops instead of f-numbers.[13] In still photography, without the need for rigorous consistency of all lenses and cameras used, slight differences in exposure are less important; however, T-stops are still used in some kinds of special-purpose lenses such as Smooth Trans Focus lenses by Minolta and Sony.

Confocalresonator

Nosepiece is a circular structure housing the objective lenses. There are holes where the different objective lenses are screwed in. To change the magnification ...

The basic approach is as follows: a second copy of the W-band signal is synthesized by picking of some light from each laser using fiber-optic couplers. A frequency detuning of Δ = + 100 MHz in this signal is implemented by shifting the optical frequency in one of the arms using an acousto-optic modulator driven by a low-noise RF signal generator (Keysight E8257D). After combining these two reference optical signals into a single optical fiber, a variable fiber delay line is inserted before the signal is down-converted using another nominally identical fast photodiode. The result is two copies of the W-band signal, separated in frequency by Δ and in time by delay τd. To complete the frequency discriminator approach, these two signals are mixed down to Δ using a W-band mixer (Hasco HWMX10-SFW), and the characteristic spectral shape is digitized using a radiofrequency spectrum analyzer (Keysight N9030B). To eliminate nulls in the frequency discriminator transfer function, several delay lengths (0.2, 1.0, 3, 4.3, 10.5, 11.6, and 14.8 km) are used22.

J. H. Dallmeyer's son, Thomas Rudolphus Dallmeyer, inventor of the telephoto lens, followed the intensity ratio terminology in 1899.[26]

The photonic resonator is fabricated from a single piece of high-purity fused silica glass. As shown in Fig. 1a, the length of the resonator is 25.4 mm, and the diameter is 15 mm, corresponding to a total volume of about 4.5 cm3. Both facets of the resonator, which is plano-convex with a 1 m radius of curvature (ROC), are super-polished to reduce scattering. To establish the resonance, high reflective dielectric layers centered at 1550 nm are deposited on both facets of the resonator. The transmission of the coating is designed to be 10 part-per-million (ppm), and the loss is <1 ppm. Figure 1b shows the measurement of the optical field ringdown (green) in reflection, and the fitting curve (red) reveals a 13.3 μs decay constant, yielding a finesse of 170,000. Combined with the measurement of the intensity of the reflection and transmission, the cavity transmission and loss (mostly from material absorption and scattering) are verified to be 7.9 and 10.5 ppm32, respectively, which are consistent with the expected values. The corresponding cavity linewidth is 24 kHz, supporting a quality factor of about 8.3 billion at this wavelength. The birefringence of the cavity leads to two polarization states with a frequency separation of about 30 MHz, which has a negligible impact on laser frequency stabilization by launching a beam with circular polarization for PDH locking6.

a Schematic. CW laser continuous-wave laser, PM phase modulation, PNM phase noise measurement. The two lasers with center frequency at f1 and f2, respectively, are locked to separated photonic resonator (PR) modes simultaneously with frequency difference at a multiple of free-spectral range (FSR). Inset: The expected phase noise of the heterodyned signal demodulated by the photodetector (PD) is below the thermorefractive (TR) noise of individual laser at low Fourier frequencies, and limited by inloop error of the PDH locking. Phase noise measurement. b Phase noise measurement. Red: phase noise at carrier frequency of 96 GHz. Orange and blue: PDH inloop error for laser 1 and 2, respectively. Gray: phase noise at a carrier frequency of 4 GHz. Green dash: thermorefractive noise.

Doeleman, S. et al. Adapting a cryogenic sapphire oscillator for very long baseline interferometry. Publ. Astron. Soc. Pac. 123, 582 (2011).

Figure 2a presents the scheme of the laser frequency stabilization on the photonic resonator and the characterization of the stability. A continuous-wave laser (CW laser 1) at 1550 nm is frequency-locked on the photonic resonator and heterodyned with the reference laser (CW laser 2). The beatnote signal is recorded by a phase noise analyzer and a frequency counter for analysis. The measured noise performance is almost exclusively attributed to the photonic resonator since the contribution from the reference laser is negligible (see Methods). Figure 2b shows the power spectral density (PSD) of the phase noise measurement (red line). For Fourier frequencies from 1 Hz to 3 kHz, the laser noise is limited by thermorefractive noise (TR, green-dashed line)27,28 arising from the monolithic fused silica spacer. Around 10 kHz, the laser noise is at −112 dBc/Hz, which is limited by the PDH detection noise, including shot noise and detector impedance noise. For frequencies above 20 kHz, the servo inloop errors from each of the two stabilized lasers are the dominant contribution; the servo bump at 750 kHz is from the reference laser, and the 500 kHz feature is from the resonator-stabilized laser. The vibration-induced noise (gray line) is derived by multiplying the vibration noise on the breadboard where the resonator is sitting, and the simulated vibration sensitivity is well below the TR noise floor. The laser linewidth is determined by integrating the PSD of the phase noise from high Fourier frequencies towards zero33. Figure 2c shows the full-width half-maximum (FWHM) laser linewidth corresponding to the integrated phase noise up to 1 rad2 is 18 Hz, dominated by the TR-noise limit at 15 Hz.

Tapley, B. D., Bettadpur, S., Ries, J. C., Thompson, P. F. & Watkins, M. M. Grace measurements of mass variability in the earth system. Science 305, 503–505 (2004).

Zhang, W. et al. Reduction of residual amplitude modulation to 1 × 10−6 for frequency modulation and laser stabilization. Opt. Lett. 39, 1980–1983 (2014).

Zhao, Q. et al. Integrated reference cavity with dual-mode optical thermometry for frequency correction. Optica 8, 1481–1487 (2021).

Rubiola, E., Salik, E., Huang, S., Yu, N. & Maleki, L. Photonic-delay technique for phase-noise measurement of microwave oscillators. J. Opt. Soc. Am. B 22, 987–997 (2005).

The authors thank Dr. Ken B. Cooper and Dr. Siamak Forouhar for helpful discussions. The reported here research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).

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To perform as a thermal-noise-limited frequency reference in the ambient environment, the vibration- and temperature-induced frequency fluctuations are two critical technical noise sources that must be significantly suppressed. With no additional vibration isolation, the vibration sensitivity of the resonator is minimized by previously reported strategies26 to achieve 10−10/g along the gravitational direction and horizontal plane. As shown in Fig. 1c, the cavity is installed in a support structure that is 25.4 mm in length, made from Teflon, and symmetrically held by four supporting teeth. According to finite element analysis, the distance between the supporting teeth and the facet of the cavity and the open angle of the tooth are chosen to be 4.35 mm and 60°, respectively. The impact of environmental temperature fluctuations on the resonator is suppressed by thermal isolation consisting of two layers and cavity support. Figure 1c shows a cross-sectional view of the enclosure. The temperature fluctuation on layer 1 (copper, 60 mm × 70 mm × 55 mm) is stabilized to mK level by using a resistive heater. Layer 2 (copper) is bolted down to layer 1 through a plate (Teflon) with three o-rings (Viton) on the bottom to reduce the thermal conductivity. The cavity support (Teflon) is installed in layer 2, and thermal damping is provided by the material and the minimization of the contact area with the resonator. The thermal isolation is equal to a second order resistor–capacitor low pass filter, and the temperature-induced frequency fluctuation is 8.5 × 10−13τ, where τ is the averaging time.

As shown in Fig. 2a, we perform PDH locking to stabilize the frequency of CW laser 1 at 1550 nm to the photonic resonator. Using a polarization-maintaining fiber coupler, only about 10 μW is sent to the resonator for frequency locking, while the majority of the laser power (several mW) is reserved for characterization or other applications. A fiber-based waveguide electro-optic modulator (EOM) implements phase modulation at 9.8 MHz to generate the locking error signal, while the laser current modulation port is used as the actuator for frequency lock. To investigate the frequency stability, CW laser 1 locked on the resonator is heterodyned with the reference laser, i.e., CW laser 2 locked on a 5-cm cubic vacuum-gap FP cavity fabricated from ultralow expansion glass. The frequency stability of the reference laser is 2 × 10−15, which is well below the TR noise of the photonic resonator. Due to the narrow cavity linewidth, the RAM-induced noise42 is substantially suppressed. Consequently, 1 ppm RAM corresponds to 0.024 Hz frequency fluctuation, which is a negligible level compared to the TR noise. In-line isolators are inserted into the locking scheme to prevent unexpected interference. The temperature of the EOM is controlled to stabilize the long-term RAM drift.

Savchenkov, A. et al. Spectral purity improvement in flickering self-injection locked lasers. IEEE J. Quantum Electron. 58, 1–9 (2022).

By 1895, Hodges contradicts Bothamley, saying that the f-number system has taken over: "This is called the f/x system, and the diaphragms of all modern lenses of good construction are so marked."[28]

A T-stop (for transmission stops, by convention written with capital letter T) is an f-number adjusted to account for light transmission efficiency (transmittance). A lens with a T-stop of N projects an image of the same brightness as an ideal lens with 100% transmittance and an f-number of N. A particular lens's T-stop, T, is given by dividing the f-number by the square root of the transmittance of that lens: T = N transmittance . {\displaystyle T={\frac {N}{\sqrt {\text{transmittance}}}}.} For example, an f/2.0 lens with transmittance of 75% has a T-stop of 2.3: T = 2.0 0.75 = 2.309... {\displaystyle T={\frac {2.0}{\sqrt {0.75}}}=2.309...} Since real lenses have transmittances of less than 100%, a lens's T-stop number is always greater than its f-number.[10]

a Schematic of frequency stabilization and phase noise measurement. CW laser continuous-wave laser, PR photonic resonator, PNM phase noise measurement, PD photodetector. b Phase noise measurement (red) representing the stability of resonator-stabilized laser in the frequency domain. Thermorefractive noise (TR, green-dashed) sets the noise floor up to 1 kHz, and the thermal Brownian noise (blue-dashed) and derived vibration-induced noise are well below the measurement. c The integrated phase noise. The phase error of 1 rad2 corresponds to 9 Hz, leading to the FWHM laser linewidth of 18 Hz.

Li, Y. et al. Photonic generation of w-band arbitrary waveforms with high time-bandwidth products enabling 3.9 mm range resolution. Optica 1, 446–454 (2014).

Argence, B. et al. Prototype of an ultra-stable optical cavity for space applications. Opt. Express 20, 25409–25420 (2012).

where f is the focal length, and D is the diameter of the entrance pupil (effective aperture). It is customary to write f-numbers preceded by "f/", which forms a mathematical expression of the entrance pupil's diameter in terms of f and N.[1] For example, if a lens's focal length were 100 mm and its entrance pupil's diameter were 50 mm, the f-number would be 2. This would be expressed as "f/2" in a lens system. The aperture diameter would be equal to f/2.

Abich, K. et al. In-orbit performance of the grace follow-on laser ranging interferometer. Phys. Rev. Lett. 123, 031101 (2019).

Most electronic cameras allow to amplify the signal coming from the pickup element. This amplification is usually called gain and is measured in decibels. Every 6 dB of gain is equivalent to one T-stop in terms of light transmittance. Many camcorders have a unified control over the lens f-number and gain. In this case, starting from zero gain and fully open iris, one can either increase f-number by reducing the iris size while gain remains zero, or one can increase gain while iris remains fully open.

Green: FFS with linear drift. Red-circle: FFS with linear drift removed, achieving 7 × 10−14. The error bar accordingly shows 1σ confidence interval.  Gray; predicted thermorefractive (TR) noise.

As in the earlier DIN and ASA film-speed standards, the ISO speed is defined only in one-third stop increments, and shutter speeds of digital cameras are commonly on the same scale in reciprocal seconds. A portion of the ISO range is the sequence

Hood, C. J., Kimble, H. J. & Ye, J. Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity. Phys. Rev. A 64, 033804 (2001).

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The f-number of the human eye varies from about f/8.3 in a very brightly lit place to about f/2.1 in the dark.[17] Computing the focal length requires that the light-refracting properties of the liquids in the eye be taken into account. Treating the eye as an ordinary air-filled camera and lens results in an incorrect focal length and f-number.

In this article, we report an advanced monolithic photonic resonator for laser stabilization that achieves thermal-noise-limited operation in the ambient environment. Optimization of cavity fabrication by using low-loss fused silica (Suprasil 3001) permits resonator finesse of 170,000. By engineering the cavity geometry and support structure, technical noise is suppressed so that the phase noise of the resonator-stabilized laser is limited only by thermorefractive noise27,28 at the close-to-carrier frequency, and achieves −112 dBc/Hz at 10 kHz. The corresponding integrated linewidth is 18 Hz, among the narrowest results reported to date for monolithic resonators. The fractional frequency stability is 7 × 10−14 after subtracting a constant 160 Hz/s linear frequency drift. This performance level is attained with the resonator installed in a compact enclosure of 0.23 L volume, without vacuum or additional vibration isolation.

Figure 3 presents the fractional frequency stability (FFS) of the resonator-stabilized laser for time domain analysis. The beatnote signal is recorded by a λ-type, dead-time-free frequency counter to compute the Allan deviation. With the linear drift (160 Hz/s), the FFS (green line) reaches <1 × 10−13 for τ < 0.08 s, and 8.5 × 10−13τ due to linear drift for τ > 0.1 s. After correcting for the linear drift in data processing, the FFS (circle-line, with error bar representing 1σ confidence interval) achieves 7 × 10−14 for 0.08 s < τ < 0.3 s, approaching the TR-noise-limit (gray) at 5 × 10−14 and dominated by residual nonlinear frequency drift for τ > 1 s.

Photographic film's and electronic camera sensor's sensitivity to light is often specified using ASA/ISO numbers. Both systems have a linear number where a doubling of sensitivity is represented by a doubling of the number, and a logarithmic number. In the ISO system, a 3° increase in the logarithmic number corresponds to a doubling of sensitivity. Doubling or halving the sensitivity is equal to a difference of one T-stop in terms of light transmittance.

Ignoring differences in light transmission efficiency, a lens with a greater f-number projects darker images. The brightness of the projected image (illuminance) relative to the brightness of the scene in the lens's field of view (luminance) decreases with the square of the f-number. A 100 mm focal length f/4 lens has an entrance pupil diameter of 25 mm. A 100 mm focal length f/2 lens has an entrance pupil diameter of 50 mm. Since the area is proportional to the square of the pupil diameter,[6] the amount of light admitted by the f/2 lens is four times that of the f/4 lens. To obtain the same photographic exposure, the exposure time must be reduced by a factor of four.

Sometimes the same number is included on several scales; for example, an aperture of f/1.2 may be used in either a half-stop[7] or a one-third-stop system;[8] sometimes f/1.3 and f/3.2 and other differences are used for the one-third stop scale.[9]

A 200 mm focal length f/4 lens has an entrance pupil diameter of 50 mm. The 200 mm lens's entrance pupil has four times the area of the 100 mm f/4 lens's entrance pupil, and thus collects four times as much light from each object in the lens's field of view. But compared to the 100 mm lens, the 200 mm lens projects an image of each object twice as high and twice as wide, covering four times the area, and so both lenses produce the same illuminance at the focal plane when imaging a scene of a given luminance.

Lasermodes pdf

Fortier, T. et al. Generation of ultrastable microwaves via optical frequency division. Nat. Photonics 5, 425–429 (2011).

The f-number is also known as the inverse relative aperture, because it is the inverse of the relative aperture, defined as the aperture diameter divided by focal length.[5] The relative aperture indicates how much light can pass through the lens at a given focal length. A lower f-number means a larger relative aperture and more light entering the system, while a higher f-number means a smaller relative aperture and less light entering the system. The f-number is related to the numerical aperture (NA) of the system, which measures the range of angles over which light can enter or exit the system. The numerical aperture takes into account the refractive index of the medium in which the system is working, while the f-number does not.

Leibrandt, D. R., Thorpe, M. J., Bergquist, J. C. & Rosenband, T. Field-test of a robust, portable, frequency-stable laser. Opt. Express 19, 10278–10286 (2011).

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In photography, stops are also a unit used to quantify ratios of light or exposure, with each added stop meaning a factor of two, and each subtracted stop meaning a factor of one-half. The one-stop unit is also known as the EV (exposure value) unit. On a camera, the aperture setting is traditionally adjusted in discrete steps, known as f-stops. Each "stop" is marked with its corresponding f-number, and represents a halving of the light intensity from the previous stop. This corresponds to a decrease of the pupil and aperture diameters by a factor of 1/√2 or about 0.7071, and hence a halving of the area of the pupil.

An f-number is a measure of the light-gathering ability of an optical system such as a camera lens. It is calculated by dividing the system's focal length by the diameter of the entrance pupil ("clear aperture").[1][2][3] The f-number is also known as the focal ratio, f-ratio, or f-stop, and it is key in determining the depth of field, diffraction, and exposure of a photograph.[4] The f-number is dimensionless and is usually expressed using a lower-case hooked f with the format f/N, where N is the f-number.

Panuski, C., Englund, D. & Hamerly, R. Fundamental thermal noise limits for optical microcavities. Phys. Rev. X 10, 041046 (2020).

Laser resonatorPdf

Piper in 1901[29] discusses five different systems of aperture marking: the old and new Zeiss systems based on actual intensity (proportional to reciprocal square of the f-number); and the U.S., C.I., and Dallmeyer systems based on exposure (proportional to square of the f-number). He calls the f-number the "ratio number", "aperture ratio number", and "ratio aperture". He calls expressions like f/8 the "fractional diameter" of the aperture, even though it is literally equal to the "absolute diameter" which he distinguishes as a different term. He also sometimes uses expressions like "an aperture of f 8" without the division indicated by the slash.

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Liu, K. et al. 36 hz integral linewidth laser based on a photonic integrated 4.0 m coil resonator. Optica 9, 770–775 (2022).

In 1867, Sutton and Dawson defined "apertal ratio" as essentially the reciprocal of the modern f-number. In the following quote, an "apertal ratio" of "1⁄24" is calculated as the ratio of 6 inches (150 mm) to 1⁄4 inch (6.4 mm), corresponding to an f/24 f-stop:

a Photograph of the photonic resonator. b The cavity ring-down (green) in reflected field when the laser frequency is swept crossing the resonance and the exponential decay fitting (red) with 13.3 μs time constant leading a finesse of 170,000. c The section view of cavity assembly, including the two-layer thermal shield and the resonator. The heater and thermistor are attached to layer 1 for temperature stabilization.