From 52ff45a7c687d502492be0fa6e54f9b99d501465 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Mon, 27 Feb 2023 15:03:21 +0100 Subject: Publish "Website adventures" part 4 about images --- .../blog/2022/website-adventures-basics/index.md | 5 +- .../2022/website-adventures-generators/index.md | 7 +- source/blog/2022/website-adventures-maths/index.md | 5 +- .../website-adventures-images/avif-reddit.avif | Bin 0 -> 59792 bytes .../blog/2023/website-adventures-images/index.md | 394 +++++++++++++++++++++ .../website-adventures-images/jpeg-arithmetic.jpg | Bin 0 -> 308893 bytes .../website-adventures-images/jpeg-huffman.jpg | Bin 0 -> 339094 bytes .../website-adventures-images/png-original.png | Bin 0 -> 1207860 bytes .../png-oxipng-zopfli.png | Bin 0 -> 973478 bytes .../2023/website-adventures-images/png-oxipng.png | Bin 0 -> 982403 bytes .../website-adventures-images/png-zopflipng.png | Bin 0 -> 975943 bytes .../website-adventures-images/simulation-full.png | Bin 0 -> 973478 bytes .../website-adventures-images/simulation-half.avif | Bin 0 -> 27563 bytes .../website-adventures-images/simulation-half.jpg | Bin 0 -> 127051 bytes .../website-adventures-images/simulation-half.png | Bin 0 -> 246421 bytes .../website-adventures-images/simulation-half.webp | Bin 0 -> 55982 bytes .../website-adventures-images/webp-lossless.webp | Bin 0 -> 766604 bytes .../2023/website-adventures-images/webp-lossy.webp | Bin 0 -> 139868 bytes .../webp-near-lossless.webp | Bin 0 -> 492606 bytes .../know/concept/modulational-instability/index.md | 2 +- 20 files changed, 405 insertions(+), 8 deletions(-) create mode 100644 source/blog/2023/website-adventures-images/avif-reddit.avif create mode 100644 source/blog/2023/website-adventures-images/index.md create mode 100644 source/blog/2023/website-adventures-images/jpeg-arithmetic.jpg create mode 100644 source/blog/2023/website-adventures-images/jpeg-huffman.jpg create mode 100644 source/blog/2023/website-adventures-images/png-original.png create mode 100644 source/blog/2023/website-adventures-images/png-oxipng-zopfli.png create mode 100644 source/blog/2023/website-adventures-images/png-oxipng.png create mode 100644 source/blog/2023/website-adventures-images/png-zopflipng.png create mode 100644 source/blog/2023/website-adventures-images/simulation-full.png create mode 100644 source/blog/2023/website-adventures-images/simulation-half.avif create mode 100644 source/blog/2023/website-adventures-images/simulation-half.jpg create mode 100644 source/blog/2023/website-adventures-images/simulation-half.png create mode 100644 source/blog/2023/website-adventures-images/simulation-half.webp create mode 100644 source/blog/2023/website-adventures-images/webp-lossless.webp create mode 100644 source/blog/2023/website-adventures-images/webp-lossy.webp create mode 100644 source/blog/2023/website-adventures-images/webp-near-lossless.webp (limited to 'source') diff --git a/source/blog/2022/website-adventures-basics/index.md b/source/blog/2022/website-adventures-basics/index.md index 64ed446..02aebef 100644 --- a/source/blog/2022/website-adventures-basics/index.md +++ b/source/blog/2022/website-adventures-basics/index.md @@ -5,7 +5,7 @@ layout: "blog" toc: true --- -Published on 2022-11-20, last updated on 2022-11-28. +Published on 2022-11-20, last updated on 2023-02-27. Making and managing this personal website has been an adventure. In this series, I go over the technical challenges I've encountered @@ -13,7 +13,8 @@ and philosophical decisions I've made, and I review some of the tools I've used along the way. After [part 1](/blog/2022/website-adventures-generators/), this is part 2, -followed by [part 3](/blog/2022/website-adventures-maths/). +followed by [part 3](/blog/2022/website-adventures-maths/) +and [part 4](/blog/2023/website-adventures-images/). diff --git a/source/blog/2022/website-adventures-generators/index.md b/source/blog/2022/website-adventures-generators/index.md index 42a6230..da7dde4 100644 --- a/source/blog/2022/website-adventures-generators/index.md +++ b/source/blog/2022/website-adventures-generators/index.md @@ -5,14 +5,15 @@ layout: "blog" toc: true --- -Published on 2022-11-15, last updated on 2022-11-28. +Published on 2022-11-15, last updated on 2023-02-27. Making and managing this personal website has been an adventure. In this series, I go over the technical challenges I've encountered and philosophical decisions I've made, and I review some of the tools I've used along the way. -This is part 1, followed by [part 2](/blog/2022/website-adventures-basics/) -and [part 3](/blog/2022/website-adventures-maths/). +This is part 1, followed by [part 2](/blog/2022/website-adventures-basics/), +[part 3](/blog/2022/website-adventures-maths/) +and [part 4](/blog/2023/website-adventures-images/). diff --git a/source/blog/2022/website-adventures-maths/index.md b/source/blog/2022/website-adventures-maths/index.md index a49b473..879a7e3 100644 --- a/source/blog/2022/website-adventures-maths/index.md +++ b/source/blog/2022/website-adventures-maths/index.md @@ -5,7 +5,7 @@ layout: "blog" toc: true --- -Published on 2022-11-28. +Published on 2022-11-28, last updated on 2023-02-27. Making and managing this personal website has been an adventure. In this series, I go over the technical challenges I've encountered @@ -13,7 +13,8 @@ and philosophical decisions I've made, and I review some of the tools I've used along the way. After [part 1](/blog/2022/website-adventures-generators/) and [part 2](/blog/2022/website-adventures-basics/), -this is part 3. +this is part 3, +followed by [part 4](/blog/2023/website-adventures-images/). diff --git a/source/blog/2023/website-adventures-images/avif-reddit.avif b/source/blog/2023/website-adventures-images/avif-reddit.avif new file mode 100644 index 0000000..686672a Binary files /dev/null and b/source/blog/2023/website-adventures-images/avif-reddit.avif differ diff --git a/source/blog/2023/website-adventures-images/index.md b/source/blog/2023/website-adventures-images/index.md new file mode 100644 index 0000000..d678362 --- /dev/null +++ b/source/blog/2023/website-adventures-images/index.md @@ -0,0 +1,394 @@ +--- +title: "Adventures in making this website:
image optimization" +date: 2023-02-27 +layout: "blog" +toc: true +--- + +Published on 2023-02-27. + +Making and managing this personal website has been an adventure. +In this series, I go over the technical challenges I've encountered +and philosophical decisions I've made, +and I review some of the tools I've used along the way. +After [part 1](/blog/2022/website-adventures-generators/), +[part 2](/blog/2022/website-adventures-basics/) +and [part 3](/blog/2022/website-adventures-maths/), +this is part 4. + + + +## Principles + +After some consideration, +I settled on the following five commandments of image optimization +for this website: + +1. **Provide the originals.** + If any scaling or lossy compression has been done + on an image displayed on the page, + wrap it in `` linking to the full-size lossless original file. + +2. **Prioritize quality**. + My server provider doesn't charge me for traffic, + and there aren't many images on this website, + so I can afford to keep the quality high. + The degradation of an 80% JPEG compared to 90% + isn't worth it just to save a few kilobytes. + You should do your own tests to decide what quality + is acceptable for the kind of images you have. + +3. **Limit the resolutions.** + It's wasteful to use a 2000px wide image + if it's only 200px wide on a user's screen. + I use the following rules to choose resolutions: + + 1. If an image is `W`px wide my screen, + its resolution should be roughly `1.5*W`px: + a 50% surplus is friendly to high-DPI screens, + and prevents scaling artefacts. + This rule also works in reverse: + if I have an `X`px file for whatever reason, + its scale should be chosen such that it takes up around `0.67*X`px on my screen. + + 2. Because my website's `` is ~720px wide for me, + the largest sensible image width is ~1080px. + But sometimes more is needed to show all details: + in that case, display a downscaled image on the page, + linking to the original. + When downscaling, always divide the width and height + by an integer divisor of both. + +4. **Use modern formats.** + The JPEG standard was released in 1992, + and in the three decades since then, + a lot of research has been done in image and video compression. + If a modern format can shrink files by 75% + with little to no change in quality, why wouldn't I use it? + +5. **Maintain compatibility.** + Modern image formats are great, but some are so modern + that almost nobody can use them, so fallbacks should be provided. + Fortunately, HTML has a built-in way to do this, see below. + +The HTML `` tag can be used to provide multiple formats +depending on what the browser supports, +or multiple resolutions depending on the user's screen size +(I don't use this feature as of writing). +`` is supported by all modern browsers, +but obviously [not IE11](https://caniuse.com/picture), +although that doesn't matter, because backwards compatibility is baked into it! +Here it is: + +```html + + + + + + + + + ... + + + +``` + +To automate rules 1, 3, 4 and 5, I wrote a Jekyll template: +[`image.html`](/code/prefetch-jekyll/tree/source/_includes/image.html). +To handle rule 3 there's a bit of logic, but the idea is as follows: + +* If the file is small enough not to need downscaling, + provide an AVIF and the original JPEG or PNG (in that order). + In my testing, WebP didn't help much for such tiny images. +* For larger images, the full-size file ends in `-full`, + and the half-size file in `-half`. The template detects this, + and then displays the downscaled AVIF, WebP and JPEG versions, + wrapped in a link to the original PNG/JPEG. + +Let me tell you how I squeezed every drop out of those PNG, JPEG, WebP and AVIF files. + + + +## Optimizing each format + +I will use the test image given below, which shows a simulation of +[modulational instability](/know/concept/modulational-instability/) +in an optical fiber. +The raw file, as exported by [Veusz](https://veusz.github.io/), +is a 1,207,860-byte (1.21MB, 1.15MiB) PNG +with a resolution of 2362x1062, or 2,508,444px² (2.51MP). +Click for lossless full-size view: + +{% include image.html file="simulation-full.png" + width="100%" alt="Modulational instability simulation results" %} + +This is representative of the kind of pictures I have on this website, +and it has a combination of high-contrast text and lines, gradual colour transitions, +and fine details (especially at the top of the left panel). +Depending on what images you want to optimize, +your mileage may vary if you take my advice. +Figure out what works for you, and I hope this post helps with that. + + + +### PNG + +Despite dating from the late 1990s, +[PNG](https://en.wikipedia.org/wiki/PNG) is still pretty solid. +Its losslessness makes it by far the largest format discussed here, +but it would be unfair to hold that against it. +For full quality, PNG is the perfect choice +before more exotic formats like TIFF. + +But not all lossless compression is the same. +Given a PNG image, you may be able to reduce its size +using [Oxipng](https://github.com/shssoichiro/oxipng), +which plays with the file's internal knobs +to squeeze every last drop out of the compression algorithms. +Let's dial all settings to eleven and see what it can do: + +```sh +$ oxipng --strip=all --alpha --filters=9 --zc=12 image.png +# 1,207,860 to 982,403 bytes: reduced by 225,457 bytes (18.7%) +# Took 2s on my system (oxipng 8.0.0) +``` + +A 16.4% reduction for losslessly compressed data is pretty impressive. +`--strip=all` removes all unneeded metadata, +`--alpha` enables transparency optimizations, +e.g. removing the A channel from an opaque RGBA image, +and `--zc=12` sets the highest [Deflate](https://en.wikipedia.org/wiki/Deflate) level. + +`--filters=9` tells Oxipng to brute-force the PNG's per-row +[filtering](https://en.wikipedia.org/wiki/PNG#Filtering) setting, +which always gave the best results in my tests. +I you want, you can instead use `--opt=max` to try all filtering approaches +(including `9`): this is slower, but may be better for some images? +I'm not sure. + +But can we go smaller? Yes, in fact we can, although it comes at a cost. +Yes, that's over 2 minutes to optimize a single 2.5MP image: + +```sh +$ oxipng --strip=all --alpha --filters=9 --zopfli image.png +# 1,207,860 to 973,478 bytes: reduced by 234,382 bytes (19.4%) +# Took 2m9s on my system (oxipng 8.0.0) +``` + +So what does `--zopfli` do? Well, [Zopfli](https://github.com/google/zopfli) +is a Google-made compressor aiming to minimize file sizes +while being Deflate-compatible. +Basically, it's [zlib](https://zlib.net/) but better. +Interestingly, Zopfli ships with its own PNG optimizer `zopflipng`... +Let's see what it can do: + +```sh +$ zopflipng --iterations=15 --filters=b -y input.png output.png +# 1,207,860 to 975,943 bytes: reduced by 231,917 bytes (19.2%) +# Took 1m3s on my system (zopfli 1.0.3) +``` + +You can increase the number of iterations if you want to, +but the returns diminish strongly, +and it wasn't enough to beat Oxipng in a reasonable time. +Time is the key word: +no, `zopflipng` doesn't beat Oxipng's Zopfli mode, +but it does save a lot of time! + +File links: +[original PNG](png-original.png), +[Oxipng (no Zopfli)](png-oxipng.png), +[Oxipng (Zopfli)](png-oxipng-zopfli.png), +[Zopflipng](png-zopflipng.png), + + + +### JPEG + +While PNG has held up well, [JPEG](https://en.wikipedia.org/wiki/JPEG) hasn't really. +It's the most popular lossy compression format, +but its distinctive damage to the image's quality +is worse than modern alternatives, +and high-quality JPEGs tend to be quite large. + +Figuring out how to optimize JPEGs is easy, +because someone else has already done the hard work for us: +[MozJPEG](https://github.com/mozilla/mozjpeg) +is a [friendly fork](https://libjpeg-turbo.org/About/Mozjpeg) +of the popular `libjpeg-turbo` package, +where Mozilla uses all the latest tech to make small, good-looking JPEGs: + +```sh +$ cjpeg -quality 90 -outfile output.jpg input.png +# 1,207,860 to lossy 339,094 bytes: reduced by 868,766 bytes (71.9%) +# Took 0.2s on my system (MozJPEG 4.1.1) +``` + +But what if you don't have a PNG, only a non-optimized JPEG? +Then you can choose between: + +```sh +$ jpegtran -copy none -outfile output.jpg input.jpg +$ jpegoptim --strip-all image.jpg # separate package +``` + +And as long as you have MozJPEG installed, +these will squeeze every drop out of your file. +If you can't use MozJPEG for some reason, +`jpegoptim` can still help a bit. + +If you're feeling adventurous, you can pass the `-arithmetic` flag +to `cjpeg` or `jpegtran` to employ [arithmetic coding](https://en.wikipedia.org/wiki/Arithmetic_coding) +instead of [Huffman coding](https://en.wikipedia.org/wiki/Huffman_coding), +which knocks a few percent off the size: + +```sh +# Don't actually do this! +$ cjpeg -quality 90 -arithmetic -outfile output.jpg input.png +# 1,207,860 to lossy 308,893 bytes: reduced by 898,967 bytes (74.4%) +# Took 0.4s on my system (MozJPEG 4.1.1) +``` + +The problem is that many image viewers (including browsers) +can't display these files because arithmetic coding was patented, +so implementations couldn't just be freely distributed or used. +The patents have expired now, but the world hasn't caught up yet. + +File links: +[normal JPEG](jpeg-huffman.jpg), +[arithmetic JPEG](jpeg-arithmetic.jpg). +Bonus points if you can open the latter. + + + +### WebP + +[WebP](https://en.wikipedia.org/wiki/WebP) is the Swiss army knife of formats, +supporting both lossless and lossy compression of static or animated images. +It's only been around for a few years, and hasn't gained so much popularity, +probably because it was immediately overshadowed by AVIF (to be discussed below). +Internally, it relies on VP8 video compression: +this provides a nice size reduction, +but the images tend to get blurred slightly, +and low-contrast areas can become blocky. + +Indeed high-quality JPEGs look better than high-quality WebPs, +but the latter tend to be less than half the size, so it isn't bad at all. +According to my tests, the best way to create a WebP is: + +```sh +$ cwebp -q 90 -m 6 -sharp_yuv -sns 100 input.png -o output.webp +# 1,207,860 to lossy 139,868 bytes: reduced by 1,067,992 bytes (88.4%) +# Took 0.3s on my system (libwebp 1.3.0, libsharpyuv 0.2.0) +``` + +The flags `-q 90` and `-m 6` set the quality to 90% +with the slowest compression method. +The most important setting is `-sharp_yuv`, +which counteracts a lot of the blurriness +and more accurately preserves the contrast, +at the cost of slightly larger files. +`-sns 100` has a big effect on file size, without much influence on quality. + +I don't use WebP's lossless mode, but according to +[this post](https://siipo.la/blog/whats-the-best-lossless-image-format-comparing-png-webp-avif-and-jpeg-xl) +it's one of the best out there, so I had a quick look, +and it's indeed pretty impressive compared to PNG: + +```sh +$ cwebp -lossless -z 9 input.png -o output.webp +# 1,207,860 to 766,604 bytes: reduced by 441,256 bytes (36.5%) +# Took 7.2s on my system (libwebp 1.3.0) +``` + +You can also pass `-near_lossless 0` to let it slightly change +pixel values for better compressibility. +This causes visible degradation, but still looks better than a lossy-mode WebP: + +```sh +$ cwebp -lossless -near_lossless 0 -z 9 input.png -o output.webp +# 1,207,860 to 492,606 bytes: reduced by 715,254 bytes (59.2%) +# Took 6.6s on my system (libwebp 1.3.0) +``` + +File links: +[lossy WebP](webp-lossy.webp), +[lossless WebP](webp-lossless.webp), +[near-lossless WebP](webp-near-lossless.webp). + + + +### AVIF + +[AVIF](https://en.wikipedia.org/wiki/AVIF), +the new kid on the block based on [AV1](https://en.wikipedia.org/wiki/AV1) video compression, +is honestly incredible: +it reduces file sizes much further than WebP, +and the final quality is so good that I sometimes struggled +to tell the difference with the original PNG. +I suspect that AVIF's success is the reason why Google is winding down +their work on [WebP 2](https://en.wikipedia.org/wiki/WebP#WebP_2) +and [JPEG-XL](https://en.wikipedia.org/wiki/JPEG_XL#Preliminary_web_browser_support). + +AV1 compression is complicated business, +so I haven't exhaustively tested all options as I did for JPEG and WebP. +Instead, I used the settings I found in +[this Reddit post](https://www.reddit.com/r/AV1/comments/o7s8hk/high_quality_encoding_of_avif_images_using/), +because I trust random strangers on the Internet: + +```sh +$ avifenc --min 0 --max 63 --speed 0 -a end-usage=q -a cq-level=31 -a color:sharpness=2 \ + -a tune=ssim -a color:enable-chroma-deltaq=1 -a color:deltaq-mode=3 -a color:aq-mode=1 \ + input.png output.avif +# 1,207,860 to lossy 59,792 bytes: reduced by 1,148,068 bytes (95.1%) +# Took 11s on my system (libavif 0.11.1) +``` + +Yes, it's slower to encode, and you can see some degradation if you look closely, +but that's a 95% compression ratio! +It looks slightly worse than the JPEG from earlier (which was 5 times as big), +but better than the lossy WebP (which was over twice the size). +Incredible. + +AVIF is quite new, so support is still lacking. +As of writing, Firefox and Chromium work perfectly, but Edge can't display AVIFs yet, +and if I try to open a file with [`imv`](https://sr.ht/~exec64/imv/), it looks messed up. + +File links: +[lossy AVIF](avif-reddit.avif). + + + +### JPEG-XL + +As of writing, Firefox can't display JPEG-XL yet, and Google has +[aborted support](https://bugs.chromium.org/p/chromium/issues/detail?id=1178058#c84) for it. +If it doesn't work on the web and I can't even test it locally, +then I definitely can't optimize it! + +It's sad, because there's evidence +(e.g. in [this post](https://chipsandcheese.com/2021/02/28/modern-data-compression-in-2021-part-2-the-battle-to-dethrone-jpeg-with-jpeg-xl-avif-and-webp/)) +that JPEG-XL is a significant improvement over AVIF +in every way: fidelity, file size, and encoding/decoding speed, +although the latter might be due to AV1's relative immaturity. +It can also losslessly recompress existing JPEGs! + + + +## Conclusion + +Of course, optimizing images is optional, +but doing so speeds up page loading for users on bad connections, +and, if your server provider bills you for traffic, +can save you a significant amount of money too. +My website doesn't contain many images, but on the pages that do, +using a modern format like AVIF reduces transfers by >80% compared to JPEG, +which has been the standard for far too long. +WebP isn't as good as AVIF, but is also a respectable choice. +JPEG-XL would be the best of all words, but sadly doesn't seem meant to be. diff --git a/source/blog/2023/website-adventures-images/jpeg-arithmetic.jpg b/source/blog/2023/website-adventures-images/jpeg-arithmetic.jpg new file mode 100644 index 0000000..eb19de6 Binary files /dev/null and b/source/blog/2023/website-adventures-images/jpeg-arithmetic.jpg differ diff --git a/source/blog/2023/website-adventures-images/jpeg-huffman.jpg b/source/blog/2023/website-adventures-images/jpeg-huffman.jpg new file mode 100644 index 0000000..3c1e9fa Binary files /dev/null and b/source/blog/2023/website-adventures-images/jpeg-huffman.jpg differ diff --git a/source/blog/2023/website-adventures-images/png-original.png b/source/blog/2023/website-adventures-images/png-original.png new file mode 100644 index 0000000..d8812c1 Binary files /dev/null and b/source/blog/2023/website-adventures-images/png-original.png differ diff --git a/source/blog/2023/website-adventures-images/png-oxipng-zopfli.png b/source/blog/2023/website-adventures-images/png-oxipng-zopfli.png new file mode 100644 index 0000000..ebbca6c Binary files /dev/null and b/source/blog/2023/website-adventures-images/png-oxipng-zopfli.png differ diff --git a/source/blog/2023/website-adventures-images/png-oxipng.png b/source/blog/2023/website-adventures-images/png-oxipng.png new file mode 100644 index 0000000..06fb22f Binary files /dev/null and b/source/blog/2023/website-adventures-images/png-oxipng.png differ diff --git a/source/blog/2023/website-adventures-images/png-zopflipng.png b/source/blog/2023/website-adventures-images/png-zopflipng.png new file mode 100644 index 0000000..ecb9ede Binary files /dev/null and b/source/blog/2023/website-adventures-images/png-zopflipng.png differ diff --git a/source/blog/2023/website-adventures-images/simulation-full.png b/source/blog/2023/website-adventures-images/simulation-full.png new file mode 100644 index 0000000..ebbca6c Binary files /dev/null and b/source/blog/2023/website-adventures-images/simulation-full.png differ diff --git a/source/blog/2023/website-adventures-images/simulation-half.avif b/source/blog/2023/website-adventures-images/simulation-half.avif new file mode 100644 index 0000000..9f0f5cf Binary files /dev/null and b/source/blog/2023/website-adventures-images/simulation-half.avif differ diff --git a/source/blog/2023/website-adventures-images/simulation-half.jpg b/source/blog/2023/website-adventures-images/simulation-half.jpg new file mode 100644 index 0000000..e1cc716 Binary files /dev/null and b/source/blog/2023/website-adventures-images/simulation-half.jpg differ diff --git a/source/blog/2023/website-adventures-images/simulation-half.png b/source/blog/2023/website-adventures-images/simulation-half.png new file mode 100644 index 0000000..d4cb347 Binary files /dev/null and b/source/blog/2023/website-adventures-images/simulation-half.png differ diff --git a/source/blog/2023/website-adventures-images/simulation-half.webp b/source/blog/2023/website-adventures-images/simulation-half.webp new file mode 100644 index 0000000..d2a0edb Binary files /dev/null and b/source/blog/2023/website-adventures-images/simulation-half.webp differ diff --git a/source/blog/2023/website-adventures-images/webp-lossless.webp b/source/blog/2023/website-adventures-images/webp-lossless.webp new file mode 100644 index 0000000..8579535 Binary files /dev/null and b/source/blog/2023/website-adventures-images/webp-lossless.webp differ diff --git a/source/blog/2023/website-adventures-images/webp-lossy.webp b/source/blog/2023/website-adventures-images/webp-lossy.webp new file mode 100644 index 0000000..c8e541b Binary files /dev/null and b/source/blog/2023/website-adventures-images/webp-lossy.webp differ diff --git a/source/blog/2023/website-adventures-images/webp-near-lossless.webp b/source/blog/2023/website-adventures-images/webp-near-lossless.webp new file mode 100644 index 0000000..b6e1d3c Binary files /dev/null and b/source/blog/2023/website-adventures-images/webp-near-lossless.webp differ diff --git a/source/know/concept/modulational-instability/index.md b/source/know/concept/modulational-instability/index.md index d646503..3c0de4b 100644 --- a/source/know/concept/modulational-instability/index.md +++ b/source/know/concept/modulational-instability/index.md @@ -198,7 +198,7 @@ then it can in turn also cause MI in its own surroundings, leading to a cascade of secondary and tertiary gain areas. This is seen above for $$z > 30 L_\mathrm{NL}$$. -What we described is "pure" MI, but there also exists +Here we described "pure" MI, but there also exists a different type caused by Raman scattering. In that case, amplification occurs at the strongest peak of the Raman gain $$\tilde{g}_R(\omega)$$, even when the parent pulse has $$\beta_2 > 0$$. -- cgit v1.2.3