It took some time. Apple wasn't always happy with the technology of OLED displays. Now, Apple has had the advantage of learning how to put an OLED display into production in the Apple Watch. That technology won't be far behind in the future iPhones.
One of the things we keep learning is that companies that compete against Apple are all too happy to roll out glitzy new technology that makes them look good, but upon a more detailed analysis proves to be not (or not completely) ready for prime time. In other words, the competing technology appears in the specification and comparison charts, to the glee of competitors, but it may not be, as Tim Cook often points out, the very best it can be.
A case in point is the transition from LCD displays in the iPhone to OLED displays. Other smartphone companies have beaten Apple into the marketplace with OLED, but Apple engineers recognize the compromises that had to be made and aren't willing to subject Apple customers to that game. Not when genuine satisfaction and visual perfection are at stake. That's the legacy of Steve Jobs at work.
We can start with Tim Cook's own public affirmations of that philosophy. In March of 2013, he said, responding to phablet pressure, before the iPhone 6 launched.
Some customers value large screen size; others value also other factors such as resolution, color quality, white balance, brightness, reflectivity, screen longevity, power consumption, portability, compatibility with apps and many things. Our competitors had made some significant trade-offs in many of these areas in order to ship a larger display. We would not ship a larger-display iPhone while these trade-offs exist.
The article linked above then goes into some of the technical details why Apple didn't, in a similar fashion, embrace OLEDs for iPhones. Several factors have come into play that now make the transition to OLED both practical and desirable.
First, OLEDs have historically been rather expensive. Next, few companies make OLED displays, namely LG, Samsung and (now) Sharp. The relationship Apple has with each of those companies, the need to have multiple sources, the production costs, the manufacturer's commitment to being able to make displays in the quantity Apple needs, and the technical performance of the OLED display all come into play.
Meanwhile, Apple has been able to develop its expertise in OLED displays with the Apple Watch. Dr. Raymond M. Soneira, President, DisplayMate Technologies Corporation, has explained the technical development of the Apple Watch's display, made by LG, in one of his research papers. Some of the relevant sections, comparing LCD to OLED smartwatches, are quoted here.
LCD: ... Manufacturing small and thin LCDs for a smart watch is very challenging because they are made up of many layers including a backlight, two light polarizers, various optical films in addition to the two glass sheets that sandwich the active Liquid Crystal layer. As a result, LCDs for watches tend to be much thicker than OLEDs, typically over 2.0 mm, and also have a wide rim because the LCD drive signals must enter along the outer edges since the visible part of the LCD display must transmit the backlight.
Moving on, Dr. Soneira summarizes the applicability of OLEDs to a smartwatch.
OLED: ... OLED displays are an ideal technology for smart watches because they are thin single layer solid-state displays with high pixels-per-inch (ppi) that can be manufactured and cut into small smart watch sized screens with resolutions of 320x320 pixels and above. They offer much better black levels, viewing angles, and power management compared with LCDs. In 2013 Samsung introduced the first OLED smart watch, the Galaxy Gear, with 320x320 pixels and full 24-bit color. The Galaxy Gear 2 was introduced in 2014, and performed very well in our Display Shoot-Out tests. These OLED displays are made on a rigid substrate like almost all smartphone displays. They are fairly thin at just 0.9 mm and can be made with a very narrow rim and bezel, which are major advantages over LCDs. But the newest Flexible OLED displays have evolved way beyond this…
Not only have technical issue apparently been resolved that Tim Cook referred to back in 2013, but now we have evidence that suppliers are ramping up to meet demand from Apple. However, this Bloomberg article emphasizes that the expected arrival of an OLED display for the iPhone is in 2017, not this year.
There appears to be a happy confluence of events occuring. Apple's 10th anniversary iPhone in 2017, the thinness of the OLED display, its lower power requirements (no backlighting) and deeper blacks, advances in display coatings, OLED production ramping and lower costs, and Apple's learning curve with the Apple watch all seem to be converging on an OLED display in next year's iPhone.
As stated above, Apple takes its time and gets the technology right. When that happens, customers tend to forget about the occasionally rushed rollouts by competitors and the result manifests itself in customer satisfaction surveys.
Of course, if pressed, anyone would admit that that's how Apple has developed its products all along. It's the waiting for that fully-baked product that hurts.